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International Conference on Structural Engineering Dynamics – ICEDyn 2009

Profile image of Miguel NevesMiguel Neves

2010, Shock and Vibration

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307 pages

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Abstract

This paper reports on the modeling and on the experimental verification of electromechanically coupled beams with varying crosssectional area for piezoelectric energy harvesting. The governing equations are formulated using the Rayleigh-Ritz method and Euler-Bernoulli assumptions. A load resistance is considered in the electrical domain for the estimate of the electric power output of each geometric configuration. The model is first verified against the analytical results for a rectangular bimorph with tip mass reported in the literature. The experimental verification of the model is also reported for a tapered bimorph cantilever with tip mass. The effects of varying cross-sectional area and tip mass on the electromechanical behavior of piezoelectric energy harvesters are also discussed. An issue related to the estimation of the optimal load resistance (that gives the maximum power output) on beam shape optimization problems is also discussed.

Figures (394)

Ficure 1: A bimorph piezoelectric energy harvester under clamped- free boundary conditions.
Ficure 1: A bimorph piezoelectric energy harvester under clamped- free boundary conditions.
TABLE 1: Geometric and material properties of the bimorph harvester. n the first case, the model is verified against the analytical results of a bimorph cantilever with tip mass reported in the iterature [13]. The experimental verification of the model is then reported for a tapered bimorph cantilever with tip mass. Finally, a discussion regarding the calculation of the optimal oad resistance (for maximum power output) is presented. The effects of varying cross-sectional area, tip mass, and estimate of optimal load resistance on the electromechanical behavior and shape optimization problems of piezoelectric energy harvesters are also discussed. It is important to mention that, in the following discussions, the power output is normalized per base acceleration (in terms of gravitational acceleration), which is assumed to be smaller than that which would cause failure in the different piezoelectric energy harvesters considered in this work.
TABLE 1: Geometric and material properties of the bimorph harvester. n the first case, the model is verified against the analytical results of a bimorph cantilever with tip mass reported in the iterature [13]. The experimental verification of the model is then reported for a tapered bimorph cantilever with tip mass. Finally, a discussion regarding the calculation of the optimal oad resistance (for maximum power output) is presented. The effects of varying cross-sectional area, tip mass, and estimate of optimal load resistance on the electromechanical behavior and shape optimization problems of piezoelectric energy harvesters are also discussed. It is important to mention that, in the following discussions, the power output is normalized per base acceleration (in terms of gravitational acceleration), which is assumed to be smaller than that which would cause failure in the different piezoelectric energy harvesters considered in this work.
and the electric current FRF is obtained by dividing the voltage FRF by the load resistance of the electrical circuit and the electrical peak power FRF (since the voltage FRF is the peak voltage FRF) is the product of the voltage and current FRFs. load, electrical power output, and relative tip motion) can be obtained from the equations of motion ((9) and (10)). The excitation is due to the harmonic motion of the base in the transverse direction, wz = Ye!’ (where wa(t) is the base displacement, Y, is its amplitude, w is the excitation frequency, and j is the unit imaginary number), and the voltage output-to-base acceleration FRF can be obtained as
and the electric current FRF is obtained by dividing the voltage FRF by the load resistance of the electrical circuit and the electrical peak power FRF (since the voltage FRF is the peak voltage FRF) is the product of the voltage and current FRFs. load, electrical power output, and relative tip motion) can be obtained from the equations of motion ((9) and (10)). The excitation is due to the harmonic motion of the base in the transverse direction, wz = Ye!’ (where wa(t) is the base displacement, Y, is its amplitude, w is the excitation frequency, and j is the unit imaginary number), and the voltage output-to-base acceleration FRF can be obtained as
TABLE 2: Geometric and material properties of the tapered bimorph harvester.
TABLE 2: Geometric and material properties of the tapered bimorph harvester.
FicurE 2: Voltage FRF (a) and tip velocity (FRF) (b) for eight values of load resistance.
FicurE 2: Voltage FRF (a) and tip velocity (FRF) (b) for eight values of load resistance.
Figure 3: Experimental setup used for the verification of relations for a tapered beam.
Figure 3: Experimental setup used for the verification of relations for a tapered beam.
FiGureE 4: Model and experimental voltage FRFs (a) and tip velocity FRFs (b) for three values of load resistance.
FiGureE 4: Model and experimental voltage FRFs (a) and tip velocity FRFs (b) for three values of load resistance.
Ficure 6: Variation of optimum load resistance with parameter P.
Ficure 6: Variation of optimum load resistance with parameter P.
Figure 5: Standard shape modification of a bimorph piezoelectric harvester by parameters P and Q.
Figure 5: Standard shape modification of a bimorph piezoelectric harvester by parameters P and Q.
Figure 7: Variation of power output (per squared based acceleration) with parameter P and tip mass.
Figure 7: Variation of power output (per squared based acceleration) with parameter P and tip mass.
Ficure 8: Variation of power output (per squared based acceleration) with parameter Q and tip mass.
Ficure 8: Variation of power output (per squared based acceleration) with parameter Q and tip mass.
Ficure 1: Single degree of freedom oscillator: (a) harmonic base excitation and (b) harmonic force applied to the mass
Ficure 1: Single degree of freedom oscillator: (a) harmonic base excitation and (b) harmonic force applied to the mass
Ficure 2: Maximum system amplitude with v = 0.025 m/s and a sine sweep of the forcing frequency.
Ficure 2: Maximum system amplitude with v = 0.025 m/s and a sine sweep of the forcing frequency.
Ficure 3: Up-sweep jumps with v = 0.025 m/s: comparison between frequency response and stress-strain curves.
Ficure 3: Up-sweep jumps with v = 0.025 m/s: comparison between frequency response and stress-strain curves.
Ficure 4: Down-sweep jumps with v = 0.025 m/s: comparison between frequency response and stress-strain curves.
Ficure 4: Down-sweep jumps with v = 0.025 m/s: comparison between frequency response and stress-strain curves.
Ficure 5: Maximum system amplitude with v = 0.075 m/s and increasing the forcing frequency.
Ficure 5: Maximum system amplitude with v = 0.075 m/s and increasing the forcing frequency.
Ficure 6: Up-sweep jumps with v = 0.075 m/s: comparison between frequency response and stress-strain curves
Ficure 6: Up-sweep jumps with v = 0.075 m/s: comparison between frequency response and stress-strain curves
Ficure 7: Down-sweep jumps with v = 0.075 m/s: comparison between frequency response and stress-strain curves
Ficure 7: Down-sweep jumps with v = 0.075 m/s: comparison between frequency response and stress-strain curves
start to occur. Therefore, linear stress-strain curve is changed to a hysteretic behavior associated with incomplete phase transformations. Note that, for frequencies smaller than @ = 0.76, the system has a linear behavior. By slightly increasing the forcing frequency, the system presents a hysteretic behav- ior. The hysteresis loop causes a significant increase of the strain, which produces a dynamical jump. By continuing to increase the forcing frequency the hysteresis loop starts to become smaller until it disappears and the system presents a linear behavior again.
start to occur. Therefore, linear stress-strain curve is changed to a hysteretic behavior associated with incomplete phase transformations. Note that, for frequencies smaller than @ = 0.76, the system has a linear behavior. By slightly increasing the forcing frequency, the system presents a hysteretic behav- ior. The hysteresis loop causes a significant increase of the strain, which produces a dynamical jump. By continuing to increase the forcing frequency the hysteresis loop starts to become smaller until it disappears and the system presents a linear behavior again.
Ficure 8: Maximum system amplitude with 6 = 0.012 and increasing the forcing frequency. TABLE 1: SMA parameters.
Ficure 8: Maximum system amplitude with 6 = 0.012 and increasing the forcing frequency. TABLE 1: SMA parameters.
FicurE 9: Up-sweep jumps with 6 = 0.012: comparison between frequency response and stress-strain curves.
FicurE 9: Up-sweep jumps with 6 = 0.012: comparison between frequency response and stress-strain curves.
Ficure 10: Up-sweep jumps with 6 = 0.012: comparison between frequency response and stress-strain curves.
Ficure 10: Up-sweep jumps with 6 = 0.012: comparison between frequency response and stress-strain curves.
Ficure 1: (a) Cable resonance with low damping (blue curve fitting); (b) SSI principle.
Ficure 1: (a) Cable resonance with low damping (blue curve fitting); (b) SSI principle.
Ficure 2: (a) ELSA cable facility with zoomed views of the SSI attachments. (b) Transducers type and positions on the cable no. 1
Ficure 2: (a) ELSA cable facility with zoomed views of the SSI attachments. (b) Transducers type and positions on the cable no. 1
FiGurE 3: (a) SMA characterisation setup; (b) SMA training curve; (c) SMA behaviour during a cable dynamic test.
FiGurE 3: (a) SMA characterisation setup; (b) SMA training curve; (c) SMA behaviour during a cable dynamic test.
Ficure 4: (a) Measurement of the static stiffness increase in cable no. 1. (b) Measured force/displacement curve.
Ficure 4: (a) Measurement of the static stiffness increase in cable no. 1. (b) Measured force/displacement curve.
Ficure 5: (a) Displacement and (b) associated spectrum at 37.8 m on the free, constraint, and SSI cable no. 1.
Ficure 5: (a) Displacement and (b) associated spectrum at 37.8 m on the free, constraint, and SSI cable no. 1.
FiGurE 6: (a) Displacement and (b) associated spectrum at 7.2 m on the free, constraint, and SSI cable no. 2.
FiGurE 6: (a) Displacement and (b) associated spectrum at 7.2 m on the free, constraint, and SSI cable no. 2.
FiGure 7: (a) Displacement and (b) associated spectrum at 22.5 (midspan) on the free, constraint, and SSI cable no. 1.
FiGure 7: (a) Displacement and (b) associated spectrum at 22.5 (midspan) on the free, constraint, and SSI cable no. 1.
Ficure 8: Equivalent stiffness and damping for the SSI cable no. 2 as compared to the restrained configuration.
Ficure 8: Equivalent stiffness and damping for the SSI cable no. 2 as compared to the restrained configuration.
Ficure 9: Displacement (a), acceleration (b), and acceleration spectrum (c) for the free and SSI cable no. 1 (F = 100 N on Ist mode)
Ficure 9: Displacement (a), acceleration (b), and acceleration spectrum (c) for the free and SSI cable no. 1 (F = 100 N on Ist mode)
Ficure 10: SSI damping effect during a free decay after a short (a) or long (b) loading period on mode number 1.
Ficure 10: SSI damping effect during a free decay after a short (a) or long (b) loading period on mode number 1.
Ficure ll: Nearly steady state response of SSI cable no. 1 at 7.2 m during a long shaking period.
Ficure ll: Nearly steady state response of SSI cable no. 1 at 7.2 m during a long shaking period.
Figure 12: Detuning effect computed on a SDoF model in function of the switch position and input frequency.
Figure 12: Detuning effect computed on a SDoF model in function of the switch position and input frequency.
Figure 13: Variation of the reduction factor with respect to the switch position (4 different tensions of the spring)
Figure 13: Variation of the reduction factor with respect to the switch position (4 different tensions of the spring)
In a generic way, the discretized structural dynamics of the two substructures, namely, the main structure and the attached friction damper beam, are given by where the nodal normal contact forces Fy, and tangential friction forces F; act as internal forces on the contact interface between main structure and damper beam. On the right- hand side, the external forces appear, namely, the two pairs of clamping forces Fy ;, Fy, and dynamic excitation loads F.,.. (later, Fy, is controlled).
In a generic way, the discretized structural dynamics of the two substructures, namely, the main structure and the attached friction damper beam, are given by where the nodal normal contact forces Fy, and tangential friction forces F; act as internal forces on the contact interface between main structure and damper beam. On the right- hand side, the external forces appear, namely, the two pairs of clamping forces Fy ;, Fy, and dynamic excitation loads F.,.. (later, Fy, is controlled).
FicurE 2: Photograph of the experimental setup with structure, damper beam, piezoelectric stack, force cell, accelerometers, and attached shaker stinger (cf. Figure 1). Figure 1: Sketch of the benchmark structure (steel, 775 mm length, 40 mm width, and 3 mm thickness) with adaptive friction damper beam (steel, 160 mm length, 40 mm width, and 3 mm thickness).
FicurE 2: Photograph of the experimental setup with structure, damper beam, piezoelectric stack, force cell, accelerometers, and attached shaker stinger (cf. Figure 1). Figure 1: Sketch of the benchmark structure (steel, 775 mm length, 40 mm width, and 3 mm thickness) with adaptive friction damper beam (steel, 160 mm length, 40 mm width, and 3 mm thickness).
adaptive screw is used for the output definition of i,,;. The nonlinear observer is of the form rel’
adaptive screw is used for the output definition of i,,;. The nonlinear observer is of the form rel’
Figure 4: (a) Normal force stiffness relation. (b) 1D Jenkins friction model. (c) Jenkins model hysteresis loop (thick) in comparison to the Coulomb friction hysteresis (thin). Note, that for increasing tangential stiffness, the hysteresis approximates the Coulomb model hysteresis.
Figure 4: (a) Normal force stiffness relation. (b) 1D Jenkins friction model. (c) Jenkins model hysteresis loop (thick) in comparison to the Coulomb friction hysteresis (thin). Note, that for increasing tangential stiffness, the hysteresis approximates the Coulomb model hysteresis.
Ficure 5: Measured and simulated FRFs for impulse excitation and Fy; = 100 N, Fy, = 6000 N (excitation and measurement close to tip on the mid-axis) and determined modal damping ratios.
Ficure 5: Measured and simulated FRFs for impulse excitation and Fy; = 100 N, Fy, = 6000 N (excitation and measurement close to tip on the mid-axis) and determined modal damping ratios.
After linearization around u,,; = 0, Fy = 0, the observer gains 1 in (5) are determined by a Kalman design procedure from the solution of the associated Riccati equation for appropriate state and measurement noise variance matrices. The obtained control loop is shown in Figure 6. Its absolute value is furthermore maximized for optimality in the Lyapunov sense. Recalling the dynamics of a 1-DOF system
After linearization around u,,; = 0, Fy = 0, the observer gains 1 in (5) are determined by a Kalman design procedure from the solution of the associated Riccati equation for appropriate state and measurement noise variance matrices. The obtained control loop is shown in Figure 6. Its absolute value is furthermore maximized for optimality in the Lyapunov sense. Recalling the dynamics of a 1-DOF system
Ficure 7: Hysteresis-optimal control: measured (top) and simulated (bottom) accelerance FRFs for controlled sine-sweep excitation with (solid) and without (dashed) control (k;}. = 2.5 - 10’ N/m). FiGure 6: Closed control loop with hysteresis-optimal control law.
Ficure 7: Hysteresis-optimal control: measured (top) and simulated (bottom) accelerance FRFs for controlled sine-sweep excitation with (solid) and without (dashed) control (k;}. = 2.5 - 10’ N/m). FiGure 6: Closed control loop with hysteresis-optimal control law.
Figure 8: Lyapunoy-type control: measured (top) and simulated (bottom) accelerance FRFs for controlled sine-sweep excitation with (solid) and without (dashed) control (e = 1/200 m/s).
Figure 8: Lyapunoy-type control: measured (top) and simulated (bottom) accelerance FRFs for controlled sine-sweep excitation with (solid) and without (dashed) control (e = 1/200 m/s).
2.2. Modal Parameter Estimation. All modern, commercia algorithms for estimating modal parameters from exper- imental input-output data utilize matrix coefficient, poly- nomial models. This general matrix coefficient, polynomia’ formulation yields essentially the same polynomial form for both time and frequency domain data. Note, however, that this notation does not mean that, for an equivalent mode order, the associated matrix coefficients are numerically equal. The size of the square, matrix coefficients ([«]), and the order of the polynomial (m) vary with the algorithm. Once the matrix coefficients ([a]) have been found, the modal frequencies (A, or z,) can be found as the roots of the matrix coefficient polynomial (see (3) or (5)) using any one of a number of numerical techniques, normally involving an eigenvalue problem of the companion matrix associated with the matrix coefficient polynomial.
