Journal ArticleDOI
Parametric resonance and nonlinear string vibrations
TLDR
In this article, it is shown that it is nonlinearity rather than damping that limits the growth of a resonantly excited mode, although damping is needed for steady-state oscillations to occur.Abstract:
Periodic changes in the tension of a taut string parametrically excite transverse motion in the string when the driving frequency is close to twice the natural frequency of any transverse normal mode of the string. The literature on this phenomenon is synthesized and extended to include the effects of damping as well as nonlinearity. It is shown that it is nonlinearity rather than damping that limits the growth of a resonantly excited mode, although damping is needed for steady-state oscillations to occur. The validity of the usual approximation that the string tension depends only on time and not on space is checked by modeling a string as point masses joined by massless linear springs. It is found that although this approximation is likely to be violated in practice, the violation does not have a significant effect on the results. The source of the disagreement in the literature for the speed of longitudinal waves in a stretched string is identified.read more
Citations
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Excitation of large-amplitude parametric resonance by the mechanical stiffness modulation of a microstructure
TL;DR: In this article, an approach for parametric excitation of large-amplitude stable oscillations of a microstructure operated by a parallel-plate electrode is presented. But the experimental results indicate that the suggested approach can be efficiently used for excitation in various types of microdevices where stable resonant operation combined with robustness and large vibrational amplitudes are desirable.
Journal ArticleDOI
Efficient primary and parametric resonance excitation of bistable resonators
TL;DR: In this article, a single crystal silicon in-plane arch microbeam is fabricated such that it can be excited axially from one of its ends by a parallel-plate electrode.
Journal ArticleDOI
Hyperbolic chaos of standing wave patterns generated parametrically by a modulated pump source.
TL;DR: There is a possibility of hyperbolic chaotic dynamics associated with the expanding circle map for spatial phases of parametrically excited standing wave patterns in a model system governed by a one-dimensional wave equation with nonlinear dissipation.
Journal ArticleDOI
Optimizing parametric oscillators with tunable boundary conditions
Harel Plat,Izhak Bucher +1 more
TL;DR: In this article, a passive dynamical system connected to one end of a taught string which parametrically couples its axial motion to transverse vibration is analyzed and it is shown that a specific selection of parameters can reduce the limiting effect of nonlinear stretching thus allowing one to excite high-order modes with small external forces.
Journal ArticleDOI
Verification of hyperbolicity for attractors of some mechanical systems with chaotic dynamics
TL;DR: In this paper, a statistical analysis of the angles of intersection of stable and unstable manifolds for mechanical systems with hyperbolic attractors of the Smale-Williams type is provided.
References
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Journal ArticleDOI
Mathieu functions, a visual approach
TL;DR: In this article, the behavior of the Mathieu functions is illustrated by using a variety of plots with representative examples taken from mechanics, and they show how they can be applied to describe standing, traveling, and rotating waves in physical systems.
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Applications of the Mathieu equation
TL;DR: The properties of the Mathieu equation are reviewed in this article in order to discuss some of the applications that have appeared in recent years, such as vibrations in an elliptic drum, the inverted pendulum, the radio frequency quadrupole, frequency modulation, stability of a floating body, alternating gradient focusing, the Paul trap for charged particles, and the mirror trap for neutral particles.
Journal ArticleDOI
Nonlinear generation of missing modes on a vibrating string
K. A. Legge,Neville H. Fletcher +1 more
TL;DR: The nonlinear transfer of energy among modes of different frequencies on a vibrating string is investigated both theoretically and experimentally in this paper, where it is shown that at least one of the end supports has finite mechanical admittance if there is to be any mode coupling.
Journal ArticleDOI
Non-linear three-dimensional large-amplitude damped free vibration of a stiff elastic stretched string
TL;DR: In this paper, the deformed shape of the central line of a cylindrical rod is determined for finite displacements with small strains of a slender homogeneous elastic rod under initial stress and the original geometrical construction which gives the cross-section's orientation during the motion is detailed.
Journal ArticleDOI
Intrinsic nonlinear effects in vibrating strings
TL;DR: In this article, the two perpendicular polarizations of transverse oscillation in stretched strings are parametrically coupled so that energy is spontaneously exchanged between the perpendicular modes, and the approximate solution of the equations of motion shows that the trajectory is an ellipse with a reverse precession.