Transversal vibrations of the axially moving sandwich belts

Inspired by literature on free transversal vibrations of one axially moving belt, we derive and solve analytically coupled partial differential equations of the transversal vibrations of an axially moving sandwich double-belt system. A numerical experiment and visualization are carried out.     

Energy transfer in the hybrid system dynamics (energy transfer in the axially moving double belt system)

First, as an introduction, using the author’s published references, a short survey of an analytical study of the energy transfer between two coupled subsystems, as well as between a linear and nonlinear oscillators of a hybrid system, in the free and forced vibrations of a different type of inter connections between subsystems is presented. Second, as author’s new research result, an analytical study of the energy transfer between two coupled like-string belts interconnected by light pure elastic layer in the axially moving sandwich double belt system, in the free vibrations is presented. On the basis of the obtained analytical expressions for the kinetic and potential energy of the belts and potential energy of the of light pure elastic distributed layer numerous conclusions are derived. In the pure linear elastic double belt system no transfer energy between different eigen modes of transversal vibrations of the axially moving double belt system, but in every from of the set of the infinite numbers eigen modes, there are transfer energy between belts. Each of the eigen modes of the free transversal vibrations are like two-frequency. The change of the potential energy of the booth belts is four frequency, and interaction part of the potential energy is one frequency in the each eigen mode. Changes of the kinetic energy of the both belts of the sandwich double axially moving bet system is two frequency like oscillatory regimes with two time multiplicities of the eineg frequencies of the corresponding eigen amplitude mode.     

Dynamical analysis of axially moving plate by finite difference method

The complex natural frequencies for linear free vibrations and bifurcation and chaos for forced nonlinear vibration of axially moving viscoelastic plate are investigated in this paper. The governing partial differential equation of out-of-plane motion of the plate is derived by Newton’s second law. The finite difference method in spatial field is applied to the differential equation to study the instability due to flutter and divergence. The finite difference method in both spatial and temporal field is used in the analysis of a nonlinear partial differential equation to detect bifurcations and chaos of a nonlinear forced vibration of the system. Numerical results show that, with the increasing axially moving speed, the increasing excitation amplitude, and the decreasing viscosity coefficient, the equilibrium loses its stability and bifurcates into periodic motion, and then the periodic motion becomes chaotic motion by period-doubling bifurcation.     

Double plate system with a discontinuity in the elastic bonding layer

The double plate system with a discontinuity in the elastic bonding layer of Winker type is studied in this paper. When the discontinuity is small, it can be taken as an interface crack between the bi-materials or two bodies (plates or beams). By comparison between the number of multifrequencies of analytical solutions of the double plate system free transversal vibrations for the case when the system is with and without discontinuity in elastic layer we obtain a theory for experimental vibration method for identification of the presence of an interface crack in the double plate system. The analytical analysis of free transversal vibrations of an elastically connected double plate systems with discontinuity in the elastic layer of Winkler type is presented. The analytical solutions of the coupled partial differential equations for dynamical free and forced vibration processes are obtained by using method of Bernoulli’s particular integral and Lagrange’s method of variation constants. It is shown that one mode vibration corresponds an infinite or finite multi-frequency regime for free and forced vibrations induced by initial conditions and one-frequency or corresponding number of multi-frequency regime depending on external excitations. It is shown for every shape of vibrations. The analytical solutions show that the discontinuity affects the appearance of multi-frequency regime of time function corresponding to one eigen amplitude function of one mode, and also that time functions of different vibration basic modes are coupled. From final expression we can separate the new generalized eigen amplitude functions with corresponding time eigen functions of one frequency and multi-frequency regime of vibrations. The English text was polished by Keren Wang.     

Transversal vibrations of double-plate systems

This paper presents an analytical and numerical analysis of free and forced transversal vibrations of an elastically connected double-plate system. Analytical solutions of a system of coupled partial differential equations, which describe corresponding dynamical free and forced processes, are obtained using Bernoulli’s particular integral and Lagrange’s method of variation constants. It is shown that one-mode vibrations correspond to two-frequency regime for free vibrations induced by initial conditions and to three-frequency regime for forced vibrations induced by one-frequency external excitation and corresponding initial conditions. The analytical solutions show that the elastic connection between plates leads to the appearance of two-frequency regime of time function, which corresponds to one eigenamplitude function of one mode, and also that the time functions of different vibration modes are uncoupled, for each shape of vibrations. It has been proven that for both elastically connected plates, for every pair of m and n, two possibilities for appearance of the resonance dynamical states, as well as for appearance of the dynamical absorption, are present. Using the MathCad program, the corresponding visualizations of the characteristic forms of the plate middle surfaces through time are presented.The English text was polished by Keren Wang.     