2.2. Modal Parameter Estimation. All modern, commercia algorithms for estimating modal parameters from exper- imental input-output data utilize matrix coefficient, poly- nomial models. This general matrix coefficient, polynomia’ formulation yields essentially the same polynomial form for both time and frequency domain data. Note, however, that this notation does not mean that, for an equivalent mode order, the associated matrix coefficients are numerically equal. The size of the square, matrix coefficients ([«]), and the order of the polynomial (m) vary with the algorithm. Once the matrix coefficients ([a]) have been found, the modal frequencies (A, or z,) can be found as the roots of the matrix coefficient polynomial (see (3) or (5)) using any one of a number of numerical techniques, normally involving an eigenvalue problem of the companion matrix associated with the matrix coefficient polynomial.
TABLE 1: Summary of modal parameter estimation algorithms.
TABLE 1: Summary of modal parameter estimation algorithms.
TABLE 2: Four corners of modal parameter estimation. order form. A recent paper explains that the two methods are theoretically equivalent [8].
TABLE 2: Four corners of modal parameter estimation. order form. A recent paper explains that the two methods are theoretically equivalent [8].
When comparing modal (base) vectors, at either the short or the long dimension, a pole-weighted vector can be constructed independent of the original algorithm used to estimate the poles and modal (base) vectors. For a given order k of the pole-weighted vector, the modal (base) vector and the associated pole can be used to formulate the pole-weighted vector as follows:
When comparing modal (base) vectors, at either the short or the long dimension, a pole-weighted vector can be constructed independent of the original algorithm used to estimate the poles and modal (base) vectors. For a given order k of the pole-weighted vector, the modal (base) vector and the associated pole can be used to formulate the pole-weighted vector as follows:
Ficure |: Eighth order, pole-weighted vector (state vector) example.
Ficure |: Eighth order, pole-weighted vector (state vector) example.
FiGure 2: Eighth order, pole-weighted vector (state vector) example, top view.
FiGure 2: Eighth order, pole-weighted vector (state vector) example, top view.
Figure 3: Extended consistency diagram, conventional version. challenging. Based upon the construction of the disc, real- valued, normal modes can be expected and the inability to resolve these modes can be instructive relative to both modal parameter estimation algorithm and autonomous procedure performance. For the interested reader, a number of realistic examples are shown in other past papers including FRF data from an automotive structure and a bridge structure [9, 16].
Figure 3: Extended consistency diagram, conventional version. challenging. Based upon the construction of the disc, real- valued, normal modes can be expected and the inability to resolve these modes can be instructive relative to both modal parameter estimation algorithm and autonomous procedure performance. For the interested reader, a number of realistic examples are shown in other past papers including FRF data from an automotive structure and a bridge structure [9, 16].
Figure 4: Extended consistency diagram, pole-weighted MAC version. Figure 4 is an example using a pole-weighted vector (state vector) method of producing a similar consistency diagram. This approach is easier to implement numerically and provides equal or better results when compared to the conventional approach for almost all cases. In this example, every estimate from every matrix coefficient polynomial solution from every algorithm is converted into a pole- weighted vector of a specific order, in this case tenth order. Then, the consistency diagram is developed by using the same vector correlation methods (MAC) [17, 18] to iden- tify consistency without a need for a separate consistency comparison for frequency and damping (since the modal frequency is included in the pole-weighted vector). A similar set of symbols, as those used in Figure 3, are used to define increased levels of vector consistency in terms of the MAC value between all pole-weighted vectors.
Figure 4: Extended consistency diagram, pole-weighted MAC version. Figure 4 is an example using a pole-weighted vector (state vector) method of producing a similar consistency diagram. This approach is easier to implement numerically and provides equal or better results when compared to the conventional approach for almost all cases. In this example, every estimate from every matrix coefficient polynomial solution from every algorithm is converted into a pole- weighted vector of a specific order, in this case tenth order. Then, the consistency diagram is developed by using the same vector correlation methods (MAC) [17, 18] to iden- tify consistency without a need for a separate consistency comparison for frequency and damping (since the modal frequency is included in the pole-weighted vector). A similar set of symbols, as those used in Figure 3, are used to define increased levels of vector consistency in terms of the MAC value between all pole-weighted vectors.
Ficure 5: Pole-weighted MAC of all consistency diagram solutions, before and after threshold applied.
Ficure 5: Pole-weighted MAC of all consistency diagram solutions, before and after threshold applied.
TABLE 3: Summary of autonomous modal parameter estimation statistics. the possibility that more than one mode has been included in a cluster. Consider in a cluster, then taking the SVD of the group of pole- weighted vectors and selecting the singular vector associated with the largest singular value. This chosen singular vector contains both the shape and the modal frequency informa- tion. The modal frequency is identified by dividing the first order portion by the zeroth order portion of the vector in a least squares sense. (Note that it is also possible to solve the frequency polynomial which would result from using the complete vector.) Also, for numerical reasons, the pole- weighted vector is actually computed in the z domain.
TABLE 3: Summary of autonomous modal parameter estimation statistics. the possibility that more than one mode has been included in a cluster. Consider in a cluster, then taking the SVD of the group of pole- weighted vectors and selecting the singular vector associated with the largest singular value. This chosen singular vector contains both the shape and the modal frequency informa- tion. The modal frequency is identified by dividing the first order portion by the zeroth order portion of the vector in a least squares sense. (Note that it is also possible to solve the frequency polynomial which would result from using the complete vector.) Also, for numerical reasons, the pole- weighted vector is actually computed in the z domain.
Ficure 6: Pole surface consistency, 2320 Hertz region.
Ficure 6: Pole surface consistency, 2320 Hertz region.
large systems, it is a nontrivial task. For this reason, a lot of formalisms have been developed. Numerical formalisms especially play a significant role as they are used in computer multibody codes which becomes more and more popular and sophisticated [9].
large systems, it is a nontrivial task. For this reason, a lot of formalisms have been developed. Numerical formalisms especially play a significant role as they are used in computer multibody codes which becomes more and more popular and sophisticated [9].
Ficure 3: Typical damper characteristic and its logarithmic approx- imation.
Ficure 3: Typical damper characteristic and its logarithmic approx- imation.
Ficure 4: (a) Multibody bus suspension system; (b) rigid-flexible multibody bus model.
Ficure 4: (a) Multibody bus suspension system; (b) rigid-flexible multibody bus model.
Figure 6: Virtual breaking performance on a slippery road. Com- parison of angular velocity of the left wheel: with and without break assist. FicureE 5: ABS/ASR/EBD signal circuit.
Figure 6: Virtual breaking performance on a slippery road. Com- parison of angular velocity of the left wheel: with and without break assist. FicureE 5: ABS/ASR/EBD signal circuit.
Ficure 8: Measurement data approximation: the inflated air spring (a) and the bump stop (b) characteristics. Ficure 7: One-quarter of an ECAS model.
Ficure 8: Measurement data approximation: the inflated air spring (a) and the bump stop (b) characteristics. Ficure 7: One-quarter of an ECAS model.
Ficure 9: Comparison between numerical and experimental vibration acceleration as a function of frequency: (a) point A; (b) point B; (c) point C; (d) point D.
Ficure 9: Comparison between numerical and experimental vibration acceleration as a function of frequency: (a) point A; (b) point B; (c) point C; (d) point D.
Ficure 10: Hybrid urban bus model-stress Von Mises’s measure- ment.
Ficure 10: Hybrid urban bus model-stress Von Mises’s measure- ment.
TABLE 1: Basic information about geometry and material properties used for modeling of supporting structure.
TABLE 1: Basic information about geometry and material properties used for modeling of supporting structure.
Ficure 1: Experimental set-up showing the wind turbine blade section mounted on the test rig with the coordinate system.
Ficure 1: Experimental set-up showing the wind turbine blade section mounted on the test rig with the coordinate system.
Ficure 3: Coherence functions for the two driving points. It is used as measure of the FRF quality. Ideally it should take value equal to 1. FiGure 2: Linearity check for one of the points on the blade. Voltage values = 0.5 V,1V,1,5 V, and 2 V.
Ficure 3: Coherence functions for the two driving points. It is used as measure of the FRF quality. Ideally it should take value equal to 1. FiGure 2: Linearity check for one of the points on the blade. Voltage values = 0.5 V,1V,1,5 V, and 2 V.
Figure 4: Estimated experimental mode shapes of the modified blade section and support structure.
Figure 4: Estimated experimental mode shapes of the modified blade section and support structure.
FicureE 5: AutoMAC matrices for experimental modal models with sensors only on the modified blade section (a), support structure (b), an blade section with support structure (c).
FicureE 5: AutoMAC matrices for experimental modal models with sensors only on the modified blade section (a), support structure (b), an blade section with support structure (c).
Ficure 6: FE model of the blade section clamped to the support structure. Yellow bulbs denote test and FE geometry correlation node mapping.
Ficure 6: FE model of the blade section clamped to the support structure. Yellow bulbs denote test and FE geometry correlation node mapping.
TABLE 2: Initial consistency of the modal model parameters. investigated. Observing the values of the MAC criterion between test and simulation modes (Figure 7), differences can be notified. They are caused by the influence of the support structure and not perfectly excited Ist bending mode. Further investigation of observed differences is presented in Section 4.
TABLE 2: Initial consistency of the modal model parameters. investigated. Observing the values of the MAC criterion between test and simulation modes (Figure 7), differences can be notified. They are caused by the influence of the support structure and not perfectly excited Ist bending mode. Further investigation of observed differences is presented in Section 4.
Figure 7: MAC matrix for test and FE simulation modal vectors of modified blade without support structure.
Figure 7: MAC matrix for test and FE simulation modal vectors of modified blade without support structure.
FIGURE 8: Frequency sensitivity matrix graphically presenting normalized magnitude of the impact of selected design variables (inputs) on he modes frequencies of interest (outputs).
FIGURE 8: Frequency sensitivity matrix graphically presenting normalized magnitude of the impact of selected design variables (inputs) on he modes frequencies of interest (outputs).
FicurE 9: Mountings of supporting structure modeled with steel pipes, steel, and bushing properties
FicurE 9: Mountings of supporting structure modeled with steel pipes, steel, and bushing properties
Ficure 10: Example of 3D scatter plot of two inputs (factors) impact on the output (response) 7th mode frequency.
Ficure 10: Example of 3D scatter plot of two inputs (factors) impact on the output (response) 7th mode frequency.
FiGurE ll: Histogram plot of 9th mode frequency distribution.
FiGurE ll: Histogram plot of 9th mode frequency distribution.
FiGuRE 12: Quadratic response surface models 3D perspective plot for the same input variables and (a) 4th mode, (b) 5th mode, and (c) 7th mode frequency.
FiGuRE 12: Quadratic response surface models 3D perspective plot for the same input variables and (a) 4th mode, (b) 5th mode, and (c) 7th mode frequency.
TABLE 4: Updated parameters and their final values. 5. Conclusions
TABLE 4: Updated parameters and their final values. 5. Conclusions
Ficure 13: MAC matrix, test versus updated FE model of the blade with flexible support.
Ficure 13: MAC matrix, test versus updated FE model of the blade with flexible support.
TABLE 5: Final consistency of the modal model parameters. representation has improved the consistency in between test and simulations. Design of experiment with response surface model study allowed successful updating of the FE model confirmed by modal assurance criterion. The comparison of experimental and numerical models clearly shows the influence of support structure flexibility.
TABLE 5: Final consistency of the modal model parameters. representation has improved the consistency in between test and simulations. Design of experiment with response surface model study allowed successful updating of the FE model confirmed by modal assurance criterion. The comparison of experimental and numerical models clearly shows the influence of support structure flexibility.
Ficure 1: Hybrid system: base-isolated structure with semiactive fluid viscous damper.
Ficure 1: Hybrid system: base-isolated structure with semiactive fluid viscous damper.
The system described by (1) can be represented in the state-space form by
The system described by (1) can be represented in the state-space form by
FicurE 2: Proposed control loop: force tracking scheme using a MPC controller with a semiactive (SA) algorithm.
FicurE 2: Proposed control loop: force tracking scheme using a MPC controller with a semiactive (SA) algorithm.
Afalk) = falk)—falk-1); A(k) = {a,(k), @g(k | k),...,4g(k+ HP - 1 | k)}t is the vector (dimension HP x 1) of measured and future disturbances (commonly assumed as constant and
Afalk) = falk)—falk-1); A(k) = {a,(k), @g(k | k),...,4g(k+ HP - 1 | k)}t is the vector (dimension HP x 1) of measured and future disturbances (commonly assumed as constant and
The optimal input force moves can be obtained as the solution of an unconstrained optimization problem (minimization) with the following cost function [24]: definite or at least one is positive definite and the other is semipositive definite.
The optimal input force moves can be obtained as the solution of an unconstrained optimization problem (minimization) with the following cost function [24]: definite or at least one is positive definite and the other is semipositive definite.
Manipulating the expressions of (9), the Kalman filter with a stationary gain is obtained by solving the corresponding Riccati equation with adequate covariance matrices: In order to find adequate MPC controller solutions, stable closed-loop systems (linear counterpart, blocks SA device, and algorithm with unitary transfer functions) should be
Manipulating the expressions of (9), the Kalman filter with a stationary gain is obtained by solving the corresponding Riccati equation with adequate covariance matrices: In order to find adequate MPC controller solutions, stable closed-loop systems (linear counterpart, blocks SA device, and algorithm with unitary transfer functions) should be
Figure 3: Simulation results for the SDOF model (f,, = 0.25 Hz; € = 10%) with a passive damper. (a) Evolution of responses versus passive device damping for 10 type 1 seismic actions and the correspondent mean curve; (b) damping coefficients (for a = 0.15) for the minimum mean peak acceleration response function of system natural frequencies under type 1 and 2 seismic actions.
Figure 3: Simulation results for the SDOF model (f,, = 0.25 Hz; € = 10%) with a passive damper. (a) Evolution of responses versus passive device damping for 10 type 1 seismic actions and the correspondent mean curve; (b) damping coefficients (for a = 0.15) for the minimum mean peak acceleration response function of system natural frequencies under type 1 and 2 seismic actions.
Figure 4: Input ground motions characteristic considered in the numerical simulations. (a) One of ten artificial accelerograms; (b) frequency spectra of 10 accelerograms; (c) response spectra of 10 accelerograms.
Figure 4: Input ground motions characteristic considered in the numerical simulations. (a) One of ten artificial accelerograms; (b) frequency spectra of 10 accelerograms; (c) response spectra of 10 accelerograms.
Ficure 5: (a) Simulation results for the SDOF model (f,, = 0.25 Hz and € = 10%) subjected to one type 1 accelerogram without devices, with passive and SAMPCVD; (b) peak values’ statistics for 10 type 1 input time series.
Ficure 5: (a) Simulation results for the SDOF model (f,, = 0.25 Hz and € = 10%) subjected to one type 1 accelerogram without devices, with passive and SAMPCVD; (b) peak values’ statistics for 10 type 1 input time series.
FiGure 6: (a) SDOF system's mean peak responses (f,, = 0.25Hz and € = 10% for SAMPCVD with HC = 1 andr = 1) evolutions witl control interval T, and prediction horizon HP, when subjected to type 1 seismic actions (10 accelerograms). (b) Ratios evolution with contro interval (relative to T, = 5 ms) for HP = 50.
FiGure 6: (a) SDOF system's mean peak responses (f,, = 0.25Hz and € = 10% for SAMPCVD with HC = 1 andr = 1) evolutions witl control interval T, and prediction horizon HP, when subjected to type 1 seismic actions (10 accelerograms). (b) Ratios evolution with contro interval (relative to T, = 5 ms) for HP = 50.
