Forced Vibration Problems Research Articles

Dynamics of a solid body on a horizontally vibrating base

The problem of forced vibrations of a heavy flat body on a horizontal base is considered. It is assumed that the base moves according to a harmonic law in the horizontal direction, and a dry friction force acts in the plane of contact between the body and the base. A solution is obtained for steady-state vibrations with two instantaneous stops and motion with two long-term stops during the vibration period. The dependences of the amplitude of the velocity and phases of vibrations of the body on the frequency of movement of the base are plotted. Keywords: support plane, dry friction, nonlinear vibrations. munitsyn@rambler.ru

Large Amplitude Free and Forced Vibrations of Functionally Graded Timoshenko Beams Using Coupled Displacement Field Method

AbstractThe large amplitude vibrations of functionally graded (FG) beams consisting of metal rich layers at the bottom, ceramic rich layers at the top, and a concentrated mass at the mid-span have been studied using coupled displacement field method. Unlike traditional methods, the coupled displacement field method reduces the 2n undetermined coefficients problem, one each for total rotation and transverse displacement distribution of the beam at n modes, to n undetermined coefficients using a coupling equation obtained from the minimization of potential energy principle. A suitable admissible function having single undetermined coefficient has been assumed for total rotation distribution and the corresponding transverse displacement distribution of the beam has been obtained at each mode for hinged-hinged and clamped-clamped FG beams. The equations of motion for large amplitude vibrations of FG beams at each mode in terms of the undetermined coefficients are derived from the conservation of total energy principle. The free vibration problem is solved using harmonic balance method whereas the forced vibration problem is solved using the Newmark-β method to obtain the time response of the undetermined coefficients and the dynamic response of the beam has been computed from the modal superposition method. The proposed coupled displacement field approach has been successfully applied for the first time to study the large amplitude vibrations of FG beams with suitable validations, and the influence of power law index, slenderness ratio, harmonic load, and concentrated mass has been investigated.

Free and Forced Vibration Analysis in Abaqus Based on the Polygonal Scaled Boundary Finite Element Method

The polygonal scaled boundary finite element method (PSBFEM) is a novel approach integrating the standard scaled boundary finite element method and the polygonal mesh technique. In this work, a user-defined element (UEL) for dynamic analysis based on the PSBFEM is developed in the general finite element software ABAQUS. We present the main procedures of interacting with Abaqus, updating AMATRX and RHS, defining the UEL element, and solving the stiffness and mass matrices through eigenvalue decomposition. Several benchmark problems of free and forced vibration are solved to validate the proposed implementation. The results show that the PSBFEM is more accurate than the FEM with mesh refinement. Moreover, the PSBFEM avoids the occurrence of hanging nodes by constructing a polygonal mesh. Thus, the PSBFEM can choose an appropriate mesh resolution for different structures ensuring accuracy and reducing calculation costs.

Forced vibration analysis of composite beams with piezoelectric layers based on the variable separation method

This paper proposes a new strategy based on the proper generalized decomposition method to solve the forced vibration problem for bi-dimensional piezoelectric composite beams. The approach considers a classical harmonic space-frequency description of the dynamic problem and redefines it by approximating the displacement and electric potential fields as a sum of separated functions of x (beam axis coordinate), z (thickness coordinate) and ω (load frequency). The methodology consists in an iterative algorithm that solves three one-dimensional linear problems at each iteration. The result is a 2D solution in frequency domain with 1D computational complexity. Several numerical tests with wide range of slenderness ratios are considered in order to assess the validity of the method. Moreover, a study of different combinations of boundary conditions is carried out. The results in terms of modal parameters and frequency response functions are analysed and compared with exact elasticity solution and finite element simulations.

