Shinichi Maruyama

Experiments on chaotic vibrations of a post-buckled cantilevered beam connected by a string to an axial spring

Experimental results are presented on chaotic vibrations of a post-buckled cantilevered beam constrained by a string. The string is stretched between the top end of the beam and an axial spring. The axial spring consists of a leaf spring with an attached mass and is fixed on a base frame close to the clamped end of the beam. The length of the string is less than that of the beam. The beam is excited by lateral periodic acceleration. By increasing the attached mass on the axial spring, nonlinear responses of the beam are examined. Nonperiodic response is observed in a typical frequency region. The response is examined by the Fourier spectrum, maximum Lyapunov exponents, Poincaré projection, and principal component analysis. Predominant chaotic response is generated by the internal resonance with a frequency ratio of one-to-three. The fundamental mode and the second mode of vibration are strongly coupled in the chaotic response. When the attached mass of the axial spring is increased, the natural frequency of the axial spring approaches the region of chaotic response. Therefore, the number of vibration modes that contribute to the chaos increases. Furthermore, the Poincaré projection of the chaotic response shows more scattered figures.

Dynamic Responses for Viscoelastic Shock Absorbers to Protect a Finger under Impact Force

This paper deals with dynamic responses of viscoelastic shock absorbers to protect a finger under impact forces by a sliding rigid object. The outline of the viscoelastic absorber is tube and the cross section of the absorber is in the shape of hollow semi circle. The sliding object having initial velocities collided with the absorbers. And the relevant restoring forces are measured using Levitation Mass Method proposed by Fujii. In this paper, we carry out numerical analysis of dynamic response for the viscoelastic absorbers under the same conditions with the experiment using LMM. The absorbers are modeled using a nonlinear concentrated spring with nonlinear hysteresis. This nonlinear spring is connected to the levitated block, which is modeled by three-dimensional finite elements. The experimental data are compared with the calculated ones using our proposed FEM.

Topology Optimization of Poroelastic Structures to Minimize Mean Sound Pressure Levels

In optimization problems that aim to minimize noise, elastic structures have been designed so that fundamental eigenfrequencies depart from excitation frequencies. Moreover, for the sake of simplicity, sound pressure responses have rarely been calculated. In this paper, we propose a new topology optimization method for the design of poroelastic material layouts that minimize sound pressure levels by sound attenuation. In this method, the surrounding air is exactly modeled, and poroelastic material is located in a space filled with air to efficiently dissipate power. The Biot’s theory is incorporated into the optimization scheme to deal with poroelastic material, and we utilize a new bi-material continuum that consists of poroelastic material combined with an equivalent representation of air in the Biot’s theory. Several design problems are presented to demonstrate that the proposed method can provide optimal layouts of poroelastic material that reduce sound pressure levels within specified frequency ranges.Copyright

Effects of a Concentrated Mass on Chaotic Vibrations of a Clamped Circular Plate With Initial Deformation

Experimental results are presented on effects of a concentrated mass on chaotic vibrations of a clamped circular plate. The plate has initial deformation due to initial deflection and initial in-plane compressive constraint at the boundary. The concentrated mass is attached on the center of the plate. Under periodic excitation, non-periodic responses with dynamic snap-through are generated on the plates. The responses are inspected by the Fourier spectrum, the Poincare projection, the maximum Lyapunov exponents and the principal component analysis. The non-periodic responses are found to be chaotic responses. The lowest mode of vibration shows the largest contribution ratio. When the concentrated mass is attached on the plate, the region of the response is shifted to the lower frequency. Furthermore, the width of the frequency region is decreased. The contribution ratio of the lowest mode slightly increases.Copyright

Numerical Simulation of Impact Responses for an Elastic Structure with a Viscoelastic Spring Including Nonlinear Hysteresis Damping

To compute dynamic characteristics of nonlinear viscoelastic springs with elastic structures having huge degree-of-freedom, Yamaguchi proposed a new fast numerical method using finite element method. In this method, restoring forces of the springs are expressed using power series of their elongation. In the expression, nonlinear hysterisis damping is introduced. Finite element for the nonlinear spring having complex coefficients is expressed and is connected to the elastic structures modeled by linear solid finite element. Further, to save computational time, the discrete equations in physical coordinate are transformed into the nonlinear ordinary coupled equations using normal coordinate corresponding to linear natural modes. In this report, the proposed method is applied to simulation for impact responses of a silicone block with an elastic structure (a S-shaped structure) by colliding with a concentrated mass. The concentrated mass has initial velocities and collides with the silicon block. Accelerations of the elastic structure and the concentrated mass are measured using Levitation Mass Method proposed by Fujii. The calculated accelerations from the proposed FEM, corresponds to the experimental ones.

