Design of shaped piezoelectric modal sensor for beam with arbitrary boundary conditions by using Adomian decomposition method

The theory for designing distributed piezoelectric modal sensors is well established for beam structures. However, the current modal sensor theory is limited in scope in that it can only be applied in the case of classical boundary conditions (i.e., either clamped, free, simply supported or sliding). In this paper a solution to the problem of finding the shape of piezoelectric modal sensors for a beam with arbitrary boundary conditions is proposed, using the Adomian decomposition method (ADM). A general expression for designing the shape of a piezoelectric modal sensor is presented, in which the output signal of the designed sensor is proportional to the response of the target mode. Other modes are filtered out. The modal sensor shape is expressed as a function of the second spatial derivative of the structural mode shape function. Based on the ADM and employing some simple mathematical operations, the closed-form series solution of the second spatial derivative of the mode shapes can be determined. Then the shapes of the designed modal sensors are obtained. Finally, some numerical examples are given to demonstrate the feasibility of the proposed modal sensors. It is shown that, for classical boundary conditions, the shapes of the modal sensors based on the ADM agree well with analytical and numerical results given in the literature. For general boundary conditions it is found that the shape of the modal sensors is influenced by the number of modes of interest because the second spatial derivatives of the mode shapes are not orthogonal to one another. The modal sensors for general boundary conditions can be considered as modal filters within a limited frequency band.     

Optimal sensor placement for spatial lattice structure based on genetic algorithms

Optimal sensor placement technique plays a key role in structural health monitoring of spatial lattice structures. This paper considers the problem of locating sensors on a spatial lattice structure with the aim of maximizing the data information so that structural dynamic behavior can be fully characterized. Based on the criterion of optimal sensor placement for modal test, an improved genetic algorithm is introduced to find the optimal placement of sensors. The modal strain energy (MSE) and the modal assurance criterion (MAC) have been taken as the fitness function, respectively, so that three placement designs were produced. The decimal two-dimension array coding method instead of binary coding method is proposed to code the solution. Forced mutation operator is introduced when the identical genes appear via the crossover procedure. A computational simulation of a 12-bay plain truss model has been implemented to demonstrate the feasibility of the three optimal algorithms above. The obtained optimal sensor placements using the improved genetic algorithm are compared with those gained by exiting genetic algorithm using the binary coding method. Further the comparison criterion based on the mean square error between the finite element method (FEM) mode shapes and the Guyan expansion mode shapes identified by data-driven stochastic subspace identification (SSI-DATA) method are employed to demonstrate the advantage of the different fitness function. The results showed that some innovations in genetic algorithm proposed in this paper can enlarge the genes storage and improve the convergence of the algorithm. More importantly, the three optimal sensor placement methods can all provide the reliable results and identify the vibration characteristics of the 12-bay plain truss model accurately.     

Numerical and experimental analysis of uncertainty on modal parameters estimated with the stochastic subspace method

Modal parameters of structures are often used as inputs for finite element model updating, vibration control, structural design or structural health monitoring (SHM). In order to test the robustness of these methods, it is a common practice to introduce uncertainty on the eigenfrequencies and modal damping coefficients under the form of a Gaussian perturbation, while the uncertainty on the mode shapes is modeled in the form of independent Gaussian noise at each measured location. A more rigorous approach consists however in adding uncorrelated noise on the time domain responses at each sensor before proceeding to an operational modal analysis. In this paper, we study in detail the resulting uncertainty when modal analysis is performed using the stochastic subspace identification method. A Monte-Carlo simulation is performed on a simply supported beam, and the uncertainty on a set of 5000 modal parameters identified with the stochastic subspace identification method is discussed. Next, 4000 experimental modal identifications of a small clamped–free steel plate equipped with 8 piezoelectric patches are performed in order to confirm the conclusions drawn in the numerical case study. In particular, the results point out that the uncertainty on eigenfrequencies and modal damping coefficients may exhibit a non-normal distribution, and that there is a non-negligible spatial correlation between the uncertainty on mode shapes at sensors of different locations.     