An approach to the solution of nonlinear forced vibration problem of structural systems reinforced with advanced materials in the presence of viscous damping

In this study, the nonlinear forced vibration of composite structural systems such as plates, panels and shells reinforced with advanced materials in the presence of linear viscous damping is investigated. Hamilton principle and von Kármán-type nonlinear theory are used to obtain the theoretical model of double-curved shells reinforced by carbon nanotubes (CNTs). The nonlinear partial differential equations are reduced to ordinary differential equations using Galerkin method. By using the multiscale method, the frequency-amplitude relation and nonlinear forced vibration frequency of structural systems are obtained for the first time. Since double-curved shells can be transformed into other structural systems such as spherical and hyperbolic-paraboloid shells, rectangular plate and cylindrical panel in special cases, the expressions for nonlinear frequencies can also be used for them. In additional, the backbone curve and the nonlinear frequency/linear frequency ratio are determined as a function of the amplitude in primary resonance for the first time. The results are verified by comparing the reliability and accuracy of the proposed formulation with those in the literature. Finally, a systematic study is aimed at controlling the influence of nonlinearity and types of distribution of CNTs on the frequencies and their quantitative and qualitative variation in the presence of external excitation and viscous damping.

Free and Forced Wave Vibration Analysis of a Timoshenko Beam/Frame Carrying a Two Degrees-of-Freedom Spring-Mass System

Abstract In this paper, free and forced vibrations of a transversely vibrating Timoshenko beam/frame carrying a discrete two-degrees-of-freedom spring-mass system are analyzed using the wave vibration approach, in which vibrations are described as waves that propagate along uniform structural elements and are reflected and transmitted at structural discontinuities. From the wave vibration standpoint, external excitations applied to a structure have the effect of injecting vibration waves to the structure. In the combined beam/frame and two-degrees-of-freedom spring-mass system, the vibrating discrete spring-mass system injects waves into the distributed beam/frame through the spring forces at the two spring attached points. Assembling the propagation, reflection, transmission, and external force injected wave relations in the beam/frame provides an analytical solution to vibrations of the combined system. In this study, the effects of rotary inertia and shear deformation on bending vibrations are taken into account, which is important when the combined structure involves a short beam element or when higher frequency modes are of interest. Numerical examples are given, with comparisons to available results based on classical vibration theories. The wave vibration approach is seen to provide a systematic and concise solution to both free and forced vibration problems in hybrid distributed and discrete systems.

Transverse Forced Vibrations of the Plates, the Dissipative Properties of Which are Described Memory Functions

The fundamentals of theoretical methods for engineering calculations of bending vibrations of thin plates, the material of which has hereditary properties, are presented. The manifestation of the hereditary properties of the material of the plate under consideration at a given stress-strain state can be judged by the relaxation core of the material. The dependence of the creep and relaxation kernels on the time difference corresponds to the fact that the “memory” of the material about the force effect produced at a given moment is determined by the elapsed time interval. In particular, this means that if the force action on the elastic-viscous body is cyclic, then the deformation of the body will also be cyclic with some phase shift. A technique for solving forced vibrations of plates under the influence of harmonic loads is proposed. An exact solution to the problem of forced vibrations of a supported rectangular plate, whose dissipative properties are described by memory functions, is constructed.

Forced vibration analysis of a fiber reinforced composite beam

In this study, forced vibration analysis of a fiber reinforced composite cantilever beam is investigated under a harmonic load. In the beam model, the Timoshenko beam theory is used. The governing equations of problem are derived by using the Lagrange procedure. In the solution of the problem the Ritz method is used and algebraic polynomials are used with the trivial functions for the Ritz method. In the solution of the forced vibration problem, the Newmark average acceleration method is used in the time history. In the numerical examples, the effects of fibre orientation angles, the volume fraction and dynamic parameters on the forced vibration response of fiber reinforced composite beam are presented and discussed.

Forced Vibration Analysis of Composite Beams Reinforced by Carbon Nanotubes.

This paper presents forced vibration analysis of a simply supported beam made of carbon nanotube-reinforced composite material subjected to a harmonic point load at the midpoint of beam. The composite beam is made of a polymeric matrix and reinforced the single-walled carbon nanotubes with their various distributions. In the beam kinematics, the first-order shear deformation beam theory was used. The governing equations of problem were derived by using the Lagrange procedure. In the solution of the problem, the Ritz method was used, and algebraic polynomials were employed with the trivial functions for the Ritz method. In the solution of the forced vibration problem, the Newmark average acceleration method was applied in the time history. In the numerical examples, the effects of carbon nanotube volume fraction, aspect ratio, and dynamic parameters on the forced vibration response of carbon nanotube-reinforced composite beams are investigated. In addition, some comparison studies were performed, with special results of published papers to validate the using formulations.

