Showing papers in "Journal of Vibration and Acoustics in 1996"

An Impedance-Based System Modeling Approach for Induced Strain Actuator-Driven Structures

Abstract: This paper presents the theoretical development and experimental verification of a system model of piezoelectric (PZT) patch actuators for induced strain actuation of two-dimensional active structures. The model includes the dynamic interaction between PZT actuators and their host structures. Analytical solutions of the output behavior of the PZT actuators have been developed based upon the actuator input impedance and the mechanical impedance of the host structures. The impedancebased model was then applied to thin plates and thin shells, and to beams. The case studies demonstrate the generality and utility of the impedance modeling approach. A simply-supported thin plate with surface-bonded PZT patches was built and tested so that the ability of the impedance model to accurately predict the dynamic performance of the actuator and the host structure has been verified. When compared with conventional static models, the impedance modeling method offers insight into the dynamic coupling of the integrated PZT/substrate systems.

Experimental Verification of the Importance of The Nonlinear Curvature in the Response of a Cantilever Beam

Abstract: An experimental and theoretical investigation into the first- and second-mode responses of a parametrically excited slender cantilever beam is presented. Inclusion of quadratic damping in the analytical model significantly improves the agreement between the experimental and theoretical results. In addition, the experimental results verify that the often ignored nonlinear curvature terms play a dominant role in the response of the first mode and that the nonlinear inertia terms play a dominant role in the response of the second mode.

The Derivatives of Repeated Eigenvalues and Their Associated Eigenvectors

Abstract: This paper considers the calculation of eigenvalue and eigenvector derivatives when the eigenvalues are repeated. An extension to Nelson’s method is used to calculate the first order derivatives of eigenvectors when the derivatives of the associated eigenvalues are also equal. The continuity of the eigenvalues and eigenvectors is discussed, and the discontinuities in the eigenvectors, when they are regarded as functions of two or more design parameters, is demonstrated. The validity of Taylor series for the eigenvalues and eigenvectors is examined and the use of these series critically assessed.

Estimation of Mass, Stiffness and Damping Matrices from Frequency Response Functions

Abstract: A frequency-domain method for estimating the mass, stiffness and damping matrices of the model of a structure is presented. The developed method is based on our previous work on the extraction of normal modes from the complex modes of a structure. A transformation matrix is obtained from the relationship between the complex and the normal frequency response functions ofa structure. The transformation matrix is employed to calculate the damping matrix of the system. The mass and the stiffness matrices are identified from the normal frequency response functions by using the least squares method. Two simulated systems are employed to illustrate the applicability of the proposed method. The results indicate that the damping matrix can be identified accurately by the proposed method. The reason for the good results is that the damping matrix is identified independently from the mass and the stiffness matrices. In addition, the robustness of the new approach to uniformly distributed measurement noise is also addressed.

Liapunov-Floquet Transformation: Computation and Applications to Periodic Systems

Abstract: In this paper, a new analysis technique in the study of dynamical systems with periodically varying parameters is presented. The method is based on the fact that all linear periodic systems can be replaced by similar linear time-invariant systems through a suitable periodic transformation known as the Liapunov-Floquet (L-F) transformation. A general technique for the computation of the L-F transformation matrices is suggested. In this procedure, the state vector and the periodic matrix of the linear system equations are expanded in terms of the shifted Chebyshev polynomials over the principal period. Such an expansion reduces the original differential problem to a set of linear algebraic equations from which the state transition matrix (STM) can be constructed over the period in closed form. Application of Floquet theory and eigenanalysis to the resulting STM yields the L-F transformation matrix in a form suitable for algebraic manipulations. The utility of the L-F transformation in obtaining solutions of both linear and nonlinear dynamical systems with periodic coefficients is demonstrated. It is shown that the application of L-F transformation to free and harmonically forced linear periodic systems directly provides the conditions for internal and combination resonances and external resonances, respectively. The application of L-F transformation to quasilinear periodic systems provides a dynamically similar quasilinear systems whose linear parts are time-invariant and the solutions of such systems can be obtained through an application of the time-dependent normal form theory. These solutions can be transformed back to the original coordinates using the inverse L-F transformation. Two dynamical systems, namely, a commutative system and a Mathieu type equation are considered to demonstrate the effectiveness of the method. It is shown that the present technique is virtually free from the small parameter restriction unlike averaging and perturbation procedures and can be used even for those systems for which the generating solutions do not exist in the classical sense. The results obtained from the proposed technique are compared with those obtained via the perturbation method and numerical solutions computed using a Runge-Kutta type algorithm.

