Chaotic Oscillations of a Post-Buckled Beam Constrained by an Axial Spring: Part 1 — Experiment

2004 ◽  
Author(s):  
Ken-ichi Nagai ◽  
Shinichi Maruyama ◽  
Kazuya Sakaimoto ◽  
Takao Yamaguchi
Keyword(s):  
Principal Component ◽  
Lateral Acceleration ◽  
Maximum Lyapunov Exponent ◽  
Chaotic Oscillations ◽  
Test Beam ◽  
Restoring Force ◽  
Concentrated Load ◽  
Chaotic Response ◽  
Buckled Beam ◽  
Exciting Frequency

Experimental results are presented on chaotic oscillations of a post-buckled beam subjected to periodic lateral acceleration. A thin steel beam of thickness 0.198mm, breath 12.7 mm and length 106mm is used as a test beam. Both ends of the beam are clamped and one end is connected to an axial spring. First, natural frequencies of the beam are measured under an axial compression. Under the post-buckled configuration of the beam, characteristics of static deflection by a concentrated load on the beam are obtained. The post-buckled beam shows the soften-and-hardening characteristics of restoring force. The frequency regions of chaotic responses are inspected. The chaotic responses around these domains are examined carefully by time histories, the Poincare´ maps, the Fourier spectra, the maximum Lyapunov exponents and the principal component analysis. The predominant chaotic responses of the beam are generated by the jump phenomena. The chaotic responses are related to the sub-harmonic resonances of 1/2 and 1/3 orders with the lowest mode of vibration. The maximum Lyapunov exponent of the former chaotic response of 1/2 order is larger than that of the latter chaotic response of 1/3 order. Onsets of the chaotic responses are also confirmed by the Poincare´ projection in the variation of exciting frequency.

Chaotic Oscillations of a Post-Buckled Beam Constrained by an Axial Spring: Part 2 — Analysis

2004 ◽  
Author(s):  
Shinichi Maruyama ◽  
Ken-ichi Nagai ◽  
Takao Yamaguchi ◽  
Kazuaki Hoshi
Keyword(s):  
Principal Component Analysis ◽  
Lyapunov Exponent ◽  
Principal Component ◽  
Component Analysis ◽  
Degree Of Freedom ◽  
Chaotic Oscillations ◽  
Harmonic Response ◽  
Chaotic Response ◽  
Buckled Beam ◽  
Modes Of Vibration

To compare with corresponding experiment, analytical results are presented on chaotic oscillations of a post-buckled beam constrained by an axial spring. The beam with an initial deflection is clamped at both ends. The beam is compressed to a post-buckled configuration by the axial spring. Then, the beam is subjected to both accelerations of gravity and periodic lateral excitation. Basic equations of motion includes geometrical nonlinearity of deflection and in-plane displacement. Applying the Galerkin procedure to the basic equation and using the mode shape function proposed by the author, a set of nonlinear ordinary differential equations is obtained with a multiple-degree-of-freedom system. Linear natural frequency due to the axial compression and restoring force of the post-buckled beam are obtained. Next, periodic responses of the beam are inspected by the harmonic balance method. Chaotic responses are obtained by the numerical integration of the Runge-Kutta-Gill method. Chaotic time responses are inspected by the Fourier spectra, the Poincare´ projections, the maximum Lyapunov exponents. Contribution of the number of modes of vibration to the chaos is also discussed by the principal component analysis. Chaotic response is generated within the sub-harmonic resonance responses of 1/2 and 1/3 orders. The maximum Lyapunov exponent corresponded to the sub-harmonic response of 1/2 order is greater than that of the sub-harmonic response of 1/3 order. By the inspection of the Lyapunov exponent on the chaotic response and the analysis with the multiple-degree-of-freedom system, more than three modes of vibration contribute to the chaos. Using the principal component analysis to the chaotic responses at multiple positions of the beam, the lowest mode of vibration contributes dominantly. Higher modes of vibration contribute to the chaos with small amount of amplitude.

Control of a Buckled Beam Subjected to a Compressive Force

2001 ◽  
Author(s):  
Hiroshi Yabuno ◽  
Kazuya Ando ◽  
Nobuharu Aoshima
Keyword(s):  
Steady State ◽  
Disturbance Observer ◽  
Coulomb Friction ◽  
Control Method ◽  
Compressive Force ◽  
Restoring Force ◽  
Velocity Feedback ◽  
Stabilization Control ◽  
Buckled Beam ◽  
Position Feedback

Abstract In this paper, we deal with a stabilization control for the buckled beam subjected to a compressive force. It is easily predicted that by applying a restoring force to the beam (P control), which is proportional to the deflection, the critical compressive force for the buckling is increased. However it is theoretically and experimentally clarified in our former study that in the neighborhood of the critical point, the effect of Coulomb friction at the supporting points is relatively increased even if it is very slight. It follows that the beam cannot be stabilized to the trivial steady state only by using the position feedback control and also velocity feedback. In this paper, we propose a stabilization control method of the beam to the trivial steady state by the aid of disturbance observer. Furthermore, the validity of the theoretically proposed method is experimentally confirmed.