A meshless collocation method based on radial basis functions for free and forced vibration analysis of functionally graded plates using FSDT

The current study presents a meshless collocation method based on radial basis functions (MC-RBF) to analyze the free and forced vibration of functionally graded plates. We used the first-order shear deformation plate theory for the displacements and strains. The governing equations are discretized via the MC-RBF method to obtain the natural frequencies. We suggested a new algorithm (modified LOOCV) to acquire the shape parameters of the radial basis function in the free and forced vibration problems. The discretized equations are converted to reduced equations via the proposed transformation matrix. The run-time of this method is significantly lower than the other methods. In particular, the accuracy of the proposed algorithm is acceptable. The discretized equations are reduced by eigenvectors and are converted to state-space form and solved using the numerical Runge-Kutta method. The various boundary conditions are taken into account, such as their clamped, simply supported, free and mixture, and various power-law indices. The results show that the MC-RBF method using a modified LOOCV algorithm is a convenient method for free and forced vibration of FG plates.

Dynamic responses of functionally graded and layered composite beams

This paper presents and compares the free and damped forced vibrations of layered and functionally graded composite beams. In the considered study, a cantilever beam subjected to a harmonic point load at the free end is investigated with layered and functionally graded materials. In the kinematics of the beam, the Timoshenko beam theory is used. The governing equations of problem are derived by using the Lagrange procedure. In the solution of the problem, the Ritz method is used. Algebraic polynomials are used with the trial functions for the Ritz method. In the obtaining of free vibration results, the eigenvalue procedure is implemented. In the solution of the damped forced vibration problem, the Newmark average acceleration method is used in the time history. In the damping effect, the Kelvin-Voigt viscoelastic model is used with the constitutive relations. In the numerical examples, the effects of material distribution parameter and dynamic parameters on the natural frequencies and forced vibration responses of functionally graded beams are obtained and compared with the results of the layered composite beam. Also, comparison studies are performed in order to validate the used formulations.

ДИНАМИЧЕСКОЕ ДОГРУЖЕНИЕ СТЕРЖНЯ ПРИ ВНЕЗАПНОМ ИЗМЕНЕНИИ СТРУКТУРЫ УПРУГОГО ОСНОВАНИЯ

A mathematical model of the dynamic process in the structurally nonlinear system «beam – two-parameter foundation», which arises as a result of a sudden change in the physical and mechanical properties of the foundation, leading to zeroing of its shear stiffness, is constructed in a structurally-nonlinear system «beam – two-parameter foundation» The solution of the static problem of bending of a beam supported on a Pasternak base serves as the initial condition for the problem of forced vibrations of a beam on a Winkler base that arose after the sudden formation of a defect. Solutions to static and dynamic problems are constructed by the method of initial parameters with the use of vectors of states of beam sections and matrices of the influence of initial parameters on the state of arbitrary sections. In the analysis of forced vibrations, the decomposition of the load and deflections of the initial static state into rows according to the forms of natural vibrations of the new state is used.

Forced Vibration Responses of Axially Functionally Graded Beams by using Ritz Method

This work presents forced vibration responses of a cantilever beam made of functionally graded material under a harmonic load. The material properties of beam vary along the axial direction. The kinematics of the beam are considered within Timoshenko beam theory. The governing equations of problem are derived by using the Lagrange procedure. In the solution of the problem the Ritz method is used and algebraic polynomials are used with the trivial functions for the Ritz method. In the solution of the forced vibration problem, the Newmark average acceleration method is used in the time history. In this study, free and forced vibration responses of the axially functionally graded beam are investigated in detail. In the numerical examples, the effects of material graduation, geometric and dynamic parameters on the free and forced vibration response of axially graded beam are investigated.