Free Vibration of Serpentine Belt Drive Systems

Abstract: The vibration of an automotive serpentine belt drive system greatly affects the perceived quality and the reliability of the system. Accessory drives with unfavorable vibration characteristics transmit excessive noise and vibration to other vehicle structures, to the vehicle occupants, and may also promote the fatigue and failure of system components. Moreover, these characteristics are a consequence of decisions made early on in the design and arrangement of the accessory drive system. The present paper focuses on fundamental modeling issues that are central to predicting accessory drive vibration. To this end, a prototypical drive is evaluated, which is composed of a driven pulley, a driving pulley, and a dynamic tensioner. The coupled equations of free response governing the discrete and continuous elements are presented herein. A closed-form solution method is used to evaluate the natural frequencies and modeshapes. Attention focuses on a key linear mechanism that couples tensioner arm rotation and transverse vibration of the adjacent belt spans. Modal tests on an experimental drive confirm the theoretical predictions.

Long term structural dynamics of mechanical systems with local nonlinearities

Abstract: This paper deals with the long term behavior of periodically excited mechanical systems consisting of linear components and local nonlinearities. The number of degrees of freedom of the linear components is reduced by applying a component mode synthesis technique. Lyapunov exponents are used to identify the character of the long term behavior of a nonlinear dynamic system, which may be periodic, quasi-periodic or chaotic. Periodic solutions are calculated efficiently by solving a two-point boundary value problem using finite differences. Floquet multipliers are calculated to determine the local stability of these solutions and to identify local bifurcation points. The methods presented are applied to a beam system supported by a one-sided linear spring, which reveals very rich, complex dynamic behavior.

Performance of Particle Dampers Under Random Excitation

Abstract: An experimental and analytical study is made of the performance of particle dampers under wide-band random excitation. A small model, provided with a nonlinear auxiliary mass damper, was used to investigate the major system parameters that influence the performance of particle dampers: total auxiliary mass ratio, particle size, container dimension, and the intensity and direction of the excitation. It is shown that properly designed particle dampers, even with a relatively small mass ratio, can considerably reduce the response of lightly damped structures. An approximate analytical solution, which is based on the concept of an equivalent single unit-impact damper, is presented. It is shown that the approximate solution can provide an adequate estimate of the root-mean-square response of the randomly excited primary system when provided with a particle damper that is operating in the vicinity of its optimum range of parameters.

Dynamic Behavior of a Drill-String: Experimental Investigation of Lateral Instabilities

Abstract: This paper focuses on laboratory tests concerned with the lateral behavior of a rod representative of part of drill-string in the area of rotary oil drilling. The original experimental set-up takes into account the curvature of the rod, mud, stabilizers and rotation speed. The lateral behavior of the drill-string subjected to the axial excitations of the bit is governed by time varying parameter equations due to torsionlateral and longitudinal-lateral couplings. The experimental results highlight the different kinds of lateral instabilities and they are compared either with existing experimental, or theoretical results. The experimental investigation described in this paper is included in a wide ranging study which also involves theory and the development ofa computer code, both briefly presented here.

