Solution Of Forced Vibration Research Articles

Damped vibration analysis of an elastically connected complex double-string system

Abstract The main purpose of the present paper is to consider theoretically damped transverse vibrations of an elastically connected double-string system. This system is treated as two viscoelastic strings with a Kelvin–Voigt viscoelastic layer between them. A theoretical analysis has been made for a simplified model of the system, in which assumed physical parameters make it possible to decouple the governing equations of motion by introducing the principal co-ordinates. Applying the method of separation of variables and the modal expansion method, exact analytical solutions for damped free and forced responses of the system subjected to arbitrarily distributed transverse continuous loads are determined in the case of arbitrary magnitude of linear viscous damping. It is important to note that the solutions obtained are explicitly expressed in terms of parameters characterizing the physical properties of the system under discussion. For the sake of completeness of the analysis, solutions for undamped free and forced vibrations are also formulated.

Vibration Analysis of Rotors Utilizing Implicit Directional Information of Complex Variable Descriptions

A complex variable description of planar motion incorporates directivity as inherent information which is therefore very convenient in vibration analysis of rotors. This paper proposes to use the directional information explicitly when the equation of motion of a rotor is formulated in complex variables. It is shown that the free vibration solution to the equation of motion formulated as such can be defined as the directional natural mode because it describes not only the shape and frequency but also the direction of the free vibration response. The directional frequency response functions (dFRFs) that have been used recently are obtained as the solution to the forced vibration solution to the equation of motion. Symmetric and anti-symmetric motions of a geometrically symmetric rigid rotor are used as examples to explain these concepts and their practical significances. The proposed approach allows clear understanding and definitions of some unique characteristics of rotor vibrations, such as the forward and backward modes, and forward and backward critical speeds, which have been often used in confusing or incorrect ways.

Analysis of a rotating elastic beam with piezoelectric films as an angular rate sensor.

The flexural vibration of an elastic beam with surface-bonded piezoelectric films rotating about its axis is studied. One-dimensional equations governing the motion of the beam are developed, including the effects of Coriolis and centrifugal forces. The equations are used in the analysis of the flexural vibration of the beam under the excitation of an alternating electric voltage. Forced vibration solution is obtained. The beam can be used as a gyroscope for detecting the angular rate of the rotation. Voltage sensitivity and its dependence on various geometric and physical parameters are examined.

Exact Analytic Solutions of Vibrations of Infinite 1–D Elastic Lines with Lumped Parameters

In this paper we will analyse the most essential drawbacks of known solutions for the problem of vibrant infinite 1–D elastic lumped lines. We will present exact analytic solutions for forced and free vibrations of semi-infinite and infinite homogeneous elastic lines. We will analyse the solutions obtained, consider their physical essence and verify these solutions to corroborate they are exact and complete.

One-dimensional equations for a piezoelectric ring and applications in a gyroscope

One-dimensional equations for coupled in-plane extensional and flexural vibrations of a piezoelectric ring are derived. The equations are used to analyze a ring piezoelectric gyroscope. Free and forced vibration solutions are obtained. Resonant frequencies and voltage sensitivity as well as their dependence on rotation rate and other parameters are examined.

F-1018 内部を数点で支持された周辺支持円板の固有振動解析(G10-1 連続体の解析)(G10 機械力学・計測制御部門一般講演)

The natural frequencies and the mode shapes of a simply supported thin circular plate with several supported points are determined. A partial integro-differential equation is solved using eigenfunctions and eigenfrequencies of a simply supported thin circular plate. If the reaction forces at the supported points for free vibration are considered as harmonically excitation forces with unknown amplitudes, a periodic solution of forced vibration of a simply supported thin circular plate is obtained. Using the conditions that the deflection must be zero at the supported points, simultaneous equations with respect to the unknown amplitudes of forces are obtained. The frequency equation is obtained from a necessary and sufficient condition for existence of the non-trivial solutions.