Ficure 7: Result for system with f,, = 0.25 Hz and € = 10% considering T, = 5 ms, HC = 1. r = 1 and d = 10. (a) System eigenvalues’ moduli (maximum values) evolution with acceleration weight; (b) system responses’ (peak values) evolution with acceleration weight when subjected to type 1 seismic actions; (c) system responses’ (mean peak values) evolution with prediction horizon for different natural frequencies (f,, = 0.25 to 1.00 Hz) for SAMPCVD, when subjected to type 1 seismic actions (10 input time accelerograms).
Ficure 7: Result for system with f,, = 0.25 Hz and € = 10% considering T, = 5 ms, HC = 1. r = 1 and d = 10. (a) System eigenvalues’ moduli (maximum values) evolution with acceleration weight; (b) system responses’ (peak values) evolution with acceleration weight when subjected to type 1 seismic actions; (c) system responses’ (mean peak values) evolution with prediction horizon for different natural frequencies (f,, = 0.25 to 1.00 Hz) for SAMPCVD, when subjected to type 1 seismic actions (10 input time accelerograms).
Ficure 8: Simulation results for the SDOF model (f,, = 0.25 Hz and € = 10%) subjected to one type 1 accelerogram employing SA system with T, = 5 ms, HP = 50, HC = 1, r = 1, and d = 10, with both VD (a) and COO (b) control algorithms.
Ficure 8: Simulation results for the SDOF model (f,, = 0.25 Hz and € = 10%) subjected to one type 1 accelerogram employing SA system with T, = 5 ms, HP = 50, HC = 1, r = 1, and d = 10, with both VD (a) and COO (b) control algorithms.
FiGurE 9: SDOF mean peak response values for the original system and ratios (relative to the original structure) for different system's natural frequencies.
FiGurE 9: SDOF mean peak response values for the original system and ratios (relative to the original structure) for different system's natural frequencies.
Ficure 10: 10-storey base-isolated building and equivalent model for analysis.
Ficure 10: 10-storey base-isolated building and equivalent model for analysis.
Ficure II: 10-storey base-isolated building responses to 10 type 1 and type 2 accelerograms. Mean peak values in terms of relative displace- ments, accelerations, base shear V, and device's force f for the systems under study. "ABLE 1: 10-storey base isolated building results. Comparison in terms of mean response peak values ratios: relative displacements, accel rations (at base and top) and base shear force V, in relation to the original structure; and peak device's force ratio f/W in relation to th tructure’s weight W.
Ficure II: 10-storey base-isolated building responses to 10 type 1 and type 2 accelerograms. Mean peak values in terms of relative displace- ments, accelerations, base shear V, and device's force f for the systems under study. "ABLE 1: 10-storey base isolated building results. Comparison in terms of mean response peak values ratios: relative displacements, accel rations (at base and top) and base shear force V, in relation to the original structure; and peak device's force ratio f/W in relation to th tructure’s weight W.
Figure 1: The commonly accepted assumption of white noise input should be thought of as loading an imaginary linear filter that produces the unknown forces. Thus the actual physical forces do not need to be white noise or have a flat spectrum.
Figure 1: The commonly accepted assumption of white noise input should be thought of as loading an imaginary linear filter that produces the unknown forces. Thus the actual physical forces do not need to be white noise or have a flat spectrum.
where b,, is the mode shape for mode n and y, is a vector describing the modal participation of the considered mode. It is important to note that the negative time part of the correlation function matrix R,_(t) is in fact a free decay because it is written as a linear combination of modal contributions (terms proportional to the mode shape times a complex exponential), whereas the positive time part R,, (7) is in fact only a free decay if it is used in its transposed form so the terms y,b2 turn into the form bv. and the response becomes proportional to the mode shapes. This means that whenever correlation functions are used as free decays, using the positive part of the correlation function matrix, the matrix must be used in its transposed form. It should also be noted that if we had not followed the definition given by (5) but instead defined the correlation matrix as R(t) = E[y(t + ny’ (t)] which is common in some presentations of random vibration theory, then because of stationarity this is equal to Ely(t)y" (t —T)].So this swaps the time of the solutions for the correlation function matrix in (17), and thus in this case the positive part of the correlation function matrix can indeed be used as free decays without taking the transpose. Other central relations in random vibrations are the modal decompositions of the correlation function and PSD function matrices. The modal decomposition of the correla- tion function matrix is due to James et al. [6]. Expressing a general response by its modal decomposition and assuming white noise input where the correlations functions all degen- erate to the Dirac delta functions it can be shown that the correlation function matrix for negative times R,_(t) and for positive times R,,, (tT) is given by
where b,, is the mode shape for mode n and y, is a vector describing the modal participation of the considered mode. It is important to note that the negative time part of the correlation function matrix R,_(t) is in fact a free decay because it is written as a linear combination of modal contributions (terms proportional to the mode shape times a complex exponential), whereas the positive time part R,, (7) is in fact only a free decay if it is used in its transposed form so the terms y,b2 turn into the form bv. and the response becomes proportional to the mode shapes. This means that whenever correlation functions are used as free decays, using the positive part of the correlation function matrix, the matrix must be used in its transposed form. It should also be noted that if we had not followed the definition given by (5) but instead defined the correlation matrix as R(t) = E[y(t + ny’ (t)] which is common in some presentations of random vibration theory, then because of stationarity this is equal to Ely(t)y" (t —T)].So this swaps the time of the solutions for the correlation function matrix in (17), and thus in this case the positive part of the correlation function matrix can indeed be used as free decays without taking the transpose. Other central relations in random vibrations are the modal decompositions of the correlation function and PSD function matrices. The modal decomposition of the correla- tion function matrix is due to James et al. [6]. Expressing a general response by its modal decomposition and assuming white noise input where the correlations functions all degen- erate to the Dirac delta functions it can be shown that the correlation function matrix for negative times R,_(t) and for positive times R,,, (tT) is given by
correlation functions and the free decays are then arranged in a Hankel matrix, Vold et al. [11, 12], Here the operating responses y(n) are given in terms of the discrete time t,, = nAt. The matrix containing the AR matrices of the free decays
correlation functions and the free decays are then arranged in a Hankel matrix, Vold et al. [11, 12], Here the operating responses y(n) are given in terms of the discrete time t,, = nAt. The matrix containing the AR matrices of the free decays
We can then estimate the observability and controllability matrices as follows: In ERA two Hankel matrices are formed, Juang and Pappa 16] and Pappa et al. [17, 18],
We can then estimate the observability and controllability matrices as follows: In ERA two Hankel matrices are formed, Juang and Pappa 16] and Pappa et al. [17, 18],
FiGure 2: Results of using FDD to identify the first three modes on the first dataset of the HCT case. The singular values of the SD matrix are shown in dotted line. In the plot the frequency lines where the mode shape vectors are estimated are indicated by an asterisk and the corresponding modal decomposition is shown in solid line.
FiGure 2: Results of using FDD to identify the first three modes on the first dataset of the HCT case. The singular values of the SD matrix are shown in dotted line. In the plot the frequency lines where the mode shape vectors are estimated are indicated by an asterisk and the corresponding modal decomposition is shown in solid line.
TABLE 2: Modal identification on dataset 4 of the Heritage Court Tower case. according to (22); the full correlation function matrix was transposed as described in Section 4 and all columns (all free decays) in the transposed correlation function matrix were then used for the identification. For the first dataset 500 discrete time lags were used in the correlation function matrix; however, for the last dataset where the modes are somewhat more difficult to identify, 950 time lags were used. For the AR, ITD, and ERA techniques a low model (corresponding to na = 2) order was used including 6 modes for dataset 1 and 8 modes for dataset 4. transform of the directly estimated correlation function matrix.
TABLE 2: Modal identification on dataset 4 of the Heritage Court Tower case. according to (22); the full correlation function matrix was transposed as described in Section 4 and all columns (all free decays) in the transposed correlation function matrix were then used for the identification. For the first dataset 500 discrete time lags were used in the correlation function matrix; however, for the last dataset where the modes are somewhat more difficult to identify, 950 time lags were used. For the AR, ITD, and ERA techniques a low model (corresponding to na = 2) order was used including 6 modes for dataset 1 and 8 modes for dataset 4. transform of the directly estimated correlation function matrix.
TABLE 1: Modal identification on dataset 1 of the Heritage Court Tower case.
TABLE 1: Modal identification on dataset 1 of the Heritage Court Tower case.
Ficure 3: Mode shapes of the HCT building found by AR identification and merging the mode shapes from the 4 data using the reference DOFs. The units of the coordinate system are meters.
Ficure 3: Mode shapes of the HCT building found by AR identification and merging the mode shapes from the 4 data using the reference DOFs. The units of the coordinate system are meters.
Ficure 1: Application of static FE-based method: loads, constraint, and connection elements applied to the detailed 3D beam model.
Ficure 1: Application of static FE-based method: loads, constraint, and connection elements applied to the detailed 3D beam model.
FicurE 2: Workflow for the creation of beam and joint FE concept models.
FicurE 2: Workflow for the creation of beam and joint FE concept models.
n this paper, a new MPC connection element is defined, which enables to create an interface between the concept D beam model and the detailed 3D joint model, as shown in Figure 3. For such purpose, the kinematic relationships between the displacements and rotations of the beam node (dependent node of the connection element) and the dis- placements of the nodes of the detailed 3D FE model of the joint at the interface section (independent nodes of the connection element) are derived in the form of a transformation matrix [R], by using a static approach based on equilibrium conditions [11]. The basic idea is to obtain a second transformation matrix [S] that defines the relationship between the total forces at the central node of the beam/joint interface and the nodal loads over the section, considering theoretical stress fields resulting from the Saint- Venant assumptions with respect to axial, bending, shear, and torsion load-cases for a beam-like structure [12]. In linear elastic field, this loads correlation can be inverted, returning the searched kinematic relationship. where {o} is the vector of nodal stresses, [S,] is the stress recovery matrix, and {F} is the vector of total forces applied to the central node. In particular, (1) for a generic rectangular cross-section can be detailed as follows: The physical meaning of each parameter is given in Table 1.
n this paper, a new MPC connection element is defined, which enables to create an interface between the concept D beam model and the detailed 3D joint model, as shown in Figure 3. For such purpose, the kinematic relationships between the displacements and rotations of the beam node (dependent node of the connection element) and the dis- placements of the nodes of the detailed 3D FE model of the joint at the interface section (independent nodes of the connection element) are derived in the form of a transformation matrix [R], by using a static approach based on equilibrium conditions [11]. The basic idea is to obtain a second transformation matrix [S] that defines the relationship between the total forces at the central node of the beam/joint interface and the nodal loads over the section, considering theoretical stress fields resulting from the Saint- Venant assumptions with respect to axial, bending, shear, and torsion load-cases for a beam-like structure [12]. In linear elastic field, this loads correlation can be inverted, returning the searched kinematic relationship. where {o} is the vector of nodal stresses, [S,] is the stress recovery matrix, and {F} is the vector of total forces applied to the central node. In particular, (1) for a generic rectangular cross-section can be detailed as follows: The physical meaning of each parameter is given in Table 1.
where {f} is the vector of total nodal forces, [S,] is the form nodal load matrix, and {o} is the vector of total nodal stresses. In extended notation, (5) can be rewritten as follows: Note that, for open sections, the forces applied on internal nodes receive the contribution of average stresses relating to both adjacent elements; instead, external nodes belong to a single element and then their values are considerably lower. Therefore, by combining (1) and (5), it is possible to obtain the relation between the total forces at the central node, {F}, and the nodal forces on the outer contour of the section {f}: To obtain the [R] matrix that relates 1D beam displace- ments and rotations and 3D displacement values at the interface, linear relations between forces and displacements are assumed. In this way, it is sufficient to transpose [S]: Since the average stresses at the interface are constant, the nodal loads for i and j can be calculated by using shape functions, yielding
where {f} is the vector of total nodal forces, [S,] is the form nodal load matrix, and {o} is the vector of total nodal stresses. In extended notation, (5) can be rewritten as follows: Note that, for open sections, the forces applied on internal nodes receive the contribution of average stresses relating to both adjacent elements; instead, external nodes belong to a single element and then their values are considerably lower. Therefore, by combining (1) and (5), it is possible to obtain the relation between the total forces at the central node, {F}, and the nodal forces on the outer contour of the section {f}: To obtain the [R] matrix that relates 1D beam displace- ments and rotations and 3D displacement values at the interface, linear relations between forces and displacements are assumed. In this way, it is sufficient to transpose [S]: Since the average stresses at the interface are constant, the nodal loads for i and j can be calculated by using shape functions, yielding
where A™ is the area of the element, N indicates the shape function, and s is the local coordinate, useful to measure the length of element.
where A™ is the area of the element, N indicates the shape function, and s is the local coordinate, useful to measure the length of element.
FiGure 3: The interface element (circled in red) between equivalent 1D beam model and detailed 3D joint model. The central node is in yellow (dependent node on the beam side) while the peripheral nodes are in red (independent nodes on the joint side).
FiGure 3: The interface element (circled in red) between equivalent 1D beam model and detailed 3D joint model. The central node is in yellow (dependent node on the beam side) while the peripheral nodes are in red (independent nodes on the joint side).
TABLE 2: Equivalent beam properties estimated by the dynamic FE- based method. model. Two different concept models of the joint have been created: in one model, the Craig-Bampton dynamic reduction was applied to the detailed 3D FE model with rigid RBE2 elements at each beam/joint interface; in the other model, the proposed MPC elements were used to connect the central node to nodes placed on the cross-section at each interface.
TABLE 2: Equivalent beam properties estimated by the dynamic FE- based method. model. Two different concept models of the joint have been created: in one model, the Craig-Bampton dynamic reduction was applied to the detailed 3D FE model with rigid RBE2 elements at each beam/joint interface; in the other model, the proposed MPC elements were used to connect the central node to nodes placed on the cross-section at each interface.
Ficure 5: Profile of detailed 3D beam model: a generic shell element k and the two nodes at interface, i and j [11]. Figure 4: Stress distribution along rectangular cross-sections, due to vertical shear (t,,, and T,, in (a)) and warping torsion (a, in (b)): the global effect of linear parts in the central node is zero.
Ficure 5: Profile of detailed 3D beam model: a generic shell element k and the two nodes at interface, i and j [11]. Figure 4: Stress distribution along rectangular cross-sections, due to vertical shear (t,,, and T,, in (a)) and warping torsion (a, in (b)): the global effect of linear parts in the central node is zero.
TABLE 3: Dynamic comparison between the original and the two-concept FE models, in terms of natural frequencies, % errors, and modal correlation factors.
TABLE 3: Dynamic comparison between the original and the two-concept FE models, in terms of natural frequencies, % errors, and modal correlation factors.
Ficure 6: Application model: cross-section geometry of beams (a), mesh of detailed 3D beam model with welding zones (b), and spot weld model, with a central Hexa solid element connected to nodes of flanges by interpolation elements (c).
Ficure 6: Application model: cross-section geometry of beams (a), mesh of detailed 3D beam model with welding zones (b), and spot weld model, with a central Hexa solid element connected to nodes of flanges by interpolation elements (c).
Ficure 7: Application case: detailed 3D FE model of the structure.
Ficure 7: Application case: detailed 3D FE model of the structure.
FiGureE 8: Concept model: before (a) and after (b) the reduction of joint in SE. The three master nodes of SE are in yellow.
FiGureE 8: Concept model: before (a) and after (b) the reduction of joint in SE. The three master nodes of SE are in yellow.
FIGURE 2: Waves in railway track as periodic structure-passing band and stopping band.
FIGURE 2: Waves in railway track as periodic structure-passing band and stopping band.
FiGurE 1: Phase waves velocities V, versus wave number dependent on the beam parameters (inequality (3)).
FiGurE 1: Phase waves velocities V, versus wave number dependent on the beam parameters (inequality (3)).
FiGure 4: Railway track with stiff type of sleepers with double-point fastening systems. Figure 3: Two kinds of sleepers with different fastening systems.
FiGure 4: Railway track with stiff type of sleepers with double-point fastening systems. Figure 3: Two kinds of sleepers with different fastening systems.
