Axially Moving Beam

The contour motion of the belt drive, i. e. the motion with the constant trajectory is addressed. The belt is considered as a closed Cosserat line whose particles have translational and rotational degrees of freedom. The problem is considered in the framework of geometrically nonlinear formulation with no restrictions on the smallness of displacements and rotations. The spatial (Eulerian) coordinate which is the arc coordinate in the actual configuration is introduced. The belt is divided into four segments: two contact segments on the pulleys and two free spans. The friction forces are assumed to obey the Coulomb law. The study is limited to the stationary case with the constant angular velocities of the pulleys and the equations in components are derived for both contact and free spans. In the contact segment two assumptions are employed to eliminate the unknown contact pressure and friction: (i) the full contact i. e. coincidence between the pulley and the belt and (ii) the stick condition i. e. the belt velocity is related to the pulley angular velocity. A nondimensional coordinate is introduced in the segments to obtain the boundary value problem with fixed boundaries. The boundary coordinates of the contact zones are the integration constants of the derived problem along with the other constants. 1 Introduction Friction belt drives were studied extensively as they are interesting from a mechanical perspective. Until recently, one-dimensional models of extensible strings were

In this paper, the stability of a simply supported axially moving elastic panel (plate undergoing cylindrical deformation) is considered. A complex variable technique and bifurcation theory are applied. As a result, variational equations and a variational principle are derived. Analysis of the variational principle allows the study of qualitative properties of the bifurcation points. Asymptotic behaviour in a small neighbourhood around an arbitrary bifurcation point is analyzed and presented. It is shown analytically that the eigenvalue curves in the (ω, V 0) plane cross both the ω and V 0 axes perpendicularly. It is also shown that near each bifurcation point, the dependence ω(V 0) for each mode approximately follows the shape of a square root near the origin. The obtained results complement existing numerical studies on the stability of axially moving materials, especially those with finite bending rigidity. From a rigorous mathematical viewpoint, the presence of bending rigidity is essential, because the presence of the fourth-order term in the model changes the qualitative behaviour of the bifurcation points. The results are applicable to both axially moving panels and axially moving beams.

A simplified model of the belt motion with small strains is proposed. The main purpose of the modeling is to show the effects arising when the line of action of friction forces is shifted to the belt’s middle axis. The prestressed shearable model of the belt is used in this study. The differential equations of the steady state motion are integrated and combined together with the boundary conditions into two nonlinear systems of algebraic equations corresponding to the different cases of the belt behavior: presence and absence of a sliding segment. The nonlinearity results from the fact that the boundaries of the contact segments are a priori unknown. The case without sliding requires introduction of a concentrated force at the point where the belt leaves the pulley. Considerable effects of the assumptions of contact characterization on the simulation results are demonstrated.

Nonlinear dynamics of one-mode approximation of an axially moving continuum such as a moving magnetic tape is studied. The system is modeled as a beam moving with varying speed, and the transverse vibration of the beam is considered. The cubic stiffness term, arising out of ®nite stretching of the neutral axis during vibration, is included in the analysis while deriving the equations of motion by Hamilton's principle. One-mode approximation of the governing equation is obtained by the Galerkin's method, as the objective in this work is to examine the low-dimensional chaotic response. The velocity of the beam is assumed to have sinusoidal ¯uctuations superposed on a mean value. This approximation leads to a parametrically excited Duf®ng's oscillator. It exhibits a symmetric pitchfork bifurcation as the axial velocity of the beam is varied beyond a critical value. In the supercritical regime, the system is described by a parametrically excited double-well potential oscillator. It is shown by numerical simulation that the oscillator has both period-doubling and intermittent routes to chaos. Melnikov's criterion is employed to ®nd out the parameter regime in which chaos occurs. Further, it is shown that in the linear case, when the operating speed is supercritical, the oscillator considered is isomorphic to the case of an inverted pendulum with an oscillating support. It is also shown that supercritical motion can be stabilised by imposing a suitable velocity variation.

Dynamics of a belt drive is analyzed using a nonlinear model of an extensible string at contour motion, in which the trajectories of particles of the belt are predetermined. The equations of string dynamics at the tight and slack spans are considered in a fixed domain by transforming into a spatial frame. Assuming the absence of slip of the belt on the surface of the pulleys, we arrive at a new model with a discontinuous velocity field and concentrated contact forces. Finite difference discretization allows numerical analysis of the resulting system of partial differential equations with delays. Example solution for the acceleration of a belt drive and an investigation of its frequency response depending on the velocity are presented and discussed.

