Nonlinear Normal Modes of a Parametrically Excited Cantilever Beam

We investigate theoretically thenonlinear normal modes of a vertical cantilever beam excited by aprincipal parametric resonance. We apply directly the method ofmultiple scales to the governing nonlinear nonautonomousintegral-partial-differential equation and associated boundary conditions.In the absence of damping, it is shown that the system has nonlinear normal modes, as defined by Rosenberg, even in the presence of the parametric excitation.We calculate the spatial correction to the linear mode shapedue to the effects of the inertia and curvature nonlinearities andthe parametric excitation. We compare the result obtained withthe direct approach with that obtained using a single-mode Galerkindiscretization.The deviation between the two predictions increases as the oscillationamplitude increases.     

Effect of Coulomb Damping on Buckling of a Two-Rod System

This paper describes a significant influence of a slight Coulomb damping on buckling, using a simple two rods system. Coulomb damping produces equilibrium regions around the well-known stable and unstable steady states under the pitchfork bifurcation which occurs in the case without Coulomb damping. Also, the stability of the states in the equilibrium regions is examined by using the phase portrait. As a consequence, due to the slight Coulomb damping, it is theoretically clarified that the states in the equilibrium regions are locally stable, even in the neighborhood of the unstable steady states under the pitchfork bifurcation in the case without Coulomb damping, i.e., even in the neighborhood of the unstable trivial steady states in the postbuckling and the unstable nontrivial steady states under the subcritical pitchfork bifurcation. Furthermore, the experimental results are in qualitative agreement with the theoretically predicted phenomena.     

Effect of Lateral Linear Stiffness on Nonlinear Characteristics of Hunting Motion of a Railway Wheelset

Railway wheelset experiences the problem of hunting above a critical speed, which is a kind of self-excited oscillation. At the critical speed, it is known that the system undergoes a subcritical Hopf bifurcation. Therefore, for clarifying the nonlinear characteristics of hunting it is very important to detect, for example, the nonlinear forces in the wheelset due to the creep forces acting between the wheels and rails, and the nonlinear component of the resorting forces by the suspensions. However, it is impossible to determine each force quantitatively. In the present paper, it is first shown, by using the center manifold theory and the method of normal form, that the nonlinear characteristics of the bifurcation in a wheelset model with two degrees of freedom are governed by a single parameter, hence each nonlinear force need not be detected when examining the nonlinear characteristics. Also, a method of determining the governing parameter from experimentally observed radiuses of the unstable limit cycle is proposed. Next, we experimentally investigate the variation of the parameter due to the presence of linear spring suspensions in the lateral direction and discuss the variation of the nonlinear characteristics of the hunting motion, which depends on the lateral stiffness. As a result, the improvement of the stability of the wheelset against the disturbance by the linear spring suspensions is clarified.     

Experimental investigation of a buckled beam under high-frequency excitation

In this paper, we investigate theoretically and experimentally dynamics of a buckled beam under high-frequency excitation. It is theoretically predicted from linear analysis that the high-frequency excitation shifts the pitchfork bifurcation point and increases the buckling force. The shifting amount increases as the excitation amplitude or frequency increases. Namely, under the compressive force exceeding the buckling one, high-frequency excitation can stabilize the beam to the straight position. Some experiments are performed to investigate effects of the high-frequency excitation on the buckled beam. The dependency of the buckling force on the amounts of excitation amplitude and frequency is compared with theoretical results. The transient state is observed in which the beam is recovered from the buckled position to the straight position due to the excitation. Furthermore, the bifurcation diagrams are measured in the cases with and without high-frequency excitation. It is experimentally clarified that the high-frequency excitation changes the nonlinear property of the bifurcation from supercritical pitchfork bifurcation to subcritical pitchfork bifurcation and then the stable steady state of the beam exhibits hysteresis as the compressive force is reversed. This work was partially supported by the Japanese Ministry of Education, Culture, Sports, Science, and Technology, under Grants-in-Aid for Scientific Research 16560377.     

Stabilization and utilization of nonlinear phenomena based on bifurcation control for slow dynamics

Mechanical systems may experience undesirable and unexpected behavior and instability due to the effects of nonlinearity of the systems. Many kinds of control methods to decrease or eliminate the effects have been studied. In particular, bifurcation control to stabilize or utilize nonlinear phenomena is currently an active topic in the field of nonlinear dynamics. This article presents some types of bifurcation control methods with the aim of realizing vibration control and motion control for mechanical systems. It is also indicated through every control method that slowly varying components in the dynamics play important roles for the control and the utilizations of nonlinear phenomena. In the first part, we deal with stabilization control methods for nonlinear resonance which is the 1/3-order subharmonic resonance in a nonlinear spring-mass-damper system and the self-excited oscillation (hunting motion) in a railway vehicle wheelset. The second part deals with positive utilizations of nonlinear phenomena by the generation and the modification of bifurcation phenomena. We propose the amplitude control method of the cantilever probe of an atomic force microscope (AFM) by increasing the nonlinearity in the system. Also, the motion control of a two link underactuated manipulator with a free link and an active link is considered by actuating the bifurcations produced under high-frequency excitation. This article is a discussion on the bifurcation control methods presented by the author and co-researchers by focusing on the actuation of the slowly varying components included in the original dynamics.     