The mass normalization of the displacement and strain mode shapes in a strain experimental modal analysis using the mass-change strategy

The classic experimental modal analysis (EMA) is a well-known procedure for determining the modal parameters. The less frequently used strain EMA is based on a response measurement using strain sensors. The results of a strain EMA are the modal parameters, where in addition to the displacement mode shapes the strain mode shapes are also identified. The strain EMA can be used for an experimental investigation of a stress–strain distribution without the need to build a dynamical model. It can also be used to determine the modal parameters when, during modal testing, a motion sensor cannot be used and so a strain sensor is used instead. The displacement and strain mode shapes that are determined with the strain EMA are not mass normalized (scaled with respect to the orthogonality properties of the mass-normalized modal matrix), and therefore some dynamical properties of the system cannot be obtained. The mass normalization can be made with the classic EMA, which requires the use of a motion sensor. In this research a new approach to the mass normalization in the strain EMA, without using a motion sensor, is presented. It is based on the recently introduced mass-change structural modification method, which is used for the mass normalization in an operational modal analysis. This method was modified in such a way that it can be used for the mass normalization in the strain EMA. The mass-normalized displacement and strain mode shapes were obtained using a combination of the proposed approach and the strain EMA. The proposed approach was validated on real structures (beam and plate).     

STRUCTURAL DAMAGE DETECTION BASED ON A MICRO-GENETIC ALGORITHM USING INCOMPLETE AND NOISY MODAL TEST DATA

This paper describes a procedure for detecting structural damage based on a micro-genetic algorithm using incomplete and noisy modal test data. As the number of sensors used to measure modal data is normally small when compared with the degrees of freedom of the finite element model of the structure, the incomplete mode shape data are first expanded to match with all degrees of freedom of the finite element model under consideration. The elemental energy quotient difference is then employed to locate the damage domain approximately. Finally, a micro-genetic algorithm is used to quantify the damage extent by minimizing the errors between the measured data and numerical results. The process may be either of single-level or implemented through two-level search strategies. The study has covered the use of frequencies only and the combined use of both frequencies and mode shapes. The proposed method is applied to a single-span simply supported beam and a three-span continuous beam with multiple damage locations. In the study, the modal test data are simulated numerically using the finite element method. The measurement errors of modal data are simulated by superimposing random noise with appropriate magnitudes. The effectiveness of using frequencies and both frequencies and mode shapes as the data for quantification of damage extent are examined. The effects of incomplete and noisy modal test data on the accuracy of damage detection are also discussed.     

Placement and dimension optimization of shunted piezoelectric patches for vibration reduction

Passive structural vibration reduction by means of shunted piezoelectric patches is addressed in this paper. We present a strategy to optimize, in terms of damping efficiency, the geometry of piezoelectric patches as well as their placement on the host elastic structure. This procedure is based on the maximization of the modal electro-mechanical coupling factor (MEMCF) of the mechanical vibration mode to which the shunt is tuned. To illustrate the method, a general analytical model of a laminated beam is proposed. Two particular configurations are investigated: (i) a beam with two collocated piezoelectric patches connected in series or in parallel to the shunt and (ii) a cantilever beam with one patch. After a modal expansion, original closed-form solutions of the MEMCF are exhibited, which enables to compute optimal values for the placement, length and thickness of the piezoelectric patches that maximize the MEMCF. A dimensionless model is used so that this study can be used to design any smart beam, whatever be its dimensions. More general results about the coupling mechanisms between the piezoelectric patches and the host structure are also raised. In particular, it is found that the patches thickness is an essential parameter and that several configurations are possible, depending on the considered vibration mode. Experiments are also proposed to validate the model.     

Free vibration analysis of moderately thick asymmetric piezoelectric adaptive cantilever beams using the distributed transfer function approach

A semi-analytical distributed transfer function (DTF) approach is proposed for the free-vibration analysis of moderately thick cantilever beams with a single surface-bonded piezoelectric patch. The asymmetric piezoelectric adaptive structure is decomposed into three segments; the first and third segments are bare beam parts before and after the patch, while the second segment contains the beam part with attached piezoelectric patch bonded to its upper surface. The theoretical formulation assumes first-order shear deformation kinematics and linear electric potential through the patch thickness with an electrode equipotential physical condition, and uses the extended Hamilton?s principle to derive the equations of motion and electromechanical boundary conditions. The latter, together with the continuity and equilibrium conditions at the segments interfaces, are then transformed into a first-order state space equation that is solved using the DTF approach. The electrodes of the piezoelectric patch are considered either in short-circuit (SC) or open-circuit (OC); this leads to two free-vibration problems to be solved for the corresponding SC and OC frequencies, from which the Electro-Mechanical Coupling Coefficient (EMCC) is post-treated. Four benchmarks from the open literature are simulated in order to validate the proposed approach. Very satisfactory correlations are obtained for all examples with maximum errors less thank 5 percent in all results. For future reference, an additional benchmark is proposed to assess the influence of the patch-to-composite host width ratio on the effective modal EMCC. It was found that the latter is mode-dependent (as expected) and decreases with increasing the former.     

