scholarly journals Closure to “Discussions of ‘Existence and Stability of Ultraharmonics and Subharmonics in Forced Nonlinear Oscillations’” (1955, ASME J. Appl. Mech., 22, pp. 279–282)

1955 ◽  
Vol 22 (2) ◽  
pp. 282
Author(s):  
T. K. Caughey
Keyword(s):  
Nonlinear Oscillations ◽  
Existence And Stability

The Existence and Stability of Ultraharmonics and Subharmonics in Forced Nonlinear Oscillations

1954 ◽  
Vol 21 (4) ◽  
pp. 327-335
Author(s):  
T. K. Caughey
Keyword(s):  
Nonlinear Oscillations ◽  
Initial Conditions ◽  
Forced Oscillations ◽  
Order System ◽  
Response Curves ◽  
Restoring Force ◽  
Amplitude Frequency Response ◽  
Existence And Stability ◽  
Linearized Theory ◽  
The Stability

Abstract A study is made of the forced oscillations of a second-order system having a small cubic nonlinearity in the restoring force. It is shown that under suitable conditions ultraharmonic or subharmonic motion exists in addition to the harmonic motion which a linearized theory would predict. By studying the stability of such motions it is shown that at points on the amplitude frequency-response curves having vertical tangents, instability and consequently “jumps” occur. A study of the dependence of the motion on the initial conditions reveals that while ultra-harmonic and harmonic motions are rather insensitive to initial conditions, the existence of subharmonic motion can be achieved only for a restricted set of initial conditions.

scholarly journals On Nonlinear Oscillations and Stability of Coupled Pendulums in the Case of a Multiple Resonance

2020 ◽  
Vol 16 (4) ◽  
pp. 607-623
Author(s):  
A.P. Markeev ◽  
◽  
T.N. Chekhovskaya ◽  
Keyword(s):  
Periodic Motion ◽  
Nonlinear Oscillations ◽  
Initial Conditions ◽  
Nonlinear Approximation ◽  
Natural Oscillations ◽  
Periodic Action ◽  
Straight Line Passing ◽  
Multiple Resonance ◽  
Horizontal Beam ◽  
Existence And Stability

The points of suspension of two identical pendulums moving in a homogeneous gravitational field are located on a horizontal beam performing harmonic oscillations of small amplitude along a fixed horizontal straight line passing through the points of suspension of the pendulums. The pendulums are connected to each other by a spring of low stiffness. It is assumed that the partial frequency of small oscillations of each pendulum is exactly equal to the frequency of horizontal oscillations of the beam. This implies that a multiple resonance occurs in this problem, when the frequency of external periodic action on the system is equal simultaneously to two its frequencies of small (linear) natural oscillations. This paper solves the nonlinear problem of the existence and stability of periodic motions of pendulums with a period equal to the period of oscillations of the beam. The study uses the classical methods due to Lyapunov and Poincaré, KAM (Kolmogorov, Arnold and Moser) theory, and algorithms of computer algebra. The existence and uniqueness of the periodic motion of pendulums are shown, its analytic representation as a series is obtained, and its stability is investigated. For sufficiently small oscillation amplitudes of the beam, depending on the value of the dimensionless parameter which characterizes the stiffness of the spring connecting the pendulums, the found periodic motion is either Lyapunov unstable or stable for most (in the sense of Lebesgue measure) initial conditions or formally stable (stable in an arbitrarily large, but finite, nonlinear approximation).

MULTIPLE-SCALE AND NUMERICAL ANALYSES FOR THE NONLINEAR OSCILLATIONS OF A GAS BUBBLE SURROUNDED BY A MAXWELL'S FLUID

2019 ◽  
Vol 46 (3) ◽  
pp. 261-275
Author(s):  
César Yepes ◽  
Jorge Naude ◽  
Federico Mendez ◽  
Margarita Navarrete ◽  
Fátima Moumtadi
Keyword(s):  
Nonlinear Oscillations ◽  
Multiple Scale ◽  
Numerical Analyses ◽  
Gas Bubble
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