Nonlinear Vibrations Of Beams Research Articles

Nonlinear vibration of beams in an axial force field

The nonlinear vibrational characteristics of beams with variable axial restraint at the boundary, and with an axial force applied at any intermediate station of the beam are studied. The behavior of the system at the time of maximum amplitude is determined by the use of successive integration and iteration scheme. Numerical results are presented for a uniform, simply supported beam, for the cases of movable and immovable hinges. For the case of an immovable hinge, the nonlinearity is always of the hardening type. In the case of a movable hinge, for low values of the imposed axial force, the nonlinearity is of the softening type. But, the softening behavior changes to a hardening one when the axial force approaches the value of the buckling load. When the results are particularized for the linear case, the agreement is very good with those available in the literature.

Non-linear Free Vibration Analysis of Stepped Beams under Gravity by Transfer Matrix Method

The non-linear free vibrations of stepped beams under gravity are analyzed by using the transfer matrix method. By using the harmonic balance method, in which a constant term and a sinusoidal harmonic term are assumed as time functions, the governing partial differential equation reduces to simultaneous ordinary non-linear differential equations for the corresponding spatial functions. The transfer matrices are derived from these equations. The numerical results for simply supported and clamped, one-stepped thickness beams with rectangular cross-section are presented and the effects of gravity, the beam geometry, the concentrated mass, and the boundary conditions on the non-linear vibration characteristics are discussed.

Non-linear free vibrations of stepped thickness beams

The non-linear free vibrations of stepped thickness beams are analyzed by assuming sinusoidal responses and using the transfer matrix method. The numerical results for clamped and simply supported, one-stepped thickness beams with rectangular cross-section are presented and the effects of the beam geometry on the non-linear vibration characteristics are discussed. The results are also compared with those obtained by a Galerkin method in which the linear mode function of the beam is used. The use of a Galerkin method seems to considerably overestimate the non-linearity of the stepped thickness beam in certain cases.

Galerkin finite element method for non-linear beam vibrations

A Galerkin finite element method is presented, for studying non-linear vibrations of beams describable in terms of moderately large bending theory. The transverse displacement term w alone is used, although several previous attempts to do the same with a Ritz element have failed. This, together with certain assumptions regarding the nature of the vibration, allows an eigenvalue-like quantity characteristic of non-linear vibration to be defined and computed for various amplitudes of vibration. The solution to the non-linear eigenvalue problem is effected in two ways. In one, the exact mode shape and the frequency corresponding to the reference amplitude of vibration are determined by solving iteratively a series of eigenvalue problems until the required convergence is obtained. In the second approach, one assumes that the mode shape does not change with the amplitude and, by a virtual work type approach in which the linear mode shape is used as the weighting vector, reduces the problem to that of a single degree of freedom system. The eigenvalue corresponding to the chosen mode is determined in a manner similar to but subtly different from Rayleigh's method. The accuracy and applicability of this approximate method is critically examined. Numerical results are presented to demonstrate that the governing differential equations of the problem do not admit variables separable solutions (time and space) in the clamped-clamped and simply supported-clamped cases.

A method of stability analysis for non-linear vibration of beams

In this paper, a method of stability analysis for the large amplitude, steady state response of a non-linear beam under periodic excitation is presented. The stability problem is investigated by studying the behavior of a small perturbation of the steady state response which results in a coupled Hill-type equation. The problem is transformed by the harmonic balance method into an eigenvalue problem of a non-symmetric matrix. The effectiveness and the accuracy of the proposed method for a Mathieu equation are examined and the application to the stability analysis of the non-linear vibrations of a beam is presented.

Non-linear free vibrations of inextensible beams

Non-linear free vibrations of inextensible clamped-free and free-free beams are analyzed by using Galerkin's method and the harmonic balance method.

Non-linear vibrations of three-layer beams with viscoelastic cores, II: Experiment

Experimental results are presented for large amplitude, forced motion of damped, three-layer beams. The beams are constructed with a viscoelastic material constrained between stiff, elastic, outer layers. The sandwich beam is axially restrained; therefore large amplitude displacements cause non-linear response. When the beam is forced at one-half of the lateral vibration resonant frequency, superharmonic response occurs. The experiment is briefly described and frequency response characteristics, spatial shapes and a measure of superharmonic response are presented. The results are compared with predictions from a previously developed theory.

Reply by Author to G. Prathap

References Mei, C., A Finite-Element Approach for Panel Flutter, AIAA Journal, Vol. 15, Aug. 1977, pp. 1107-1110. Mei, C., Nonlinear Vibration of Beams by Matrix Displacement Method, AIAA Journal, Vol. 10, March 1972, pp. 355-357. Mei, C., Finite Element Displacement Method for Large Amplitude Free Flexural Vibrations of Beams and Plates, International Journal of Computers and Structures, Vol. 3, Jan. 1973, pp. 163-174.

Nonlinear vibration of beams with variable axial restraint

Nonlinear vibration of beams with variable axial restraint

Nonlinear vibrations of beams considering shear deformation and rotary inertia

Nonlinear vibrations of beams considering shear deformation and rotary inertia

Non-linear vibrations of three-layer beams with viscoelastic cores I. Theory

Approximate equations of motion are developed for large amplitude motions of three-layer axially restrained unsymmetrical beams with viscoelastic cores. The external force consists of a constant plus an oscillatory term. The combination of this form of forcing and the large amplitude motions cause the beam to respond at multiples of the forcing frequency. This can lead to difficulties in the complex modulus approach to viscoelasticity. These are overcome here through use of hereditary integrals and their relationships with complex moduli. Theoretical results on the frequency response of clamped, symmetrical beams are compared with earlier experimental work. On the whole, reasonable agreement is found.

