A Preliminary Investigation of the Dynamic Stability of Flexible Manipulators Performing Repetitive Tasks

1986 ◽  
Vol 108 (3) ◽  
pp. 206-214 ◽  
Author(s):  
D. A. Streit ◽  
C. M. Krousgrill ◽  
A. K. Bajaj
Keyword(s):  
Parametric Excitation ◽  
Floquet Theory ◽  
Equations Of Motion ◽  
Preliminary Investigation ◽  
Zero Solution ◽  
Flexible Manipulator ◽  
Governing Equations ◽  
Repetitive Tasks ◽  
The Stability ◽  
Two Degree Of Freedom

The governing equations of motion for the compliant coordinates describing a flexible manipulator performing repetitive tasks contain parametric excitation terms. The stability of the zero solution to these equations is investigated using Floquet theory. Analytical and numerical results are presented for a two-degree-of-freedom model of a manipulator with one prismatic joint and one revolute joint.

An Application of the Poincare´ Map to the Stability of Nonlinear Normal Modes

1980 ◽  
Vol 47 (3) ◽  
pp. 645-651 ◽  
Author(s):  
L. A. Month ◽  
R. H. Rand
Keyword(s):  
Floquet Theory ◽  
Poincaré Map ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Theory Approach ◽  
Poincare Map ◽  
Linearized Stability ◽  
Nonlinear Normal ◽  
The Stability ◽  
Two Degree Of Freedom

The stability of periodic motions (nonlinear normal modes) in a nonlinear two-degree-of-freedom Hamiltonian system is studied by deriving an approximation for the Poincare´ map via the Birkhoff-Gustavson canonical transofrmation. This method is presented as an alternative to the usual linearized stability analysis based on Floquet theory. An example is given for which the Floquet theory approach fails to predict stability but for which the Poincare´ map approach succeeds.

Nonlinear Dynamic of Cantilever Functionally Graded Plates Under the Thermalmechanical Loads

2009 ◽  
Author(s):  
Yu-xin Hao ◽  
Wei Zhang ◽  
Jian-hua Wang
Keyword(s):  
Nonlinear System ◽  
Functionally Graded Materials ◽  
Nonlinear Dynamic ◽  
Equations Of Motion ◽  
Functionally Graded ◽  
Degree Of Freedom ◽  
Mode Shapes ◽  
Governing Equations ◽  
Graded Materials ◽  
Two Degree Of Freedom

An analysis on nonlinear dynamic of a cantilevered functionally graded materials (FGM) plate which subjected to the transverse excitation in the uniform thermal environment is presented for the first time. Materials properties of the constituents are graded in the thickness direction according to a power-law distribution and assumed to be temperature dependent. In the framework of the Third-order shear deformation plate theory, the nonlinear governing equations of motion for the functionally graded materials plate are derived by using the Hamilton’s principle. For cantilever rectangular plate, the first two vibration mode shapes that satisfy the boundary conditions is given. The Galerkin’s method is utilized to discretize the governing equations of motion to a two-degree-of-freedom nonlinear system under combined thermal and external excitations. By using the numerical method, the two-degree-of-freedom nonlinear system is analyzed to find the nonlinear responses of the cantilever FGMs plate. The influences of the thermal environments on the nonlinear dynamic response of the cantilevered FGM plate are discussed in detail through a parametric study.

Dynamic analysis of a microelectromechanical systems resonant gyroscope excited using combination parametric resonance

2006 ◽  
Vol 220 (9) ◽  
pp. 1463-1479 ◽  
Author(s):  
B J Gallacher ◽  
J S Burdess
Keyword(s):  
Perturbation Analysis ◽  
Parametric Excitation ◽  
Microelectromechanical Systems ◽  
Equations Of Motion ◽  
Multiple Time ◽  
Excitation Scheme ◽  
Electrostatically Actuated ◽  
Modes Of Vibration ◽  
The Stability ◽  
Electrostatic Coupling

This paper investigates the application of parametric excitation to a resonant microelectromechanical systems (MEMS) gyroscope. The modal equations of motion of an electrostatically actuated ring are derived and shown to be coupled via the electrostatic stiffness. Such electrostatic coupling between in-plane modes of vibration permits parametric instabilities that may be exploited in a novel excitation scheme. A multiple time scale perturbation method is used to analyse the response of the ring gyroscope to the combination parametric excitations with the principal objective of separating the drive and response frequencies of the ring gyroscope. As pairs of flexural modes of the perfect ring are degenerate, the combination excitation between distinct modes demand the ring to be analysed as a four degree of freedom system. Slight mis-tuning between the otherwise degenerate modes is incorporated in the perturbation analysis. The results of the perturbation analysis are subsequently used to determine the stability boundaries for a typical ring gyroscope when excited using a sum combination resonance between the flexural modes of order 2 and 5. In this case, the ratio of the drive and response frequencies is approximately 10:1. Drive and sense configurations that enable effective parametric excitation of a desired mode are investigated. Simulation of the oscillator scheme is achieved using MATLAB Simulink and this validates the perturbation analysis. Agreement between the models within 10 per cent is demonstrated.