Nonlinear dynamics of axially moving beam with coupled longitudinal–transversal vibrations

In this study, the nonlinear vibrations of an axially moving beam are investigated by considering the coupling of the longitudinal and transversal motion. The Galerkin method is used to truncate the governing partial differential equations into a set of coupled nonlinear ordinary differential equations. By detuning the axially velocity, the exact parameters with which the system may turn to internal resonance are detected. The method of multiple scales is applied to the governing equations to study the nonlinear dynamics of the steady-state response caused by the internal–external resonance. The saturation and jump phenomena of such system have been reported by investigating the nonlinear amplitude–response curves with respect to external excitation, internal, and external detuning parameters. The longitudinal external excitation may trigger only longitudinal response when excitation amplitude is weak. However, beyond the critical excitation amplitude, the response energy will be transferred from the longitudinal motion to the transversal motion even the excitation is employed on the longitudinal direction. Such energy transfer due to saturation has the potential to be used in the vibration suppression.     

Nonlinear vibrations and stability of an axially moving beam with an intermediate spring support: two-dimensional analysis

The nonlinear coupled longitudinal-transverse vibrations and stability of an axially moving beam, subjected to a distributed harmonic external force, which is supported by an intermediate spring, are investigated. A?case of three-to-one internal resonance as well as that of non-resonance is considered. The equations of motion are obtained via Hamilton??s principle and discretized into a set of coupled nonlinear ordinary differential equations using Galerkin??s method. The resulting equations are solved via two different techniques: the pseudo-arclength continuation method and direct time integration. The frequency-response curves of the system and the bifurcation diagrams of Poincaré maps are analyzed.     

Non-linear dynamics of the sandwich double circular plate system

Multi-frequency vibrations of a system of two isotropic circular plates interconnected by a visco-elastic layer that has non-linear characteristics are considered. The considered physical system should be of interest to many researches from mechanical and civil engineering. The first asymptotic approximation of the solutions describing stationary and no stationary behavior, in the regions around the two coupled resonances, is the principal result of the authors. A series of the amplitude-frequency and phase-frequency curves of the two frequency like vibration regimes are presented. That curves present the evolution of the first asymptotic approximation of solutions for different non-linear harmonics obtained by changing external excitation frequencies through discrete as well as continuous values. System of the partial differential equations of the transversal oscillations of the sandwich double circular plate system with visco-non-linear elastic layer, excited by external, distributed, along plate surfaces, excitation are derived and approximately solved for various initial conditions and external excitation properties. System of differential equations of the first order with respect to the amplitudes and the corresponding number of the phases in the first asymptotic averaged approximation are derived for different corresponding multi-frequency non-linear vibration regimes. These equations are analytically and numerically considered in the light of the stationary and no stationary resonant regimes, as well as the multi-non-linear free and forced mode mutual interactions, number of the resonant jumps.     

Dynamic analysis of viscoelastic sandwich bems

Dynamic analysis of sandwich beams is performed. The external layers are modeled as beams on the basis of the Timoshenko model, and the internal layer possesses the characteristics of Winkler's viscoelastic one-directional base. The free vibrations are described by a homogeneous system of conjugate partial differential equations. After separation of variables in the system of differential equations, the boundary problem is solved and four complex sequences—a frequency sequence and a sequence of free-vibration modes—are obtained. The orthogonality of the complex modes of the free vibration is demonstrated. The free-vibration problem is solved under arbitrary initial conditions. The results of computations conducted for different sandwich beams with different parameters are presented. Pedagogical University, Bydgoszcz, Poland. Published in Prikladnaya Mekhanika, Vol. 36, No. 5, pp. 122–130, May, 2000.     