On the instability of nonlinear vibrations of free-free beams

It is the purpose of this note to discuss the nonexistence of global solutions to a nonlinear partial differential equation which governs the motion of a free-free beam. In [1] it was shown, via reduction to an ordinary differential equation that higher modes are not possible. We give a fairly simple criterion for the nonexistence of solutions directly from the partial differential equation.

Nonlinear Vibration of Beams Using a Form-Function Approximation

The nonlinear response and stability of a vibrating, buckled beam is analyzed using a form-function approximation. Such an approximation differs from the usual linear series approximations in that the unknown parameter appears nonlinearly. This new approximation has two major advantages. Since all harmonics (in both space and time) are represented, the dominant harmonics are “singled out” in the solution, thus making it very efficient. The form-function representation also permits an insight into system behavior not found in other methods. Knowledge of the form parameter shows explicitly how the form of the response changes with the system parameters, e.g., forcing function magnitude and frequency, material constants, etc.

Nonlinear Torsional Vibrations of Thin-Walled Beams of Open Section

An attempt has been made to derive and solve the governing differential equation of large amplitude torsional vibrations of simply supported doubly symmetric thin-walled beams of open section. Graphs indicating the influence of large amplitudes on nonlinear period of torsional vibrations for various nondimensional beam constants are presented.

A new type of non-linear approximation with application to the duffing equation

A new type of trial solution which differs from the usual linear combination of approximating functions is considered. It involves modifying the approximating functions with “form functions;” functions containing undetermined parameters appearing non-linearly, the proper choice of which provide a closer approximation to the large local curvatures which appear in some non-linear problems. In this paper the “form function” approximation is demonstrated for steady-state solutions of the Duffing equation. This equation arises in the problem of non-linear vibration of buckled beams and plates. It is shown that the stability behavior of these steady-state solutions is governed by a Hill equation. It is found that the “form function” approximation gives noticeably better numerical results than, for example, those given by the harmonic balance method. The method also provides additional insight into the non-linear behavior, particularly in the low frequency response region.

Literature Review : NONLINEAR VIBRATION OF BEAMS WITH TIME-DEPENDENT BOUNDARY CONDITIONS Aravamudan, K. S. and Murthy, P.N. Intl. J. Nonlinear Mech. 8(3), 195-212 (June 1973) 3 refs Refer to Abstract No. 73-1453

Literature Review : NONLINEAR VIBRATION OF BEAMS WITH TIME-DEPENDENT BOUNDARY CONDITIONS Aravamudan, K. S. and Murthy, P.N. Intl. J. Nonlinear Mech. 8(3), 195-212 (June 1973) 3 refs Refer to Abstract No. 73-1453

Non-linear and parametric vibration of thin-walled beams of equal angle-section

The linear coupling of the various types of motion of thin-walled beams of equal angle-section was discussed in reference [1]. Here their interaction due to non-linear effects is studied. In addition to the usual displacements in the axial direction, a second-order effect is introduced. The mutual interaction of two types of motion at a time is studied. This is done by retaining quadratic non-linearities while ignoring cubic non-linearities in the derived differential equations of motion, so obtaining pairs of (quadratically) coupled non-linear partial differential equations in the relevant dependent variables. Application of Galerkin's approximation gives a set of ordinary differential equations with periodic coefficients. The regions of instability are evaluated, and are expressed in non-dimensional terms. Experimental tests on a series of cantilever beams show that torsional and cross-sectional modes are parametrically excited by support motion in either the axial or the transverse direction. When the forcing frequency approaches one of the natural frequencies (longitudinal or transverse), it is shown that the parametric instability region becomes substantially wider. The effect on the instability region due to changes in the basic dimensions of the beam is also discussed.

Non-linear vibration of beams with time-dependent boundary conditions

The present paper deals with the influence of large amplitudes on inextensional and extensional vibration of slender uniform elastic beams, when the boundaries are constrained to undergo a time-dependent displacement. Employing a fixed inertial co-ordinate system and a Lagrangian description of motion, the governing equations are formulated using energy principles, assuming the motion to be extensional. The solutions are attempted by a regular perturbation technique with finite transform method and a Galerkin's method extended to the present problem using Mindlin's transformations. The stability of periodic solutions are discussed by transforming the variational equation to the standard form of Hills' and Mathieus' equations, for the inextensional and extensional cases, respectively.

Nonlinear transverse vibrations of beams with properties that vary along the length

The method of multiple scales is used to analyze the nonlinear transverse vibrations of inhomogeneous beams including the effects of transverse shear and rotary inertia. First-order uniform perturbation expansions are derived for the displacements, shear force, and frequency in terms of the ratios of maximum transverse displacement to cross-sectional radius of gyration and cross-sectional radius of gyration to wave length.

Nonlinear Transverse Vibrations of Inhomogeneous Beams

The method of multiple scales is used to analyze the nonlinear transverse vibrations of inhomogeneous beams including the effects of transverse shear and rotary inertia. First-order uniform perturbation expansions are derived for the displacements, shear force, and frequency in terms of the ratios of maximum transverse displacement to cross-sectional radius of gyration and cross-sectional radius of gyration to wavelength.