Dynamics of a rod undergoing a longitudinal impact by a body

A longitudinal elastic impact caused by a body on a thin rod is considered. The results of theoretical, finite element, and experimental approaches to solving the problem are compared. The theoretical approach takes into account both the propagation of longitudinal waves in the rod and the local deformations described in the Hertz model. This approach leads to a differential equation with a delayed argument. The three-dimensional dynamic problem is considered in terms of the finite element approach in which the wave propagation and local deformation are automatically taken into account. A benchmark test of these two approaches showed a complete qualitative and satisfactory quantitative agreement of the results concerning the contact force and the impact time. In the experiments, only the impact time was determined. The comparison of the measured impact time with the theoretical and finite element method’s results was satisfactory. Owing to the fact that the tested rod was relatively short, the approximate model with two degrees of freedom was also developed to calculate the force and the impact time. The problem of excitation of transverse oscillation after the rebound of the impactor off the rod is solved. For the parametric resonance, the motion has a character of beats at which the energy of longitudinal oscillation is transferred into the energy of transverse oscillation and vice versa. The estimate for the maximum possible amplitude of transverse oscillation is obtained.     

Displacement of a rigid rod in a viscous medium

The spatial displacement of an absolutely rigid vertical rod in a viscous medium is considered; the rod is hinged at the upper end to a platform moving on the surface of the viscous medium over a specified curvilinear trajectory. A load is attached to the lower end of the rod. Using a variational Lagrangian equation, a nonlinear system of ordinary differential equations in terms of the angles of rod rotation determining the position of points of this rod at any time is obtained. As an example, the problem is solved for conditions of acceleration, uniform motion, and deceleration of the reference point moving over a trajectory consisting of rectilinear sections and portions of circles. The differential equations obtained may be used in determining the position of rod-type elements suspended in a viscous medium.Donetsk. Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 21, pp. 108–112, 1990.     

Multi-pulse chaotic motions of a rotor-active magnetic bearing system with time-varying stiffness

In this paper, we investigate the Shilnikov type multi-pulse chaotic dynamics for a rotor-active magnetic bearings (AMB) system with 8-pole legs and the time-varying stiffness. The stiffness in the AMB is considered as the time-varying in a periodic form. The dimensionless equation of motion for the rotor-AMB system with the time-varying stiffness in the horizontal and vertical directions is a two-degree-of-freedom nonlinear system with quadratic and cubic nonlinearities and parametric excitation. The asymptotic perturbation method is used to obtain the averaged equations in the case of primary parametric resonance and 1/2 subharmonic resonance. It is found from the numerical results that there are the phenomena of the Shilnikov type multi-pulse chaotic motions for the rotor-AMB system. A new jumping phenomenon is discovered in the rotor-AMB system with the time-varying stiffness.     

A qualitative analysis of the subharmonic oscillations of a parametrically excited flexible rod

A non-autonomous non-linear dynamical system with a small parameter that describes the parametric oscillations of a flexible rod with three static equilibrium positions is obtained. The generating equation of this model is a dynamical system in a plane with a separatrix loop. The qualitative analysis presented includes an investigation of the stability and bifurcation of subharmonic motions at resonance energy levels.     

Oscillations of a Prismatic Rod

Equations are obtained for the longitudinal and transverse oscillations of a prismatic rod that differ from the standard equations of structural mechanics (strength of materials). The difference lies both in the appearance of new terms in the equations of motion corresponding to the Timoshenko theory, and in a refinement of the boundary conditions and the conditions on the rigidity of the rod for transverse oscillations. The oscillations of a vertical cantilevered rod when its base moves horizontally from a state of rest are examined and a comparison is made with the classical theory of transverse (bending) oscillations of prismatic rods.     

Stability of a pit lifting vessel on a periodic system of vertical beams

Stability is considered for the unperturbed motion of a lifting vessel in two-sided guides in a vertical pit shaft, where allowance is made for the inertial behavior of the guides. A Hill equation is derived from the equations for the perturbed motion of the center of mass of the vessel and the elastic vibrations in the guides after certain simplifications. Numerical studies have been made on the stability of the solutions, and a region has been constructed for the main parametric resonance in this system.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 20, pp. 109–112, 1989.     

Synthesis of a control in a non-linear dynamical system based on decomposition

A non-linear controllable dynamical system with many degrees of freedom, described by Lagrange equations of the second kind, is considered. Geometric constraints are imposed on the magnitudes of the controls. It is assumed that, in the equations of motion, the kinetic energy matrix is close to a certain constant diagonal matrix. It is possible, for example, to reduce the equations of motion of robots, the drives of which have large gear ratios, to a system of this kind. A problem is formulated on the transfer of a system in a finite time from a specified initial state to a final state with zero velocities. The method of decomposition [1] is used to construct the equations. Sufficient conditions are found subject to which the maximum values of the non-linear terms in the equations of motion do not exceed the permissible magnitudes of the controls. In this case, non-linearities are treated as limited perturbations and the system is decomposed into independent, linear, second-order subsystems. A feedback control is specified for these subsystems which guarantees that each of them is brought into the final state for any permissible perturbations. The control has a simple structure. Applications of the proposed approach to problems in the control of manipulating robots are considered.     

