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.     

Vibration analysis of harmonically excited non-linear system using the method of multiple scales

An analytical method is presented for evaluation of the steady state periodic behavior of non-linear systems. This method is based on the substructure synthesis formulation and a multiple scales procedure, which is applied to the analysis of non-linear responses. A complex non-linear system is divided into substructures, of which equations are approximately transformed to modal co-ordinates including non-linear term under the reasonable procedure. Then, the equations are synthesized into the overall system and the solution of the non-linear system can be obtained. Based on the method of multiple scales, the proposed procedure reduces the size of large-degree-of-freedom problem in solving the non-linear equations. Feasibility and advantages of the proposed method are illustrated by the application of the analytic procedure to the non-linear rotating machine system as a large mechanical structure system. Results obtained are reported to be an efficient approach with respect to non-linear response prediction when compared with other conventional methods.     

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.     

Semi-Inverse Solution of a Pure Beam Bending Problem in Gradient Elasticity Theory: The Absence of Scale Effects

The semi-inverse solutions of pure beam bending problems within the three-dimensional formulation of gradient elasticity theory as exact tests for the problem of estimating the efficient bending stiffness of so-called scale-dependent thin beams and plates due to the necessity of modeling sensing devices are presented. It is shown that the solutions within the gradient elasticity theory give classic beam bending stiffnesses and demonstrate the invalidity of the widespread results and estimates obtained in the past 15 years during study of scale effects within the gradient beam theories, according to which the relative bending stiffness grows by a hyperbolic law with decreasing thickness.     

Resonant vibration control of rotating beams

Rotating structures, like e.g. wind turbine blades, may be prone to vibrations associated with particular modes of vibration. It is demonstrated, how this type of vibrations can be reduced by using a collocated sensor-actuator system, governed by a resonant controller. The theory is here demonstrated by an active strut, connecting two cross-sections of a rotating beam. The structure is modeled by beam elements in a rotating frame of reference following the beam. The geometric stiffness is derived in a compact form from an initial stress formulation in terms of section forces and moments. The stiffness, and thereby the natural frequencies, of the beam depend on the rotation speed and the controller is tuned to current rotation speed to match the resonance frequency of the selected mode. It is demonstrated that resonant control leads to introduction of the intended level of damping in the selected mode and, with good modal connectivity, only very limited modal spill-over is generated. The controller acts by resonance and therefore has only a moderate energy consumption, and successfully reduces modal vibrations at the resonance frequency.     

Strongly nonlinear beats in the dynamics of an elastic system with a strong local stiffness nonlinearity: Analysis and identification

We consider a linear cantilever beam attached to ground through a strongly nonlinear stiffness at its free boundary, and study its dynamics computationally by the assumed-modes method. The nonlinear stiffness of this system has no linear component, so it is essentially nonlinear and nonlinearizable. We find that the strong nonlinearity mostly affects the lower-frequency bending modes and gives rise to strongly nonlinear beat phenomena. Analysis of these beats proves that they are caused by internal resonance interactions of nonlinear normal modes (NNMs) of the system. These internal resonances are not of the classical type since they occur between bending modes whose linearized natural frequencies are not necessarily related by rational ratios; rather, they are due to the strong energy-dependence of the frequency of oscillation of the corresponding NNMs of the beam (arising from the strong local stiffness nonlinearity) and occur at energy ranges where the frequencies of these NNMs are rationally related. Nonlinear effects start at a different energy level for each mode. Lower modes are influenced at lower energies due to larger modal displacements than higher modes and thus, at certain energy levels, the NNMs become rationally related, which results in internal resonance. The internal resonances of NNMs are studied using a reduced order model of the beam system. Then, a nonlinear system identification method is developed, capable of identifying this type of strongly nonlinear modal interactions. It is based on an adaptive step-by-step application of empirical mode decomposition (EMD) to the measured time series, which makes it valid for multi-frequency beating signals. Our work extends an earlier nonlinear system identification approach developed for nearly mono-frequency (monochromatic) signals. The extended system identification method is applied to the identification of the strongly nonlinear dynamics of the considered cantilever beam with the local strong nonlinear stiffness at its free end.     