Transient Response Technique Applied to Active Magnetic Bearing Machinery During Rotor Drop

Abstract: The active magnetic bearing (AMB) is a relatively new technology which has many advantages compared with conventional bearing design. In an AMB system. the rolling-element back-up bearings are indispensable to protect the magnetic bearing rotor and stator, and other stationary seals along the rotor shqft. In this paper, a theoretical formulation is proposed and solved numerically to examine the transient response of the flexible rotor, from the time just previous to when the AMB shuts down and including the rotor drop onto the back-up bearing. The backward whirl of the rotor, which may lead to the destructive damage of the machinery, has been analytically predicted at very light support damping and very high support damping. Also, the vibration due to the nonlinearity of the contact point geometry has been included in the analysis. The influence of the support damping on the displacement of the disk and also the contact force between the journal and the inner-race of the back-up bearing have been computed for various rotor system parameters. By comparing these results with the optimum support damping for the simple flexible rotor model, it is shown that this support damping optimization can be applicable for specifying the required optimum range of support damping for the back-up bearings of AMB systems.

Stability and limit cycles of parametrically excited, axially moving strings

Abstract: Tension fluctuations are the dominant source of excitation in automotive belts. In particular designs, these fluctuations may parametrically excite large amplitude transverse belt vibrations and adversely impact belt life. This paper evaluates an efficient discrete model ofa parametrically excited translating belt. The efficiency derives from the use of translating string eigenfunctions as a basis for a Galerkin discretization of the equations of transverse belt response. Accurate and low-order models lead to simple closed-form solutions for the existence and stability of limit cycles near parametric instability regions. In particular, simple expressions are found for the stability boundaries of the general nth-mode principal parametric instability regions and the first summation and difference parametric instability regions. Subsequent evaluation of the weakly nonlinear equation of motion leads to an analytical expression for the amplitudes (and stability) of nontrivial limit cycles that exist around the nth-mode principal parametric instability regions. Example results highlight important conclusions concerning the response of automotive belt drives.

Muffler Performance Studies Using a Direct Mixed-Body Boundary Element Method and a Three-Point Method for Evaluating Transmission Loss

Abstract: In this paper, a single-domain boundary element method is presented for muffler analysis. This method is based on a direct mixed-body boundary integral formulation recently developed for acoustic radiation and scattering from a mix of regular and thin bodies. The main feature of the mixed-body integral formulation is that it can handle all kinds of complex internal geometries, such as thin baffles, extended inlet/ outlet tubes, and perforated tubes, without using the tedious multi-domain approach. The variables used in the direct integral formulation are the velocity potential (or sound pressure) on the regular wall surfaces, and the velocity potential jump (or pressure jump ) on any thin-body or perforated surfaces. The linear impedance boundary condition proposed by Sullivan and Crocker (1978) for perforated tubes is incorporated into the mixed-body integral formulation. The transmission loss is evaluated by a new method called the three-point method. Unlike the conventional four-pole transfer-matrix approach that requires two separate computer runs for each frequency, the three-point method can directly evaluate the transmission loss in one single boundary-element run. Numerical results are compared to existing experimental data for three different muffler configurations.

Pendulum as Vibration Absorber for Flexible Structures: Experiments and Theory

Abstract: The dynamic response ofa beam-tip mass-pendulum system subjected to a sinusoidal excitation is investigated. A simple pendulum mounted to a tip mass of a beam is used as a vibration absorber. The nonlinear equations of motion are developed to investigate the autoparametric interaction between the first two modes of the system. The nonlinear terms appear due to the curvature of the beam and the coupling effect between the beam and pendulum. Complete energy transfer between modes is shown to occur when the beam frequency is twice the pendulum frequency. Experimental results are compared with a theoretical solution obtained using numerical integration. The experimental results are in qualitative agreement with the theory.

On Nonlinear Normal Modes of Systems With Internal Resonance

Abstract: A complex-variable invariant-manifold approach is used to construct the normal modes of weakly nonlinear discrete systems with cubic geometric nonlinearities and either a one-to-one or a three-to-one internal resonance. The nonlinear mode shapes are assumed to be slightly curved four-dimensional manifolds tangent to the linear eigenspaces of the two modes involved in the internal resonance at the equilibrium position. The dynamics on these manifolds is governed by three first-order autonomous equations. In contrast with the case of no internal resonance, the number of nonlinear normal modes may be more than the number of linear normal modes. Bifrcations of the calculated nonlinear normal modes are investigated.

