Analytical results are presented on chaotic vibrations of a post-buckled L-shaped beam with an axial constraint. The L-shaped beam is composed of two beams which are a horizontal beam and a vertical beam. The two beams are firmly connected with a right angle at each end. The beams joint with the right angle is attached to a linear spring. The other ends are firmly clamped for displacement. The L-shaped beam is compressed horizontally via the spring at the beams joint. The L-shaped beam deforms to a post-buckled configuration. Boundary conditions are required with geometrical continuity of displacements and dynamical equilibrium with axial force, bending moment, and share force, respectively. In the analysis, the mode shape function proposed by the senior author is introduced. The coefficients of the mode shape function are fixed to satisfy boundary conditions of displacements and linearized equilibrium conditions of force and moment. Assuming responses of the beam with the sum of the mode shape function, then applying the modified Galerkin procedure to the governing equations, a set of nonlinear ordinary differential equations is obtained in a multiple-degree-offreedom system. Nonlinear responses of the beam are calculated under periodic lateral acceleration. Nonlinear frequency response curves are computed with the harmonic balance method in a wide range of excitation frequency. Chaotic vibrations are obtained with the numerical integration in a specific frequency region. The chaotic responses are investigated with the Fourier spectra, the Poincaré projections, the maximum Lyapunov exponents and the Lyapunov dimension. Applying the procedure of the proper orthogonal decomposition to the chaotic responses, contribution of vibration modes to the chaotic responses is confirmed. The following results have been found: The chaotic responses are generated with the ultra-subharmonic resonant response of the two-third order corresponding to the lowest mode of vibration. The Lyapunov dimension shows that three modes of vibration contribute to the chaotic vibrations predominantly. The results of proper orthogonal decomposition confirm that the three modes contribute to the chaos, which are the first, second, and third modes of vibration. Moreover, the results of the proper orthogonal decomposition are evaluated with velocity which is equivalent to kinetic energy. Higher modes of vibration show larger contribution to the chaotic responses, even though the first mode of vibration has the largest contribution ratio.
Turbine blades are now being used under increasingly severe conditions in order to increase the thermal efficiency of gas turbines. Friction dampers are often used to reduce the vibration of the blade and improve the plant reliability. This is a general study dealing with resonance passing where the natural frequency of the turbine blade coincides with the frequency of specific harmonic excitation forces while increasing the turbine rotation speed. Asynchronous components of excitation forces are also considered in addition to the synchronous components caused by specific harmonic excitation forces. In this study, a new method for predicting the characteristics of nonlinear vibration under excitation force including both synchronous and asynchronous force components is developed. In order to investigate the effect of additional asynchronous loading, time history response analyses considering nonlinear vibration using simulated turbine blades were conducted. Results showed that friction damper slip can be induced by the presence of the additional asynchronous excitation force components even for low values of synchronous excitation force. It is shown that it is possible to use a calibration factor to predict vibration characteristics considering friction slipping by estimating the ratio of the total excitation force to the single harmonic excitation force. To verify the effect of asynchronous excitation force and the validity of the proposed correction method, verification tests were conducted experimentally. The experimental results show that friction slipping occurred under small harmonic excitation force when there was asynchronous excitation force and show good agreement with the numerical results. Moreover, the validity of the proposed method which corrects the dynamic characteristics obtained using of the first order harmonic balance method is confirmed.
Experimental results are presented on non 艮 inear vibrations of a plate carrying a conoentrated mass with mixed bQundary of rectangular and circular shapes . The plate with the mixed boundary is simply supported fbr deflection . Under periodic lateral excitation , chaetic responses are obtained in a specific frequcncy region . The chaotic responses are examined with the Fourier spectra and the maximum Lyapunov exponents . Applying the principal component analysis , contribution ratio of each vibration mode is clarified. It is found that chaotic responses of the plate without the concentrated mass are gencrated from the intcrnal resonant response between the lowest and sixth modes . With the concentrated mass , chaotic responses are generated from the combination resonant response between the lowest and fburth modes ofvibration , Increasing the concentrated mass , time variation ofcontribution ratio corresponding to each vibration mode becomes smalL Key We rds :Nonlinear Vibration , Vibration ofContinuous System , plate , Chaos , Cou led Vibration
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