Dynamic Response of Cantilever FGM Cylindrical Shell

2011 ◽  
Vol 130-134 ◽  
pp. 3986-3993 ◽  
Author(s):  
Yu Xin Hao ◽  
Wei Zhang ◽  
L. Yang ◽  
J.H. Wang
Keyword(s):  
Cylindrical Shell ◽  
Volume Fraction ◽  
Nonlinear Oscillations ◽  
Equations Of Motion ◽  
Functionally Graded ◽  
Governing Equations ◽  
Temperature Dependent ◽  
Graded Materials ◽  
First Order ◽  
Two Degree Of Freedom

An analysis on the nonlinear dynamics of a cantilever functionally graded materials (FGM) cylindrical shell subjected to the transversal excitation is presented in thermal environment.Material properties are assumed to be temperature-dependent. Based on the Reddy’s first-order shell theory,the nonlinear governing equations of motion for the FGM cylindrical shell are derived using the Hamilton’s principle. The Galerkin’s method is utilized to discretize the governing partial equations to a two-degree-of-freedom nonlinear system including the quadratic and cubic nonlinear terms under combined external excitations. It is our desirable to choose a suitable mode function to satisfy the first two modes of transverse nonlinear oscillations and the boundary conditions for the cantilever FGM cylindrical shell. Numerical method is used to find that in the case of non-internal resonance the transverse amplitude are decreased by increasing the volume fraction index N.

Basic Study on the Seismic Response of the High-Speed-Moving Vehicle Considering Passengers' Dynamics

2012 ◽  
Vol 452-453 ◽  
pp. 1200-1204
Author(s):  
Atsuhiko Shintani ◽  
Tomohiro Ito ◽  
Yudai Iwasaki
Keyword(s):  
High Speed ◽  
Equations Of Motion ◽  
Seismic Excitations ◽  
Lagrangian Equation ◽  
Basic Study ◽  
Moving Vehicle ◽  
Seismic Input ◽  
The Stability ◽  
Risk Rate ◽  
Two Degree Of Freedom

The stability of the high-speed running vehicle subjected to seismic excitations considering passengers' dynamics are considered. A vehicle consists of one body, two trucks and four wheel sets. A passenger is modeled by simple two degree of freedom vibration system. The equations of motion of the vehicle and passengers are calculated by Lagrangian equation of motion. Combining two models, the behavior of the vehicle subjected to actual seismic input considering passengers' dynamics are calculated by numerical simulation. The stability of the vehicle is evaluated by using the risk rate of rollover. We investigate the possibility of the rollover of the vehicle. We focus on the effect of the dynamic characteristics of the human and the number of the passengers when the vehicle is subjected to the seismic excitation.

An Approximate Analysis of Quasi-Periodic Systems Via Floquét Theory

2017 ◽  
Vol 13 (2) ◽  
Author(s):  
Ashu Sharma ◽  
S. C. Sinha
Keyword(s):  
Parametric Excitation ◽  
Periodic System ◽  
Floquet Theory ◽  
Picard Iteration ◽  
Periodic Systems ◽  
Transition Matrices ◽  
Frequency Spectra ◽  
Periodic Coefficients ◽  
Stability Diagrams ◽  
The Stability

Parametrically excited linear systems with oscillatory coefficients have been generally modeled by Mathieu or Hill equations (periodic coefficients) because their stability and response can be determined by Floquét theory. However, in many cases, the parametric excitation is not periodic but consists of frequencies that are incommensurate, making them quasi-periodic. Unfortunately, there is no complete theory for linear dynamic systems with quasi-periodic coefficients. Motivated by this fact, in this work, an approximate approach has been proposed to determine the stability and response of quasi-periodic systems. It is suggested here that a quasi-periodic system may be replaced by a periodic system with an appropriate large principal period and thus making it suitable for an application of the Floquét theory. Based on this premise, a systematic approach has been developed and applied to three typical quasi-periodic systems. The approximate boundaries in stability charts obtained from the proposed method are very close to the exact boundaries of original quasi-periodic equations computed numerically using maximal Lyapunov exponents. Further, the frequency spectra of solutions generated near approximate and exact boundaries are found to be almost identical ensuring a high degree of accuracy. In addition, state transition matrices (STMs) are also computed symbolically in terms of system parameters using Chebyshev polynomials and Picard iteration method. Stability diagrams based on this approach are found to be in excellent agreement with those obtained from numerical methods. The coefficients of parametric excitation terms are not necessarily small in all cases.