Modal coupling effects in the free vibration of elastically interconnected beams

The problem of free vibration of a uniform beam elastically interconnected to a cantilevered beam, representing an idealized launch vehicle aeroelastic model in a wind tunnel, is studied. With elementary beam theory modelling, numerical results are obtained for the frequencies, mode shapes and the generalized modal mass of this elastically coupled system, for a range of values of the spring constants and cantilevered beam stiffness and inertia values. The study shows that when the linear springs are supported at the nodal points corresponding to the first free-free beam mode, the modal interaction comes primarily from the rotational spring stiffness. The effect of the linear spring stiffness on the higher model modes is also found to be marginal. However, the rotational stiffness has a significant effect on all the predominantly model modes as it couples the model deformations and the support rod deformations. The study also shows that through the variations in the stiffness or the inertia values of the cantilever beam affect only the predominantly cantilever modes, these variations become important because of the fact that the cantilevered support rod frequencies may come close to, or even cross over, the predominantly model mode frequencies. The results also bring out the fact that shifting of the support points away from the first mode nodal points has a maximum effect only on the first model mode.     

An automatic mode pairing strategy using an enhanced modal assurance criterion based on modal strain energies

In the context of finite element model updating using output-only vibration test data, natural frequencies and mode shapes are used as validation criteria. Consequently, the correct pairing of experimentally obtained and numerically derived natural frequencies and mode shapes is important. In many cases, only limited spatial information is available and noise is present in the measurements. Therefore, the automatic selection of the most likely numerical mode shape corresponding to a particular experimentally identified mode shape can be a difficult task. The most common criterion for indicating corresponding mode shapes is the modal assurance criterion. Unfortunately, this criterion fails in certain cases and is not reliable for automatic approaches.In this paper, the purely mathematical modal assurance criterion will be enhanced by additional physical information from the numerical model in terms of modal strain energies. A numerical example and a benchmark study with experimental data are presented to show the advantages of the proposed energy-based criterion in comparison to the traditional modal assurance criterion.     

Mode shape expansion using perturbed force approach

A new approach for expanding incomplete experimental mode shapes is presented which considers the modelling errors in the analytical model and the uncertainties in the vibration modal data measurements. The proposed approach adopts the perturbed force vector that includes the effect of the discrepancy in mass and stiffness between the finite element model and the actual tested dynamic system. From the developed formulations, the perturbed force vector can be obtained from measured modal data and is then used for predicting the unmeasured components of the expanded experimental mode shapes. A special case that does not require the experimental natural frequency in the mode shape expansion process is also discussed. A regularization algorithm based on the Tikhonov solution incorporating the generalized cross-validation method is employed to filter out the influence of noise in measured modal data on the predictions of unmeasured mode components. The accuracy and robustness of the proposed approach is verified with respect to the size of measured data set, sensor location, model deficiency and measurement uncertainty. The results from two numerical examples, a plane frame structure and a thin plate structure, show that the proposed approach has the best performance compared with the commonly used existing expansion methods, and can reliably produce the predictions of mode shape expansion, even in the cases with limited modal data measurements, large modelling errors and severe measurement noise.     

应用长度极化压电陶瓷33模态的 半主动减振技术研究*

本文提出利用长度方向极化的压电材料的33模态来实现半主动减振。论文以横梁为例,通过理论分析和有限元仿真,对比研究了当压电材料分别连接31, 33两种模态对应的最佳分流电路时,压电材料两种模态在横梁的共振频率附近的减振效果。结果表明33模态比31模态具有更高的减振效率。此外,鉴于33模态存在极化长度有限的问题,仿真分析了压电材料的尺寸和位置对减振效果的影响。在此基础之上,提出了一个利用压电材料33模态的多模态减振的组合设计,对横梁的前三个模态起到了很好地减振作用。相对31模态而言,横梁的每个振动模态均有约15dB的减振提升。     

Spatial filters in structural control

This paper discusses the use of modal filters in structural control. Discrete piezoelectric array sensors are first discussed and their lack of roll-off due to spatial aliasing is pointed out. In the second part, a new porous distributed electrode concept is introduced, which allows the effective piezoelectric coefficient to be tailored in two dimensions.     

