Kimihiko Yasuda

Spatial Motion of a String Subjected to Two Harmonic Excitations With Different Frequencies

This paper studies spatial motion of a stretched string subjected to two harmonic excitations with different frequencies. Two cases are considered, one in which both excitations are in a plane, and one in which they are perpendicular to each other. First, theoretical analysis by the method of multiple time scales is conducted. It is found that in the former case sub-combination resonance and summed-and-differential combination resonance of whirling type can occur, and in the latter case sub-combination resonances parallel and perpendicular to the excitations and of whirling type can occur. Then, experiment is conducted. In the experiment, it was confirmed that the same types of motions as those predicted in the theoretical analysis occured. In addition, it was found that for the case in which the excitations are in a plane the response is sensitive to the difference between the natural frequencies in the two perpendicular directions. Finally, effect of the difference between the natural frequencies is studied.Copyright

Numerical Integration Method to Determine Periodic Solutions of Nonlinear Systems

A new method to obtain periodic solution of nonlinear systems is proposed by combining secant method with numerical integration. This method can be applied to nonlinear systems with discontinuous characteristics without special treatment. To check the validity of the proposed method, we applied this method to nonlinear systems with discontinuous characteristics, such as piecewise linear spring system, system with Coulomb’s friction and impact damper system. Comparing the results by the proposed method with the results by simple numerical integration or experimental results, the validity of the proposed method is confirmed.Copyright

Nonstationary vibration of a rotating shaft with nonlinear spring characteristics during acceleration through a major critical speed. A discussion by the asymptotic method and the FFT method.

Nonstationary vibrations of a rotating shaft with nonlinear spring characteristics are investigated. Firstly we obtain the first-order approximate solution by the asymptotic method, paying attention to the nonlinear components in the polar coordinate expression, and clarify that only the isotropic nonlinear component influences this solution. Next, we propose the complex-FFT method where nonstationary vibration wave data obtained from numerical integration of the equations of motion are treated as complex numbers. By this method, we can extract the desired vibration component and obtain its amplitude variation curve. Comparing these curves and those of the asymptotic method, we show that the curve obtained by the asymptotic method has comparatively large quantitative error. In addition, we clarify that the anisotropic nonlinear components which do not appear in the first approximate solution of the asymptotic method cause higher-order vibration components in the nonstationary wave.

Proposition of an incremental transfer matrix method for nonlinear vibration analysis.

A new transfer matrix method is proposed which can be used to analyze steady-state responses of nonlinear multi-degree-of-freedom systems. For proposing the method, the quantities describing the dynamic state of the system are expressed in the form of Fourier series, and formulation is made with respect to the increments in the Fourier coefficients. As numerical examples, the method is applied to the analysis of harmonic as well as subharmonic oscillations of a three-degree-of-freedom system. By comparing the results of the method with those of numerical integration of the equations of motion, the validity of the proposed method is confirmed.

A nonlinear vibration analysis of membranes with various shapes by the boundary element method.

The nonlinear vibrations of membranes with various shapes are analyzed by the boundary element method. In order that the application of the boundary element method may be practical in such nonlinear dynamic problems, a treatment that produces modal equations from the governing nonlinear partial differential equations is employed. In this way, it is shown that the original problem is reduced to solving successively several easier problems, and that the modal equations derived thus enable the treatment of various nonlinear oscillations. As an example of the application of this treatment, rectangular and circular membranes are first analyzed, the results of which are compared with other published data to confirm the validity of the treatment. Then, trapezoidal and elliptical membranes are analyzed, and some typical nonlinear oscillations such as subharmonic and summed-and differential harmonic oscillations are shown to occur.