Perturbation Analysis of Diffeomorphisms in Contact Dynamical System

2012 ◽  
Vol 529 ◽  
pp. 264-267
Author(s):  
Da Wei Sun
Keyword(s):  
Dynamical System ◽  
Perturbation Analysis ◽  
Contact System ◽  
Hamiltonian Mechanics ◽  
Perturbation Techniques ◽  
Small Perturbations

This paper studies the perturbations of strictly contact diffeomorphisms in contact dynamical system. By constructing new lifting methods for contact system and using some perturbation techniques in Hamiltonian mechanics, this paper proves that there exists an arbitrary small perturbations such that the corresponding function of the strictly contact isotopy does not equal to a constant at any time.

Determination of Instability Boundaries for a Highly Flexible Prismatic-Jointed Robot Performing Repetitive Maneuvers

1991 ◽  
Author(s):  
Keith W. Buffinton
Keyword(s):  
Floquet Theory ◽  
Equations Of Motion ◽  
Perturbation Techniques ◽  
Axial Motion ◽  
The Third ◽  
Straightforward Application ◽  
Method Yield ◽  
Robotic Devices ◽  
Axially Moving

Abstract Presented in this work are the equations of motion governing the behavior of a simple, highly flexible, prismatic-jointed robotic manipulator performing repetitive maneuvers. The robot is modeled as a uniform cantilever beam that is subject to harmonic axial motions over a single bilateral support. To conveniently and accurately predict motions that lead to unstable behavior, three methods are investigated for determining the boundaries of unstable regions in the parameter space defined by the amplitude and frequency of axial motion. The first method is based on a straightforward application of Floquet theory; the second makes use of the results of a perturbation analysis; and the third employs Bolotin’s infinite determinate method. Results indicate that both perturbation techniques and Bolotin’s method yield acceptably accurate results for only very small amplitudes of axial motion and that a direct application of Floquet theory, while computational expensive, is the most reliable way to ensure that all instability boundaries are correctly represented. These results are particularly relevant to the study of prismatic-jointed robotic devices that experience amplitudes of periodic motion that are a significant percentage of the length of the axially moving member.

Derivation of the Explicit Solution of the Inverse Involute Function and its Applications

1992 ◽  
Author(s):  
Harry Hui Cheng
Keyword(s):  
Series Solution ◽  
Asymptotic Series ◽  
Explicit Solutions ◽  
Series Solutions ◽  
Perturbation Techniques ◽  
Pocket Calculator ◽  
Practical Applications ◽  
Involute Gears ◽  
Simple Expansion ◽  
Explicit Series Solutions

Abstract The involute function ε = tanϕ – ϕ or ε = invϕ, and the inverse involute function ϕ = inv−1(ε) arise in the tooth geometry calculations of involute gears, involute splines, and involute serrations. In this paper, the explicit series solutions of the inverse involute function are derived by perturbation techniques in the ranges of |ε| < 1.8, 1.8 < |ε| < 5, and |ε| > 5. These explicit solutions are compared with the exact solutions, and the expressions for estimated errors are also developed. Of particular interest in the applications are the simple expansion ϕ = inv−1(ε) = (3ε)1/3 – 2ε/5 which gives the angle ϕ (< 45°) with error less than 1.0% in the range of ε < 0.215, and the economized asymptotic series expansion ϕ = inv−1 (ε) = 1.440859ε1/3 – 0.3660584ε which gives ϕ with error less than 0.17% in the range of ε < 0.215. The four, seven, and nine term series solutions of ϕ = inv−1 (ε) are shown to have error less than 0.0018%, 4.89 * 10−6%, and 2.01 * 10−7% in the range of ε < 0.215, respectively. The computation of the series solution of the inverse involute function can be easily performed by using a pocket calculator, which should lead to its practical applications in the design and analysis of involute gears, splines, and serrations.

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