Complex eigensolutions of coupled flexural and longitudinal modes in a beam with inclined elastic supports with non-proportional damping

Structure borne vibration and noise in an automobile are often explained by representing the full vehicle as a system of elastically coupled beam structures representing the body, engine cradle and body subframe where the engine is often connected to the chassis via inclined viscoelastic supports. To understand more clearly the interactions between a beam structure and isolators, this article examines the flexural and longitudinal motions in an elastic beam with intentionally inclined mounts (viscoelastic end supports). A new analytical solution is derived for the boundary coupled Euler beam and wave equations resulting in complex eigensolutions. This system is demonstrated to be self-adjoint when the support stiffness matrices are symmetric; thus, the modal analysis is used to decouple the equations of motion and solve for the steady state, damped harmonic response. Experimental validation and computational verifications confirm the validity of the proposed formulation. New and interesting phenomena are presented including coupled rigid motions, modal properties for ideal angled roller boundaries, and relationships between coupling and system modal loss factors. The ideal roller boundary conditions when inclined are seen as a limiting case of coupled longitudinal and flexural motions. In particular, the coupled rigid body motions illustrate the influence of support stiffness coupling on the eigenvalues and eigenfunctions. The relative modal strain energy concept is used to distinguish the contribution of longitudinal and flexural deformation modes. Since the beam is assumed to be undamped, the system damping is derived from the viscoelastic supports. The support damping (for a given loss factor) is shown to be redistributed between the system modes due to the inclined coupling mechanisms. Finally, this article provides valuable insight by highlighting some technical issues a real-life designer faces when balancing modeling assumptions such as rigid or elastic formulations, proportional or non-proportional damping, and coupling terms in multidimensional joint properties.     

A FINITE ELEMENT FORMULATION FOR COUPLING RIGID AND FLEXIBLE BODY DYNAMICS OF ROTATING BEAMS

The work presented in this paper is based on an existing comprehensive formulation for rotating flexible systems. In the existing formulation the flexible degrees of freedom (d.o.f.) are represented by an analytically computed modal basis and the coupling matrices between the rigid- and the flexible-body d.o.f. are developed based on the analytical modal representation of the flexible d.o.f. In this paper, the existing formulation is generalized for rotating beams by representing the flexible d.o.f. either as physical d.o.f. of a finite element formulation or as a set of retained and internal d.o.f. of a Craig-Bampton formulation. The coupling matrices between the rigid-body rotation and the flexible d.o.f. are developed accordingly. The non-linear effects from the work done by the centrifugal forces are included in the formulation. Finite element shape functions of a beam element in a three-dimensional space and finite element shape functions for solid elements are employed for deriving the coupling terms between the rigid-body d.o.f. and the physical d.o.f. An additional transformation is required and performed when the right-body d.o.f. are coupled with the internal and the retained d.o.f. of a Craig-Bampton formulation. The coupled system of equations is solved in the time domain by combining the Newmark method for time integration and the Newton-Raphson method for solving the non-linear system of equations within each time step. Analyses are performed for a flexible rotating beam in order to validate the development. An analytical solution is compared with the new formulations that represent the rotating beam flexibility with the physical d.o.f. of beam or solid elements. The analytical solution is also compared to the formulation that represents the flexible d.o.f. in terms of retained and internal d.o.f. of a Craig-Bampton formulation. Very good correlation between the analytical and numerical results is observed.     

Modal balancing of flexible rotors with bow and distributed unbalance

Unbalance and bow are found to be one of the most common causes of synchronous machinery vibrations in rotating systems. Concentrated lumped mass models are adopted in most of the finite element approach for modeling unbalances and subsequent balancing in rotating systems. But this assumption may not be appropriate for long slender rotors with unbalances distributed along the length of the rotor. A polynomial curve for eccentricity distribution with finite element modeling is used to identify the distributed unbalance. The unbalance eccentricity distributions are estimated using the measured vibration responses at a speed below the balancing speed. Modal correction mass required to balance a rotor at its first bending critical speed, having both distributed unbalance and bow is computed knowing the amplification factor at critical speed. The rotor is balanced at its first bending critical speed using modal balancing method in a single trial run and using a single balancing plane. The method thus avoids multiple trial runs required for modal balancing of flexible rotors. This method is verified on an experimental rotor having both bow and unbalance. The concept of quantifying the distributed unbalance using ‘Norm’ of eccentricity polynomial function is also introduced for the first time.     