The dynamic stiffness method is extended to large amplitude free and forced vibrations of frames. When the steady state vibration is concerned, the time variable is replaced by the frequency parameter in the Fourier series sense and the governing partial differential equations are replaced by a set of ordinary differential equations in the spatial variables alone. The frequency-dependent shape functons are generated approximately for the spatial discretization. These shape functions are the exact solutions of a beam element subjected to mono-frequency excitation and constant axial force to minimize the spatial discretization errors. The system of ordinary differential equations is replaced by a system of non-linear algebraic equations with the Fourier coefficients of the nodal displacements as unknowns. The Fourier nodal coefficients are solved by the Newtonian algorithm in an incremental manner. When an approximate solution is available, an improved solution is obtained by solving a system of linear equations with the Fourier nodal increments as unknowns. The method is very suitable for parametric studies. When the excitation frequency is taken as a parameter, the free vibration response of various resonances can be obtained without actually computing the linear natural modes. For regular points along the response curves, the accuracy of the gradient matrix (Jacobian or tangential stiffness matrix) is secondary (cf. the modified Newtonian method). However, at the critical positions such as the turning points at resonances and the branching points at bifurcations, the gradient matrix becomes important. The minimum number of harmonic terms required is governed by the conditions of completeness and balanceability for predicting physically realistic response curves. The evaluations of the newly introduced mixed geometric matrices and their derivatives are given explicitly for the computation of the gradient matrix.

The setting of a looped drive belt on two equal pulleys is considered. The belt is modelled as a Cosserat rod, and a geometrically nonlinear model with account for tension and transverse shear is applied. The pulleys are considered as rigid bodies, and the belt–pulley contact is assumed to be frictionless. The problem has two axes of symmetry; therefore, the boundary value problem for the system of ordinary differential equations is formulated and solved for a quarter of the belt. The considered part consists of two segments, which are the free segment without the loading and the contact segment with the full frictionless contact. The introduction of a dimensionless material coordinate at both segments leads to a ninth-order system of ordinary differential equations. The boundary value problem for this system is solved numerically by the shooting method and finite difference method. As a result, the belt shape including the rotation angle, forces, moments, and the contact pressure a...

In this paper, the Incremental Harmonic Balance (IHB) method is formulated for the nonlinear vibration analysis of axially moving beams. The Galerkin method is used to discretize the governing equations. A high dimensional model that can take nonlinear model coupling into account is derived. The forced response of an axially moving strip with internal resonance between the first two transverse modes is studied. Particular attention is paid to the fundamental, superharmonic and subharmonic resonance as the excitation frequency is close to the first, second or one-third of the first natural frequency of the system. Numerical results reveal the rich and interesting nonlinear phenomena that have not been presented in the existent literature on the nonlinear vibration of axially moving media.

A simplified model of the belt motion with small strains is proposed. The main purpose of the modeling is to show the effects arising when the line of action of friction forces is shifted to the belt’s middle axis. The prestressed shearable model of the belt is used in this study. The differential equations of the steady state motion are integrated and combined together with the boundary conditions into two nonlinear systems of algebraic equations corresponding to the different cases of the belt behavior: presence and absence of a sliding segment. The nonlinearity results from the fact that the boundaries of the contact segments are a priori unknown. The case without sliding requires introduction of a concentrated force at the point where the belt leaves the pulley. Considerable effects of the assumptions of contact characterization on the simulation results are demonstrated.

This study is motivated by issues in belt dynamics and paper forming where the oscillatory motion of the contact point between the belt and pulley or the web and roll is important. The objective is to evaluate the influence of the nonlinear contact boundary conditions on the dynamics of the belt or web. To accomplish this objective, two models are analyzed using perturbation methods. The first model represents a string on an elastic foundation (an approximation of the paper forming process) and the second model represents a tensioned beam (accessory drive belt). The first correction to the fundamental natural frequency and the dynamic belt/web length are determined as a function of the governing parameters. For both models, the change in belt length is found to be proportional to the radius of the pulley/roll and the initial belt/web energy.