A Novel Approach for Decoupling of Dynamical Equations of Flexible Manipulators

1999 ◽  
Author(s):  
Ali Meghdari ◽  
Farbod Fahimi
Keyword(s):  
Multibody Systems ◽  
Equations Of Motion ◽  
Flexible Manipulator ◽  
Flexible Manipulators ◽  
Dynamical Equations ◽  
First Order ◽  
Novel Approach ◽  
Elastic Multibody Systems ◽  
Two Degree Of Freedom ◽  
Generalized Coordinate

Abstract Recent advances in the study of dynamics of elastic multibody systems, specially the flexible manipulators, indicate the need and importance of decoupling the equations of motion. In this paper, an improved method for deriving elastic generalized coordinates is presented. In this regard, the Kane’s equations of motion for elastic multibody systems are considered. These equations are in the generalized form and may be applied to any desired holonomic system. Flexibility in choosing generalized speeds in terms of generalized coordinate derivatives in Kane’s method is used. It is shown that proper choice of a congruency transformation between generalized coordinate derivatives and generalized speeds leads to a series of first order decoupled equations of motion for holonomic elastic multibody systems. Furthermore, numerical implementation of the decoupling technique using congruency transformation is discussed and presented via simulation of a two degree of freedom flexible manipulator.

An Approximate Analysis of Quasi-Periodic Systems via Floquét Theory

2017 ◽  
Author(s):  
Ashu Sharma ◽  
Subhash C. Sinha
Keyword(s):  
Parametric Excitation ◽  
Periodic System ◽  
Floquet Theory ◽  
Periodic Systems ◽  
Frequency Spectra ◽  
Periodic Coefficients ◽  
Approximate Approach ◽  
Varying Coefficients ◽  
P Transformation ◽  
The Stability

Parametrically excited systems are generally represented by a set of linear/nonlinear ordinary differential equations with time varying coefficients. In most cases, the linear systems have been modeled by Mathieu or Hill equations (periodic coefficients) because their stability and response can be determined by Floquét theory. However, in many cases the parametric excitation is not periodic but consists of frequencies that are incommensurate, making them quasi-periodic. Unfortunately, there is no complete theory for linear dynamic systems with quasi-periodic coefficients. Motivated by this fact, in this work, an approximate approach has been proposed to determine the stability and response of quasi-periodic systems. Although Floquét theory is applicable only to periodic systems, it is suggested here that a quasi-periodic system may be replaced by a periodic system with an appropriate large principal period and thus making it suitable for an application of the Floquét theory. Based on this premise, a systematic approach has been developed and applied to two typical quasi-periodic systems. The approximate boundaries in stability charts obtained from the proposed method are extremely close to the exact boundaries of the original quasi-periodic equations. The exact boundaries are detected by computing the maximal Lyapunov exponents. Further, the frequency spectra of solutions generated near approximate and exact boundaries are found to be almost identical ensuring a high degree of accuracy. The coefficients of the parametric excitation terms are not necessarily small in all cases. ‘Instability loops’ or ‘Instability pockets’ that appear in the stability diagram of Meissner’s equation are also observed in one case presented here. The proposed approximate approach would allow one to construct Lyapunov-Perron (L-P) transformation matrices that reduce quasi-periodic systems to systems whose linear parts are time-invariant. The L-P transformation would pave the way for controller design and bifurcation analysis of quasi-periodic systems.

Parametric Stability of a Two-Degree-of-Freedom Machine System: Part I—Equations of Motion and Stability

1987 ◽  
Vol 109 (2) ◽  
pp. 210-215 ◽  
Author(s):  
R. I. Zadoks ◽  
A. Midha
Keyword(s):  
Equations Of Motion ◽  
Monodromy Matrix ◽  
Degree Of Freedom ◽  
Computationally Efficient ◽  
Machine System ◽  
Stable Operating ◽  
System Part ◽  
The Stability ◽  
Nonlinear Equations Of Motion ◽  
Two Degree Of Freedom

An important question facing a designer is whether a certain machine system will have a stable operating condition. To date, the investigations which deal with this question have been scarce. This study treats an elastic two-degree-of-freedom system with position-dependent inertia and external forcing. In Part I, the nonlinear equations of motion are derived and linearized about the system’s steady-state rigid-body response. The stability of the linearized equations is examined using Floquet theory, and a computationally efficient method for approximating the monodromy matrix is presented. A specific example is proposed and the results are presented in Part II of this paper.

Analysis of Aerodynamics Behavior of FGM Cylindrical Panel

2013 ◽  
Vol 364 ◽  
pp. 118-123
Author(s):  
Li Yang ◽  
Yu Xin Hao
Keyword(s):  
Material Properties ◽  
Effective Properties ◽  
Equations Of Motion ◽  
Cylindrical Panel ◽  
Aerodynamic Load ◽  
Governing Equations ◽  
External Excitation ◽  
Third Order ◽  
Power Law Function ◽  
Two Degree Of Freedom

An analysis on the aerodynamics behavior of a FGM cylindrical panel in air flow subjected to an external excitation is presented in this paper. Material properties of the constituents are assumed to be temperature-independent and the effective properties of FGM panel are graded according to a simple power law function in terms of the volume fractions in thickness direction. Based on von Karman hypothesis and Hamilton’s principle, the nonlinear governing equations of motion are derived. Galerkin’s method is then utilized to obtain a two-degree-of-freedom nonlinear system including the quadratic and cubic nonlinear terms. A third-order piston theory is applied for the aerodynamic load. The bifurcations are presented for FGM panel by numerical simulation to show the influences of in-plane load on the nonlinear dynamics.

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