Ficure 6: Rotation of three sleepers with double-point fastening during motion in the zone behind the bridge abutment. FicureE 5: Scheme of track in transition zone with two types of sleepers and fastening systems.
Ficure 6: Rotation of three sleepers with double-point fastening during motion in the zone behind the bridge abutment. FicureE 5: Scheme of track in transition zone with two types of sleepers and fastening systems.
Figure 7: Vertical displacement of the sleepers with double-point fastening during motion in the zone behind the rigid base.
Figure 7: Vertical displacement of the sleepers with double-point fastening during motion in the zone behind the rigid base.
Figure 8: Comparison of rotation of two sleepers with different weight and fastening systems—vehicle speed 20 m/s.
Figure 8: Comparison of rotation of two sleepers with different weight and fastening systems—vehicle speed 20 m/s.
FicurE 9: Rail rotation in the tracks with different sleepers and fastening systems at the vehicle speed 20 m/s.
FicurE 9: Rail rotation in the tracks with different sleepers and fastening systems at the vehicle speed 20 m/s.
Ficure ll: Track mistuning by the change of sleeper spacing (rotation of every second sleeper). Ficure 10: Scheme of rack with the change of sleeper spacing, distances d; = d, + Aj.
Ficure ll: Track mistuning by the change of sleeper spacing (rotation of every second sleeper). Ficure 10: Scheme of rack with the change of sleeper spacing, distances d; = d, + Aj.
Ficure 13: Vertical displacements of the vehicle/track contact points at the speed 30 m/s [1], 12].
Ficure 13: Vertical displacements of the vehicle/track contact points at the speed 30 m/s [1], 12].
Figure 14: Vertical displacements of the vehicle/track contact points at the speed 50 m/s [1], 12].
Figure 14: Vertical displacements of the vehicle/track contact points at the speed 50 m/s [1], 12].
Figure 12: Track “Y-shaped” sleepers spacing as visible above.
Figure 12: Track “Y-shaped” sleepers spacing as visible above.
Ficure 16: Vertical displacements of rails registered 180 cm in front of the contact points of the buggy for classic and “Y”-type track at the speed 40 m/s [12]. References Ficure 15: Vertical displacements of rails registered 120 cm in front of the contact points of the buggy for classic and “Y”-type track at the speed 40 m/s [12].
Ficure 16: Vertical displacements of rails registered 180 cm in front of the contact points of the buggy for classic and “Y”-type track at the speed 40 m/s [12]. References Ficure 15: Vertical displacements of rails registered 120 cm in front of the contact points of the buggy for classic and “Y”-type track at the speed 40 m/s [12].
We apply the following nondimensional PDEs governing dynamics of our plate:
We apply the following nondimensional PDEs governing dynamics of our plate:
Figure 1: Plate computational scheme.
Figure 1: Plate computational scheme.
TABLE 1: Plate bifurcations.
TABLE 1: Plate bifurcations.
After reduction to the normal form, (7)-(9) are solved via the fourth-order Runge-Kutta method, where on each time step we need to solve a large system of linear algebraic equations regarding time, and a time step is yielded by the Runge principle. Equations (7) are supplemented with boundary condi- tions (4), which have the following difference representation (flexible nonstretched (uncompressed) ribs support):
After reduction to the normal form, (7)-(9) are solved via the fourth-order Runge-Kutta method, where on each time step we need to solve a large system of linear algebraic equations regarding time, and a time step is yielded by the Runge principle. Equations (7) are supplemented with boundary condi- tions (4), which have the following difference representation (flexible nonstretched (uncompressed) ribs support):
Figure 2: Chart of the character of plate vibrations.
Figure 2: Chart of the character of plate vibrations.
The position vector of the satellite mass center with respect to the inertial reference system (x, y, z) (see Figure 1) is Details of the equations of motion derivation can be found in [12], where L is the Lagrangian of the system, the generalized coordinates are V and a, R is the Rayleigh dissipation function, tT, is the internal torque, and 1, is the external torque. Assume that R,T,, T,,w, V are given by
The position vector of the satellite mass center with respect to the inertial reference system (x, y, z) (see Figure 1) is Details of the equations of motion derivation can be found in [12], where L is the Lagrangian of the system, the generalized coordinates are V and a, R is the Rayleigh dissipation function, tT, is the internal torque, and 1, is the external torque. Assume that R,T,, T,,w, V are given by
Ficure 1: Satellite model with slosh dynamics pendulum analogous mechanical system. As a result, the mass of the fuel velocity is
Ficure 1: Satellite model with slosh dynamics pendulum analogous mechanical system. As a result, the mass of the fuel velocity is
Figure 5: Poles position during the estimation.
Figure 5: Poles position during the estimation.
Ficure 1: Largest span of analyzed footbridges (circular ribbon footbridge not included).
Ficure 1: Largest span of analyzed footbridges (circular ribbon footbridge not included).
Figure 2: Main structural deck types for footbridges in Portugal.
Figure 2: Main structural deck types for footbridges in Portugal.
Ficure 3: Main typologies of footbridges in Portugal.
Ficure 3: Main typologies of footbridges in Portugal.
Ficure 4: Frequencies of lst mode in the three directions (T, L, and V) as a function of largest span: (a) to (c) all typologies for V, T, L, and (d) V + L; (e) to (h) vertical for steel box-girders, truss structures, RC structures, and other structures, respectively. Red curves are explained in Section 3.3.
Ficure 4: Frequencies of lst mode in the three directions (T, L, and V) as a function of largest span: (a) to (c) all typologies for V, T, L, and (d) V + L; (e) to (h) vertical for steel box-girders, truss structures, RC structures, and other structures, respectively. Red curves are explained in Section 3.3.
Ficure 5: Comparison of frequencies for steel box-girders and RC footbridges.
Ficure 5: Comparison of frequencies for steel box-girders and RC footbridges.
Ficure 6: Correlation of V and L frequencies for all footbridges.
Ficure 6: Correlation of V and L frequencies for all footbridges.
TABLE 2: Frequency range (Hz) for different patterns of movement (adapted from [14]). one attends to the frequency of movement, resulting from a speed of 0.5 m/s to 0.8 m/s (for slow walk) to 3.5 m/s (for jogging-slow running) with a step size from 0.65 m for slow walk and not exceeding 1.7 m for fast running (Table 2).
TABLE 2: Frequency range (Hz) for different patterns of movement (adapted from [14]). one attends to the frequency of movement, resulting from a speed of 0.5 m/s to 0.8 m/s (for slow walk) to 3.5 m/s (for jogging-slow running) with a step size from 0.65 m for slow walk and not exceeding 1.7 m for fast running (Table 2).
FiGure 7: (a) Force-time typical diagram for different movements: left-step frequency < 2.2 Hz, right-step frequency > 2.2 Hz; (b) variation in time-space imposed by the walking movement [15].
FiGure 7: (a) Force-time typical diagram for different movements: left-step frequency < 2.2 Hz, right-step frequency > 2.2 Hz; (b) variation in time-space imposed by the walking movement [15].
Ficure 8: Amplitude values in mg of the measured maximum motion at mid-span. Series 1: noise; Series 2: 1 person walking at “normal” pace (2 Hz); Series 3: 1 person jumping from 30 cm; Series 4: excitation close to resonant conditions.
Ficure 8: Amplitude values in mg of the measured maximum motion at mid-span. Series 1: noise; Series 2: 1 person walking at “normal” pace (2 Hz); Series 3: 1 person jumping from 30 cm; Series 4: excitation close to resonant conditions.
Ficure 9: Amplitude values in mg of the measured maximum motion at mid-span (T; L; V): 1 person walking at “normal” pace (2 Hz) in different typologies (Series 2).
Ficure 9: Amplitude values in mg of the measured maximum motion at mid-span (T; L; V): 1 person walking at “normal” pace (2 Hz) in different typologies (Series 2).
TABLE 3: Amplitude values in mg of the measured maximum motion at mid-span for a group of 10 footbridges in the transverse (T), longitudinal (L), and vertical (V) directions.
TABLE 3: Amplitude values in mg of the measured maximum motion at mid-span for a group of 10 footbridges in the transverse (T), longitudinal (L), and vertical (V) directions.
Figure 10: (a) Allowable vertical accelerations as a function of frequency of vertical mode [26], (b) frequencies to be avoided [14, 17]
Figure 10: (a) Allowable vertical accelerations as a function of frequency of vertical mode [26], (b) frequencies to be avoided [14, 17]
TABLE 4: Acceleration limits recommended in some codes (Figure 10(a)). Zivanovié et al. [11], Heinemayer et al.[29], SETRA [30], and HIVOSS Project [2] are three important references summarizing topics and restrictions for comfort and safety of footbridges. More recent advancements were produced and are presented in publications such as Footbridge-2008 [31] and Footbridge-2011 [32]. one instrument), the agreement is very good, also for high frequency modes.
TABLE 4: Acceleration limits recommended in some codes (Figure 10(a)). Zivanovié et al. [11], Heinemayer et al.[29], SETRA [30], and HIVOSS Project [2] are three important references summarizing topics and restrictions for comfort and safety of footbridges. More recent advancements were produced and are presented in publications such as Footbridge-2008 [31] and Footbridge-2011 [32]. one instrument), the agreement is very good, also for high frequency modes.
TABLE 5: Comparison of frequencies (Hz) between measurements in situ and the analytical model. TABLE 6: Peak acceleration amplitudes for different tests.
TABLE 5: Comparison of frequencies (Hz) between measurements in situ and the analytical model. TABLE 6: Peak acceleration amplitudes for different tests.
T Values of vertical accelerations exceeding the less stringent limit (93.8 mg); *the most stringent limit (66.7 mg); “values exceeding the code ones for horizontal vibrations (20 mg, EN 1990 [20]).
T Values of vertical accelerations exceeding the less stringent limit (93.8 mg); *the most stringent limit (66.7 mg); “values exceeding the code ones for horizontal vibrations (20 mg, EN 1990 [20]).
TABLE 7: Comparison of frequencies (Hz) between measurements in situ and the analytical model.
TABLE 7: Comparison of frequencies (Hz) between measurements in situ and the analytical model.
Figure H: Steel box-girder structure under study: (a) analytical model; (b) view of column, deck, and stairway.
Figure H: Steel box-girder structure under study: (a) analytical model; (b) view of column, deck, and stairway.
TABLE 8: Peak acceleration amplitudes for different tests. ' Values exceeding the code limits for horizontal vibration (20 mg). EN-1990 [20]).
TABLE 8: Peak acceleration amplitudes for different tests. ' Values exceeding the code limits for horizontal vibration (20 mg). EN-1990 [20]).
Ficure 12: Comparison of amplitudes at mid-span (vertical accel- eration) between in situ measurements and the analytical model: 1 person walking at various speeds (steel box-girder structure). Red circles: analytical model; blue squares; in situ measurements; vertical line at 2.2 Hz separates “walk” from “running”
Ficure 12: Comparison of amplitudes at mid-span (vertical accel- eration) between in situ measurements and the analytical model: 1 person walking at various speeds (steel box-girder structure). Red circles: analytical model; blue squares; in situ measurements; vertical line at 2.2 Hz separates “walk” from “running”
TABLE 10: Acceleration measured at 1/2 span for the normal walking. carried out at the same location but in a different epoch of the year.
TABLE 10: Acceleration measured at 1/2 span for the normal walking. carried out at the same location but in a different epoch of the year.
TABLE 9: Comparison of frequencies (Hz) between the analytical model and measurements in situ. of lateral lock-in where feedback of lateral forces cancels out the positive action of damping of the structure, resulting in unbounded growth of response [34]:
TABLE 9: Comparison of frequencies (Hz) between the analytical model and measurements in situ. of lateral lock-in where feedback of lateral forces cancels out the positive action of damping of the structure, resulting in unbounded growth of response [34]:
FiGurE 14: Comparison of amplitudes at mid-span (vertical accel- eration) between in situ measurements and the analytical model: one person walking at various speeds (RC structure). Red circles: analytical model; blue squares: in situ measurements; vertical line at 2.2 Hz separates “walk” from “running”
FiGurE 14: Comparison of amplitudes at mid-span (vertical accel- eration) between in situ measurements and the analytical model: one person walking at various speeds (RC structure). Red circles: analytical model; blue squares: in situ measurements; vertical line at 2.2 Hz separates “walk” from “running”
Figure 13: RC structure under study: (a) analytical model; (b) deck cross section; (c) view of column, deck, and stairway.
Figure 13: RC structure under study: (a) analytical model; (b) deck cross section; (c) view of column, deck, and stairway.
Ficure 15: (a) Vision of the CC Vasco da Gama (Calatrava) structure; and (b) analytical model.
Ficure 15: (a) Vision of the CC Vasco da Gama (Calatrava) structure; and (b) analytical model.
Ficure 16: First vibration mode with f = 0.84 Hz with predominant vertical movement: (a) front view; (b) from above; (c) lateral view.
Ficure 16: First vibration mode with f = 0.84 Hz with predominant vertical movement: (a) front view; (b) from above; (c) lateral view.
Ficure 17: RC footbridge in Faro: (a) front view; (b) cross-section with precast (I + slab + I); (c) central column (longitudinal view) (d) column-beam connection (units in m).
Ficure 17: RC footbridge in Faro: (a) front view; (b) cross-section with precast (I + slab + I); (c) central column (longitudinal view) (d) column-beam connection (units in m).
Ficure 18: Modes of vibration: (a) Ist longitudinal; (b) 2nd tranversal; (c) 3rd vertical, central span.
Ficure 18: Modes of vibration: (a) Ist longitudinal; (b) 2nd tranversal; (c) 3rd vertical, central span.
TABLE 11: Comparison of frequencies (Hz) between measurements in situ and the analytical model.
TABLE 11: Comparison of frequencies (Hz) between measurements in situ and the analytical model.
Ficure 1: Physical model of unbalanced rotating system with one SMA coil spring (k, c).
Ficure 1: Physical model of unbalanced rotating system with one SMA coil spring (k, c).
Ficure 3: Transformation temperatures of the NiTi SMA wire obtained by DSC.
Ficure 3: Transformation temperatures of the NiTi SMA wire obtained by DSC.
Ficure 2: Typical frequency response for the system under unbal- anced rotating.
Ficure 2: Typical frequency response for the system under unbal- anced rotating.
Ficure 4: Elastic modulus as a function of frequency for the NiTi SMA wire at three temperatures: 30°C, 50°C, and 70°C.
Ficure 4: Elastic modulus as a function of frequency for the NiTi SMA wire at three temperatures: 30°C, 50°C, and 70°C.
Ficure 6: Behavior of the elastic modulus of the NiTi SMA wire with the change of excitation frequency.
Ficure 6: Behavior of the elastic modulus of the NiTi SMA wire with the change of excitation frequency.
Ficure 5: Loss factor as a function of frequency for the NiTi SMA wire at three temperatures: 30°C, 50°C, and 70°C.
Ficure 5: Loss factor as a function of frequency for the NiTi SMA wire at three temperatures: 30°C, 50°C, and 70°C.
Ficure 7: Behavior of the loss factor of the NiTi SMA wire with the change of excitation frequency.
Ficure 7: Behavior of the loss factor of the NiTi SMA wire with the change of excitation frequency.
TABLE 2: Main parameters of the experimental test bench.
TABLE 2: Main parameters of the experimental test bench.
TABLE 1: Stiffness of the NiTi SMA spring actuator as a function of temperature.
TABLE 1: Stiffness of the NiTi SMA spring actuator as a function of temperature.
Ficure 9: Theoretical and experimental stiffness of the NiTi SMA spring as a function of temperature. Figure 8: Details of thermomechanical cycling for training the NiTi SMA spring. (a) Cooling (~3°C); (b) delay; (c) heating by electrical current (~80°C).
Ficure 9: Theoretical and experimental stiffness of the NiTi SMA spring as a function of temperature. Figure 8: Details of thermomechanical cycling for training the NiTi SMA spring. (a) Cooling (~3°C); (b) delay; (c) heating by electrical current (~80°C).