Stability and Controllability of Euler-Bernoulli Beams With Intelligent Constrained Layer Treatments

Abstract: This paper studies the stability and controllability of Euler-Bernoulli beams whose bending vibration is controlled through intelligent constrained layer (ICL) damping treatments proposed by Baz (1993) and Shen (1993, 1994). First of all, the homogeneous equation of motion is transformed into a first order matrix equation in the Laplace transform domain. According to the transfer junction approach by Yang and Tan (1992), existence of nontrivial solutions of the matrix equation leads to a closed-form characteristic equation relating the control gain and closed-loop poles of the system. Evaluating the closed-form characteristic equation along the imaginary axis in the Laplace transform domain predicts a threshold control gain above which the system becomes unstable. In addition, the characteristic equation leads to a controllability criterion for ICL beams. Moreover, the mathematical structure of the characteristic equation facilitates a numerical algorithm to determine root loci of the system. Finally, the stability and controllability of Euler-Bernoulli beams with ICL are illustrated on three cantilever beams with displacement or slope feedback at the free end.

Nonlinear Coupled Vibration Response of Serpentine Belt Drive Systems

Abstract: This theoretical and experimental study identifies a key nonlinear mechanism that promotes strongly coupled dynamics of serpentine belt drive systems. Attention is focused on a prototypical three-pulley system that contains the essential features of automotive serpentine drives having automatic (spring-loaded) tensioners. A theoretical model is presented that describes pulley and tensioner arm rotations, and longitudinal and transverse belt vibration response. A recent investigation demonstrates that infinitesimal belt stretching creates a linear mechanism that couples transverse belt vibration to tensioner arm rotation. Here, it is further demonstrated that finite belt stretching creates a nonlinear mechanism that may lead to strong coupling between pulley/tensioner arm rotation and transverse belt vibration, in the presence of an internal resonance. Theoretical and experimental results confirm the existence of this nonlinear coupling mechanism. In particular, it is shown that very large transverse belt vibrations can result from small resonant torque pulses applied to the crankshaft or accessory pulleys. These large amplitude transverse vibrations are particularly sensitive to seemingly small changes in the rotational mode characteristics.

An Energy-Based Parametric Control Approach for Structural Vibration Suppression via Semi-Active Piezoelectric Networks

Abstract: A structural vibration control concept, using piezoelectric materials shunted with real-time adaptable electrical networks, has been investigated. The variable resistance and inductance in an external RL circuit are used as control inputs. An energy-based parametric control scheme is created to reduce the total system energy (the main structure mechanical energy plus the electrical and mechanical energies of the piezoelectric material and electrical circuit) while minimizing the energy flowing into the main structure. Stability of the closed-loop system is proved. The performance of the controller is examined through analyzing a beam example. It is shown that the structure energy level and vibration amplitude can be suppressed effectively.

Distributed Parameter Modal Filtering Using Smart Sensors

Abstract: This paper considers the design of distributed parameter modal sensors called smart sensors, with a particular emphasis on filtering the combination of appropriately weighted vibration modes providing a specific performance index in control strategy. First, with a two-dimensional distributed parameter sensor using a PVDF film, the necessary and sufficient condition for sensing the transformed modes of a structure is derived. Then, by considering the practicability of the two-dimensional sensors, an alternative approach based upon one-dimensional smart sensors is presented. It is found that the latter approach holds the necessary condition for sensing the transformed mode. This problem is overcome by introducing multiple one-dimensional smart sensors. Moreover, the design procedure for the multiple one-dimensional smart sensors for measuring the transformed mode is established. Finally, an experiment is conducted, demonstrating the validity of the smart sensors.

