Subharmonic Oscillation Of Order Research Articles

Nonlinear subharmonic oscillation of orthotropic graphene-matrix composite

We investigate the nonlinear responses of graphene-matrix composite to harmonic and subharmonic resonances. Assuming anisotropic mechanical properties of graphene, we derive size-dependent governing equations of motion for graphene resting on a matrix based on the von Karman hypotheses and nonlocal elasticity theory. Response of graphene oscillation under a uniform pressure is obtained using the averaging method. We study the effects of length scale as well as the presence of the elastic matrix on harmonic and subharmonic oscillations of graphene. Our results reveal that subharmonic oscillation of order 1/3 can occur when the ratio of excitation to natural frequencies exceeds three. Also, the subharmonic oscillation of the system is triggered in an appropriate initial condition.

453 弾性支持はりにおける1/2次分数調波振動(OS6-1 非線形振動,OS6 振動解析と制振)

453 弾性支持はりにおける1/2次分数調波振動(OS6-1 非線形振動,OS6 振動解析と制振)

Forced Oscillations of a Rotating Shaft with Nonlinear Spring Characteristics and Internal Damping. Subharmonic Oscillation of Order 1/2 and Entrainment.

Forced Oscillations of a Rotating Shaft with Nonlinear Spring Characteristics and Internal Damping. Subharmonic Oscillation of Order 1/2 and Entrainment.

The 1/2-subharmonic and summed-and-differential harmonic oscillations in a nonlinear rectilinear vibratory system. The variations of resonance curves and the occurrence of unstable vibration due to quartic nonlinear terms.

This paper deals theoretically with a subharmonic oscillation of order 1/2 and some sort of summed-and-differential harmonic oscillations in a rectilinear vibratory system with nonlinear spring characteristics, which are expressed by the terms of the second, third and fourth powers of coordinates. Especially, the effects of quartic terms on these oscillations caused by the nonlinear terms up to the third power of coordinates are investigated. It is clarified that quartic terms have qualitatively different effects from those of quadratic terms, and that unstable vibrations occur due to quartic terms. These kinds of unstable vibrations have been observed experimentally in a rotating shaft system by the authors of this paper.

Nonlinear Forced Oscillations of a Shallow Spherical Shell

Axisymmetric forced oscillations of a shallow spherical shell subjected to harmonic excitation are investigated. Based on modal equations derived from the von Karman dynamic equations, a theoretical analysis is conducted of the harmonic oscillation as well as the subharmonic oscillation of order 1/2. The theoretical analysis reveals that in certain cases the response curve of the harmonic oscillation is of soft spring type, that the subharmonic oscillation of order 1/2 can occur, and that the characters of the subharmonic oscillation are greatly influenced by internal resonance. An experimental analysis is also conducted with use of a shallow spherical steel shell. The experimental analysis confirms the validity of the theoretical analysis.

On the Subharmonic Oscillations of Unsymmetrical Shafts

When a rotating shaft is supported by single-row deep groove ball bearings, the restoring force of the shaft has nonlinear spring characteristics. In the case of an unsymmetrical shaft, the subharmonic oscillation of order 1/2 of the forward precession shows unique features owing to the influence of some of the nonlinear components which have no effects on such an oscillation of a symmetrical shaft. By changing the assembly of the experimental apparatus and the degree of unbalance, we could obtain three kinds of resonance curves concerning this subharmonic oscillation. They are the following: a stable resonance curve of the hard spring type, an unstable region where unstable vibration with frequency 1/2·ω(ω: angular velocity of the shaft) appears, and the case where no subharmonic oscillation appears. In the analysis of such an unsymmetrical shaft system with nonlinear characteristics, the method utilizing normal coordinates cannot be used. We can explain such phenomena theoretically by the method proposed in this paper.

Theoretical discussions on vibrations of a rotating shaft with nonlinear spring characteristics

Various kinds of nonlinear forced oscillations may appear, when the restoring force of a rotating shaft has nonlinear spring characteristics. A rotating shaft system with gyroscopic moments acting does not experience rectilinear lateral vibrations of the shaft but a whirling type of motion. The nonlinear spring characteristics of the shaft are assumed to be distributed two-dimensionally and polar coordinates are used for their representation. Nonlinear spring characteristics expressed in polar coordinates may be characterized by a component with a constant value and other components whose magnitudes vary 1, 2, 3, 4, ... times, during a single whirl of the shaft around its equilibrium position. This type of representation gives a clear description of the phenomena of nonlinear forced oscillations and aids in the prediction of their occurrences. The present discussion centers on the subharmonic oscillation of order 1/3 of forward precession as a representative case. Other kinds of nonlinear oscillations are discussed briefly. Experimental results of previous reports may clearly be explained in the light of the results of this paper.

Subharmonic Oscillation of Order 1/3 in Roller Chain : 1st Report, Theoretical Analysis of the System Having Multiple Degrees of Freedom

The subharmonic oscillations of order 1/3 in a roller chain regarded as a system with multiple degrees of freedom are analysed under the boundary conditions that its one end is fixed and the other end is subject to lateral forced displacement. Such oscillations are excited by the fluctuations of tension, namely of the extension of the chain during vibration. The solutions for these subharmonic oscillations are obtained by applying the harmonic balance method. It is clear that only two kinds of these solutions can be realized with the limited regions of stability. Further, the modes of these vibrations and the fluctuation patterns of tension are determined and characters of their stable steady vibrations are discussed.