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

2009 ◽  
Author(s):  
Shinichi Maruyama ◽  
Ken-ichi Nagai ◽  
Kota Muto ◽  
Takao Yamaguchi
Keyword(s):  
Principal Component Analysis ◽  
Vibration Mode ◽  
Harmonic Balance Method ◽  
Principal Component ◽  
Component Analysis ◽  
Axial Displacement ◽  
Chaotic Vibrations ◽  
Vibrating System ◽  
Chaotic Response ◽  
Cantilevered Beam

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.

Chaotic Oscillations of a Shallow Cylindrical Shell-Panel With a Concentrated Elastic-Support

2000 ◽  
Author(s):  
Takao Yamaguchi ◽  
Ken-ichi Nagai
Keyword(s):  
Cylindrical Shell ◽  
Numerical Solutions ◽  
Harmonic Balance Method ◽  
Type Equation ◽  
Inertia Force ◽  
Chaotic Oscillations ◽  
Shallow Cylindrical Shell ◽  
Chaotic Response ◽  
Modes Of Vibration ◽  
Shell Panel

Abstract This paper presents numerical solutions on chaotic oscillations of a shallow cylindrical shell-panel excited by a periodic acceleration. The shell with rectangular boundary is simply supported along all edges, and the center of the shell is supported by an elastic spring. The Donnell-Mushtari-Vlasov type equation is used with the modification of an inertia force. The governing equation is reduced to a nonlinear differential equation of a multi-degree-of-freedom system by the Bubnov-Galerkin procedure. To estimate regions of the chaotic response, periodic solutions of steady state response are first calculated by the harmonic balance method. Next, time evolutions of the chaotic motion are obtained numerically by the Runge-Kutta-Gill method. The chaotic response accompanied with a dynamic snap-through is identified both by means of Lyapunov exponents and Poincaré projections. For the shell with a spring, the Lyapunov dimension is smaller than for the case without the spring. Multiple modes of vibration contributes to the generation of chaos, in paticular, the higher modes of vibration are significant.

On the Understanding of Chaos in Duffings Equation Including a Comparison With Experiment

1986 ◽  
Vol 53 (1) ◽  
pp. 5-9 ◽  
Author(s):  
E. H. Dowell ◽  
C. Pezeshki
Keyword(s):  
Theoretical Model ◽  
Initial Value Problem ◽  
Physical Experiment ◽  
Chaotic Oscillations ◽  
External Excitation ◽  
Initial Value ◽  
Comparison Of Results ◽  
Buckled Beam ◽  
Comparison With Experiment ◽  
Forced Excitation

The dynamics of a buckled beam are studied for both the initial value problem and forced external excitation. The principal focus is on chaotic oscillations due to forced excitation. In particular, a discussion of their relationship to the initial value problem and a comparison of results from a theoretical model with those from a physical experiment are presented.

Driving Style Clustering using Naturalistic Driving Data

2019 ◽  
Vol 2673 (6) ◽  
pp. 176-188 ◽  
Author(s):  
Kuan-Ting Chen ◽  
Huei-Yen Winnie Chen
Keyword(s):  
Traffic Safety ◽  
Principal Component ◽  
Lateral Acceleration ◽  
Underlying Structure ◽  
Driver Assistance Systems ◽  
Chi Square ◽  
Driving Style ◽  
Driving Experience ◽  
Partitioning Around Medoids ◽  
Using Data

Knowledge of driving styles may contribute to traffic safety, riding experience, and support the design of advanced driver-assistance systems or highly automated vehicles. This study explored the possibility of identifying driving styles directly from driving parameters using data from the Strategic Highway Research Program 2 database. Partitioning Around Medoids method was implemented to cluster driving styles based on 14 variables derived from time series records. Principal component analysis was then conducted to understand the underlying structure of the clusters and provide visualization to aid interpretation. Three clusters of driving styles were identified, for which the influential differentiating factors are speed maintained, lateral acceleration maneuver, braking, and longitudinal acceleration. Chi-square test of homogeneity was performed to compare the proportions of trips assigned to the three driving style clusters across levels of each driver attribute (age, gender, driving experience, and annual mileage). The results showed that all four attributes examined had an impact on how the trips were clustered, thus suggesting that the clusters capture individual differences in driving styles to some extent. While our results demonstrate the potential of naturalistic vehicle kinematics in capturing individuals’ driving styles, it was also possible that the identified clusters were classifying mostly drivers’ transient behaviors rather than habitual driving styles. More vehicle parameters and information about road conditions are necessary to obtain deeper insights into driving styles.

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