In this article, we provide an overview of a selection of topics of current interest in nonlinear dynamics and vibrations of mechanical systems. Specifically, we cover the traditional topics of structural dynamics, rotating systems, vibration control, vehicle dynamics, and machining, as well as some topics that have emerged more recently, namely micro- and nano-electromechanical systems, piecewise smooth systems, and structural health monitoring and system identification. In the introductory and closing sections, we offer our perspective on the state of affairs in nonlinear dynamics and its utility in the study of mechanical systems.

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Observations, analysis and understanding of nonlinear rotordynamic phenomena observed in aircraft gas turbine engines and other high-speed rotating machinery over the course of the author's career are described. Included are observations of sum-and-difference frequency response; effects of roller bearing clearance; relaxation oscillations; subharmonic response; chaotic response; and other generic nonlinear responses such as superharmonic and ultra-subharmonic response.

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The present state and future of “nonlinear dynamics” is explained in this review. First, chaotic vibrations of nonlinear beams are used as a material to demonstrate our present understanding of chaos, compared with the situation of the early stage of its research. Second, two topics, microcantilever dynamics of tapping-mode atomic force microscopy and design of Earth to the Moon transfer trajectories of spacecrafts, are chosen for describing the importance and usefulness of “nonlinear dynamics” in new technologies. Moreover, two applications of “nonlinear dynamics” to biped walking robots and nonlinear optimal control are briefly addressed.

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We formulate and solve parametric resonance problems for one- and multiple degrees of freedom systems in three-dimensional space of physical parameters: excitation frequency, amplitude, and viscous damping coefficient assuming that the last two parameters are small. The main result obtained here is that we find the instability domains (simple and combination parametric resonances) as half-cones in three-parameter space with the use of eigenfrequencies and eigenmodes of the corresponding conservative system.

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This paper presents a geometric approach to holonomic mechanical systems in the context of Lagrangian systems. First, it is shown how a standard regular Lagrangian system can be established in the context of symplectic geometry; namely, how a symplectic structure on the tangent bundle of a configuration manifold can be induced from the canonical symplectic structure on the cotangent bundle by using the Legendre transformation and also how a second-order Lagrangian vector field can be developed by an energy function associated to a given Lagrangian as well as the induced symplectic structure on the tangent bundle. Second, it is demonstrated that Lagrangian systems with holonomic constraints can be formulated in the context of the induced symplectic structure by combining constraint distributions with the second-order Lagrangian vector field. Further, it is shown how the standard Lagrangian system can be also understood in the context of an induced Dirac structure on the tangent bundle. Finally, a holonomic Lagrangian system is illustrated by a planar linkage system, together with a local expression of differential algebraic equations (DAE) of index 3.

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This paper is concerned with nonlinear model order reduction for electro-mechanical systems described by port-Hamiltonian formulae. A novel weighted balacend realization and model order reduction procedure is proposed which preserves port-Hamiltonian structure as well as stability, reachability and observability of the original system. This implies that one can utilize the intrinsic physical properties such as physical energy and the corresponding dissipativity for the reduced order model. Further, the proposed method reduces the computational effort in solving partial differential equations for nonlinear balanced realization. A numerical simulation shows how the proposed method works.

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In this paper, a dynamical model of a motorcycle, which consists of four rigid bodies with nine degrees of freedom, is presented. In this model, the cross-sectional shape of the tire is described as a half-circle and its deformation is taken into account. By taking account of the tire slip condition or the tire nonslip condition in the longitudinal direction of the wheels, each equation of motion is derived. Also, by carrying out simulations, it is verified that the responses to the front steering impulsive torque are in good agreement with the results obtained using commercial software. Moreover, the longitudinal friction force and the lateral force in a turning maneuver are analyzed.

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In the rotating machinery, such as automobile engines, the driving torque changes periodically and torsional vibrations occur. In this study, the dynamic characteristics of centrifugal pendulum vibration absorbers which are used to suppress torsional vibrations are investigated both theoretically and experimentally. In the theoretical analysis, the nonlinear characteristics are taken into consideration under the assumption that the pendulums vibrate with large amplitude. It is clarified that, although the centrifugal pendulum has remarkable effects on suppressing harmonic vibration, it induces large amplitude harmonic vibrations, the second and third order superharmonic resonances, and unstable vibrations of harmonic type under some condition,. Moreover, this paper proposes various methods to suppress these secondarily induced vibrations, and show that it is possible to suppress torsional vibrations to the substantially zero amplitude-level in the whole rotational speed range.