FiGcure 10: Experimental test bench: (1) SMA spring; (2) unbalanced motor; (3) minicooler; (4) electrical heating system (Joule effect); (5) accelerometer; (6) impact hammer; (7) LabVIEW Interface.
FiGcure 10: Experimental test bench: (1) SMA spring; (2) unbalanced motor; (3) minicooler; (4) electrical heating system (Joule effect); (5) accelerometer; (6) impact hammer; (7) LabVIEW Interface.
TABLE 3: Experimental damping factor of the mechanical system as a function of temperature. Table 4 allows verification of the validity of the relation- ship between the loss factor and the damping factor, 4 = 2¢.q- As this relationship is valid only when the system is in resonance, the loss factor (14) of the NiTi SMA spring was calculated from the simplification of (13), considering
TABLE 3: Experimental damping factor of the mechanical system as a function of temperature. Table 4 allows verification of the validity of the relation- ship between the loss factor and the damping factor, 4 = 2¢.q- As this relationship is valid only when the system is in resonance, the loss factor (14) of the NiTi SMA spring was calculated from the simplification of (13), considering
TABLE 4: Relationship between damping factor and loss factor for the resonance condition. the ratio of frequency equal to 1 (in resonance condition). Consider
TABLE 4: Relationship between damping factor and loss factor for the resonance condition. the ratio of frequency equal to 1 (in resonance condition). Consider
Ficure 12: Performance of the temperature control system for the SMA spring.
Ficure 12: Performance of the temperature control system for the SMA spring.
FicurE lH: Representation of the temperature control system for the SMA spring actuator.
FicurE lH: Representation of the temperature control system for the SMA spring actuator.
Ficure 13: Impulse response of the mechanical system at 25°C.
Ficure 13: Impulse response of the mechanical system at 25°C.
Ficure 16: Time response of the mechanical system in the resonance condition with temperature variation and SMA spring initially in the martensitic phase (blue curve; f,, = 10.83 Hz). Ficure 15: Damping factor of the mechanical system as a function of temperature.
Ficure 16: Time response of the mechanical system in the resonance condition with temperature variation and SMA spring initially in the martensitic phase (blue curve; f,, = 10.83 Hz). Ficure 15: Damping factor of the mechanical system as a function of temperature.
Ficure 14: Impulse response of the mechanical system at 70°C.
Ficure 14: Impulse response of the mechanical system at 70°C.
FiGure 17: Time response of the mechanical system in the resonance condition with temperature variation and SMA spring initially in the austenitic phase (red curve; f,, = 12.70 Hz).
FiGure 17: Time response of the mechanical system in the resonance condition with temperature variation and SMA spring initially in the austenitic phase (red curve; f,, = 12.70 Hz).
Ficure 19: Loss factor and damping factor of the mechanical system as a function of temperature. Figure 18: Time response of the mechanical system in resonance for the SMA spring at three temperatures.
Ficure 19: Loss factor and damping factor of the mechanical system as a function of temperature. Figure 18: Time response of the mechanical system in resonance for the SMA spring at three temperatures.
The null subspace (NSA) and enhanced-PCA methoc (EPCA) proposed in [3, 4], respectively, are variant method: of the PCA method obtained by exploiting Hankel matrices of the dynamical system [8]. The data-driven block Hanke! matrix is defined in (3), where 2i is a user-defined numbe! of row blocks, each block contains m rows (number o: measurement sensors), and j is the number of column: (practically j = N — 2i+ 1). The Hankel matrix H, ,; consist: of 2im rows and is split into two equal parts of i block rows which represent past and future data, respectively. Comparec to the observation matrix X, the Hankel matrix is buil using time-lagged vibration signals and not instantaneou: representations of responses. This enables taking into accoun' time correlations between measurements when current date depend on past data. Therefore, the objective pursued here ir using block Hankel matrices rather than observation matrice: is to improve the sensitivity of the detection method: where the subscripts of H, 5, denote the subscript of the first and last element of the first column in the block Hankel matrix.
The null subspace (NSA) and enhanced-PCA methoc (EPCA) proposed in [3, 4], respectively, are variant method: of the PCA method obtained by exploiting Hankel matrices of the dynamical system [8]. The data-driven block Hanke! matrix is defined in (3), where 2i is a user-defined numbe! of row blocks, each block contains m rows (number o: measurement sensors), and j is the number of column: (practically j = N — 2i+ 1). The Hankel matrix H, ,; consist: of 2im rows and is split into two equal parts of i block rows which represent past and future data, respectively. Comparec to the observation matrix X, the Hankel matrix is buil using time-lagged vibration signals and not instantaneou: representations of responses. This enables taking into accoun' time correlations between measurements when current date depend on past data. Therefore, the objective pursued here ir using block Hankel matrices rather than observation matrice: is to improve the sensitivity of the detection method: where the subscripts of H, 5, denote the subscript of the first and last element of the first column in the block Hankel matrix.
FicureE 1: Angle @ formed by active subspaces according to the reference and current states, due to a dynamic change.
FicureE 1: Angle @ formed by active subspaces according to the reference and current states, due to a dynamic change.
TABLE |: Description of the damage scenarios according to the cutting sections shown in Figure 2.
TABLE |: Description of the damage scenarios according to the cutting sections shown in Figure 2.
TABLE 2: Change in the eigenfrequencies (identified by SSI). with an elastic modulus of 42700 N/mm” and a measured compressive strength of 58.3 N/mm* (quality control of man- ufacturer). The quality of the reinforcement is St 1470/1670 and the corresponding elastic modulus 205000 N/mm”. In the upper section of the panel, there are 4 wires with a diameter of 5mm and in the lower Section 12 wires with a diameter of 7 mm. Before testing, the concrete at the bottom side in the middle of the slab along axis C (Figure 6(b)) was removed, as shown in Figure 6(a), to give access to the reinforcement for the later procedures of cutting tendons.
TABLE 2: Change in the eigenfrequencies (identified by SSI). with an elastic modulus of 42700 N/mm” and a measured compressive strength of 58.3 N/mm* (quality control of man- ufacturer). The quality of the reinforcement is St 1470/1670 and the corresponding elastic modulus 205000 N/mm”. In the upper section of the panel, there are 4 wires with a diameter of 5mm and in the lower Section 12 wires with a diameter of 7 mm. Before testing, the concrete at the bottom side in the middle of the slab along axis C (Figure 6(b)) was removed, as shown in Figure 6(a), to give access to the reinforcement for the later procedures of cutting tendons.
FiGuRE 5: Damage detection results using EPCA. FiGuRE 4: Location of the sensors on the bridge deck.
FiGuRE 5: Damage detection results using EPCA. FiGuRE 4: Location of the sensors on the bridge deck.
FiGurE 7: Measurement setup: impact point (101-145 and 201-245) and accelerometer positions (Ref. 1-Ref. 3).
FiGurE 7: Measurement setup: impact point (101-145 and 201-245) and accelerometer positions (Ref. 1-Ref. 3).
TABLE 3: Damage scenarios.
TABLE 3: Damage scenarios.
Ficure 8: EPCA detection for the RC slab (always unloaded state).
Ficure 8: EPCA detection for the RC slab (always unloaded state).
TABLE 4: Description of damages. TABLE 5: The shift of the first eigenfrequency (Hz) from the intact state until before the collapse (always unloaded state).
TABLE 4: Description of damages. TABLE 5: The shift of the first eigenfrequency (Hz) from the intact state until before the collapse (always unloaded state).
Let us consider the Frequency Response Functions (FRFs) H*(w) for a single input at location s:
Let us consider the Frequency Response Functions (FRFs) H*(w) for a single input at location s:
Ficure 9: EPCA detection for the PrC slab (always unloaded state).
Ficure 9: EPCA detection for the PrC slab (always unloaded state).
Ficure 10: Comparison of mode-shapes obtained by SSI and the sensitivity analysis (x: position of support).
Ficure 10: Comparison of mode-shapes obtained by SSI and the sensitivity analysis (x: position of support).
Ficure 11: Damage localization in the RC slab for damage #2”.
Ficure 11: Damage localization in the RC slab for damage #2”.
Ficure 12: Damage localization in the PrC slab for damage #2", based on mode 2.
Ficure 12: Damage localization in the PrC slab for damage #2", based on mode 2.
Ficure 1: A schematic representation (a) and the actual realization (b) of the experimental setup.
Ficure 1: A schematic representation (a) and the actual realization (b) of the experimental setup.
FiGurE 2: The experimental data excerpt (8192 samples, sampling frequency 192 kHz) used in the ZAM analysis for the uncracked, 7%, 20%, and 45% of the crack depth cases ((a) to (d)).
FiGurE 2: The experimental data excerpt (8192 samples, sampling frequency 192 kHz) used in the ZAM analysis for the uncracked, 7%, 20%, and 45% of the crack depth cases ((a) to (d)).
FiGure 3: Results from the ZAM analysis of the experimental data of Figure 2 for 0% (a), 7% (c), 20% (b), and 45% (d) crack size, respectively.
FiGure 3: Results from the ZAM analysis of the experimental data of Figure 2 for 0% (a), 7% (c), 20% (b), and 45% (d) crack size, respectively.
Ficure 4: The amplitude fluctuation of the ZAM transform at the corresponding VAM frequencies, that is, 31392 Hz (a), 31300 Hz (b), and 31208 Hz (c) for each crack size (0%, 7%, 20%, and 45%), respectively.
Ficure 4: The amplitude fluctuation of the ZAM transform at the corresponding VAM frequencies, that is, 31392 Hz (a), 31300 Hz (b), and 31208 Hz (c) for each crack size (0%, 7%, 20%, and 45%), respectively.
Ficure 8: The damage index (DI) derived as the mean value of the sDI for the case of the 1/|ZAM,(t, f)| (Figure 7, first row)-DIz,.4 and the |FFT(f)| (Figure 7, fourth row)-DI,p;. The increase in the sensitivity of the DI,,,, over the DIppy is evident.
Ficure 8: The damage index (DI) derived as the mean value of the sDI for the case of the 1/|ZAM,(t, f)| (Figure 7, first row)-DIz,.4 and the |FFT(f)| (Figure 7, fourth row)-DI,p;. The increase in the sensitivity of the DI,,,, over the DIppy is evident.
TABLE 1: Modal properties of the simulated 6-DOF system. been used also to validate an innovative automated OMA procedure in [9]. The selected real test cases are representative of modal identification problems typically encountered in civil engineering and characterized by different degree of difficulty. Reference modal parameters have been extracted from these records by well-established techniques, such as frequency domain decomposition (FDD) [31] and stochastic subspace identification (SSD) [1, 32], which have provided very consistent estimates. Figure 1: The benchmark 6-DOF system.
TABLE 1: Modal properties of the simulated 6-DOF system. been used also to validate an innovative automated OMA procedure in [9]. The selected real test cases are representative of modal identification problems typically encountered in civil engineering and characterized by different degree of difficulty. Reference modal parameters have been extracted from these records by well-established techniques, such as frequency domain decomposition (FDD) [31] and stochastic subspace identification (SSD) [1, 32], which have provided very consistent estimates. Figure 1: The benchmark 6-DOF system.
FiGuRE 2: Sensitivity of computational time (sample plot). The assessment of the influence of t and p on the modal identification results can provide effective hints to ensure accurate modal estimates or to reduce the computational time with little or no accuracy losses. To this aim, the simulated dataset has been processed by SOBI, and the identified modes in the range 0-5 Hz have been compared with the theoretical values for different settings of t and p. The cumulative frequency scatter J; and the cumulative discrepancy between corresponding mode shapes J,
FiGuRE 2: Sensitivity of computational time (sample plot). The assessment of the influence of t and p on the modal identification results can provide effective hints to ensure accurate modal estimates or to reduce the computational time with little or no accuracy losses. To this aim, the simulated dataset has been processed by SOBI, and the identified modes in the range 0-5 Hz have been compared with the theoretical values for different settings of t and p. The cumulative frequency scatter J; and the cumulative discrepancy between corresponding mode shapes J,
TABLE 2: Test cases, modal identification results, and comparisons. SOBI-based automated OMA procedures for vibration-based SHM. It is worth pointing out that the possibility of automati- cally setting the analysis parameters without any preliminary calibration is a fundamental requirement for the development of automated OMA procedures. An effective control of computational efforts is possible by appropriate setting of t, taking into account that it negatively affects the accuracy of modal parameter estimates beyond the limit value of 1E — 4. Thus, the present paper provides a contribution towards the development of innovative automated OMA procedures able to satisfy widely accepted target criteria reported in the literature [6, 7, 9]. However, automated OMA based on SOBI is out of the scope of the paper.
TABLE 2: Test cases, modal identification results, and comparisons. SOBI-based automated OMA procedures for vibration-based SHM. It is worth pointing out that the possibility of automati- cally setting the analysis parameters without any preliminary calibration is a fundamental requirement for the development of automated OMA procedures. An effective control of computational efforts is possible by appropriate setting of t, taking into account that it negatively affects the accuracy of modal parameter estimates beyond the limit value of 1E — 4. Thus, the present paper provides a contribution towards the development of innovative automated OMA procedures able to satisfy widely accepted target criteria reported in the literature [6, 7, 9]. However, automated OMA based on SOBI is out of the scope of the paper.
Ficure 4: Illustration of the rule of thumbs for setting of p.
Ficure 4: Illustration of the rule of thumbs for setting of p.
Ficure 3: Sensitivity of overall accuracy (a) and mode shape accuracy (b) to p and t.
Ficure 3: Sensitivity of overall accuracy (a) and mode shape accuracy (b) to p and t.
Figure 5: Validation of the proposed rule for setting of p (t = 10° real dataset: “School of Engineering in Naples”).
Figure 5: Validation of the proposed rule for setting of p (t = 10° real dataset: “School of Engineering in Naples”).
Figure 1: Typical section airfoil.
Figure 1: Typical section airfoil.
FiGcure 2: Freeplay nonlinearity.
FiGcure 2: Freeplay nonlinearity.
2.1. Representation in State Space Form. In order to obtain a nonlinear state space model, a rational function is used to write the aerodynamic forces in time domain. Jones in 1940 solved this problem using a rational function approximation to approximate unsteady aerodynamic loads for the typical section [24]. The expressions in (2) are rearranged ina matrix form and the nonlinear function f,) is introduced such that the stiffness is defined as a nonlinear structural stiffness matrix K,,; given by
2.1. Representation in State Space Form. In order to obtain a nonlinear state space model, a rational function is used to write the aerodynamic forces in time domain. Jones in 1940 solved this problem using a rational function approximation to approximate unsteady aerodynamic loads for the typical section [24]. The expressions in (2) are rearranged ina matrix form and the nonlinear function f,) is introduced such that the stiffness is defined as a nonlinear structural stiffness matrix K,,; given by
where s is the Laplace variable, ),, is the number of lag terms, f; is the jth lag parameter (j = 1,...,m,g), and b is the aerodynamic semichord. In this paper Roger’s approach is used. This approxi- mation involves identifying every matrix Q; shown in (5) using a Least Square algorithm as proposed by Roger [29] and summarized in [30]. The approximation contains a polynomial part representing the forces on the typical section acting directly connected to the displacements u(t) and their first and second derivatives. Also, this equation has a rational part representing the influence of the wake acting on the section with a time delay. Consider
where s is the Laplace variable, ),, is the number of lag terms, f; is the jth lag parameter (j = 1,...,m,g), and b is the aerodynamic semichord. In this paper Roger’s approach is used. This approxi- mation involves identifying every matrix Q; shown in (5) using a Least Square algorithm as proposed by Roger [29] and summarized in [30]. The approximation contains a polynomial part representing the forces on the typical section acting directly connected to the displacements u(t) and their first and second derivatives. Also, this equation has a rational part representing the influence of the wake acting on the section with a time delay. Consider
where M, and D, are, respectively, the aeroelastic mass and damping matrices, K,;,:) is the nonlinear aeroelastic stiffness matrix, B is the input matrix, I is an identity matrix, and 0 is a matrix of zeros with appropriated dimension.
where M, and D, are, respectively, the aeroelastic mass and damping matrices, K,;,:) is the nonlinear aeroelastic stiffness matrix, B is the input matrix, I is an identity matrix, and 0 is a matrix of zeros with appropriated dimension.