A shear deformable beam moving along a straight path is considered as an idealization of the problem of stationary operation of a belt drive. The partial contact with a traveling surface results in the shear deformation of the beam. The tangential contact force grows near the end of the contact zone. Assuming perfect adhesion of the lower fiber of the beam to the traveling surface (no slip), we analytically demonstrate the necessity of accounting for concentrated contact forces and jump conditions, which is important for modeling the belt-pulley interaction. Along with dynamic effects, we further consider a frictional model with zones of stick and slip contact and demonstrate its convergence to the results with perfect adhesion at growing maximal friction force.

The present paper analyzes the dynamic behavior of a simply supported beam subjected to an axial transport of mass. The Galerkin method is used to discretize the problem; a high dimensional system of ordinary differential equations with linear gyroscopic part and cubic nonlinearities is obtained. The system is studied in the sub and super-critical speed ranges with emphasis on the stability and the global dynamics that exhibits special features after the first bifurcation. A sample case of a physical beam is developed and numerical results are presented concerning the convergence of the series expansion, linear subcritical behavior, bifurcation analysis and stability, and direct simulation of global postcritical dynamics. A homoclinic orbit is found in a high dimensional phase space and its stability and collapse are studied.

In this paper, the dynamic stability of a power transmission belt excited by an eccentric pulley is investigated. A theoretical model is developed to predict the belt response: simply supported boundary conditions are considered, neglecting the pulley curvature, and including the effect of the lower belt span. The transverse displacement field is expanded into sine series and the Galerkin method is applied to reduce the partial differential equation (PDE) into a set of ordinary differential equations. In order to forecast the belt response, the elastic characteristics only of the belt must be provided to the theoretical model. An experimental investigation is performed on a belt-pulley system with a pulley eccentricity; a laser displacement transducer is used to measure the transverse displacement. The combination of a direct and a parametric excitation is analyzed in detail. Interesting post-critical nonlinear dynamic behaviors are found: sub-harmonic responses and quasi-periodic motions seem to coexist, depending on the initial conditions. Experiments confirm the numerical results, thus validating the present theoretical model.

In this paper, the Incremental Harmonic Balance (IHB) method is formulated for the nonlinear vibration analysis of axially moving beams. The Galerkin method is used to discretize the governing equations. A high dimensional model that can take nonlinear model coupling into account is derived. The forced response of an axially moving strip with internal resonance between the first two transverse modes is studied. Particular attention is paid to the fundamental, superharmonic and subharmonic resonance as the excitation frequency is close to the first, second or one-third of the first natural frequency of the system. Numerical results reveal the rich and interesting nonlinear phenomena that have not been presented in the existent literature on the nonlinear vibration of axially moving media.

In this paper, the dynamic stability of a power transmission belt excited by an eccentric pulley is investigated. A theoretical model is developed to predict the belt response: simply supported boundary conditions are considered, neglecting the pulley curvature, and including the effect of the lower belt span. The transverse displacement field is expanded into sine series and the Galerkin method is applied to reduce the partial differential equation (PDE) into a set of ordinary differential equations. In order to forecast the belt response, the elastic characteristics only of the belt must be provided to the theoretical model. An experimental investigation is performed on a belt-pulley system with a pulley eccentricity; a laser displacement transducer is used to measure the transverse displacement. The combination of a direct and a parametric excitation is analyzed in detail. Interesting post-critical nonlinear dynamic behaviors are found: sub-harmonic responses and quasi-periodic motions seem to coexist, depending on the initial conditions. Experiments confirm the numerical results, thus validating the present theoretical model.

In this paper, the e!ect of pulley eccentricity on the vibration of a power transmission belt is experimentally investigated and a theoretical model is developed for validation and identi"cation purposes. This eccentricity gives rise to dangerous operating conditions for the system, especially when it excites the frequency range of the belt resonances. As expected, the system shows a hardening non-linear behavior. Moreover, the #uctuation of the belt tension, due to the pulley eccentricity, gives rise to a parametric instability that can cause catastrophic failures of the structure. Laser displacement transducers are used to measure transversal vibrations of the belt. An approximate analytical solution of the travelling beam dynamics is developed in order to model the system and reproduce the experimental data. Comparisons between analytical solution and experimental data allow the identi"cation of the unknown parameters of the analytical model. A validation of the identi"ed model is performed by comparing analytical and experimental data in di!erent operating conditions.

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 of a 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.