Van der Pol type self-excited micro-cantilever probe of atomic force microscopy

A technique for dimensional reduction of nonlinear delay differential equations (DDEs) with time-periodic coefficients is presented. The DDEs considered here have a canonical form with at most cubic nonlinearities and periodic coefficients. The nonlinear terms are multiplied by a perturbation parameter. Perturbation expansion converts the nonlinear response problem into solutions of a series of nonhomogeneous linear ordinary differential equations (ODEs) with time-periodic coefficients. One set of linear nonhomogeneous ODEs is solved for each power of the perturbation parameter. Each ODE is solved by a Chebyshev spectral collocation method. Thus we compute a finite approximation to the nonlinear infinite-dimensional map for the DDE. The linear part of the map is the monodromy operator whose eigenvalues characterize stability. Dimensional reduction on the map is then carried out. In the case of critical eigenvalues, this corresponds to center manifold reduction, while for the noncritical case resonance conditions are derived. The accuracy of the nonlinear Chebyshev collocation map is demonstrated by finding the solution of a nonlinear delayed Mathieu equation and then a milling model via the method of steps. Center manifold reduction is illustrated via a single inverted pendulum including both a periodic retarded follower force and a nonlinear restoring force. In this example, the amplitude of the limit cycle associated with a flip bifurcation is found analytically and compared to that obtained from direct numerical simulation. The method of this paper is shown by example to be applicable to systems with strong parametric excitations.     

Stabilization of the Parametric Resonance of a Cantilever Beam by Bifurcation Control with a Piezoelectric Actuator

In this work, bifurcation control using a piezoelectric actuator isimplemented to stabilize the parametric resonance induced in acantilever beam. The piezoelectric actuator is attached to the surfaceof the beam to produce a bending moment in the beam. The dimensionlessequation of motion for the beam with the piezoelectric actuator on itssurface is derived and the modulation equations for the complexamplitude of an approximate solution are obtained using the method ofmultiple scales. We then acquire the bifurcation set that expresses theboundary of the stable and unstable regions. The bifurcation set ischaracterized by the modulation equations. Next, we determine the orderof feedback gains to modify these modulation equations. By actuating thepiezoelectric actuator under the appropriate feedback, bifurcationcontrol is carried out resulting in the shift of the bifurcation set andthe expansion of the stable region. The main characteristic of thestabilization method introduced above is that the work done by thepiezoelectric actuator is zero in the state where the parametricresonance is stabilized. Thus zero power control is realized in such astate. Experimental results show the validity of the proposedstabilization method for the parametric resonance induced in thecantilever beam.     

Buckling of a beam subjected to electromagnetic force and its stabilization by controlling the perturbation of the bifurcation

For a beam subjected to electromagnetic force, magnetoelastic buckling due to the increase of such force is theoretically investigated by taking account of the nonlinearity of the electromagnetic force and the elastic force of the beam. Using Liapunov-Schmidt method and center manifold theory, the equilibrium space, the bifurcation set and the bifurcation diagram are theoretically derived. Also, the effect of the higher modes other than the buckling mode on the mode shape of the postbuckling state is discussed. Furthermore, a control method to stabilize the magnetoelastic buckling is proposed, and the unstable equilibrium state of the beam in the postbuckling state, i.e., the straight position of the beam, is stabilized by controlling the perturbation of the bifurcation.     

Bifurcation Control of Parametrically Excited Duffing System by a Combined Linear-Plus-Nonlinear Feedback Control

For a parametrically excited Duffing system we propose a bifurcation control method in order to stabilize the trivial steady state in the frequency response and in order to eliminate jump in the force response, by employing a combined linear-plus-nonlinear feedback control. Because the bifurcation of the system is characterized by its modulation equations, we first determine the order of the feedback gain so that the feedback modifies the modulation equations. By theoretically analyzing the modified modulation equations, we show that the unstable region of the trivial steady state can be shifted and the nonlinear character can be changed, by means of the bifurcation control with the above feedback. The shift of the unstable region permits the stabilization of the trivial steady state in the frequency response, and the suppression of the discontinuous bifurcation due to the change of the nonlinear character allows the elimination of the jump in the quasistationary force response. Furthermore, by performing numerical simulations, and by comparing the responses of the uncontrolled system and the controlled one, we clarify that the proposed bifurcation control is available for the stabilization of the trivial steady state in the frequency response and for the reduction of the jump in the nonstationary force response.