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A coupled inverted pendula model of competition and cooperation is proposed to develop a purely mechanical implementation comparable to the Lotka-Volterra competition model. It is shown numerically that the proposed model can produce the four stable equilibriums analogous to ecological coexistence, two states of dominance, and scramble. The authors also propose two types of open-loop strategies to switch the equilibriums. The proposed strategies can be associated with an attack and a counter attack of agents through a metaphor of martial arts.

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Analytical results are presented on contribution of multiple modes of vibration to chaotic responses of a post-buckled clamped beam constrained by an axial spring. Introducing the mode shape function proposed by the senior author and applying the Galerkin procedure to the governing equation of the beam, a set of nonlinear ordinary differential equations in amultiple-degree-of-freedom system is obtained. Chaotic time responses are integrated numerically. Responses of the beam subjected to periodic lateral acceleration are investigated by comparing with the relevant experimental results. Dominant chaotic responses are generated within the frequency ranges of the subharmonic resonance of 1/2 and 1/3 orders. The maximum Lyapunov exponent of the chaotic response corresponding to the sub-harmonic resonance of 1/2 order is greater than that of the chaos with the sub-harmonic resonance of 1/3 order. The analytical results of the chaotic responses have remarkable agreement with that of the experimental results. The Lyapunov dimension and the Poincaré projection of the chaotic responses predict that more than three modes of vibration contribute to the chaos based on the calculation from the equation of multiple-degree-of-freedom system. The principal component analysis shows that the lowest vibration mode contributes dominantly. Higher modes of vibration contribute to the chaos with small amount of amplitude.

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This paper presents a method for analysis of steady-state vibration of a beam with breathing cracks, which open and close during vibration. There are several papers treating problems of vibration analysis of a beam with breathing cracks. However, due to their treatments of the condition which determines the switch between the open and closed states of the crack, it is difficult for one to obtain steady-state vibration efficiently by methods such as the incremental harmonic balance method. Since opening and closing of a breathing crack depends on the sign of the bending moment, or the curvature, of the beam, the key point to this problem is explicit treatment of the bending moment. The mixed variational principle allows one to use deflection as well as bending moment as primary variables in the governing equation. In this paper a governing equation of a beam with breathing cracks is derived by a finite element procedure based on the mixed variational principle. Then, the derived governing equations are solved by combining the iteration method and the harmonic balance method. Finally, examples of analysis by the presented method are given.

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Experimental results and analytical results are presented on chaotic vibrations of a clamped-supported beam with a concentrated mass. The beam is elastically compressed by an axial spring at the simply supported end and is excited by lateral periodic acceleration. In the experiment, periodic and chaotic vibrations are detected under several conditions of the axial compression. In the analysis, the governing equation is reduced to nonlinear differential equations of a multiple-degree-of-freedom system by the Galerkin procedure. The nonlinear periodic responses are calculated by the harmonic balance method. The chaotic responses are numerically integrated by the Runge-Kutta-Gill method. The chaotic responses of the beam are examined with the Fourier spectra, the Poincaré projections and the maximum Lyapunov exponents and the principal component analysis. Under a specific axial compression with post-buckled state of the beam, the chaotic vibrations dominated by dynamic snap-through are generated by the ultra-sub-harmonic resonance response of 2/3order of the fundamental vibration mode. The number of pre-dominant vibration modes that contribute to the chaos is found to be three. Decreasing the axial compression, the chaotic vibrations are induced by the internal resonance response between the second and the fundamental mode of vibration. The number of predominant vibration modes that contribute to the chaos is found to be two or three. Both results of the experimental and the analysis agree remarkably with each other in detail.

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This paper presents a new experimental identification technique for nonlinear continuous vibratory systems. The governing equations of motion of a nonlinear continuous system are described by a set of nonlinear partial differential equations and boundary conditions. Determining both of them simultaneously is a quite difficult task. Thus, one has to discretize the governing equations of motion, and reduce the order of the equations as much as possible. In analysis of nonlinear vibratory systems, it is known that one can reduce the order of the system by using the nonlinear normal modes preserving the effect of the nonlinearity accurately. The nonlinear normal modes are description of motion as nonlinear functions of the coordinates for analysis. In identification, if one can express the data as nonlinear functions of the coordinates for identification, it is expected that accurate mathematical model with minimum degrees of freedom can be determined. Based on this idea, this paper proposes an identification technique which uses nonlinear principal component analysis by a neural network. Applicability of the proposed technique is confirmed by numerical simulation.