F-1017 内部を数点で支持された周辺固定円板の固有振動解析(G10-1 連続体の解析)(G10 機械力学・計測制御部門一般講演)

The natural frequencies and the mode shapes of circular plate with clamped edge supported inside several points are determined. A partial integro-differential equation is solved using eigenfunctions and eigenfrequencies of circular plate with clamped edge. If the reaction forces at the supported points for free vibration are considered as harmonically excitation forces with unknown amplitudes, a periodic solution of forced vibration of circular plate with clamped edge is obtained. Using the conditions that the deflection must be zero at the supported points, simultaneous equations with respect to the unknown amplitudes of forces are obtained. The frequency equation is obtained from a necessary and sufficient condition for existence of the non-trivial solutions.

FREE VIBRATION ANALYSIS OF COMPLETELY FREE RECTANGULAR PLATES BY THE SUPERPOSITION–GALERKIN METHOD

The superposition-Galerkin method for analyzing the free vibration of thin isotropic and orthotropic plates as well as transverse-shear deformable plates was introduced in recent years. It has an advantage over the traditional superposition method in that it gives equally accurate results but requires much less work on the part of the analyst. Unfortunately, it has not been possible up to this time to apply it to plates with free edge conditions. This was due to mixed derivatives appearing in the formulation of free edge boundary conditions. In this paper it is shown how, with the superposition of specially selected sets of forced vibration solutions (building blocks), the above limitations are avoided. While the technique is applied here to the analysis of fully symmetric modes, only, it is demonstrated how the superposition-Galerkin method can be applied to any of the above plate problems regardless of the combination of free edge boundary conditions to be imposed.

TRANSVERSE VIBRATIONS OF ELASTICALLY CONNECTED DOUBLE-STRING COMPLEX SYSTEM, PART II: FORCED VIBRATIONS

This paper analyzes the forced transverse vibrations of an elastically connected double-string complex continuous system. The general solutions of forced vibrations of strings subjected to arbitrarily distributed continuous loads are found by using the method of expansion in a series of the mode shape functions. Different cases of exciting loadings are analyzed. The action of stationary harmonic loads and moving concentrated forces is considered. Vibrations caused by the harmonic exciting forces are discussed, and conditions of resonance and dynamic vibration absorption are formulated. Thus the string-type dynamic absorber can be used to suppress the excessive vibrations of corresponding string systems. The dynamic string-type damper is a new concept of a continuous dynamic vibration absorber (CDVA). It is shown that a corresponding two-degree-of-freedom discrete system is an analogue of an elastically connected double-string complex system. Theoretical analysis is illustrated by a numerical example.

Analysis of a ceramic bimorph piezoelectric gyroscope

The mechanism of piezoelectric transformers for producing high voltage is incorporated into a ceramic beam bimorph piezoelectric gyroscope for generating high voltage sensing signal. For the analysis of the gyroscope, one-dimensional equations of beam piezoelectric bimorphs are derived from the three-dimensional equations of linear piezoelectricity. The equations are employed in the analysis of the bimorph gyroscope. Free and forced vibration solutions are obtained showing high voltage sensitivity.

EXACT SOLUTION FOR VIBRATION ANALYSIS OF RECTANGULAR CABLE NETWORKS WITH PERIODICALLY DISTRIBUTED SUPPORTS

Abstract Rectangular cable networks supported by periodically distributed posts can be regarded as a structure with bi-periodicity in two orthogonal directions. By considering an equivalent system with cyclic bi-periodicity and applying the double U-transformation technique twice, the harmonic vibration equation can be uncoupled into a set of single variable equations, and that leads to the exact solutions. As an example, a square cable network with a 6×6 mesh and 2×2 internal supports is considered. The solutions of natural and forced vibrations are worked out by using the formulas obtained in the present study.

