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.     

极点配置法弹性平板有源隔声系统优化

对混响声场中的弹性平板有源隔声系统进行优化。根据激励频率范围确定受控模态阶次,在模态空间中建立系统降阶方程,基于极点配置方法,采用分布式系统增加受控模态的阻尼,降低低频共振声传输。同时设计模态滤波器,为控制器提供所需的状态信息。为提高控制效能,本文对传感器和作动器布放位置进行优化,尝试不同极点配置方案,并对耦合控制与独立模态控制方法的隔声效果进行比较。仿真结果显示,极点配置法有源隔声可以有效降低共振声传输,优化布放和独立模态控制方式下,控制力明显降低。优化后的有源隔声系统效能有所提升。      

ROBUST VIBRATION CONTROL OF A BEAM USING THE H∞-BASED CONTROLLER WITH MODEL ERROR COMPENSATOR

The vibration control of a flexible beam subjected to arbitrary, unmeasurable disturbance forces is investigated in this paper. The beam is analyzed by using modal expansion theorem. The independent modal space control is adopted for the active vibration control. Discrete sensors and actuators are used here. The modal filters are used as the state estimator to obtain the modal co-ordinates and modal velocities for the state feedback control. Because of the existence of the disturbance forces, the vibration control only with the state feedback control law cannot suppress the vibration well. The method of disturbance forces cancellation is then added in the feedback loop. In order to implement the disturbance forces cancellation, the unknown disturbance forces must be observed. The model error compensator is employed to observe the unknown disturbance modal forces for the direct cancellation. After the implementation of the disturbance modal forces cancellation, there are still some residual disturbance modal forces which excite the beam. The disturbance attenuation problem is of concern in the design of the state feedback control law. For ensuring that influence of the residual disturbance modal forces is reduced to an acceptable level, the robust static H_∞ state feedback controller is designed. The vibration control performances of the feedback control with the H_∞ controller and the disturbance forces cancellation are discussed.     

Controllability and observability measures of structural vibration and acoustic radiation

This paper studies the controllability and observability measures of structural vi?bration and acoustic radiation and proposes a measure of modal observability based on acoustic pressure outputs and measures of controllability and observability of acoustic radiation by combining the measures of modal controllability and observability of vibration modes with corresponding modal acoustic radiation efficiencies. A plate with control force inputs and vibrational response outputs and acoustic pressure outputs is involved to show the modal controllability and observability measures of vibration modes and measures of controllability and observabil?ity of structural vibration and acoustic radiation. Observability measures based on vibrational response outputs and acoustic pressure outputs are compared and the emphasis is on the mea?sures based on the acoustic pressure outputs. Finally the application and usefulness of the controllability and observability measures to active control of structural vibration and acous?tic radiation are illustrated and compared. It shows that the controllability and observability measures provide useful guidance for active control design especially in optimal actuator and sensor placement.     

Active vibration control for structural-acoustic coupling system of a 3-D vehicle cabin model

This paper presents an active vibration control system for use with structural-acoustic coupling system using piezoelectric actuators and piezoelectric sensors. For modelling a complicated 3-D vehicle cabin model, the structural-acoustic coupling system is analyzed by combining the structural data from modal testing with the acoustic data from the finite element method. Through the structural-acoustic analysis program, the control plate and the control modes are selected, which are most effective for attenuating its noise. A robust LQG controller with two sensor signal filters is designed to remove the experimental problems such as the spillover effect due to uncontrolled modes. The robust LQG controller for the structural-acoustic coupling system can reduce the interior noise of the cavity as well as the structural vibration of the cabin.     

Analysis of nonlinear steady state vibration of a multi-degree-of-freedom system using component mode synthesis method

In this paper, an analytical approach for nonlinear forced vibration of a multi-degree-of-freedom system is proposed using the component mode synthesis method. The whole system is divided into some components and a nonlinear modal equation of each component is derived using the free-interface vibration modes. The modal equations of all components and the conjunction conditions are solved simultaneously, and then the modal responses of components are derived. Finally, the dynamic responses of the whole system can be obtained. The degrees of freedom of modal equations can be reduced when the lower vibration modes are only adopted in each component. As a numerical example, a nine-degree-of-freedom system is considered, in which all spring have cubic type nonlinearity. As a result, it is shown that when there are no rigid modes in components, the compliance by the proposed method agrees very well with the exact one even if the lower vibration modes of components are only adopted. The other hand, in the case with rigid modes in components, the compliance has a little error compared with the exact result. It is recognized that the method proposed is very effective in the case without rigid modes in components for the actual application.     