Efficient modal control strategies for active control of vibrations

Some efficient strategies for the active control of vibrations of a beam structure using piezoelectric materials are described. The control algorithms have been implemented for a cantilever beam model developed using finite element formulation. The vibration response of the beam to an impulse excitation has been calculated numerically for the uncontrolled and the controlled cases. The essence of the method proposed is that a feedback force in different modes be applied according to the vibration amplitude in the respective modes i.e., modes having lesser vibration may receive lesser feedback. This weighting may be done on the basis of either displacement or energy present in different modes. This method is compared with existing methods of modal space control, namely the independent modal space control (IMSC), and modified independent modal space control (MIMSC). The method is in fact an extension of the modified independent space control with the addition that it proposes to use the sum of weighted multiple modal forces for control. The proposed method results in a simpler feedback, which is easy to implement on a controller. The procedure is illustrated for vibration control of a cantilever beam. The analytical results show that the maximum feedback control voltage required in the proposed method is further reduced as compared to existing methods of IMSC and MIMSC for similar vibration control. The limitations of the proposed method are discussed.     

Control of a rotating cantilever beam using a torque actuator and a distributed piezoelectric polymer actuator

A torque actuator and a distributed piezoelectric polymer (PVDF) actuator are utilized for control of a rotating cantilever flexible beam. The torque control contains proportional and derivative (PD) feedback for rigid motion control and a PVDF actuator control for vibration damping. Unlike previous approaches in the literature in which the angular velocity feedback was utilized, in this study we propose to use the linear velocity feedback (L-type) in our controller design for feasible implementation and avoiding modal truncation. The stability of the system with the L-type control has been analyzed, using the concept of a virtual joint model. The advantage of the proposed scheme lies in easy implementation, avoidance of modal truncation, efficient suppression of the dominant mode of vibration, and allowing high-speed motions. Numerical examples demonstrate the effectiveness of the proposed approach.     

Condition assessment of reinforced concrete beams using dynamic data measured with distributed long-gage macro-strain sensors

A new condition assessment strategy of reinforced concrete (RC) beams is proposed in this paper. This strategy is based on frequency analysis of the dynamic data measured with distributed long-gage macro-stain sensors. After extracting modal macro-strain, the reference-based damage index is theoretically deducted in which the variations of modal flexural rigidity and modal neutral axis height are considered. The reference-free damage index is also presented for comparison. Both finite element simulation and experiment investigations were carried out to verify the proposed method. The manufacturing procedure of long-gage fiber Bragg grating (FBG) sensor chosen in the experiment is firstly presented, followed by an experimental study on the essential sensing properties of the long-gage macro-strain sensors and the results verify the excellent sensing properties, in particular the measurement accuracy and dynamic measuring capacity. Modal analysis results of a concrete beam show that the damage appearing in the beam can be well identified by the damage index while the vibration testing results of a RC beam show that the proposed method can not only capture small crack initiation but its propagation. It can be concluded that distributed long-gage dynamic macro-strain sensing technique has great potential for the condition assessment of RC structures subjected to dynamic loading.     

Direct mode-shape expansion of a spatially incomplete measured mode by a hybrid-vector modification

A direct estimation method for expanding incomplete experimental mode shapes is presented. The approach adopts a hybrid vector which includes measured data at master degrees of freedom (dofs) and constant values at slave dofs. The constant values are refined by a set of mode-correction factors. Modelling errors between the analytical model and tested structure are also considered by introducing a series of model-correction factors. Initial-guess values of the mode-correction factors are used to decouple the coupled constructed equations, and an iterative technique for solving these equations is proposed. The results from a five-degree-of-freedom mass–spring system indicate that the proposed approach provided a better performance than the commonly used existing expansion methods and can reliably estimate unmeasured components of mode shapes, even in cases with limited modal measurements and severe measurement noise. The performance of the proposed method was also investigated using real measurements from a steel cantilever-beam experiment. Experimental data were measured by 20 accelerometers mounted at the cantilever beam: among these accelerometers, three of these were assumed to be measured, and the others were used to check the estimation accuracy of the proposed method. The results show that the unmeasured components in the mode shapes were properly estimated by implementing the proposed method, even for high-frequency modes.     