Coupled bending-torsion vibrations of rotating blades of asymmetric aerofoil cross section with allowance for shear deflection and rotary inertia by use of the Reissner method

Theoretical natural frequencies and modal shapes of the first five modes of vibration are presented for a rotating blade of asymmetric aerofoil cross section, with allowance for shear deflection and rotary inertia. Frequency equations for a rotating blade with asymmetry in one plane are developed by using the Ritz process, in two ways: namely, by proceeding according to the Reissner method and according to the classical potential energy method. In both cases shape functions for the bending moment, shearing force, twisting moment, bending slope, elastic twist and deflection are developed in series form. The results obtained are compared with those existing in the literature; it is found that the Reissner method approach yields more rapid convergence than does the classical potential energy method.     

Vibration characteristics of a rotating flexible arm with ACLD treatment

In this paper, the vibration behavior and control of a clamped–free rotating flexible cantilever arm with fully covered active constrained layer damping (ACLD) treatment are investigated. The arm is rotating in a horizontal plane in which the gravitational effect and rotary inertia are neglected. The stress–strain relationship for the viscoelastic material (VEM) is described by a complex shear modulus while the shear deformations in the two piezoelectric layers are neglected. Hamilton's principle in conjunction with finite element method (FEM) is used to derive the non-linear coupled differential equations of motion and the associated boundary conditions that describe the rigid hub angle rotation, the arm transverse displacement and the axial deformations of the three-layer composite. This refined model takes into account the effects of centrifugal stiffening due to the rotation of the beam and the potential energies of the VEM due to extension and bending. Active controllers are designed with PD for the piezosensor and actuator. The vibration frequencies and damping factors of the closed-loop beam/ACLD system are obtained after solving the characteristic complex eigenvalue problem numerically. The effects of different rotating speed, thickness ratio and loss factor of the VEM as well as different controller gain on the damped frequency and damping ratio are presented. The results of this study will be useful in the design of adaptive and smart structures for vibration suppression and control in rotating structures such as rotorcraft blades or robotic arms.     

System identification-guided basis selection for reduced-order nonlinear response analysis

Reduced-order nonlinear simulation is often times the only computationally efficient means of calculating the extended time response of large and complex structures under severe dynamic loading. This is because the structure may respond in a geometrically nonlinear manner, making the computational expense of direct numerical integration in physical degrees of freedom prohibitive. As for any type of modal reduction scheme, the quality of the reduced-order solution is dictated by the modal basis selection. The techniques for modal basis selection currently employed for nonlinear simulation are ad hoc and are strongly influenced by the analyst's subjective judgment. This work develops a reliable and rigorous procedure through which an efficient modal basis can be chosen. The method employs proper orthogonal decomposition to identify nonlinear system dynamics, and the modal assurance criterion to relate proper orthogonal modes to the normal modes that are eventually used as the basis functions. The method is successfully applied to the analysis of a planar beam and a shallow arch over a wide range of nonlinear dynamic response regimes. The error associated with the reduced-order simulation is quantified and related to the computational cost.     

IDENTIFICATION OF MODAL PARAMETERS FROM MEASURED OUTPUT DATA USING VECTOR BACKWARD AUTOREGRESSIVE MODEL

In this paper, a modal identification system that is based on the vector backward autoregressive (VBAR) model has been developed for the identification of natural frequencies, damping ratios and mode shapes of structures from measured output data. The modal identification using forward autoregressive approach has some problems in discriminating the structure modes from spurious modes. On the contrary, the VBAR approach provides a determinate boundary for the separation of system modes from spurious modes, and an eigenvalue filter for the selection of physical modes is existent in the proposed method. For convenience of application, the backward state equation established from VBAR model is transformed into a forward state equation, which is termed as transformed VFAR model in this paper. In addition, the extraction of equivalent system matrix of state equation of motion for structures from the transformed VFAR model has been developed, and then the normal modes can be calculated from the identified equivalent system matrix. Two examples of modal identification are carried out to demonstrate the availability and effectiveness of the proposed backward approach: (1) Numerical modal identification for a three-degree-of-freedom dynamic system with noise level in 20% of r.m.s of measured output data; (2) experimental modal identification of a cantilever beam. Finally, to show the advantage of the proposed VBAR approach on the selection of physical modes, the modal identification by stochastic subspace method was performed. The results from both methods are compared.     