Revisiting the Moving Mass Problem: Onset of Separation Between the Mass and Beam

Abstract: In the moving mass problem, the interaction force between a moving mass and structure obviously depends on the velocity of moving mass and the flexibility of structure. Thus, in some situations, the interaction force may become zero to change its sign, which implies the onset of the separation between the moving mass and structure. Most investigations on this subject have missed or ignored the possibility of the onset of separation in solving the dynamic responses of structures excited by moving masses. Hence, this paper investigates the onset of the separation between the moving mass and beam, and then takes into account its effect in calculating the interaction forces and also in calculating the dynamic responses of the beams considered herein. It is shown that the separation between the moving mass and structure can occur more easily and has unnegligible effects on the dynamic responses of the structure as the mass ratio (M/ml) increases, especially at high velocity of moving mass. Thus, for accurate prediction of the dynamic response of structure excited by a moving mass, the effect of separation must be taken into account in the analysis.

Practical Limitations in Achieving Shaped Modal Sensors With Induced Strain Materials

Abstract: The purpose of this work is to explore practical limitations associated with the design of distributed modal sensors from induced strain materials for two-dimensional structures and illuminate difficulties associated with positioning sensors on structures in general. Results from this study indicate that a true modal sensor cannot be realized on a two-dimensional structure unless the boundary conditions are all pinned. This result stems from the fact that the sensor aperture must be orthogonal with respect to the structural mode (eigenfunction) and the curvature of the structural mode to render a distributed modal filter. The only class of functions satisfying this condition are sinusoidal functions. In addition, positioning of distributed strain sensing material is shown to be critical to the performance of the sensor over a specified bandwidth, specifically in the vicinity of structural resonances.

A Method for Calculating the Dynamics of Rotating Flexible Structures, Part 1: Derivation

Abstract: The problem of calculating the vibrations of rotating structures has challenged analysts since it was observed that the use of traditional modal approaches may incorrectly lead to the prediction of infinite deformation when rotation rates exceed the first natural frequency Much recently published work on beams has shown that such predictions are artifacts of incorporating incomplete kinematics into the analysis, but only simple structures such as individual beams and plates are addressed The authors present a new approach to analyzing rotating flexible structures that applies to the rotation of general linear (unjointed) structures, using a system of nonlinearly coupled deformation modes This technique, tentatively named a Method of Quadratic Components, utilizes a nonlinear configuration space in which all kinematic constraints are satisfied up to second order

Parametric Control of Structural Vibrations via Adaptable Stiffness Dynamic Absorbers

Abstract: An energy-based algorithm is developed for dynamic absorbers with adaptable stiffness to suppress structural vibrations via real-time parametric control actions. A controller with multi-objective fuzzy logic is created to reduce the main structure energy while constraining the total system energy. To ensure stability, an adaptive-passive supervisor is designed to provide guidelines for implementing the control law. It is proved that the system using this supervisor is globally stable in the sense that all signals involved are bounded. The performance of the controller is demonstrated on a beam example. It is shown that the structure energy level and vibration amplitude can be suppressed effectively.

The Timoshenko Beam on an Elastic Foundation and Subject to a Moving Step Load, Part 2: Transient Response

Abstract: The transient response of a simply supported semi-infinite Timoshenko beam on an elastic foundation to a moving step load is determined. The response is found from summing the solutions to two mutually complementary sets of governing equations. The first solution is a particular solution to the forced equations of motion. The second solution is a solution to a set of homogeneous equations of motion and nonhomogeneous boundary conditions so formulated as to satisfy the initial and boundary conditions of the actual problem when the two solutions are summed. As a particular solution, the steady-state solution is used which is the motion that would appear stationary to an observer traveling with the load. Steady-state solutions were developed in Part 1 of this article for all load speeds greater than zero. The solution to the homogeneous equations of motion is developed here in Part 2. It is shown that the latter solution can be obtained by numerical integration using the method of characteristics. Particular attention is given to the cases when the load travels at the critical speeds consisting of the minimum phase velocity ofpropagating harmonic waves and the sonic speeds. It is shown that the solution to the homogeneous equations combines with the steady-state solution in such a manner that the beam displacements are continuous and bounded for all finite times at all load speeds including the critical speeds. Numerical results are presented for the critical load speed cases.