IMPROVEMENT OF THE SEMI-ANALYTICAL METHOD, FOR DETERMINING THE GEOMETRICALLY NON-LINEAR RESPONSE OF THIN STRAIGHT STRUCTURES: PART II—FIRST AND SECOND NON-LINEAR MODE SHAPES OF FULLY CLAMPED RECTANGULAR PLATES

In a previous series of papers, a semi-analytical model based on Hamilton's principle and spectral analysis has been developed for geometrically non-linear free vibrations occurring at large displacement amplitudes of clamped-clamped beams and fully clamped rectangular homogeneous and composite plates. In Part I of this series of papers, concerned with geometrically non-linear free and forced vibrations of various beams, a practical simple “multi-mode theory”, based on the linearization of the non-linear algebraic equations, written in the modal basis, in the neighbourhood of each resonance has been developed. Simple explicit formulae, ready and easy to use for analytical or engineering purposes have been derived, which allows direct calculation of the basic function contributions to the first three non-linear mode shapes of the beams considered. Also, various possible truncations of the series expansion defining the first non-linear mode shape have been considered and compared with the complete solution, which showed that an increasing number of basic functions has to be used, corresponding to increasingly sized intervals of vibration amplitudes; starting from use of only one function, i.e., the first linear mode shape, corresponding to very small amplitudes, for which the linear theory is still valid, and ending by the complete series, involving six functions, corresponding to maximum vibration amplitudes at the beam middle point up to once the beam thickness. For higher amplitudes, a complementary second formulation has been developed, leading to reproduction of the known results via the solution of reduced linear systems of five equations and five unknowns. The purpose of this paper is to extend and adapt the approach described above to the geometrically non-linear free vibration of fully clamped rectangular plates in order to allow direct and easy calculation of the first, second and higher non-linear fully clamped rectangular plate mode shapes, with their associated non-linear frequencies and non-linear bending stress patterns. Also, numerical results corresponding to the first and second non-linear modes shapes of fully clamped rectangular plates with an aspect ratio α=0·6 are presented. Data concerning the higher non-linear modes, the aspect ratio effect, and the forced vibration case will be presented later.     

Broadband energy harvesting via magnetic coupling between two movable magnets

Harvesting energy from ambient mechanical vibrations by the piezoelectric effect has been proposed for powering microelectromechanical systems and replacing batteries that have a finite life span. A conventional piezoelectric energy harvester （PEH） is usually designed as a linear resonator, and suffers from a narrow operating bandwidth. To achieve broadband energy harvesting, in this paper we introduce a concept and describe the realization of a novel nonlinear PEH. The proposed PEH consists of a primary piezoelectric cantilever beam coupled to an auxiliary piezoelectric cantilever beam through two movable magnets. For predicting the nonlinear response from the proposed PEH, lumped parameter models are established for the two beams. Both simulation and experiment reveal that for the primary beam, the introduction of magnetic coupling can expand the operating bandwidth as well as improve the output voltage. For the auxiliary beam, the magnitude of the output voltage is slightly reduced, but additional output is observed at off-resonance frequencies. Therefore, broadband energy harvesting can be obtained from both the primary beam and the auxiliary beam.     

Distributed piezoelectric modal sensors for circular plates

In this note, we deal with finding the shape of distributed piezoelectric modal sensors for circular plates with polar symmetric boundary conditions. The problem is treated by an optimization approach, where a binary function is used to model the design variable: the polarization profile of the piezoelectric layer. The numerical procedure proposed here allows us to find polarization profiles which take on two values only, i.e. either positive or negative polarization, that isolate particular vibration modes in the frequency domain.     

VIBRATIONS OF NON-UNIFORM CONTINUOUS BEAMS UNDER MOVING LOADS

The dynamic behavior of multi-span non-uniform beams transversed by a moving load at a constant and variable velocity is investigated. The continuous beam is modelled using Bernoulli-Euler beam theory. The solution is obtained by using both the modal analysis method and the direct integration method. The natural frequencies and mode shapes used in the solution of this problem are obtained exactly by deriving the exact dynamic stiffness matrices for any polynomial variation of the cross-section along the beam using the exact element method. The mode shapes are expressed as infinite polynomial series. Using the exact mode shapes yields the exact solution for general variation of the beam section in case of constant and variable velocity. Numerical examples are presented in order to demonstrate the accuracy and the effectiveness of the present study, and the results are compared to previously published results.