Application of discrete-time variable structure control in the vibration reduction of a flexible structure

This paper studies the application of using the discrete-time variable structure control method to reduce the vibration of the flexible structure. The structure is subjected to arbitrary, unmeasurable disturbance forces. The concept of independent modal space control is adopted, and the system is studied by the discrete-time model. Here, discrete sensors and actuators are used. We choose the modal filters as the state estimator to obtain the modal co-ordinates and modal velocities for the modal space control. A discrete-time variable structure controller with a disturbance force observer is adopted due to its distinguished robustness property of insensitiveness to parameter uncertainties and external disturbances. The included disturbance force observer can observe the unknown disturbance modal forces, which are used in the discrete-time variable structure control law to cancel out the excitations. The upperbound limitations of the unknown disturbances in the variable structure control, therefore, are no longer needed. The switching surface, in the discrete-time variable structure control system, is designed in an optimal sense. That is, along the switching surface, the cost function of the states is minimized. The investigation of this research focuses on the optimal switching surface design and the control performances of the discrete-time variable structure controller. The performance of estimating the disturbance modal forces and the robustness property of the control law are also discussed.     

结构振动和声辐射的可控性和可观性指标

结构振动和声辐射的可控性和可观性指标对有源控制系统作动器/传感器的布置具有重要意义。对结构振动和声辐射的可控性和可观性指标进行了研究,提出了基于声压输出的振动模态可观性指标,在振动模态可控性和可观性指标的基础上,结合振动模态的声辐射效率,提出了结构声辐射的可控性和可观性指标和基于声压输出的可观性指标。以板结构为例,对结构振动和声辐射的可控性和可观性指标进行了计算分析和讨论,比较了基于声压输出和基于振动响应输出的振动模态可观性,重点研究了基于声压输出的振动模态可观性和结构振动及声辐射的可观性特点,最后对指标值在结构声有源控制中输入(输出)位置选择上的应用进行了讨论和比较,通过数值仿真对指标的有效性进行了验证。      

Virtual sensors for active noise control in acoustic-structural coupled enclosures using structural sensing: part II--Optimization of structural sensor placement

The work proposed an optimization approach for structural sensor placement to improve the performance of vibro-acoustic virtual sensor for active noise control applications. The vibro-acoustic virtual sensor was designed to estimate the interior sound pressure of an acoustic-structural coupled enclosure using structural sensors. A spectral-spatial performance metric was proposed, which was used to quantify the averaged structural sensor output energy of a vibro-acoustic system excited by a spatially varying point source. It was shown that (i) the overall virtual sensing error energy was contributed additively by the modal virtual sensing error and the measurement noise energy; (ii) each of the modal virtual sensing error system was contributed by both the modal observability levels for the structural sensing and the target acoustic virtual sensing; and further (iii) the strength of each modal observability level was influenced by the modal coupling and resonance frequencies of the associated uncoupled structural/cavity modes. An optimal design of structural sensor placement was proposed to achieve sufficiently high modal observability levels for certain important panel- and cavity-controlled modes. Numerical analysis on a panel-cavity system demonstrated the importance of structural sensor placement on virtual sensing and active noise control performance, particularly for cavity-controlled modes.     

Stationary response of combined linear dynamic systems to stationary excitation

The stationary response of a broad class of combined linear systems to stationary random excitation is determined by the normal mode method. The systems are characterized by a viscously damped simple beam (or string, membrane, thin plate or shell, etc.) connected at discrete points to a multiplicity of viscously damped linear oscillators and/or masses. The solution of the free vibration problem by way of Green functions and the deterministic forced vibration problem by modal analysis for both proportional and non-proportional damping is reviewed. The orthogonality relation for the natural modes of vibration is used to derive a unique relationship between the cross-spectral density functions of the applied forces and the cross-spectral density functions of the generalized forces. Finally, the response spectral density functions and the mean square responses of the beam and oscillators are derived in closed form, exact for the proportionally damped system and approximate for the non-proportionally damped system.     