Vibration analysis of timoshenko beams with non-homogeneity and varying cross-section

In this paper, an analytic solution for free and forced vibrations of stepped Timoshenko beams is presented and used for the approximate analysis of generally non-uniform Timoshenko beams. In the case of free vibrations, the frequency equation is expressed in terms of some initial parameters at one end of the beam; while in the case of forced vibrations, the solution may be obtained by solving a set of algebraic equations with only two unknowns. Several examples are presented to illustrate the validity and accuracy of the analysis.

Vibration of multi-degree-of-freedom systems with non-proportional viscous damping

Abstract In this paper, free vibration, steady-state vibration and transient vibration of multi-degree-of-freedom systems with non-proportional viscous damping are presented. Natural frequencies and the corresponding damping ratios are obtained by solving the complex eigenvalue problem with complex roots. Then the amplitudes of free vibration and amplitude of steady-state solution of forced vibration are obtained by solving linear algebraic equations. The transient vibration problems are analyzed by using the superposition principle and convolutional integral. Numerical results of amplitudes of vibration and natural frequencies of two-degree-of-freedom systems with different values of coefficients of viscosity are presented. It is very interesting to note that in addition to the overdamped and underdamped cases as in a single-degree-of-freedom system, there exists a degenerated case. Under these conditions (degenerated cases) the number of natural frequencies may be less than the number of degrees of freedom. Numerical results of forced vibration, steady-state solution and transient vibration under different forcing functions are also analyzed. It is found that the effects of viscous damping are very significant in vibration.

Random vibrations of cantilevered composite beams with torsion-bending coupling

The forced-vibration problem of a layered composite cantilevered beam which includes the effects of torsion and warping is solved using a third-order shear deformation theory. The solution of the free-vibration problem in conjunction with the solution of its adjoint is used to solve for the forced response of the beam. This forced-vibration solution is used to calculate the displacements, zero-crossing rates and normal and transverse shear stress responses for beams subjected to temporally and spatially varying random transverse loads. The effect of including or excluding the effects of shear deformation on the calculated response quantities is examined for beams with different configurations. The results clearly show that the shear deformation effects cannot be ignored in the calculation of response and reliability of composite beams. Although the results are presented only for beams of uniform cross-section, the eigenfunctions developed here can be effectively used as comparison functions in a Rayleigh-Ritz type analysis of nonuniform beams.

Evaluation of Time Integration Methods. 3rd Report. Error of Forced Vibration Solution to the Input of Arbitray Forcing Functions.

The authors have been conducting research on the evaluation of errors caused by time integration methods. They proposed, in the previous paper, a method of evaluating errors in the solution of time integration methods when the harmonic input is applied to the first-or second-order system. The method is extended, in this paper, to the case where an arbitrary forcing function is applied to the system. Based on this method, a conventional method to evaluate the error in the numerical solution of forced vibration is also proposed in this paper.

Approximate Analysis of the Forced Vibration Response of Plates

An approximate method is presented for the forced vibration analysis of plates. It is applicable in excitation frequency ranges close to resonances. A displacement shape for the plate in the resonance region is assumed, which is either an exact or approximate representation for the corresponding free vibration mode shape. The response amplitude is determined from a proper energy balance. The method is demonstrated for two types of plates—simply supported rectangular and clamped circular—subjected to uniform transverse exciting pressure. Special considerations are indicated for cases when degenerate or closely spaced resonant frequencies are present. Both viscous and material damping are treated. Numerical comparisons between approximate and exact forced vibration solutions are made to demonstrate the accuracy of the method.

Two-mode nonlinear vibration of orthotropic plates using method of multiple scales