Chaotic Vibrations on a Coupled Vibrating System of a Post-Buckled Cantilevered Beam and an Axial Vibrating Body Connected With a Stretched String

Analytical results are presented on chaotic vibrations on a coupled vibrating system of a post-buckled cantilevered beam and an axial vibrating body connected with a stretched string. The string is stretched between the top end of the cantilevered beam and the axial vibrating body which consists of a mass and a spring. As an initial axial displacement is applied to the spring, the beam is buckled by the tensile force of the string. The main scope of this paper is to investigate the effects of the axial inertia of the vibrating body on the chaotic vibrations of the system. The dynamical model involves nonlinear geometrical coupling between the deformation and the axial force of the beam at the boundary. Furthermore, the problem includes the static buckling and the nonlinear vibration. By using the mode shape function, which was proposed by the senior author, as a coordinate function of the governing equations, nonlinear ordinary differential equations in multiple-degree-of-freedom system are derived by the modified Galerkin procedure. Periodic responses of the beam are calculated with the harmonic balance method, while chaotic responses are integrated numerically. Chaotic time responses are inspected with the Fourier spectra, the Poincare projections, the maximum Lyapunov exponents and the principal component analysis. Chaotic responses are generated from the sub-harmonic resonance responses of 1/2 and 1/3 orders. The results of the principal component analysis shows that the lowest mode of vibration contributes to the chaotic response dominantly, while the second mode of vibration also contribute to the chaos with small amount of amplitude. Inspection of the kinetic energy of each vibration mode shows that the vibration mode with large axial displacement is also dominant in the chaotic response.Copyright

Thickness Optimization for Multilayered Structures Located on the Coupling Surface Between a Structure and an Acoustic Cavity

This paper describes a new design method for optimizing the thickness distribution of a multilayered structure located on the coupling surface between a structure and an acoustic cavity. The design method is based on the concept of a topology optimization method incorporating a transfer matrix for the multilayered structure that includes a poroelastic media layer. The one-dimensional transfer matrix adopted here is an approximate representation addressing vibro-acoustic effects inherent in a multilayered structure over a range of low frequencies, and balances calculation times and desired accuracy. In this study, the problem of minimizing the acoustic pressure within the cavity over the prescribed frequency range is formulated under the constraint of the total weight of the design domain. The thickness of the poroelastic media layer at each nodal point is chosen as the design variables. Numerical results show that acoustic response is significantly reduced by an optimal thickness distribution having a total weight equal to or less than that of the initial thickness.

Chaotic Oscillations of a Suspended Curved Panel with Square Boundary

Experimental results and analytical results are presented on chaotic oscillations of a suspended curved panel with square boundary. In the experiment, the panel is simply supported along all edges. The configuration of the panel is deformed by the gravity force and in-plane elastic constraint. Nonlinear periodic responses of the panel are examined under periodic lateral acceleration. Chaotic responses are observed and measured for modal inspection at four points of the panel simultaneously. In the analysis, the configuration of the panel is assumed as a suspended curved form with double arcs parallel to each edge. The Donnell type equation with lateral inertia force is introduced. 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 are inspected with the Fourier spectra, the Poincare projections, the maximum Lyapunov exponents and the principal components by the Karhunen-Loeve-transformation. Both results of the experiment and the analysis show that the chaotic responses are generated within the frequency range of the same ratio to the each lowest natural-frequency. Poincare projections of the both results coincide fairly well in detail. The number of modes generated in the chaos is counted as three by the maximum Lyapunov exponent and the Lyapunov dimension. The principal component analysis shows the predominant contribution of the lowest mode of vibration.

Eigen Frequencies of Two-Dimensional Beam Structures Subjected to Large Static Deformations due to Initial Axial Displacement (FEM Analysis with Geometric Nonlinearity)

Eigen frequencies for two dimensional beam structures are investigated under initial axial displacements. When the beam structures are consisted of extremely thin flexible components, the shapes of the structures are changed due to the initial axial displacement. And then, their eigen frequencies are affected by the axial displacements. To analyze these phenomena, discrete equations using finite elements with consideration of geometrical nonlinearity are derived as cubic simultaneous nonlinear differential equations. First, large static deformations of the structures due to initial axial displacement are calculated using the proposed FEM. Linear natural frequencies for the deformed structures are investigated secondly. In the numerical analysis, both ends of the beam structures are clamped. The calculated results for straight beams with initial compressive displacement using the FEM are consistent with the theoretical results carried out by the authors previously. Further, the influences of initial axial displacement on eigen frequencies of the plane structure which is comprised of five straight beams are clarified.

Analysis on Nonlinear Coupled Vibrations of a Clamped Beam with a Tip-Mass Including Axial Inertia

Analytical results are presented on nonlinear coupled vibrations of a clamped beam with a tip-mass constrained by an axial spring. First, governing equations of the thin beam are derived including both effects of the axial inertia force and the geometrical nonlinearity of the beam. The modified Galerkin procedure is applied to the governing equation by introducing a coordinate function for the axial displacement considering the quadratic nonlinear coupling with the deflection. Nonlinear periodic responses are calculated with the harmonic balance method. Frequency responses of the principal resonance of the fundamental vibration mode are compared by changing the mass attached to the beam end. When the mass is not attached to the beam, the frequency response agrees well with the result neglecting the axial inertia. As the mass is increased, the response curve in comparatively large amplitude shifts to the lower frequency range, owing to the axial inertia of the tip-mass. The analytical result of the frequency response, taking account of the axial inertia, agrees well with the relevant experimental result of a post-buckled beam formerly presented, which verifies our analytical results.