Modeling and simulating of V-shaped piezoelectric micro-cantilevers using MCS theory considering the various surface geometries

Atomic force microscopy (AFM) is widely used as a tool in studying surfaces and mechanical properties of materials at nanoscale. This paper deals with mechanical and vibration analysis of AFM vibration in the non-contact and tapping modes for V-shaped piezoelectric micro-cantilever (MC) with geometric discontinuities and cross section variation in the air ambient. In the vibration analysis, Euler-Bernoulli beam theory based on modified couple stress (MCS) theory has been used. The governing equation of motion has been derived by using Hamilton's principle. By adopting finite element method (FEM), the MC differential equation has been solved. Damping matrix was considered in the modal space. Frequency response was obtained by using Laplace transform, and it has been compared with experimental results. Newmark algorithm has been used based on constant average acceleration to analyze time response of MC, and then time response results in the vibration mode, far from the sample surface have been compared with experimental data. In vicinity of sample surface, MC is influenced by various nonlinear forces between the probe tip and sample surface, including van der Waals, contact, and capillary forces. Time response was examined at different distances between MC base and sample surface, and the best distance was selected for topography. Topography results of different types of roughness showed that piezoelectric MC has been improved in the air ambient. Topography showed more accurate forms of roughness, when MC passes through sample surface at higher frequencies. The surface topography investigation for tapping and non-contact modes showed that using of these two modes are suitable for topography.     

Active modal control simulation of vibro-acoustic response of a fluid-loaded plate

Active modal control simulation of vibro-acoustic response of a fluid-loaded plate is presented. The active modal control of the vibro-acoustic response is implemented using piezoelectric actuators/sensors. The active modal damping is added to the coupled system via negative velocity feedback. The feedback gain between the piezoelectric actuators/sensors for the modal control is obtained using the in-vacuo modal matrix and the incompressible fluid-loaded modal matrix. The modal control performance of structural vibration and acoustic radiation of a baffled plate is numerically studied. It is shown that the proposed method increases the modal damping ratio and achieves reduction in the mean square velocity and the sound power for given modes of the fluid-loaded plate.     

OPTIMAL GAIN DISTRIBUTION FOR TWO-DIMENSIONAL MODAL TRANSDUCER AND ITS IMPLEMENTATION USING MULTI-LAYERED PVDF FILMS

Using finite element techniques to optimize the spatial gain distribution of PVDF film, we developed a modal transducer for specific modes to perform real-time vibration control of integrated smart structures. This method makes it possible to design the modal transducer for two-dimensional structure with arbitrary geometry and boundary conditions. As a practical means for implementation, the gain distribution was approximated by optimizing electrode patterns, lamination angles, and relative poling directions of the multi-layered PVDF transducer. This corresponds to the approximation of a continuous function using discrete values. A genetic algorithm was used in the optimization of the electrode pattern and lamination angle of each PVDF layer. For this purpose, the continuous value of the lamination angle was encoded into discrete values using binary 5-bit strings. Validity of the proposed concept was demonstrated experimentally. A modal sensor for the first and second modes of cantilevered composite plate was designed using two layers of PVDF films. The experimental results show that spillover signals by residual modes were successfully reduced using the optimized multi-layered PVDF sensor. The actuator was designed also using two layers of PVDF films to minimize the system energy in the control modes. Real-time vibration control system was successfully realized using the optimized sensor, actuator, and a discrete LQG controller. Closed-loop test showed that modal peaks of the first and second modes were reduced by amounts of 13 and 4 dB respectively.     

A modal disturbance estimator for vibration suppression in nonlinear flexible structures

This paper presents a control strategy for the suppression of vibration due to unknown disturbance forces in large, nonlinear flexible structures. The control action proposed, based on the modal approach, consists of two contributions. The first is the well-known Independent Modal-Space Control, which increases system damping and improves its behavior close to the resonance frequencies. The second is a disturbance estimator, which calculates the modal components of the external forces acting on the system and compensates for them using actuator forces. The system modal coordinates, required by both logics, are estimated through a modal state observer.The proposed control logic is tested on a flexible boom. The paper reports the numerical and experimental results both for the linear and nonlinear (large motion) boom configuration.     

Modal analysis by free vibration response only for discrete and continuous systems

This work aims to develop the algorithm for modal analysis by free vibration response only (MAFVRO), in particular for the general or non-proportional viscous damping system model. If the structural displacement or acceleration response due to free vibration can be measured, the system response matrices, including the displacement, velocity and acceleration, can be obtained through numerical differential or integration methods. These response matrices can then be applied to the developed MAFVRO method to determine the structural modal parameters. The numerical differential and integration methods are introduced and adopted to establish the modal parameter prediction program for the non-proportional damping model of MAFVRO. This work also shows the applications of MAFVRO to the multiple degree-of-freedom (mdof) systems and the cantilever beam, respectively. Both the discrete and continuous systems are demonstrated for the feasibility of the MAFVRO algorithm. The developed method uses the free vibration output response only and can obtain the structural modal parameters successfully.     

