The resonant response of a simple harmonic half-oscillator

Reid, Charles E.; Whineray, S.

doi:10.1016/0375-9601(95)00038-5

Cited by 4 publications

(5 citation statements)

References 5 publications

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“…A very similar figure was obtained in the experiments reported in Reid & Whineray (1995). This figure displays two of the features we commented upon earlier.…”

Section: Introductionsupporting

confidence: 89%

“…The second reason is that the above system is much easier to analyse mathematically, and by looking for exact solutions of the motion we may obtain much insight into the more general case. The air bed experiments described in Reid & Whineray (1995) show close agreement between the predictions of this model and the results of a practical demonstration. Furthermore, this simple model gives very good insights into the behaviour of the much more complicated system of the vibro-impacting cantilever beam Foale & Bishop (1994), or an impacting cylindrical tube (Goyda Teh 1989), both of which which oscillate in several modes and which have energy exchange between the modes at impact.…”

Section: Introductionsupporting

confidence: 69%

“…The windows of existence around the odd integer values of uj help to explain this. Indeed, Reid & Whineray (1995) found double impact motions in their experiments and expressed some surprise about the wide range of frequency values for which such behaviour was present. Some early studies of such orbits were made by Kotera & Peterka (1981) who wrote down the defining equations.…”

Section: Introductionmentioning

confidence: 98%

“…Some of the earliest studies of periodically forced impact oscillators were numerical and analogue computer investigations by Peterka and co-workers (Peterka & Vacik 1981Peterka 1974a, 6; Kotera & Peterka 1981, 1984Peterka Sz Kotera 1982) with more recent work being done by Shaw & Holmes (1983), Shaw (1985a, 6, c), Thompson & Stewart (1986), Foale & Bishop (1992, Nordmark (1991), Budd Sz Dux (1994a, Budd et a l . (1996) and Lamba & Budd studies of impact systems on air beds have been made in Reid & Whineray (1995) and of impacting cantilever beams (which oscillate in several modes) in Cusumano et al (1994) and Mullin (1996). Popp (1995) and Popp & Stelter (1989 have also examined these systems with respect to controlling impact oscillators.…”

Section: Introductionmentioning

confidence: 99%

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Double impact orbits of periodically forced impact oscillators

Budd

Lee

1996

Proc. R. Soc. Lond. A

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We consider a periodically forced oscillator which impacts against a rigid constraint. The forcing is both sinusoidal and non-sinusoidal and is made up of a combination of the first and second harmonics of a Fourier series. A simple restitution law is used to describe the impact. This paper centres around the analysis of double-impact periodic solutions, that is motions which repeat after every second impact. Solutions for these double-impact orbits are found, including non-physical orbits and unstable orbits. Behaviour typical of nonlinear dynamical systems, such as period-doubling bifurcations are detected and grazing bifurcations, which are unique to discontinuous dynamical systems are also observed. A combination of analytical and numerical techniques are used to uncover the behaviour of the system at odd multiples of the natural frequency, where numerical experiments tell us that stable double-impact periodic solutions exist. The persistence of these stable double-impact orbits are also examined as parameters are varied. We conclude by presenting some numerical calculations of two more general cases of impacting systems.

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“…A very similar figure was obtained in the experiments reported in Reid & Whineray (1995). This figure displays two of the features we commented upon earlier.…”

Section: Introductionsupporting

confidence: 89%

Section: Introductionsupporting

confidence: 69%

Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Double impact orbits of periodically forced impact oscillators

Budd

Lee

1996

Proc. R. Soc. Lond. A

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“…Both of the methods employed the complex-valued kernels. The response of physical system subjected to harmonic loading has been extensively studied (Reid et al 1995). Many studies on response analysis in resonant systems can be found.…”

Section: Introductionmentioning

confidence: 99%

Computation of dynamic stiffness and flexibility for arbitrarily shaped two-dimensional membranes

Chen¹,

Chung²

2002

Structural Engineering and Mechanics

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In this paper, dynamic stiffness and flexibility for circular membranes are analytically derived using an efficient mixed-part dual boundary element method (BEM). We employ three approaches, the complex-valued BEM, the real-part and imaginary-part BEM, to determine the dynamic stiffness and flexibility. In the analytical formulation, the continuous system for a circular membrane is transformed into a discrete system with a circulant matrix. Based on the properties of the circulant, the analytical solutions for the dynamic stiffness and flexibility are derived. In deriving the stiffness and flexibility, the spurious resonance is cancelled out. Numerical aspects are discussed and emphasized. The problem of numerical instability due to division by zero is avoided by choosing additional constraints from the information of real and imaginary parts in the dual formulation. For the overdetermined system, the least squares method is considered to determine the dynamic stiffness and flexibility. A general purpose program has been developed to test several examples including circular and square cases.

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Numerical Computations

Trott

2006

The Mathematica GuideBook for Numerics

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The resonant response of a simple harmonic half-oscillator

Cited by 4 publications

References 5 publications

Double impact orbits of periodically forced impact oscillators

Double impact orbits of periodically forced impact oscillators

Computation of dynamic stiffness and flexibility for arbitrarily shaped two-dimensional membranes

Numerical Computations

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