TABLE 1: Physical and geometric properties of the 2D airfoil.
TABLE 1: Physical and geometric properties of the 2D airfoil.
Ficure 4: Linear flutter analysis: V-f diagram. Ficure 3: The nonlinear function f,,; represented in a particular range of B(t).
Ficure 4: Linear flutter analysis: V-f diagram. Ficure 3: The nonlinear function f,,; represented in a particular range of B(t).
3.1. Closed-Loop Takagi-Sugeno Controller. In this section a feedback control strategy is used to control the amplitude of oscillation in the control surface. The feedback force is obtained by applying a feedback gain to the state systems states, such that the control inputs are given by u,(t) = —Gx(t), where G is the feedback gain matrix. The feedback gain is calculated using linear matrix inequality to solve Lyapunov’s function for stability, in this case
3.1. Closed-Loop Takagi-Sugeno Controller. In this section a feedback control strategy is used to control the amplitude of oscillation in the control surface. The feedback force is obtained by applying a feedback gain to the state systems states, such that the control inputs are given by u,(t) = —Gx(t), where G is the feedback gain matrix. The feedback gain is calculated using linear matrix inequality to solve Lyapunov’s function for stability, in this case
Ficure 5: Linear flutter analysis: V-g diagram.
Ficure 5: Linear flutter analysis: V-g diagram.
Figure 8: Linear equivalent stiffness from the HBM.
Figure 8: Linear equivalent stiffness from the HBM.
FiGuRE 9: LCO frequencies obtained from the HBM.
FiGuRE 9: LCO frequencies obtained from the HBM.
Ficure ll: Case 1: control surface rotation (|6] = 0.4° and V = 12.5 m/s). Ficure 10: Values of f,: actual versus computed using TS approach.
Ficure ll: Case 1: control surface rotation (|6] = 0.4° and V = 12.5 m/s). Ficure 10: Values of f,: actual versus computed using TS approach.
FiGure 12: Case 1: control surface rotation (|6| = 0.6° and V = 12.5 m/s). 4.1. Linear Flutter Boundary. Figure 1 illustrates the model and its physical and geometric properties are presented in Table 1. Figures 4 and 5 show, respectively, the classical V- f and V-g diagrams for the linear flutter solution. According to these results flutter speed is equal to 12.7 m/s.
FiGure 12: Case 1: control surface rotation (|6| = 0.6° and V = 12.5 m/s). 4.1. Linear Flutter Boundary. Figure 1 illustrates the model and its physical and geometric properties are presented in Table 1. Figures 4 and 5 show, respectively, the classical V- f and V-g diagrams for the linear flutter solution. According to these results flutter speed is equal to 12.7 m/s.
Ficure 14: Case: phase plan (|5| = 0.6° and V = 12.5 m/s). to 0.001 seconds. To compute the control gain, a column input matrix was used to represent an actuator applying force on the control surface (B,. = {0 0 1}7). Also, it was considered that the parameter g is equal to 1.0 for all cases. However, in particular for experimental applications where limitations exist on the actuator force, the parameter g can be conveniently chosen between ]0, 1[. functions, such that
Ficure 14: Case: phase plan (|5| = 0.6° and V = 12.5 m/s). to 0.001 seconds. To compute the control gain, a column input matrix was used to represent an actuator applying force on the control surface (B,. = {0 0 1}7). Also, it was considered that the parameter g is equal to 1.0 for all cases. However, in particular for experimental applications where limitations exist on the actuator force, the parameter g can be conveniently chosen between ]0, 1[. functions, such that
Ficure 13: Case 1: phase plan (|6| = 0.4° and V = 12.5 m/s).
Ficure 13: Case 1: phase plan (|6| = 0.4° and V = 12.5 m/s).
Ficure 15: Case 2: control surface rotation (|6| = 0.4° and V = 13.1 m/s).
Ficure 15: Case 2: control surface rotation (|6| = 0.4° and V = 13.1 m/s).
FiGurE 17: Case 2: zoom selection area for the control surface rotation (|6| = 0.6° and V = 13.1 m/s). FiGurE 16: Case 2: control surface rotation (|6| = 0.6° and V = 13.1 m/s).
FiGurE 17: Case 2: zoom selection area for the control surface rotation (|6| = 0.6° and V = 13.1 m/s). FiGurE 16: Case 2: control surface rotation (|6| = 0.6° and V = 13.1 m/s).
Ficure 19: Case 2: phase plan (|5| = 0.6° and V = 13.1 m/s).
Ficure 19: Case 2: phase plan (|5| = 0.6° and V = 13.1 m/s).
Ficure 18: Case 2: phase plan (|6| = 0.4° and V = 13.1 m/s).
Ficure 18: Case 2: phase plan (|6| = 0.4° and V = 13.1 m/s).
Ficure 21: Illustrative scheme of integration process using Hénon’s technique. FiGurRE 20: Case 2: control force (|6| = 0.6° and V = 13.1 m/s).
Ficure 21: Illustrative scheme of integration process using Hénon’s technique. FiGurRE 20: Case 2: control force (|6| = 0.6° and V = 13.1 m/s).
and the linearised equations would remain coupled (unlike (5)) and consequently the gains in (6) would take a different form, as Clearly, there are a greater number of control gains in (8) than in (6), which means that there is more control flexibility, which might be used, for example, to assign the eigenvectors as well as the eigenvalues. This may be readily achieved using methods such as that presented in [11] and is illustrated through a numerical example later on.
and the linearised equations would remain coupled (unlike (5)) and consequently the gains in (6) would take a different form, as Clearly, there are a greater number of control gains in (8) than in (6), which means that there is more control flexibility, which might be used, for example, to assign the eigenvectors as well as the eigenvalues. This may be readily achieved using methods such as that presented in [11] and is illustrated through a numerical example later on.
and the coordinate system which maps the original nonlinear system to the partially linearised system may be expressed as expressions to those obtained for the complete input-output case presented above. Equation (2) is now rewritten as
and the coordinate system which maps the original nonlinear system to the partially linearised system may be expressed as expressions to those obtained for the complete input-output case presented above. Equation (2) is now rewritten as
As with the full output feedback case, it is now necessary to choose actual inputs so that the nonlinearity is eliminated. This is achieved by which is identical to the full output feedback case, except of course that there are now only m outputs. Further (n-m) coordinates are needed and are chosen as coefficients of the orthonormal basis of the null space of Gi. nim) 8O that
As with the full output feedback case, it is now necessary to choose actual inputs so that the nonlinearity is eliminated. This is achieved by which is identical to the full output feedback case, except of course that there are now only m outputs. Further (n-m) coordinates are needed and are chosen as coefficients of the orthonormal basis of the null space of Gi. nim) 8O that
Ficure 1: The two deflection patterns assumed for the flexible wing.
Ficure 1: The two deflection patterns assumed for the flexible wing.
A simplification in the matrix subscripts previously defined in [13] has been made here. The control force distribution matrix is given (as a function of air speed) by where f, and f, are control surface (flap) angles and T,,, is a control torque applied directly to the pylon-engine assembly. The nonlinear internal force is given by
A simplification in the matrix subscripts previously defined in [13] has been made here. The control force distribution matrix is given (as a function of air speed) by where f, and f, are control surface (flap) angles and T,,, is a control torque applied directly to the pylon-engine assembly. The nonlinear internal force is given by
The aeroservoelastic matrices are given in general terms in [13] and expressed here in terms of specific parameter values in consistent units (to three significant figures) as and the nonlinear time-domain response to initial conditions is produced at an air speed just above the flutter speed.
The aeroservoelastic matrices are given in general terms in [13] and expressed here in terms of specific parameter values in consistent units (to three significant figures) as and the nonlinear time-domain response to initial conditions is produced at an air speed just above the flutter speed.
FiGurE 3: Steady-state LCO in 9, response.
FiGurE 3: Steady-state LCO in 9, response.
Figure 2: V-w and V-¢ plots for the wing-pylon-engine model.
Figure 2: V-w and V-¢ plots for the wing-pylon-engine model.
Ficure 4: Decaying response, just below linear flutter speed.
Ficure 4: Decaying response, just below linear flutter speed.
Figure 5: Feedback linearisation: response at 80ms™' (assumed- modes coordinates).
Figure 5: Feedback linearisation: response at 80ms™' (assumed- modes coordinates).
Ficure 6: Control surface deflection angles and actuator torques for exact feedback linearization.
Ficure 6: Control surface deflection angles and actuator torques for exact feedback linearization.
Ficure 7: Feedback linearisation with cancellation of nonlinearity only and with eigenvector assignment: response at 80 ms" (assumed- modes coordinates).
Ficure 7: Feedback linearisation with cancellation of nonlinearity only and with eigenvector assignment: response at 80 ms" (assumed- modes coordinates).
Ficure 8: Control surface deflection angles and actuator torques for exact feedback linearisation: cancellation of nonlinearity only, with assignment of eigenvectors. As before, feedback linearisation successfully places the poles to the desired values and the transient response shown in Figure 10 is obtained. It is evident from the third subplot in Figure 10 that the coordinate q,, eventually stabilises because the zero dynamics of the system are stable. When a +40% error in K,, is incorporated and the closed-loop response is simulated based on the same feedback parameters instability does not occur (as in the 3130 case) but a degradation in performance is observed in Figure 11 where it is seen that the control fails to eliminate an LCO.
Ficure 8: Control surface deflection angles and actuator torques for exact feedback linearisation: cancellation of nonlinearity only, with assignment of eigenvectors. As before, feedback linearisation successfully places the poles to the desired values and the transient response shown in Figure 10 is obtained. It is evident from the third subplot in Figure 10 that the coordinate q,, eventually stabilises because the zero dynamics of the system are stable. When a +40% error in K,, is incorporated and the closed-loop response is simulated based on the same feedback parameters instability does not occur (as in the 3130 case) but a degradation in performance is observed in Figure 11 where it is seen that the control fails to eliminate an LCO.
where T is defined in (18) and where the erroneous nonlinear parameter is given by Equation (31) shows that the nonlinear parameter error results in an input to the closed-loop system. It may be readily shown, for the 3130 system, that and the second-order equation of the closed-loop system is then cast as
where T is defined in (18) and where the erroneous nonlinear parameter is given by Equation (31) shows that the nonlinear parameter error results in an input to the closed-loop system. It may be readily shown, for the 3130 system, that and the second-order equation of the closed-loop system is then cast as
Ficure 9: Feedback-linearisation with nonlinear parameter error: response at 80 ms".
Ficure 9: Feedback-linearisation with nonlinear parameter error: response at 80 ms".
Figure 10: Feedback-linearisation: response at 97.5 ms! (assumed-modes coordinates). Thus, the initially assumed value Kr of the nonlinear param- eter is continually updated at each time step. Effectively, this increases the dimension of the state vector by 1, as Kr becomes part of the system state. Now, choosing P such that for arbitrary Q > 0,
Figure 10: Feedback-linearisation: response at 97.5 ms! (assumed-modes coordinates). Thus, the initially assumed value Kr of the nonlinear param- eter is continually updated at each time step. Effectively, this increases the dimension of the state vector by 1, as Kr becomes part of the system state. Now, choosing P such that for arbitrary Q > 0,
Ficure ll: Feedback-linearisation with nonlinear parameter error: response at 97.5 ms‘.
Ficure ll: Feedback-linearisation with nonlinear parameter error: response at 97.5 ms‘.
FIGURE 12: 3130 system with adaptive feedback linearisation: response at 80 ms!
FIGURE 12: 3130 system with adaptive feedback linearisation: response at 80 ms!
Ficure 13: 2120 system with feedback linearisation: response at 97.5 ms’.
Ficure 13: 2120 system with feedback linearisation: response at 97.5 ms’.
Control forces are applied to the wing-pylon-engine system by means of control surfaces. It is assumed in this work that two control surfaces (different from [14]) are available, the first (closest to the wing root) spanning 85% of the length of the wing and the second spanning the remaining length (the contribution of the control surfaces to the dynamics of the overall system is neglected). The widths of the first and second control surfaces are assumed to be 20% and 33.33% of the chord length, respectively. These particular dimensions have been chosen so as to optimise the distribution of work performed by each control surface. In addition to the ailerons, it is assumed that a separate actuator is available to apply a torque T,, directly on the engine rotational degree of freedom. The forcing vector is found to be
Control forces are applied to the wing-pylon-engine system by means of control surfaces. It is assumed in this work that two control surfaces (different from [14]) are available, the first (closest to the wing root) spanning 85% of the length of the wing and the second spanning the remaining length (the contribution of the control surfaces to the dynamics of the overall system is neglected). The widths of the first and second control surfaces are assumed to be 20% and 33.33% of the chord length, respectively. These particular dimensions have been chosen so as to optimise the distribution of work performed by each control surface. In addition to the ailerons, it is assumed that a separate actuator is available to apply a torque T,, directly on the engine rotational degree of freedom. The forcing vector is found to be
Evidently, the internal dynamics expressions in (B.4) are nonlinear. Now, the zero dynamics may be obtained by setting to zero the coordinates corresponding to the external dynamics (i.e. the partially linearised system). In this case, the external coordinates are Z,, Z,, Z;, Z, so that the zero dynamics are given by
Evidently, the internal dynamics expressions in (B.4) are nonlinear. Now, the zero dynamics may be obtained by setting to zero the coordinates corresponding to the external dynamics (i.e. the partially linearised system). In this case, the external coordinates are Z,, Z,, Z;, Z, so that the zero dynamics are given by
The roots are found to be complex with negative real parts, thus revealing the type of equilibrium point to be a stable focus. The nature of the above equilibrium point may be determined by examining the eigenvalues of (B.6), evaluated at the equilibrium point,
The roots are found to be complex with negative real parts, thus revealing the type of equilibrium point to be a stable focus. The nature of the above equilibrium point may be determined by examining the eigenvalues of (B.6), evaluated at the equilibrium point,
The objective function contains the residuals between the target frequency eigenvalues (A ;) with both the eigenvalues associated to the system natural frequencies (A,) and the MRQA (u;). The a; and b; are adequate weighting factors. In the synthesis defined by (6)-(8) we will adjust the design variables and thus the stiffness and mass matrices, such that yu; and A, will both converge to the natural prescribed target frequency eigenvalues, A ;- For instance, admitting that during the optimization iterations the MRQA will converge to the target frequency eigenvalue (u; — r ;) implies that the mode shape will also converge to @;; that is, we will havep; — @;. The MRQA can be used to define the following first state- ment of our synthesis problem, where we want to generate a structure that will have its eigenvalues and mode shapes the closest possible to the prescribed target pairs (A;,;), j = 1,...,s as follows:
The objective function contains the residuals between the target frequency eigenvalues (A ;) with both the eigenvalues associated to the system natural frequencies (A,) and the MRQA (u;). The a; and b; are adequate weighting factors. In the synthesis defined by (6)-(8) we will adjust the design variables and thus the stiffness and mass matrices, such that yu; and A, will both converge to the natural prescribed target frequency eigenvalues, A ;- For instance, admitting that during the optimization iterations the MRQA will converge to the target frequency eigenvalue (u; — r ;) implies that the mode shape will also converge to @;; that is, we will havep; — @;. The MRQA can be used to define the following first state- ment of our synthesis problem, where we want to generate a structure that will have its eigenvalues and mode shapes the closest possible to the prescribed target pairs (A;,;), j = 1,...,s as follows:
TABLE |: Target frequencies (Case 1). TABLE 2: Target mode shapes (Case 1).
TABLE |: Target frequencies (Case 1). TABLE 2: Target mode shapes (Case 1).