AN EXTENSION OF COMPLEX MODES FOR THE RESOLUTION OF FINITE-ELEMENT POROELASTIC PROBLEMS

A new modal synthesis technique inspired from complex modes is presented in this paper. The purpose of this method is to find an alternative to the weakness of usual modal techniques in order to improve resolution of finite-element poroelastic problems. Firstly, the construction of modes and the synthesis for a general forcing is explained. Secondly, the approach is applied to a one-dimensional poroelastic problem. Finally, the approach is numerically tested and its performance is investigated by comparison with solutions obtained with a direct resolution of the system.     

Modal correlation approaches for general second-order systems: Matching mode pairs and an application to Campbell diagrams

Modal correlation is well developed for undamped and proportionally damped vibrating systems. It is less well defined for generally damped linear systems. This paper addresses the fundamental problem of comparing two general second-order linear systems through modal information. It considers precisely the problem of how to achieve matching of modes (mode pairs).There are several possible motivations for modal correlation of which the most important is probably the model updating application. In that application, one set of modes derives from a numerical model and the other from measured data. This paper focuses mainly on a different application—constructing Campbell diagrams for rotating machines. There are two significant differences here: (a) the two sets of modes being compared at any one time are from the same numerical model but for different spin speeds and (b) there is generally a strong distinction between the left and right modes of the system. Without some modal correlation approach, the Campbell diagram is constructed simply as a set of points on the frequency-speed graph. With modal correlation, the eigenvalue problem can be solved at far fewer speeds and the points can be joined meaningfully.A dimensionless (n×n) modal-matching array is produced whose entries indicate which pairs of modes from the first system best correlate with any particular pair of modes from the second system. The presented work is motivated mainly by the application of developing Campbell diagrams for rotating machines by means which are more effective than simply plotting a large set of discrete points. Wider applications of this paper include model updating procedures where mode pairs must be matched initially to ensure convergence towards the exact system.     

Vibration control of beams on elastic foundation under a moving vehicle and random lateral excitations

The formulation of three-dimensional dynamic behavior of a Beam On Elastic Foundation (BOEF) under moving loads and a moving mass is considered. The weight of the vehicle is modeled as a moving point load, however the effect of the lateral excitation is considered by modeling: (case 1) a lateral moving load with random intensity for wind excitation and (case 2) a moving mass just in lateral direction of the beam for earthquake excitation. A Dirac-delta function is used to describe the position of the moving load and the moving mass along the beam. The beam foundations are considered as elastic Winkler-type in two perpendicular transverse directions. This model is proposed to investigate the bending response of the rails under the effect of traveling vehicle weight while a random excitation such as earthquake or wind takes place. The results showed the importance of considering the effect of earthquake/wind actions as in bending stress of the beam on elastic foundations. The effect of different regions (different support stiffness) and different velocities of the vehicle on the response of the beam are investigated in mentioned directions. At the end, a linear optimal control algorithm with displacement–velocity feedback is proposed as a solution to suppress the response of BOEFs. By the method of modal analyses and taking into account enough number of vibration modes, state-space equation is obtained, then sufficient number of actuators was chosen for each direction. Stochastic analyses were performed in lateral direction in order to illustrate a comprehensive view for the response of the beam under the random moving load in both controlled and uncontrolled systems. Furthermore, the efficiency of control algorithm on critical velocities is verified by parametric analyses in the vertical direction with the constant moving load for different regions.     