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In this paper, an incremental transfer stiffness coefficient method is developed for analyzing the periodic steady-state vibrations of large-scale structures with localized strong nonlinear elements. This method is created by combining the harmonic balance and transfer stiffness coefficient methods using the incremental method. Firstly, the structure under consideration was separated into linear and nonlinear elements. Then, the inner degrees of freedom of the linear elements were eliminated from a process of successive approximation of the solution by applying the transfer stiffness coefficient method. Subsequently, the computational technique of the transfer influence coefficient method was applied to the above process to achieve a dramatic reduction in the computational burden. As a simple example, an algorithm based on this method was formulated to analyze in-plane flexural forced vibrations of a straight-line beam structure supported by nonlinear base elements.

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In computing a periodic steady-state vibration generated in a large-scale nonlinear structure by the incremental transfer stiffness coefficient method suggested in a previous paper, the stable and unstable solutions can be computed without distinction. Thus, the stability of the obtained solution must be examined. However, it is very difficult to analyze the stability of the solution of the nonlinear system with high degree of freedom. To overcome this difficulty, a method to reduce the dimension of the equation used for stability analysis without spoiling the accuracy is developed by applying the concept of modal analysis to the variational equation used for the stability analysis. Two types of modal matrices are considered in the reduction of dimension, and a method is proposed to extract the modes that dominate the stability of the solution. The validity of the incremental transfer stiffness coefficient method and the method of stability analysis using the reduction model is confirmed by numerical computational results.

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In a previous paper, incremental transfer stiffness coefficient method was developed in order to analyze the periodic steady-state vibrations of a large-scale structure having locally strong nonlinear elements. By this method, the computation cost of this method can be reduced markedly from the iterative computation process of approximate solution. In this paper, an algorithm based on the presented method is formulated to analyze the longitudinal, flexural and torsional coupled vibration of a three-dimensional tree structure supported by nonlinear base support elements. In addition, stability analysis using a reduction model is applied to the periodic solution obtained for the three-dimensional tree structure. The validity of the incremental transfer stiffness coefficient method and the method of stability analysis using the reduction model for three-dimensional tree structures is confirmed by the numerical computational results.

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This study investigates the nonlinear vibrations of an elastic structure, with a liquid-filled cylindrical tank, which is subjected to a vertical sinusoidal excitation. This structure-tank system behaves as an autoparametric system. Modal equations governing the coupled motions of the structure and liquid sloshing are derived when the natural frequency of the structure is equal to twice the natural frequency of the first axisymmetric sloshing mode. Van der Pol's method is applied to the modal equations to determine the theoretical resonance curves. The theoretical results can be concluded as follows: (1) As the liquid level decreases, the resonance curve for the liquid sloshing changes from a soft spring type to a hard spring type. (2) The structure's resonance curve flattens out at small amplitude when the liquid level is appropriate. (3) Amplitude-modulated motions appear for the negative and positive values of the internal resonance ratio's deviation (the detuning parameter) in the high and low liquid levels, respectively. (4) Furthermore, the results of the bifurcation analysis, Poincaré maps and Lyapunov exponents reveal that amplitude-modulated motions and chaotic oscillations can occur in the system. In experiments, the theoretical resonance curves were quantitatively in agreement with the experimental data.

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A theoretical and experimental investigation was conducted into the spatial behavior of a flexible pipe through which fluid flows and which is built-in at one end and has an asymmetric spring-support at the other end. Planar, non-planar and beating type vibrations occur in such a system and depend on the flow velocity and spring coefficients. Equations governing amplitudes and the phase were derived and used to clarify numerically the above specific cases. Corresponding actual pipe displacements were then measured experimentally using an image processing system employing two CCD cameras. Qualitative agreement was demonstrated between the experimental and theoretical results.

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This paper discusses a mechanism of a self-excited vibration of a cleaning blade in a laser printer. We present a coupled-mode flutter model using a finite element model. A stability analysis based on the proposed model is carried out. From this result, it is clarified that two modes couple each other with increasing coefficient of friction. At that time, the natural frequency of the coupled-mode is corresponding to the frequency of the self-excited vibration. This root locus reveals a typical argand diagram for coupled-mode flutter of an undamped system via so-called Hamiltonian-Hopf bifurcation. Furthermore, we discuss about the steady-state amplitude of the self-excited vibration. First, we present a nonlinear amplitude equation by extracting the unstable modes with introduction of the adjoint vector to the eigenvector of the system. Second, from consideration about the nonlinear term which is able to restrain an increasing of amplitude, we decide the nonlinear term referring to the Rayleigh's equation. Then, the unstable mode solution obtained by the method of multiple scales is reconstructed in 2-DOF system by referring to Herrmann and Roussellet's method in pipes conveying fluid. Finally, we present the theoretical equation of the steady-state amplitude. We reveal a validity of our study by a comparison between an experiment and a numerical simulation of some modified blades.