Free bending vibrations of adhesively bonded orthotropic plates with a single lap joint

Abstract: This study is concerned with the free bending vibrations of two rectangular, orthotropic plates connected by an adhesively bonded lap joint. The influence of shear deformation and rotatory inertia in plates are taken into account in the equations according to the Mindlin plate theory. The effects of both thickness and shear deformations in the thin adhesive layer are included in the formulation. Plates are assumed to have simply supported boundary conditions at two opposite edges. However, any boundary conditions can be prescribed at the other two edges. First, equations of motion at the overlap region are derived. Then, a Levy-type solution for displacements and stress resultants are used to formulate the problem in terms of a system of first order ordinary differential equations. A revised version of the Transfer Matrix Method together with the boundary and continuity conditions are used to obtain the frequency equation of the system. The natural frequencies and corresponding mode shapes are obtained for identical and dissimilar adherends with different boundary conditions. The effects of some parameters on the natural frequencies are studied and plotted.

Use of dFRFs for Diagnosis of Asymmetric/Anisotropic Properties in Rotor-Bearing System

Abstract: The diagnostic method, which utilizes the dFRFs defined in the stationary and rotating coordinate systems, is tested with a laboratory flexible rotor-bearing system, in order to verify its effectiveness in detection of the asymmetry in shaft and the anisotropy in stator. The experimental results indicate that the dFRFs can be effectively used for the diagnosis of anisotropy and/or asymmetry in rotor systems by the investigation of two kinds of dFRF estimates using the complex input and output signals defined in the stationary and rotating coordinate systems.

Dynamic Analysis of Folded Plate Structures

Abstract: A spectral element method for analyzing wave propagation in multiply connected plates oriented at arbitrary angles is presented. These plate elements may be joined to model structures with stiffeners, open or closed thin-walled tubes, corrugated plates, channels, and ducts. The element models exactly both in-plane and out-of-plane responses over large domains. Results for point impact of a plate with stringers and a thin-walled box beam show excellent agreement with finite element solutions.

Application of the Taylor Transform to Nonlinear Vibration Problems

Abstract: A new transformation technique called the Taylor transform is introduced in this study. A great advantage of the Taylor transform over the integral transform is its ability to solve the nonlinear differential equation. Two nonlinear problems-Van der Pol's equation and the motion behavior of an earthquake isolation system-are solved by the present method. The results compare very well with the measured data and the results obtained from other analytical methods.

Torsional Instabilities in a System Incorporating a Hooke’s Joint

Abstract: Dynamic stability analysis of a driveline which incorporates a Hooke ’s joint is presented in this paper. In particular, torsional instabilities due to fluctuating angular velocity ratio across the joint are examined. The method of averaging is used to establish the critical speed ranges by considering the linearized equations which govern the torsional motion of the system. Within these ranges, parametric instability characterized by exponential build up of response amplitudes of the torsional modes occur. Closedform conditions for onset of sub-harmonic as well as sum-type combination resonance have been established. The significance of the sum-type combination resonance in a typical diesel hydraulic locomotive driveline has been demonstrated. Difference-type combination resonance has been shown to be absent. The instability conditions indicate the range of driveshaft speeds to be avoided during the design of a driveline which employs a Hooke’s joint.

Selection of Component Modes for Craig-Bampton Substructure Representations

Abstract: Three measures of modal dynamic importance are studied for the purpose of ranking Craig-Bampton substructure fixed interface mode shapes based upon their contribution to forces at the substructure interface, modal velocity, or modal displacement. These measures can be employed to identify mode shapes which are dynamically important and thus should be retained in a reduced analytical representation, or identified during a modal survey of the substructure. The first method considered, Effective Interface Mass, has been studied previously. However, new results are presented showing the relation between Effective Interface Mass and a commonly used control dynamics measure of modal importance called approximate balanced singular values. In contrast to the general case of approximate balanced singular values, Effective Interface Mass always gives an absolute measure of the dynamic importance of mode shapes. The EIM method is extended to consider modal velocity and modal displacement outputs. All three measures are applied to a simple substructure called the General Purpose Spacecraft. It is shown that each of these measures provides an efficient method for ranking the dynamic importance of Craig-Bampton fixed interface modes such that a reduced representation will accurately reproduce the substructure's response in the frequency range of interest.