A Note on a Nonlinear Model of a Piezoelectric Rod

If piezoceramics are excited by weak electric fields a nonlinear behavior can be observed, if the excitation frequency is close to a resonance frequency of the system. To derive a theoretical model nonlinear constitutive equations are used, to describe the longitudinal oscillations of a slender piezoceramic rod near the first resonance frequency. Hamilton's principle is used to receive a variational principle for the piezoelectric rod. Introducing a Rayleigh Ritz ansatz with the eigenfunctions of the linearized system to approximate the exact solution leads to nonlinear ordinary differential equations. These equations are approximated with the method of harmonic balance. Finally it is possible to calculate the amplitudes of the displacements numerically. As a result it is shown, that the Duffing type nonlinearities found in measurements can be described with this model.     

基于AP法的一类非线性机械动力系统的求解

用渐近摄动法分别一类机械系统的非线性运动控制方程1∶1、1∶2内共振主参数共振-1/2亚谐共振情况进行摄动分析,得到系统的平均方程.结果发现用渐近摄动法求得1∶2内共振的平均方程中会漏掉某些非线性项,而且内共振比值越大漏掉的项越多,由此可以看出渐近摄动法不适用于求解多模态之间内共振比值大的非线性动力学系统.     

Nonlinear response of a beam under distributed moving contact load

In this paper, the nonlinear behavior of a one-dimensional model of the disc brake pad is examined. The contact normal force between the disc brake pad lining and rotor is represented by a second order polynomial of the relative displacement between the two elastic bodies. The frictional force due to the sliding motion of the rotor against the stationary pad is modeled as a distributed follower-type axial load with time-dependent terms. By Galerkin discretization, the equation governing the transverse motion of the beam model is reduced to a set of extended Duffing system with quasi-periodically modulated excitations. Retaining the first two vibration modes in the governing equations, frequency response curves are obtained by applying a two-dimensional spectral balance method. For the first time, it is predicted that nonlinearity resulting from the contact mechanics between the disc brake pad lining and rotor can lead to a possible irregular motion (chaotic vibration) of the pad in the neighborhood of simple and parametric resonance. This chaotic behavior is identified and quantitatively measured by examining the Poincaré maps, Fourier spectra, and Lyapunov exponents. It is also found that these chaotic motions emerge as a result of successive Hopf bifurcations characterized by the torus breakdown and torus doubling routes as the excitation frequency varies. Various aspects of the numerical difficulties in the solution of the nonlinear equations are also discussed.     

Instability of parametrically second- and third-subharmonic resonances governed by nonlinear Shrödinger equations with complex coefficients

A theoretical analysis of the parametric harmonic response of two resonant modes is made based on a cubic nonlinear system. The analysis based on the method of multiple scales. Two types of the modified nonlinear Schrödinger equations with complex coefficients are derived to govern the resonance wave. One of these equations contains the first derivatives in space for a complex-conjugate type as well as a linear complex-conjugate term that is valid in the second-harmonic resonance cases. The second parametric equation contains a complex-conjugate type which is valid at the third-subharmonic resonance case. Estimates of nonlinear coefficients are made. The resulting equations have an interesting in many dynamical and physical cases. Temporal modulational method is confirmed to discuss the stability behavior at both parametric second- and third-harmonic resonance cases. Furthermore, the Benjamin–Feir instability is discussed for the sideband perturbation. The instability behavior at the sharp resonance is examined and the existence of the instability is found.     

两自由度非线性振动系统主参数激励下的分岔分析

本文给出了参数激励作用下两自由度非线性振动系统,在1:2内共振条件下主参数激励低阶模态的非线性响应.采用多尺度法得到其振幅和相位的调制方程,分析发现平凡解通过树枝分岔产生耦合模态解,采用Melnikov方法研究全局分岔行为,确定了产生Smale马蹄型混沌的参数值.     

On the averaging method for rod equations with quadratic non-linearity

The averaging method is used to approximate solutions of systems of linearly coupled, (quadratic) non-linear dispersive wave equations, which describe extensional–torsional dynamics of a rod. Existence and uniqueness results are established. Error estimates confirm the asymptotic validity of the approximation method on a long time-scale. The linear couplings between the equations imply that resonance can occur inside a single mode of the solution, but energy can also be transferred to other modes.     