Superharmonic vibrations of sandwich beams with fibre composite core layer based on the multiple scale method

This paper deals with the secondary vibration problem in the superharmonic case near the harmonic excitation of 1/3w1, arising from the vibration nonlinearity that characterizes the slender and less damping laminated beam with composite material core. For this aim the multiple scale method in conjunction with the higher order zigzag theories are used to obtain the resonance responses. In the present work the nonlinear forced vibration problem of sandwich beams under harmonic excitation is solved by the multiples scales method, based by the introduction of an artificial parameter with higher order expansions, to control the nonlinear analytical solutions. The application of this method demonstrates the sensitivity of the sandwich beams with viscoelastic composite layer to the secondary superharmonic vibrations. Following, parametric study is conducted to demonstrate the vulnerability of the laminated structures to the superharmonic vibrations and to reduce as far as possible the amplitude vibrations achieved by more appropriated structural design. The results reveal the effect of the slenderness of the sandwich beams on the hardening changes. In the other hand the results demonstrate the importance of fibre orientation angle to reduce as far as possible the amplitude responses of the sandwich structures in superharmonic vibration case.

Harmonic Standing-Wave Excitations of Simply-Supported Thick-Walled Hollow Elastic Circular Cylinders: Exact 3D Linear Elastodynamic Response

The forced-vibration response of a simply-supported isotropic thick-walled hollow elastic circular cylinder subjected to two-dimensional harmonic standing-wave excitations on its curved surfaces is studied within the framework of linear elastodynamics. Exact semi-analytical solutions for the steady-state displacement field of the cylinder are constructed using recently-published parametric solutions to the Navier-Lame equation. Formal application of the standing-wave boundary conditions generates three parameter-dependent $6\times6$ linear systems, each of which can be numerically solved in order to determine the parametric response of the cylinder's displacement field under various conditions. The method of solution is direct and demonstrates a general approach that can be applied to solve many other elastodynamic forced-response problems involving isotropic elastic cylinders. As an application, and considering several examples, the obtained solution is used to compute the steady-state frequency response in a few specific low-order excitation cases. In each case, the solution generates a series of resonances that are in exact correspondence with a unique subset of the natural frequencies of the simply-supported cylinder. The considered problem is of general theoretical interest in structural mechanics and acoustics and more practically serves as a benchmark forced-vibration problem involving a thick-walled hollow elastic cylinder.

A Wave-Based Vibration Analysis of a Finite Timoshenko Locally Resonant Beam Suspended with Periodic Uncoupled Force-Moment Type Resonators

This paper first employs and develops an exact wave-based vibration analysis approach to investigate a finite Timoshenko beam carrying periodic two-degree-of-freedom (2-DOF) uncoupled force-moment type resonators. In the approach, vibrations are described as structural waves that propagate along uniform structural elements and reflected and transmitted at structural discontinuities. Each uncoupled force-moment type resonator is considered as a cell which injects waves into the distributed beam through the transverse force and the bending moment at the attached point. By assembling wave relations of the cells into the beam, the forced vibration problem of the locally resonant (LR) structure is turned to be the solution to a related set of matrix equations. In addition, the parametric analysis provides an efficient method to obtain wide low-frequency range band-gaps. Accuracy of the proposed wave-based vibration analysis approach is demonstrated by the simulated and measured results of two sets of beam-like resonator samples.

Geometric Non-Linear Analysis of Auxetic Hybrid Laminated Beams Containing CNT Reinforced Composite Materials.

In the current work, a novel hybrid laminate with negative Poisson’s ratio (NPR) is developed by considering auxetic laminate which is composed of carbon nanotube-reinforced composite (CNTRC) and fiber-reinforced composite (FRC) materials. The maximum magnitude of out-of-plane NPR is identified in the case of (20 F/20 C/−20 C/20 C) S laminate as well. Meanwhile, a method for the geometric non-linear analysis of hybrid laminated beam with NPR including the non-linear bending, free, and forced vibrations is proposed. The beam deformation is modeled by combining higher-order shear-deformation theory (HSDT) and large deflection theory. Based on a two-step perturbation approach, the asymptotic solutions of the governing equations are obtained to capture the linear and non-linear frequencies and load-deflection curves. Moreover, a two-step perturbation methodology in conjunction with fourth-order Runge–Kutta method is employed to solve the forced-vibration problem. Several key factors, such as CNT distribution, variations in the elastic foundation, and thermal stress, are considered in the exhaustive analysis. Theoretical results for some particular cases are given to examine the geometric non-linearity behavior of hybrid beam with NPR as well as positive Poisson’s ratio (PPR).