The use of the mass augmented objective function of (9) instead of (6) may lead to a compromise solution in which the system mass is minimized but the adjustment between the frequencies and mode shapes to their prescribed values may not be the best. Thus, the range constraints are added ((10)-(11)) so that the problem statement is now given by (9)- (13). Range constraints are used instead of strict equality constraints for two reasons. Firstly, satisfaction of equality of frequencies and mode shapes to their prescribed target values may not be possible depending on the design variables used for the synthesis [3]. Also, because the numerical optimiza- tion solution tends to be harder for strict equality constraints, even for the case where they are realizable. Here the multi- pliers p < 1 and q > 1 are parameters defining the ranges; for example, p = 1-6 and q = 1+, where 6 is adjusted during the optimization, departing from say 6 = 0.1, and closing the range with say 6 = 0.0001. Experience with simple cases now shows that good solutions can be obtained adjust- ing the ranges smoothly, by means of solving a sequential optimization with decreasing ranges such that in the ith opti- mization problem 6” = r 6°), where r < 1; for example, 0.1 < r < 0.5. Because of the new constraints we can choose the weights a; and b; to be null, so defining a cleaner mass- j only objective function. objective function and also range constraints for the frequen- cies eigenvalues and the MRQAs:
The use of the mass augmented objective function of (9) instead of (6) may lead to a compromise solution in which the system mass is minimized but the adjustment between the frequencies and mode shapes to their prescribed values may not be the best. Thus, the range constraints are added ((10)-(11)) so that the problem statement is now given by (9)- (13). Range constraints are used instead of strict equality constraints for two reasons. Firstly, satisfaction of equality of frequencies and mode shapes to their prescribed target values may not be possible depending on the design variables used for the synthesis [3]. Also, because the numerical optimiza- tion solution tends to be harder for strict equality constraints, even for the case where they are realizable. Here the multi- pliers p < 1 and q > 1 are parameters defining the ranges; for example, p = 1-6 and q = 1+, where 6 is adjusted during the optimization, departing from say 6 = 0.1, and closing the range with say 6 = 0.0001. Experience with simple cases now shows that good solutions can be obtained adjust- ing the ranges smoothly, by means of solving a sequential optimization with decreasing ranges such that in the ith opti- mization problem 6” = r 6°), where r < 1; for example, 0.1 < r < 0.5. Because of the new constraints we can choose the weights a; and b; to be null, so defining a cleaner mass- j only objective function. objective function and also range constraints for the frequen- cies eigenvalues and the MRQAs:
TABLE 3: Optimal design variables (Case 1). TABLE 4: Optimal frequencies and MRQA (Case 1).
TABLE 3: Optimal design variables (Case 1). TABLE 4: Optimal frequencies and MRQA (Case 1).
TABLE 5: Optimal mode shapes (Case 1).
TABLE 5: Optimal mode shapes (Case 1).
TABLE 6: Sequential optimization iteration history (Case 1).
TABLE 6: Sequential optimization iteration history (Case 1).
The optimal values of the six design variables used to minimize the mass and satisfy the target frequencies and mode shapes are presented in Table 3 and correspond to the total mass of 188.46kg, from which 141.68 kg is made of concentrated masses. The optimal beam has a depth of 0.075 m. The optimal solution is remarkable from the point of view of reduction of the structural and nonstructural masses, when compared to the reference design where the beam depth is 0.020 m and the nonstructural mass is 380 kg.
The optimal values of the six design variables used to minimize the mass and satisfy the target frequencies and mode shapes are presented in Table 3 and correspond to the total mass of 188.46kg, from which 141.68 kg is made of concentrated masses. The optimal beam has a depth of 0.075 m. The optimal solution is remarkable from the point of view of reduction of the structural and nonstructural masses, when compared to the reference design where the beam depth is 0.020 m and the nonstructural mass is 380 kg.
TABLE 7: Target frequencies (Case 2). TABLE 8: Target mode shapes (Case 2).
TABLE 7: Target frequencies (Case 2). TABLE 8: Target mode shapes (Case 2).
TABLE 9: Optimal design variables results (Case 2A).
TABLE 9: Optimal design variables results (Case 2A).
TABLE 10: Optimal frequencies and MRQA (Case 2A).
TABLE 10: Optimal frequencies and MRQA (Case 2A).
TABLE 14: Optimal mode shapes (Case 2B).
TABLE 14: Optimal mode shapes (Case 2B).
TABLE 13: Optimal frequencies and MRQA (Case 2B).
TABLE 13: Optimal frequencies and MRQA (Case 2B).
TABLE 12: Optimal design variables results (Case 2B).
TABLE 12: Optimal design variables results (Case 2B).
TABLE 15: Target frequencies (Case 3).
TABLE 15: Target frequencies (Case 3).
TABLE 11: Optimal mode shapes (Case 2A).
TABLE 11: Optimal mode shapes (Case 2A).
FiGure 5: Optimal and target mode shapes (Case 2B).
FiGure 5: Optimal and target mode shapes (Case 2B).
TABLE 17: Optimal frequencies and MRQA (Case 3). TABLE 18: Optimal mode shapes (Case 3).
TABLE 17: Optimal frequencies and MRQA (Case 3). TABLE 18: Optimal mode shapes (Case 3).
TABLE 16: Sequential optimization iteration history (Case 3).
TABLE 16: Sequential optimization iteration history (Case 3).
Ficure 1: Complex Morlet wavelet function. 3.2. Numerical Implementation. In practice numerical imple- mentation is composed of several steps. Firstly, the autopower functions of input and output signals have to be defined as to avoid the above problem. In practice, the random Gaussian white noise N is added to the input and output time signals to form biased (noisy) inputs x,(t) and outputs y,(t) as
Ficure 1: Complex Morlet wavelet function. 3.2. Numerical Implementation. In practice numerical imple- mentation is composed of several steps. Firstly, the autopower functions of input and output signals have to be defined as to avoid the above problem. In practice, the random Gaussian white noise N is added to the input and output time signals to form biased (noisy) inputs x,(t) and outputs y,(t) as
FiGuRE 2: Impact excitation given in the time, frequency, and combined time-frequency domains.
FiGuRE 2: Impact excitation given in the time, frequency, and combined time-frequency domains.
FIGURE 3: Chirp excitation given in the time, frequency, and combined time-frequency domains.
FIGURE 3: Chirp excitation given in the time, frequency, and combined time-frequency domains.
FicurE 4: White noise excitation given in the time, frequency, and combined time-frequency domains.
FicurE 4: White noise excitation given in the time, frequency, and combined time-frequency domains.
Figure 5: Random impact excitation in the time, frequency, and combined time-frequency domains.
Figure 5: Random impact excitation in the time, frequency, and combined time-frequency domains.
Ficure 6: Schematic diagram of the 2-DOF system used in numer- ical simulations.
Ficure 6: Schematic diagram of the 2-DOF system used in numer- ical simulations.
FiGurE 7: Simulated results obtained for the impact excitation in the time, frequency, and combined time-frequency domains.
FiGurE 7: Simulated results obtained for the impact excitation in the time, frequency, and combined time-frequency domains.
Figure 8: Simulated results obtained for the chirp excitation in the time, frequency, and combined time-frequency domains.
Figure 8: Simulated results obtained for the chirp excitation in the time, frequency, and combined time-frequency domains.
FIGURE 9: Simulated results obtained for the white noise excitation in the time, frequency, and combined time-frequency domains.
FIGURE 9: Simulated results obtained for the white noise excitation in the time, frequency, and combined time-frequency domains.
FIGURE 10: Simulated results obtained for the repeated impact excitation in the time, frequency, and combined time-frequency domains.
FIGURE 10: Simulated results obtained for the repeated impact excitation in the time, frequency, and combined time-frequency domains.
FiGure Ll: Experimental time-variant test rig.
FiGure Ll: Experimental time-variant test rig.
FicurE 13: Experimental results obtained for the repeated impact excitation in combined time-frequency domain. Figure 12: Experimental results obtained for the white noise excitation in the combined time-frequency domain.
FicurE 13: Experimental results obtained for the repeated impact excitation in combined time-frequency domain. Figure 12: Experimental results obtained for the white noise excitation in the combined time-frequency domain.
Ficure 1: A typical vehicle-bridge interaction system. 2.1. Equation of Motion. When the vehicle moves from one end of the bridge to the other end at a constant speed, both the bridge and the vehicle will vibrate vertically. A vector u,(t) is used to denote the vertical displacements of a series of nodes in the finite element model of the bridge. Its first and second derivatives with respect to time ¢, that is, u(t) and ii,(t), are, respectively, the vertical velocity and acceleration of the corresponding nodes. The symbols q,(t) and q,(t) denote the vertical displacement of the wheel and the car body, respectively. As they interact with each other by the contact force, the vibration of the vehicle is influenced by the vibration of the bridge and vice versa. So this is a coupled vibration system. It is assumed that the mass of vehicle is insignificant compared to that of the bridge. The governing equation of motion derived using the fully computerized method is expressed as
Ficure 1: A typical vehicle-bridge interaction system. 2.1. Equation of Motion. When the vehicle moves from one end of the bridge to the other end at a constant speed, both the bridge and the vehicle will vibrate vertically. A vector u,(t) is used to denote the vertical displacements of a series of nodes in the finite element model of the bridge. Its first and second derivatives with respect to time ¢, that is, u(t) and ii,(t), are, respectively, the vertical velocity and acceleration of the corresponding nodes. The symbols q,(t) and q,(t) denote the vertical displacement of the wheel and the car body, respectively. As they interact with each other by the contact force, the vibration of the vehicle is influenced by the vibration of the bridge and vice versa. So this is a coupled vibration system. It is assumed that the mass of vehicle is insignificant compared to that of the bridge. The governing equation of motion derived using the fully computerized method is expressed as
2.2. Modeling of Roughness. The random road surface rough- ness of the bridge can be described by a kind of zero-mean, real-valued, and stationary Gaussian process as [19] in which w, and w, are the lower and upper cut-off spatial frequencies, respectively. The power spectral density function S,(w,) can be expressed in terms of the spatial frequency w, of the road surface roughness as
2.2. Modeling of Roughness. The random road surface rough- ness of the bridge can be described by a kind of zero-mean, real-valued, and stationary Gaussian process as [19] in which w, and w, are the lower and upper cut-off spatial frequencies, respectively. The power spectral density function S,(w,) can be expressed in terms of the spatial frequency w, of the road surface roughness as
Ficure 3: Vertical acceleration of the vehicle running on intact and damaged bridges.
Ficure 3: Vertical acceleration of the vehicle running on intact and damaged bridges.
Ficure 2: A typical profile of roughness
Ficure 2: A typical profile of roughness
Ficure 4: Damage indicator when damage is inflicted at (a) fourth element and (b) tenth element.
Ficure 4: Damage indicator when damage is inflicted at (a) fourth element and (b) tenth element.
Ficure 5: Identified equivalent Young’s modulus from first part for damage at (a) fourth element and (b) tenth element
Ficure 5: Identified equivalent Young’s modulus from first part for damage at (a) fourth element and (b) tenth element
Figure 6: Identified equivalent Young’s modulus from second part for damage at tenth element.
Figure 6: Identified equivalent Young’s modulus from second part for damage at tenth element.
Ficure 1: Destroyed reinforced concrete building by blast load [15]. structures. When a response of a building from blast load is considered, natural period of vibration of the structure is the vital parameter for a given explosion. Ductile elements made of steel and reinforced concrete absorb a lot of strain energy [3]. The effects of blast on reinforced concrete and steel structures have been widely studied by many researchers. To the knowledge of the author, the effects of corrosion with blast loads on reinforced concrete buildings have not been studied. Therefore, in this study, different blast load scenarios were performed for uncorroded and corroded reinforced concrete buildings to investigate the effect of blast loads with corrosion. Performance levels of the reinforced concrete buildings were obtained under the effect of blast loads. The impacts of the blast waves on the surface of the structural members were simulated.
Ficure 1: Destroyed reinforced concrete building by blast load [15]. structures. When a response of a building from blast load is considered, natural period of vibration of the structure is the vital parameter for a given explosion. Ductile elements made of steel and reinforced concrete absorb a lot of strain energy [3]. The effects of blast on reinforced concrete and steel structures have been widely studied by many researchers. To the knowledge of the author, the effects of corrosion with blast loads on reinforced concrete buildings have not been studied. Therefore, in this study, different blast load scenarios were performed for uncorroded and corroded reinforced concrete buildings to investigate the effect of blast loads with corrosion. Performance levels of the reinforced concrete buildings were obtained under the effect of blast loads. The impacts of the blast waves on the surface of the structural members were simulated.
where a, is the ambient air pressure ahead of wave, y is the specific heat ratio, and p is the density of air. The reflected pressure, P,, was then calculated by following equations: In this study, four different explosion distances (i-e., 6 m, 12m, 18m, and 24m) were defined with having the same amount of 150 kg TNT. The mass specific energy for TNT was equal to 4520 kJ/kg.
where a, is the ambient air pressure ahead of wave, y is the specific heat ratio, and p is the density of air. The reflected pressure, P,, was then calculated by following equations: In this study, four different explosion distances (i-e., 6 m, 12m, 18m, and 24m) were defined with having the same amount of 150 kg TNT. The mass specific energy for TNT was equal to 4520 kJ/kg.
FiGure 3: (a) Blast wave pressure-time graph. (b) Blast load and comparison [4].
FiGure 3: (a) Blast wave pressure-time graph. (b) Blast load and comparison [4].
Ficure 4: (a) Stress-strain relation of reinforced concrete [5]. (b) Stress-strain relation of steel bars [6].
Ficure 4: (a) Stress-strain relation of reinforced concrete [5]. (b) Stress-strain relation of steel bars [6].
Ficure 6: Modelled reinforced concrete building. FIGURE 5: Force-deformation relationship of a plastic hinge.
Ficure 6: Modelled reinforced concrete building. FIGURE 5: Force-deformation relationship of a plastic hinge.
FiGure 7: Blast load with a 6 m explosion: (a) uncorroded shear wall and (b) corroded shear wall.
FiGure 7: Blast load with a 6 m explosion: (a) uncorroded shear wall and (b) corroded shear wall.
Ficure 8: Blast load with 18m explosion distance: (a) uncorroded shear wall and (b) corroded shear wall.
Ficure 8: Blast load with 18m explosion distance: (a) uncorroded shear wall and (b) corroded shear wall.
Ficure 9: Blast load with a 24 m of explosion distance: (a) uncor- roded shear wall and (b) corroded shear wall.
Ficure 9: Blast load with a 24 m of explosion distance: (a) uncor- roded shear wall and (b) corroded shear wall.
Figure 1: Sandwich beam with three layers.
Figure 1: Sandwich beam with three layers.
Equation (8) shows that the dynamic behavior of a sandwich beam is similar to that of a homogeneous beam in terms of mode shapes, while the natural frequencies are different. This feature was used for the design of a damage detection method applicable to sandwich beams. with A = aL, which permits calculating the 1; values for i vibrations modes. Multiplying the expression of «* with L’ and substituting the values of A; and the angular frequencies w;, consequently the natural frequencies of the undamaged cantilever beam are obtained as follows:
Equation (8) shows that the dynamic behavior of a sandwich beam is similar to that of a homogeneous beam in terms of mode shapes, while the natural frequencies are different. This feature was used for the design of a damage detection method applicable to sandwich beams. with A = aL, which permits calculating the 1; values for i vibrations modes. Multiplying the expression of «* with L’ and substituting the values of A; and the angular frequencies w;, consequently the natural frequencies of the undamaged cantilever beam are obtained as follows:
Ficure 2: (a) Mode shapes, curvature, and natural frequency shift curves for bending mode 7, (b) the behavior of slices placed on characteristic point on the beam, and (c) the representation of function y for the same bending vibration mode.
Ficure 2: (a) Mode shapes, curvature, and natural frequency shift curves for bending mode 7, (b) the behavior of slices placed on characteristic point on the beam, and (c) the representation of function y for the same bending vibration mode.