Nonlinear dynamics of an axially moving Timoshenko beam with an internal resonance

This paper investigates the nonlinear forced dynamics of an axially moving Timoshenko beam. Taking into account rotary inertia and shear deformation, the equations of motion are obtained through use of constitutive relations and Hamilton’s principle. The two coupled nonlinear partial differential equations are discretized into a set of nonlinear ordinary differential equations via Galerkin’s scheme. The set is solved by means of the pseudo-arclength continuation technique and direct time integration. Specifically, the frequency-response curves of the system in the subcritical regime are obtained via the pseudo-arclength continuation technique; the bifurcation diagrams of Poincaré maps are obtained by means of direct time integration of the discretized equations. The resonant response is examined, for the cases when the system possesses a three-to-one internal resonance and when not. Results are shown through time traces, phase-plane portraits, and fast Fourier transforms (FFTs). The results indicate that the system displays a wide variety of rich dynamics.     

Nonlinear dynamic response of axially moving,stretched viscoelastic strings

The dynamical response of axially moving, partially supported, stretched viscoelastic belts is investigated analytically in this paper. The Kelvin–Voigt viscoelastic material model is considered and material, not partial, time derivative is employed in the viscoelastic constitutive relation. The string is considered as a three part system: one part resting on a nonlinear foundation and two that are free to vibrate. The tension in the belt span is assumed to vary periodically over a mean value (as it occurs in real mechanisms), and the corresponding equation of motion is derived by applying Newton’s second law of motion for an infinitesimal element of the string. The method of multiple scales is applied to the governing equation of motion, and nonlinear natural frequencies and complex eigenfunctions of the system are obtained analytically. Regarding the resonance case, the limit-cycle of response is formulated analytically. Finally, the effects of system parameters such as axial speed, excitation characteristics, viscousity and foundation modulus on the dynamical response, natural frequencies and bifurcation points of system are presented.     

Thermo-mechanical nonlinear vibration analysis of a spring-mass-beam system

Thermo-mechanical vibrations of a simply supported spring-mass-beam system are investigated analytically in this paper. Taking into account the thermal effects, the nonlinear equations of motion and internal/external boundary conditions are derived through Hamilton’s principle and constitutive relations. Under quasi-static assumptions, the equations governing the longitudinal motion are transformed into functions of transverse displacements, which results in three integro-partial differential equations with coupling terms. These are solved using the direct multiple-scale method, leading to closed-form solutions for the mode functions, nonlinear natural frequencies and frequency–response curves of the system. The influence of system parameters on the linear and nonlinear natural frequencies, mode functions, and frequency–response curves is studied through numerical parametric analysis. It is shown that the vibration characteristics depend on the mid-plane stretching, intra-span spring, point mass, and temperature change.     

Three-dimensional nonlinear planar dynamics of an axially moving Timoshenko beam

The three-dimensional nonlinear planar dynamics of an axially moving Timoshenko beam is investigated in this paper by means of two numerical techniques. The equations of motion for the longitudinal, transverse, and rotational motions are derived using constitutive relations and via Hamilton’s principle. The Galerkin method is employed to discretize the three partial differential equations of motion, yielding a set of nonlinear ordinary differential equations with coupled terms. This set is solved using the pseudo-arclength continuation technique so as to plot frequency-response curves of the system for different cases. Bifurcation diagrams of Poincaré maps for the system near the first instability are obtained via direct time integration of the discretized equations. Time histories, phase-plane portraits, and fast Fourier transforms are presented for some system parameters.     

Coupled global dynamics of an axially moving viscoelastic beam

The nonlinear global forced dynamics of an axially moving viscoelastic beam, while both longitudinal and transverse displacements are taken into account, is examined employing a numerical technique. The equations of motion are derived using Newton′s second law of motion, resulting in two partial differential equations for the longitudinal and transverse motions. A two-parameter rheological Kelvin–Voigt energy dissipation mechanism is employed for the viscoelastic structural model, in which the material, not partial, time derivative is used in the viscoelastic constitutive relations; this gives additional terms due to the simultaneous presence of the material damping and the axial speed. The equations of motion for both longitudinal and transverse motions are then discretized via Galerkin’s method, in which the eigenfunctions for the transverse motion of a hinged-hinged linear stationary beam are chosen as the basis functions. The subsequent set of nonlinear ordinary equations is solved numerically by means of the direct time integration via modified Rosenbrock method, resulting in the bifurcation diagrams of Poincaré maps. The results are also presented in the form of time histories, phase-plane portraits, and fast Fourier transform (FFTs) for specific sets of parameters.     