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Self-synchronized phenomena generated in rotor-type oscillators mounted on a straight-line spring-mass system are investigated experimentally and analytically. In the present study, we examine the occurrence region and pattern of self-synchronization in two types of coupled oscillators: rigidly coupled oscillators and elastically coupled oscillators. It is clarified that the existence regions of stable solutions are governed mainly by the linear natural frequency of each spring-mass system. The results of numerical analysis confirm that the self-synchronized solutions of the elastically coupled oscillators correspond to those of the rigidly coupled oscillators. In addition, the results obtained in the present study are compared with the previously reported results for a metronome system and a moving apparatus and the different properties of the phenomena generated in the rotor-type oscillators and the pendulum-type oscillators are shown in terms of the construction of branches of self-synchronized solution and the stability.

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In a previous report, the overall properties of self-synchronized phenomena generated in rotor-type oscillators were experimentally and analytically clarified using two types of systems constructed from oscillators and coupled mass-blocks. The relationship between the stable self-synchronized solutions and the linear natural frequencies of a spring-mass system for each system was also examined. In an effort to clarify the mechanism behind the occurrences of the self-synchronized phenomena, an investigation that is based on the nonlinear vibration characteristics of the systems should be conducted. Nonlinear normal modes have the potential to be useful tools for such an investigation, because the nonlinear normal modes and the self-synchronized phenomena are both periodic motions in nonlinear systems with many degrees of freedom. However, there is a very important difference: the former are free motions in conservative systems and the latter are self-excited motions in nonconservative systems. Thus, a definite relationship between the two is not obvious. This report examines this relationship using the same systems employed in the previous report. The computational results demonstrate that many characteristics of the self-synchronized phenomena can be explained by the nonlinear normal modes.

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Usage of self-excitation as an excitation method for a cantilever probe in atomic force microscopy (AFM) has been proposed to improve the low quality factor Q in liquid environments. To realize non-contact mode AFM, it is necessary to reduce the amplitude of the self-excited cantilever probe. For this study, the self-excited oscillation of the cantilever probe is generated by the angular velocity feedback. In addition, the small steady state amplitude is achieved using nonlinear feedback proportional to the squared deflection angle and the angular velocity. Regarding the microcantilever probe as a microcantilever beam, we present the equation of motion, which incorporates the geometrical nonlinear effect. The averaged equation is derived by applying the method of multiple scales and the bifurcation diagram is described theoretically. Results clarify that increasing the nonlinear feedback gain can reduce the cantilever-probe amplitude. Using an AFM that we produced, we demonstrate the nonlinear dynamics of a “van der Pol” type of self-excited cantilever. The steady state amplitude of the self-excited oscillation was 8 nm.

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Parametric resonance occurring in many mechanical systems has a special resonance mechanism compared with external resonance, and it is usually produced under the excitation, whose direction is not parallel to the motion direction of the system. In this paper, we clarify that parametric resonance can be induced also under the excitation parallel to the motion direction of the system in the case when the system is subjected to asymmetric restoring force. The dependency of the unstable region of the trivial steady state on the magnitude of the asymmetric component is investigated and the steady state under the parametric excitation is examined by nonlinear analysis. Furthermore, we use a simple experimental apparatus of a system subjected to magnetic asymmetric nonlinear restoring force and discuss the validity of the theoretical results.

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The secondary system such as pipings, tanks and other mechanical equipment is installed in the primary system such as building. The secondary system has many nonlinear characteristics. The friction or collision characteristic, which is observed in mechanical supports and joints, is the most common nonlinear characteristic. In this paper, estimation method of the first excursion probability of the secondary system with friction or gap, subjected to earthquake excitation is shown. In aseismic design, the maximum response is usually used. Then, the tolerance level is normalized by the maximum response. When the tolerance level is normalized by the maximum response of the secondary system with friction or gap, the first excursion probability does not depend on nonlinear parameters.

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