Mathematical Modeling and Simulation of a Flexible Shaft-Flexible Link System With End Mass

In this study, the equation of motion of a single link flexible robotic arm with end mass, which is driven by a flexible shaft, is obtained by using Hamilton's principle. The physical system is considered as a continuous system. As a first step, the kinetic energy and the potential energy terms and the term for work done by the nonconservative forces are established. Applying Hamilton's principle the variations are calculated and the time integral is constructed. After a series of mathematical manipulations the coupled equations of motion of the physical system and the related boundary conditions are obtained. Numerical solutions of equations of motion are obtained and discussed for verification of the model used.     

Three-dimensional non-linear oscillations of a rod with hinged supports

The free and forced flexural oscillations of a rod with hinged supports are investigated analytically and numerically. The geometrical non-linearity due to the change in the length of the central line of the rod accompanying its three-dimensional motion is taken into account. The oscillations of a rod with different natural frequencies in two mutually perpendicular directions as a consequence of the variance in the flexural stiffnesses of the rod or the stiffnesses of the supports in the different directions, are considered. It is shown in the case of natural oscillations that, together with two planar forms of motion, a form exists when a certain threshold value is exceeded, which corresponds to the motion of the cross-sections of the rod in a circle. The amplitude-frequency and phase-frequency characteristics of the system are constructed and qualitatively investigated in the neighbourhood of the principal resonance.     

Dynamics of an Euler Elastica Pipe with Internal Flow of Fluid

The equations of motion for a cantilever Euler elastica pipe are deduced applying a generalized set of Lagrange equations for non-material volumes. Based on an exact planar nonlinear beam theory the strain energy for the pipe is derived. The classical Lagrange terms and the additional terms due to moving mass entering and exiting the system result in a set of nonlinear equations of motion for the cantilever pipe with internal flow. A possible dimensional reduction and comparison to existing works are performed. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Characterization of a nonlinear MEMS-based piezoelectric resonator for wideband micro power generation

Micro-scale piezoelectric unimorph beams with attached proof masses are the most prevalent structures in MEMS-based energy harvesters considering micro fabrication and natural frequency limitations. In doubly clamped beams a nonlinear stiffness is observed as a result of midplane stretching effect which leads to amplitude-stiffened Duffing resonance. In this study, a nonlinear model of a doubly clamped piezoelectric micro power generator, taking into account geometric nonlinearities including stretching and large curvatures, is investigated. The governing nonlinear coupled electromechanical partial differential equations of motion are determined by exploiting Hamilton's principle. A semi-analytical approach implementing the perturbation method of multiple scales is used to solve the nonlinear coupled differential equations and analyze the primary and superharmonic resonances. Results indicate that operational bandwidth of the nonlinear harvester is enhanced considerably with respect to linear models. Moreover considerable amount of power is generated due to occurrence of superharmonic resonances. This yields to extraction of energy at subharmonics of the natural frequency which is crucially important in MEMS-based harvesters.     

Active vibration control of unbalanced flexible rotor–shaft systems parametrically excited due to base motion

Rotor vibrations caused by large time-varying base motion are of considerable importance as there are a good number of rotors, e.g., the ship and aircraft turbine rotors, which are often subject to excitations, as the rotor base, i.e. the vehicle, undergoes large time varying linear and angular displacements as a result of different maneuvers. Due to such motions of the base, the equations of vibratory motion of a flexible rotor–shaft relative to the base (which forms a non-inertial reference frame) contains terms due to Coriolis effect as well as inertial excitations (generally asynchronous to rotor spin) generated by different system parameters. Such equations of motion are linear but time-varying in nature, invoking the possibility of parametric instability under certain frequency–amplitude combinations of the base motion. An investigation of active vibration control of an unbalanced rotor–shaft system on moving bases is attempted in this work with electromagnetic control force provided by an actuator consisting of four electromagnetic exciters, placed on the stator in a suitable plane around the rotor–shaft. The actuator does not levitate the rotor or facilitate any bearing action, which is provided by the conventional suspension system. The equations of motion of the rotor–shaft continuum are first written with respect to the non-inertial reference frame (the moving base in this case) including the effect of rotor internal damping. A conventional model for the electromagnetic exciter is used. Numerical simulations performed on the flexible rotor–shaft modelled using beam finite elements shows that the control action is successful in avoiding the parametric instability, postponing the instability due to internal material damping and reducing the rotor response relative to the rigid base significantly, with sufficiently low demand of control current in comparison with the bias current in the actuator coils.     

Resonances in linear one-degree-of-freedom systems with piecewise-constant parameters

A technique based on the composition of elementary phase fluxes is proposed for investigating parametric resonance in systems with “large” perturbations, described by second-order linear differential equations with periodic piecewise-constant coefficients. A monodromy matrix is given and a parametric resonance criterion is indicated, which takes into account the possibility of multiple multipliers and the action of dissipative forces. When there is a two-stage dependence of the coefficients on time during one period, regions of parametric resonance are obtained for different types of linear mechanical systems with one degree of freedom.