Free vibration analysis of a rotating hub–functionally graded material beam system with the dynamic stiffening effect

A comprehensive dynamic model of a rotating hub–functionally graded material (FGM) beam system is developed based on a rigid–flexible coupled dynamics theory to study its free vibration characteristics. The rigid–flexible coupled dynamic equations of the system are derived using the method of assumed modes and Lagrange's equations of the second kind. The dynamic stiffening effect of the rotating hub–FGM beam system is captured by a second-order coupling term that represents longitudinal shrinking of the beam caused by the transverse displacement. The natural frequencies and mode shapes of the system with the chordwise bending and stretching (B–S) coupling effect are calculated and compared with those with the coupling effect neglected. When the B–S coupling effect is included, interesting frequency veering and mode shift phenomena are observed. A two-mode model is introduced to accurately predict the most obvious frequency veering behavior between two adjacent modes associated with a chordwise bending and a stretching mode. The critical veering angular velocities of the FGM beam that are analytically determined from the two-mode model are in excellent agreement with those from the comprehensive dynamic model. The effects of material inhomogeneity and graded properties of FGM beams on their dynamic characteristics are investigated. The comprehensive dynamic model developed here can be used in graded material design of FGM beams for achieving specified dynamic characteristics.     

The role of non-linear theories in transient dynamic analysis of flexible structures

Transient dynamic analysis of flexible structures undergoing large motions is considered. For rotating structures, it is explicitly shown that appropriate account of the influence of centrifugal force on the bending stiffness requires the use of a geometrically non-linear (at least second-order) beam theory. Use of a first-order (linearized) linear beam theory results in a spurious loss of bending stiffness. For a rotating plane beam, a set of linear partial differential equations of motion—that includes all inertia effects (Coriolis, centrifugal, acceleration of revolution) and coupling between extensional and flexural deformations—is derived from the fully non-linear beam theory by consistent linearization. The analysis is subsequently extended to the more general case of a plate, accomodating shear deformation, and undergoing a general three-dimensional rotating motion. The discretization process of the resulting linear equations of motion for the beam and the plate is also discussed.     

Aeroelastic flutter of a disk rotating in an unbounded acoustic medium

The linear aeroelastic stability of an unbaffled flexible disk rotating in an unbounded fluid is investigated by modeling the disk-fluid system as a rotating Kirchhoff plate coupled to the irrotational motions of a compressible inviscid fluid. A perturbed eigenvalue formulation is used to compute systematically the coupled system eigenvalues. Both a semi-analytical and a numerical method are employed to solve the fluid boundary value problem. The semi-analytical approach involves a perturbation series solution of the dual integral equations arising from the fluid boundary value problem. The numerical approach is a boundary element method based on the Hadamard finite part. Unlike previous works, it is found that a disk with zero material damping destabilizes immediately beyond its lowest critical speed. Upon the inclusion of small disk material damping, the flutter speeds become supercritical and increase with decreasing fluid density. The competing effects of radiation damping into the surrounding fluid and disk material damping control the onset of flutter at supercritical speed. The results are expected to be relevant for the design of rotating disk systems in data storage, turbomachinery and manufacturing applications.     

NON-LINEAR DYNAMIC MODEL OF AN INEXTENSIBLE ROTATING FLEXIBLE ARM SUPPORTED ON A FLEXIBLE BASE

A mathematical model for a flexible arm undergoing large planar flexural deformations, continuously rotating under the effect of a hub torque and supported by a flexible base is developed. The position of a typical material point along the span of the arm is described using the inertial reference frame via a transformation matrix from the body co-ordinate system, which is attached to the flexible root of the rotating arm. The condition of inextensibility is employed to relate the axial and transverse deflections of the material point, within the beam body co-ordinate system. The position and velocity vectors obtained, after imposing the inextensibility conditions, are used in the kinetic energy expression while the exact curvature is used in the potential energy. Lagrangian dynamics in conjunction with the assumed modes method is utilized to derive, directly, the non-linear equivalent temporal equations of motion. The resulting non-linear model, which is composed of four coupled non-linear ordinary differential equations, is discussed, simulated and the results of this simulation are presented. The effects of the base flexibility are explored by comparing the resulting simulation results, for various flexibility coefficients, with previously published works of the authors. Moreover, the numerical results show that the base flexibility has a very important effect on the stability of rotating flexible arms that should be accounted for when simulating such systems.