Model independent control of lightly damped noise/vibration systems

Feedforward control is a popular strategy of active noise/vibration control. In well-damped noise/vibration systems, path transfer functions from actuators to sensors can be modeled by finite impulse response (FIR) filters with negligible errors. It is possible to implement noninvasive model independent feedforward control by a recently proposed method called orthogonal adaptation. In lightly damped noise/vibration systems, however, path transfer functions have infinite impulse responses (IIRs) that cause difficulties in design and implementation of broadband feedforward controllers. A major source of difficulties is model error if IIR path transfer functions are approximated by FIR filters. In general, active control performance deteriorates as model error increases. In this study, a new method is proposed to design and implement model independent feedforward controllers for broadband in lightly damped noise/vibration systems. It is shown analytically that the proposed method is able to drive the convergence of a noninvasive model independent feedforward controller to improve broadband control in lightly damped noise/vibration systems. The controller is optimized in the minimum H2 norm sense. Experiment results are presented to verify the analytical results.     

Radiation clusters and the active control of sound transmission into a symmetric enclosure

Radiation cluster control is proposed for the purpose of attenuating harmonic sound transmission into a symmetric enclosure using error signals derived from structural vibration sensors. The approach falls into a category of middle authority control, which is between LAC (low authority control: structural modal control) and HAC (high authority control: radiation modal control), possessing the benefit of practicality over LAC, while providing high control performance and some flexibility of control gain assignment similar to HAC. The structure of a radiation cluster control system is outlined, showing that it is possible to control a target cluster without affecting the other clusters. A design procedure for the radiation cluster control system is then proposed. Numerical results are also presented.     

An algorithm for the optimal design of passive vibration controllers for flexible systems

A gradient algorithm is developed for the optimal design of discrete passive dampers in the vibration control of a class of flexible (distributed parameter) systems. A complete mathematical development is presented for slender beams in flexural vibration. The algorithm systematically seeks to make the modal damping and natural frequencies of the system reach a set of preassigned values. Single damper and the two damper control examples indicate that the proposed algorithm converges faster than the Davison method used in reference [1] for those cases.     

Estimation of interfacial forces in time domain for linear systems

A structural path rank ordering process under transient excitations requires a good knowledge of the interfacial path forces, which are difficult to directly measure. Four time domain methods to estimate the interfacial forces are proposed and comparatively evaluated with application to linear time-invariant, proportionally damped discrete systems. First, the transient response is derived by modal analysis and a direct time domain technique to calculate the interfacial forces is outlined. Next, the frequency domain estimation methods, based on the sub-system concept are reviewed, and an inverse Fourier transform scheme is introduced. An indirect method of estimating interfacial force in transient state is then developed through an inverse procedure of modal analysis. The sub-system approach is employed to obtain the interfacial forces based on the forced vibration response of the original system and modal data of the sub-system. Finally, an approximate time domain scheme is suggested that could be used only if the system properties are known or precisely estimated. Although the proposed indirect methods are designed for eventual experimental applications, this article provides numerical feasibility studies via a simple source-path-receiver system (with parallel vibration paths) that has five translational degrees of freedom. The proposed methods are compared under ideal impulse force excitation input and a periodic sawtooth load (without and with Gaussian noise) to observe the starting transients as well as subsequent motions and interfacial forces. Preliminary comparisons with a laboratory experiment are very promising.     

Reduction of sound transmission into a circular cylindrical shell using distributed vibration absorbers and Helmholtz resonators

A modal expansion method is used to model a cylindrical enclosure excited by an external plane wave. A set of distributed vibration absorbers (DVAs) and Helmholtz resonators (HRs) are applied to the structure to control the interior acoustic levels. Using an impedance matching method, the structure, the acoustic cavity, and the noise reduction devices are fully coupled to yield an analytical formulation of the structural kinetic energy and acoustic potential energy of a treated cylindrical cavity. Lightweight DVAs and small HRs tuned to the natural frequencies of the targeted structural and acoustic modes, respectively, result in significant acoustic and structural attenuation when the devices are optimally damped. Simulations show that significant interior noise reduction can only be achieved by adding damping to both structural and acoustic modes, which are resonant in the frequency bandwidth of interest. In order to be independent of the azimuth angle of the excitation and to avoid unwanted modal interactions, the devices are distributed evenly around the cylinder in rings. This treatment can only achieve good performance if the structure and the acoustic cavity are lightly damped.