Fast evaluation of friction forces in traction-drive contacts including thermal effects

Summary  In automotive traction drives, power is transmitted by friction forces. The friction forces result from the shear stresses developed in lubricated and highly loaded contacts between rolling bodies. Due to the kinematics of a traction drive, shear velocities occur in both the rolling direction and perpendicular to it. Due to these shear velocities and by normal pressure, the lubricant is forced to build up shear stresses. The increase of the shear stresses may be modelled by a nonlinear viscous element. The describing differential equations are coupled by the equivalent shear stress, which defines the nonlinear behaviour of the element. A fast method is described to evaluate the coupled differential equations. By using a known analytical approximation for the equivalent shear stress, the differential equations are decoupled and can be solved analytically. In an iterative procedure the equivalent shear stress is updated, and the complete solution is found. The iterative method is extended to account for thermal effects in the contact. Received 17 June 1999; accepted for publication 26 October 1999     

Stability and bifurcations of an axially moving beam with an intermediate spring support

The forced non-linear vibrations of an axially moving beam fitted with an intra-span spring-support are investigated numerically in this paper. The equation of motion is obtained via Hamilton??s principle and constitutive relations. This equation is then discretized via the Galerkin method using the eigenfunctions of a hinged-hinged beam as appropriate basis functions. The resultant non-linear ordinary differential equations are then solved via either the pseudo-arclength continuation technique or direct time integration. The sub-critical response is examined when the excitation frequency is set near the first natural frequency for both the systems with and without internal resonances. Bifurcation diagrams of Poincaré maps obtained from direct time integration are presented as either the forcing amplitude or the axial speed is varied; as we shall see, a sequence of higher-order bifurcations ensues, involving periodic, quasi-periodic, periodic-doubling, and chaotic motions.     

Modified Mindlin plate theory and shear locking-free finite element formulation

The basic equations of the Mindlin theory are specified as starting point for its modification in which total deflection and rotations are split into pure bending deflection and shear deflection with bending angles of rotation, and in-plane shear angles. The equilibrium equations of the former displacement field are split into one partial differential equation for flexural vibrations. In the latter case two differential equations for in-plane shear vibrations are obtained, which are similar to the well-known membrane equations. Rectangular shear locking-free finite element for flexural vibrations is developed. For in-plane shear vibrations ordinary membrane finite elements can be used. Application of the modified Mindlin theory is illustrated in a case of simply supported square plate. Problems are solved analytically and by FEM and the obtained results are compared with the relevant ones available in the literature.     

Two-dimensional rheological element in modelling of axially moving viscoelastic web

To model the axially moving viscoelastic web material a two-dimensional rheological element is used in this paper. This model is formed by elastic region and viscoelastic region. Using two-dimensional rheological model and the plate theory the differential equation of motion in the form of the eighth-order linear partial differential equation that governs the transverse vibrations of the system is derived. The Galerkin method is applied to simplify the governing equation into two-order truncated system defined by the set of ordinary differential equations. Numerical investigations of dynamic stability of the paper web were carried out. The effects of the transport speed and the internal damping on the dynamic behaviour of the axially moving web are presented in this paper.     

Combination,principal parametric and internal resonances of an accelerating beam under two frequency parametric excitation

This study analyses the nonlinear transverse vibration of an axially moving beam subject to two frequency excitation. Focus has been made on simultaneous resonant cases i.e. principal parametric resonance of first mode and combination parametric resonance of additive type involving first two modes in presence of internal resonance. By adopting the direct method of multiple scales, the governing nonlinear integro-partial differential equation for transverse motion is reduced to a set of nonlinear first order ordinary partial differential equations which are solved either by means of continuation algorithm or via direct time integration. Specifically, the frequency response plots and amplitude curves, their stability and bifurcation are obtained using continuation algorithm. Numerical results reveal the rich and interesting nonlinear phenomena that have not been presented in the existent literature on the nonlinear dynamics of axially moving systems.     

Melting heat transfer in steady laminar flow over a moving surface

The steady laminar boundary layer flow and heat transfer from a warm, laminar liquid flow to a melting surface moving parallel to a constant free stream is studied in this paper. The continuity, momentum and energy equations, which are coupled nonlinear partial differential equations are reduced to a set of two nonlinear ordinary differential equations, before being solved numerically using the Runge–Kutta–Fehlberg method. Results for the skin friction coefficient, local Nusselt number, velocity profiles as well as temperature profiles are presented for different values of the governing parameters. Effects of the melting parameter, moving parameter and Prandtl number on the flow and heat transfer characteristics are thoroughly examined. It is found that the problem admits dual solutions.