Nonlinear forced oscillation of a rectangular orthotropic plate subjected to uniform harmonic excitation is solved using the method of multiple scales. The governing equations are based on the von Karman type geometrical nonlinearity, and the effect of damping is included. The general multimode solution is developed for simply supported boundary conditions, and the solution is specialized for two-symmetric modes analysis. The primary resonances and the subharmonic and superharmonic secondary resonances are studied in detail. HIN laminated composite panels subjected to transverse periodic loadings can encounter deflections of the order of panel thickness or even higher. The effect of these periodic excitations on the panel can be very severe. Responses of this kind cannot be predicted by linear theory. Consequently, the need to study large deflections using nonlinear methods of analysis is of paramount importance. The formulation of the equations governing the fundamen- tal kinematic behavior of the laminated composite plates in the presence of the von Karman geometrical nonlinearity is attributed to Whitney and Leissa.1 Based on these equations, various methods have been developed to solve nonlinear free and forced vibrations of composite panels. A good survey on mainly nonlinear free and forced vibrations of isotropic plates is given in a book by Nayfeh and Mook.2 The most compre- hensive work on geometrically nonlinear analysis of both static and dynamic behavior of the laminated panels through 1972 is collected in a book by Chia. 3 Bert4 has conducted a survey on the dynamics of composite panels for the period of 1979-81. A review of literature on linear vibrations of plates can be found in a review paper by Sathyamoorthy.5 Relatively few investigations have been reported on the nonlinear forced vibration of isotropic or composite panels under harmonic excitations. Yamaki6 presented a one-term solution for free and forced vibrations of the rectangular plates, using Galerkin's method. Lin7 studied the response of a nonlinear flat panel to periodic and randomly varying load- ings. Nonlinear forced vibrations of beams and rectangular plates were studied by Eisely8 using a single-mode Galerkin's method in conjunction with the Linstedt-Duffing perturbation technique. Free and forced response of beams and plates undergoing large-amplitude oscillations using the Ritz averag- ing method were studied by Srinivasan.9 Bennett10 studied the nonlinear vibration of simply supported angle-ply laminated plates by considering the instability regions of the response of such plates subjected to harmonic excitations. Nonlinear free and forced vibration of a circular plate with clamped bound-

A two-dimensional theory for high-frequency vibrations of piezoelectric crystal plates with or without electrodes

Two-dimensional equations of motion of successively higher-order approximations for piezoelectric crystal plates with triclinic symmetry are deduced from the three-dimensional equations of linear piezoelectricity by expansion in series of trigonometric functions of the thickness coordinate of the plate. These equations, complemented by two additional relations: one, the usual relation of face tractions to the mass of electrodes, and the other relating face charges to face potentials and face displacements, can accommodate either the traction and charge boundary conditions at the faces of the plate without electrodes or the traction and potential boundary conditions at the faces of the plate with electrodes. Dispersion curves are obtained from the first- to fourth-order approximate plate equations for a rotated 45° Y-cut lithium tantalate plate without electrodes, and these curves are compared with those from the frequency equation of the three-dimensional equations with close agreement. Solutions of forced vibrations of an AT-cut quartz plate with electrodes are obtained from the first-order plate equations complemented by the two additional relations. It is shown that cutoff frequencies of the fundamental thickness-shear vibrations from the approximate equations are identical to the ‘‘exact’’ ones obtained from the three-dimensional equations.

Material damping of cantilever beams

Abstract In this paper, a method for estimating material damping in a cantilever beam is reported. This method shows that the damping can be calculated in terms of the stress distribution functions for each mode of vibration and the damping-stress function. The stress distribution function is determined by the deformation in the cantilever beam as given by the solution for undamped forced vibration. The function obtained by Lazan is employed as the damping-stress function. This approach therefore gives the damping of the beam when incorporated in machine structures undergoing vibration. By using this approach to the cantilever beam problem one theoretically derives the relationship between the loss factor and the maximum stress amplitude in each mode of vibration. It is found that the relationship between the loss factor and the maximum stress amplitude is very similar for each mode of vibration when the modes vibrate with equal stress amplitude.

Tapered circular beam as dynamic vibration absorber

The study of vibration absorption is extended by developing the theory for a continuous, uniformly tapered, circular beam. The free- and steady-state forced vibration solutions are developed for the beam supported by a central mass. The condition which results in vibration absorption is discussed. An expression is given relating beam parameters to the frequency of the input force which is to be absorbed. Also, the beam deflection amplitude during absorption is discussed.