Ficure 3: Comparison of relations used to express the rigidity loss due to damage versus the dimensionless damage depth.
Ficure 3: Comparison of relations used to express the rigidity loss due to damage versus the dimensionless damage depth.
Ficure 5: Damage severities versus dimensionless damage depth for (a) the steel beam and (b) the sandwich beam. Figure 4: Sandwich beam with damage. The algorithm can be used for all beams with any support type, simply choosing the adequate mode shape curvatures. Our researches revealed that the number of elements ®; involved in the analysis should be from six to ten; more elements are especially necessary when there are modes for which the measured results are not reliable. by dividing them by the highest value of the series. The mathematical formulation is presented as follows:
Ficure 5: Damage severities versus dimensionless damage depth for (a) the steel beam and (b) the sandwich beam. Figure 4: Sandwich beam with damage. The algorithm can be used for all beams with any support type, simply choosing the adequate mode shape curvatures. Our researches revealed that the number of elements ®; involved in the analysis should be from six to ten; more elements are especially necessary when there are modes for which the measured results are not reliable. by dividing them by the highest value of the series. The mathematical formulation is presented as follows:
Ficure 7: Damage location index for damage placed at x/L = 0.3, x/L = 0.5, and x/L = 0.7, respectively.
Ficure 7: Damage location index for damage placed at x/L = 0.3, x/L = 0.5, and x/L = 0.7, respectively.
Ficure 8: Experimental stand.
Ficure 8: Experimental stand.
Ficure 9: Identification of the crack location using the Minkowski Distance.
Ficure 9: Identification of the crack location using the Minkowski Distance.
TABLE 1: Measured natural frequencies and corresponding shifts for the analyzed beam.
TABLE 1: Measured natural frequencies and corresponding shifts for the analyzed beam.
Ficure 1: (a) Complete structure. At the top, the reinforcement plate is highlighted. The travelling mass and the counterweight are visible on the left and on the right, respectively. (b) Pendulum with the travelling mass.
Ficure 1: (a) Complete structure. At the top, the reinforcement plate is highlighted. The travelling mass and the counterweight are visible on the left and on the right, respectively. (b) Pendulum with the travelling mass.
Figure 2: (a) Actual value of the radial acceleration. (b) Measured radial accelerations, by the capacitive and the ICP sensor.
Figure 2: (a) Actual value of the radial acceleration. (b) Measured radial accelerations, by the capacitive and the ICP sensor.
FIGURE 3: Comparisons between actual accelerations and corrected capacitive estimates.
FIGURE 3: Comparisons between actual accelerations and corrected capacitive estimates.
Figure 4: Fixed-mass case. (a) Viscous coefficient as a function of time, during a swinging decay. (b) Damping force as a function of @ at each cycle.
Figure 4: Fixed-mass case. (a) Viscous coefficient as a function of time, during a swinging decay. (b) Damping force as a function of @ at each cycle.
FiGurRE 5: Moving mass. (a, b) Case 1. (c, d) Case 2. (e, f) Case 3. (a, c, e) Mass position along the beam. (b, d, f) Swinging amplitude ove time.
FiGurRE 5: Moving mass. (a, b) Case 1. (c, d) Case 2. (e, f) Case 3. (a, c, e) Mass position along the beam. (b, d, f) Swinging amplitude ove time.
Ficure 6: Moving-mass Case 1. Estimates of the damping factors, compared with the overall estimate of (3) (a) and with the updated valu of (7) (b).
Ficure 6: Moving-mass Case 1. Estimates of the damping factors, compared with the overall estimate of (3) (a) and with the updated valu of (7) (b).
Figure 7: Estimates of the damping factors, compared with the updated value of (7). Moving-mass Case 2 (a) and Case 3 (b).
Figure 7: Estimates of the damping factors, compared with the updated value of (7). Moving-mass Case 2 (a) and Case 3 (b).
To show the relation between (10) and (14), compute the estimate of the ith variable x; for a fixed mixture component k using (10). For clarity, the component index k is omitted. Consider 4. Damage Detection Using the mixture model for damage detection introduces an issue of residual scaling, because each class may have a differ- ent error variance (15). Therefore, the residual of each variable within each class is divided with the corresponding standard deviation, which is the square root of (15). Also the data dimensionality may be too high for statistical reliability (curse of dimensionality). Therefore, the first principal component scores [10] of residual (7) are used for damage detection. Control charts [ll] are used for damage detection. The control chart used in this study is the Shewhart chart [11], and the plotted variable is the subgroup mean of successive obser- vations. It is believed that the robustness of damage detection increases, because (1) additional variability due to environ- mental or operational influences can be removed, in this paper using a nonlinear model; (2) PCA is applied to the residuals avoiding the curse of dimensionality; and (3) con- trol charts utilize averaging for better statistical reliability.
To show the relation between (10) and (14), compute the estimate of the ith variable x; for a fixed mixture component k using (10). For clarity, the component index k is omitted. Consider 4. Damage Detection Using the mixture model for damage detection introduces an issue of residual scaling, because each class may have a differ- ent error variance (15). Therefore, the residual of each variable within each class is divided with the corresponding standard deviation, which is the square root of (15). Also the data dimensionality may be too high for statistical reliability (curse of dimensionality). Therefore, the first principal component scores [10] of residual (7) are used for damage detection. Control charts [ll] are used for damage detection. The control chart used in this study is the Shewhart chart [11], and the plotted variable is the subgroup mean of successive obser- vations. It is believed that the robustness of damage detection increases, because (1) additional variability due to environ- mental or operational influences can be removed, in this paper using a nonlinear model; (2) PCA is applied to the residuals avoiding the curse of dimensionality; and (3) con- trol charts utilize averaging for better statistical reliability.
Ficure 1: Training data (a) and test data (b) with a change in mean. The identified GMM model is shown in red.
Ficure 1: Training data (a) and test data (b) with a change in mean. The identified GMM model is shown in red.
TABLE 1: Parameters of the three linear components (22).
TABLE 1: Parameters of the three linear components (22).
Ficure 3: Shewhart control chart for damage detection.
Ficure 3: Shewhart control chart for damage detection.
Ficure 2: Log-likelihood with the penalty term (a) and an identified GMM model (in red) converged to a local maximum (b).
Ficure 2: Log-likelihood with the penalty term (a) and an identified GMM model (in red) converged to a local maximum (b).
Figure 4: Training data (a) and test data (b) with a change in mean. The identified GMM model is shown in red.
Figure 4: Training data (a) and test data (b) with a change in mean. The identified GMM model is shown in red.
Figure 5: Log-likelihood with the penalty term (a) and Shewhart control chart for damage detection (b).
Figure 5: Log-likelihood with the penalty term (a) and Shewhart control chart for damage detection (b).
Ficure 6: Identified lowest natural frequencies of the Z24 Bridge.
Ficure 6: Identified lowest natural frequencies of the Z24 Bridge.
Ficure 7: Correlation between the four lowest natural frequencies of the Z24 Bridge. Blue symbols indicate the training data.
Ficure 7: Correlation between the four lowest natural frequencies of the Z24 Bridge. Blue symbols indicate the training data.
Ficure 8: Log-likelihood including the penalty term.
Ficure 8: Log-likelihood including the penalty term.
FiGurE 9: Shewhart control charts for damage detection using GMM (a) and a corresponding linear model (b).
FiGurE 9: Shewhart control charts for damage detection using GMM (a) and a corresponding linear model (b).
TABLE |: States of the aircraft panel.
TABLE |: States of the aircraft panel.
Ficure 1: Probabilistic neural network architecture.
Ficure 1: Probabilistic neural network architecture.
Figure 2: Aluminum aircraft panel equipped with eight PZT patches.
Figure 2: Aluminum aircraft panel equipped with eight PZT patches.
TABLE 2: Probabilistic neural network for damage classification.
TABLE 2: Probabilistic neural network for damage classification.
Figure 4: Final degree of pertinence for the aircraft panel.
Figure 4: Final degree of pertinence for the aircraft panel.
Figure 5: Aluminum aircraft window containing ten PZT patches.
Figure 5: Aluminum aircraft window containing ten PZT patches.
TABLE 3: Classification of test set of probabilistic neural networks for each PZT patch in the aircraft panel. TABLE 4: Optimization results of the Gustafson-Kessel algorithm for the aircraft panel.
TABLE 3: Classification of test set of probabilistic neural networks for each PZT patch in the aircraft panel. TABLE 4: Optimization results of the Gustafson-Kessel algorithm for the aircraft panel.
PZTs failed to detect damage and thus made the classification impossible. The PZT3 and PZT4 correctly identified the damage 1; nevertheless the damage 2 was impossible to be distinguished from the Baseline. Finally, the PZT5, PZT7, and PZT8 managed to correctly classify the two types of damage with a degree of pertinence greater than 80%. was used. Due to the size and complexity of the structure, ten PZT patches were considered in the experiment. This number of PZT patches was arbitrary since no preliminary study was performed to optimize the test configuration. Since the beginning of the tests, the PZT10 showed poor stability and repeatability and has therefore been ignored in the test. To simulate two different types of damage in the structure, two experiments were performed as follows. First, a weight was added to the structure as shown in Figure 5(b). Second, after the mentioned weight was removed, one of the clamps (located close to the PZT2) was removed (Figure 5(c)). For every state of the structure 200 measurements were made as shown in Table 5. For every measurement 200 points were taken.
PZTs failed to detect damage and thus made the classification impossible. The PZT3 and PZT4 correctly identified the damage 1; nevertheless the damage 2 was impossible to be distinguished from the Baseline. Finally, the PZT5, PZT7, and PZT8 managed to correctly classify the two types of damage with a degree of pertinence greater than 80%. was used. Due to the size and complexity of the structure, ten PZT patches were considered in the experiment. This number of PZT patches was arbitrary since no preliminary study was performed to optimize the test configuration. Since the beginning of the tests, the PZT10 showed poor stability and repeatability and has therefore been ignored in the test. To simulate two different types of damage in the structure, two experiments were performed as follows. First, a weight was added to the structure as shown in Figure 5(b). Second, after the mentioned weight was removed, one of the clamps (located close to the PZT2) was removed (Figure 5(c)). For every state of the structure 200 measurements were made as shown in Table 5. For every measurement 200 points were taken.
TABLE 5: States of the aircraft window. TABLE 6: Classification of test set of probabilistic neural networks for each PZT patch of the aircraft window
TABLE 5: States of the aircraft window. TABLE 6: Classification of test set of probabilistic neural networks for each PZT patch of the aircraft window
TABLE 7: Optimization results of the Gustafson-Kessel algorithm for the aircraft window.
TABLE 7: Optimization results of the Gustafson-Kessel algorithm for the aircraft window.
beam, was able to detect the damage 2 without errors; this success is due to the fact that this sensor is close to the clamp position. The PZT8 and PZT9 detected only the clamp removal without errors, with an overall error percentage of less than 20%. Finally, the PZT7, which was bonded directly onto the panel and close to the removed clamp, was able to identify all states with an error percentage of 1.25%. In this case, nine probabilistic networks were imple- mented (one for each PZT) to analyze this structure as shown in Table 2. The results obtained with the test set for each one of the nine probabilistic neural networks are presented in Table 6.
beam, was able to detect the damage 2 without errors; this success is due to the fact that this sensor is close to the clamp position. The PZT8 and PZT9 detected only the clamp removal without errors, with an overall error percentage of less than 20%. Finally, the PZT7, which was bonded directly onto the panel and close to the removed clamp, was able to identify all states with an error percentage of 1.25%. In this case, nine probabilistic networks were imple- mented (one for each PZT) to analyze this structure as shown in Table 2. The results obtained with the test set for each one of the nine probabilistic neural networks are presented in Table 6.
4. Conclusion of pertinence are shown in Figure 7. The results of the PZT1, PZT3, PZT4, PZT5, and PZT6 confirm once again that these PZT patches failed to detect damage, which made the classification impossible due to the position of the patches on the structure. The PZT2, PZT8, and PZT9 correctly classified damage 2, with a degree of pertinence of 99%. However, the damage 1 was impossible to discriminate from the baseline. Finally, for the PZT7, the Gustafson-Kessel algorithm was able to correctly classify all measurements with a degree of pertinence greater than 95%.
4. Conclusion of pertinence are shown in Figure 7. The results of the PZT1, PZT3, PZT4, PZT5, and PZT6 confirm once again that these PZT patches failed to detect damage, which made the classification impossible due to the position of the patches on the structure. The PZT2, PZT8, and PZT9 correctly classified damage 2, with a degree of pertinence of 99%. However, the damage 1 was impossible to discriminate from the baseline. Finally, for the PZT7, the Gustafson-Kessel algorithm was able to correctly classify all measurements with a degree of pertinence greater than 95%.
TABLE |: Properties of the CFRP plate. between neighbouring pairs and external sources. The input signal was also generated with the PULSE output port. The piezosystem EPA-104 amplifier boosted the PULSE output signal to the actuator. To minimize the EMI and electrical noise, all equipment, connection boxes, cable shields, and the frame were connected to an independent ground system. To
TABLE |: Properties of the CFRP plate. between neighbouring pairs and external sources. The input signal was also generated with the PULSE output port. The piezosystem EPA-104 amplifier boosted the PULSE output signal to the actuator. To minimize the EMI and electrical noise, all equipment, connection boxes, cable shields, and the frame were connected to an independent ground system. To
FiGure I: Plate under study and experimental arrangement.
FiGure I: Plate under study and experimental arrangement.
Figure 2: Dimensions of the CFRP plate and positions of the damage (P1, P2, and P3) and the actuator (A1).
Figure 2: Dimensions of the CFRP plate and positions of the damage (P1, P2, and P3) and the actuator (A1).
Figure 3: FRFs used as examples of the dependency of SE with the amplitude distribution. Their values of SE are 3.163, 3.108, and 3.443.
Figure 3: FRFs used as examples of the dependency of SE with the amplitude distribution. Their values of SE are 3.163, 3.108, and 3.443.
Ficure 4: Examples of experimental FRFs. The examples correspond to the sensor 1 with the damage in position P1. In the figure, the adde mass is indicated.
Ficure 4: Examples of experimental FRFs. The examples correspond to the sensor 1 with the damage in position P1. In the figure, the adde mass is indicated.
Ficure 5: Variation of D, with attached mass. (a) Damage in position P1. (b) Damage in position P2. (c) Damage in position P3.
Ficure 5: Variation of D, with attached mass. (a) Damage in position P1. (b) Damage in position P2. (c) Damage in position P3.
enhancing some frequencies and weakening others. So, the SE of the structure will change in the presence of damage. TABLE 3: Added masses and comparison with the total mass of the plate.
enhancing some frequencies and weakening others. So, the SE of the structure will change in the presence of damage. TABLE 3: Added masses and comparison with the total mass of the plate.
TABLE 2: Properties and positions of piezoelectric sensors and actuators.
TABLE 2: Properties and positions of piezoelectric sensors and actuators.
Ficure 6: Variation of SE with the added mass in position 1 for: (a) Sensor SI; (b) Sensor S2; (c) Sensor S3; (d) Sensor S4.
Ficure 6: Variation of SE with the added mass in position 1 for: (a) Sensor SI; (b) Sensor S2; (c) Sensor S3; (d) Sensor S4.
Ficure 7: Variation of SE with the added mass in position 2 for (a) Sensor S1; (b) Sensor S2; (c) Sensor S3; (d) Sensor S4.
Ficure 7: Variation of SE with the added mass in position 2 for (a) Sensor S1; (b) Sensor S2; (c) Sensor S3; (d) Sensor S4.
Figure 8: Variation of SE with the added mass in position 3 for (a) Sensor SI; (b) Sensor S2; (c) Sensor S3; (d) Sensor S4.
Figure 8: Variation of SE with the added mass in position 3 for (a) Sensor SI; (b) Sensor S2; (c) Sensor S3; (d) Sensor S4.

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