Liquid Vibrations in Cylindrical Tanks with and Without Baffles Under Lateral and Longitudinal Excitations

The paper is devoted to issues of estimating free surface elevations in rigid cylindrical fluid-filled tanks under external loadings. The possibility of baffles installation is provided. The liquid vibrations caused by lateral and longitudinal harmonic loadings are under consideration. Free, forced and parametrical vibrations are examined. Modes of the free liquid vibrations are considered as basic functions for the analysis of forced and parametric vibrations. The modes of the free liquid vibrations in baffled and un-baffled cylindrical tanks are received by using single-domain and multi-domain boundary element methods. Effects of baffle installation are studied. The problems of forced vibrations are reduced to solving the systems of second order ordinary differential equations. For parametric vibrations the system of Mathieu equations is obtained. The numerical simulation of free surface elevations at different loadings and baffle configurations is accomplished. Beat phenomena effects are considered under lateral harmonic excitations. The phenomenon of parametric resonance is examined under longitudinal harmonic excitations.

Liquid vibrations in circular cylindrical tanks with and without baffles under horizontal and vertical excitations

The paper is devoted to issues of numerical simulation of free surface elevations in rigid circular cylindrical fluid-filled tanks under external vertical and horizontal loadings with usage boundary element methods. The liquid in reservoirs is assumed to be ideal and incompressible, and its flow is irrotational. The influence of horizontal and vertical baffle installation is considered and damping effects due baffles are estimated. The influence of elasticity effects is considered. Free, forced and parametrical vibrations are examined. Modes of the free liquid vibrations present basic functions for analysis of forced and parametric vibrations. The modes and frequencies of the free liquid vibrations in baffled and un-baffled circular cylindrical tanks are received by using single- and multi-domain boundary element methods. The new effective procedure is developed for numerical solution of resolving system of singular integral equations. The problems of forced vibrations are reduced to solving the systems of second order ordinary differential equations. For parametric vibrations the system of Mathieu equations with non-zero right-hand parts is obtained. The numerical simulation of different kinds of loadings and baffle configurations is accomplished. The instability conditions of the fluid-filled tank subjected to both horizontal and vertical excitations are considered.

On the Lamb problem: forced vibrations in a homogeneous and isotropic elastic half-space

The problem of vibrations generated in a homogeneous and isotropic elastic half-space by spatially concentrated forces, known in Seismology as (part of) the Lamb problem, is formulated here in terms of Helmholtz potentials of the elastic displacement. The method is based on time Fourier transforms, spatial Fourier transforms with respect to the coordinates parallel to the surface (in-plane Fourier transforms) and generalized wave equations, which include the surface values of the functions and their derivatives. This formulation provides a formal general solution to the problem of forced elastic vibrations in the homogeneous and isotropic half-space. Explicit results are given for forces derived from a gradient, localized at an inner point in the half-space, which correspond to a scalar seismic moment of the seismic sources. Similarly, explicit results are given for a surface force perpendicular to the surface and localized at a point on the surface. Both harmonic time dependence and time $$\delta $$ -pulses are considered (where $$\delta $$ stands for the Dirac delta function). It is shown that a $$\delta $$ -like time dependence of the forces generates transient perturbations which are vanishing in time, such that they cannot be viewed properly as vibrations. The particularities of the generation and the propagation of the seismic waves and the effects of the inclusion of the boundary conditions are discussed, as well as the role played by the eigenmodes of the homogeneous and isotropic elastic half-space. Similarly, the distinction is highlighted between the transient regime of wave propagation prior to the establishment of the elastic vibrations and the stationary-wave regime.