Subharmonic Oscillation of Order 1/3 of Roller Chain : 1st Report, Theoretical Analysis of the System Having Multiple Degrees of Freedom

Subharmonic Oscillation of Order 1/3 of Roller Chain : 1st Report, Theoretical Analysis of the System Having Multiple Degrees of Freedom

Theoretical Discussions on Vibrations of a Rotating Shaft with Nonlinear Spring Characteristics

Various kinds of nonlinear forced oscillations may appear, when the restoring force of a rotating shaft has nonlinear spring characteristics. A rotating shaft system with gyroscopic moments acting does not experience rectilinear lateral vibrations of the shaft but a whirling type of motion. The nonlinear spring characteristics of the shaft are assumed to be distributed two-dimensionally and polar coordinates are used for their representation. Nonlinear spring characteristics expressed in polar coordinates may be characterized by a component with a constant value and other components whose magnitudes vary 1, 2, 3, 4, ... times, during a single whirl of the shaft around its equilibrium position. This type of representation gives a clear description of the phenomena of nonlinear forced oscillations and aids in the prediction of their occurrences. The present discussion centers on the subharmonic oscillation of order 1/3 of forward precession as a representative case. Other kinds of nonlinear oscillations are discussed briefly. Experimental results of previous reports may clearly be explained in the light of the results of this paper.

Oscillations of a Rotating Shaft with Symmetrical Nonlinear Spring Characteristics

A rotating shaft system composed of a circular disc, an elastic shaft and two bearings is usually subject to some gyroscopic effect and has four natural frequencies p1p4 which correspond to four modes of whirling motion. Of these four, the first two are of forward precession (p1>p2>0) and the other two are of backward precession (0>p3>p4). We consider a vertical shaft system, in which a shaft is supported by a double-row self-aligning ball bearing at the upper end and by a single-row deep groove ball bearing at the lower end. If the two center lines of the upper and lower bearings are well aligned and if the shaft is situated at the middle of the angular clearance of the lower bearing, the elastic restoring force of the shaft has symmetrical nonlinear spring characteristics. For this rotating shaft system, summed-and-differential harmonic oscillations of the types [2p2p3], [2p2-p4], [p2-2p3], [p2-2p4], and [p2-p3-p4], and a subharmonic oscillation of order 1/3 of the type [3p2] can occur. In this paper, these nonlinear forced whirling oscillations are studied experimentally. Furthermore, it is pointed out that the polar coordinates are suitable for representing the nonlinear restoring force characteristics in these whirling oscillations.

On the Critical Speed of a Shaft of Sub-harmonic Oscillation

Since single-row radial ball bearings have angular a shaft supported by them has a non-linear and non-symmetrical spring characteristic provided the bearing center lines of both bearing pedestals are not in alignment. In a small angular clearance, a shaft is freely supported, and the inclination angle of the deflection curve of the end of the shaft grows larger than a small angular clearance, then the shaft is not freely supported. When bearing center lines of both pedestals are not in alignment, the equilibrium position of a shaft is not located at the middle position of a small angular clearance, then the shaft has a non-symmetrical non-linearity. Therefore, sub-harmonic oscillation of order 1/2 can occur. In fact, the whirl of shaft with the mode of sub-harmonic oscillation of 1/2 appears at a rotating speed higher than the major critical speed.

Subharmonic Oscillations in Nonlinear Systems

This paper deals with the subharmonic oscillations which occur in systems with nonlinear restoring force. It is first shown that the order of the subharmonics has a close connection to the form of the nonlinear characteristics. The subharmonic oscillation of order ⅓, i.e., the oscillation whose fundamental frequency is one-third that of the applied force, is then particularly investigated for the cases in which the nonlinear characteristics are expressed by (1) cubic and (2) quintic functions. In both cases the stability problem of the periodic solution is discussed in detail according to the stability criterion given previously by the author. The analysis reveals that in the latter case (2) the second higher-harmonic of the subharmonic, i.e., the oscillation of order ⅔ builds up with negative damping. This is verified by experiments in which the oscillation of order ⅔ causes the collapse of the original subharmonic oscillation.

Forced Oscillations with Nonlinear Restoring Force

Forced oscillations with nonlinear restoring force are studied in transient states as well as in steady states. The original differential equation characterized by a nonlinear term is transformed under certain restrictions to the following differential equation of the first order: dy/dx=Y(x,y)/X(x,y). Following Poincaré and Bendixson, the singularities and the integral curves of the above equation are discussed, the former being correlated with the periodic states of oscillations, the latter with the transient states of oscillations. The stability of the periodic solutions is determined in accordance with that of the singular points, viz., according to the roots of the characteristic equation. The integral curves yield the relationship between the given initial conditions and the periodic solutions. Thus, once the initial conditions are prescribed, we can foresee the final periodic states which are started with those conditions. With this method of investigation, we have first studied the harmonic oscillations presented in Part I and then the subharmonic oscillation of order ⅓ given in Part II. In both cases the theoretical results are compared with the experimental measurements carried out for an electric circuit containing a saturable iron core, and the satisfactory agreement is found between them.