A Geometric Approach for Establishing Dynamic Buckling Loads of Autonomous Potential Two-Degree-of-Freedom Systems

1999 ◽  
Vol 66 (1) ◽  
pp. 55-61 ◽  
Author(s):  
A. N. Kounadis
Keyword(s):  
Geometric Approach ◽  
Energy Balance Equation ◽  
Dynamic Buckling ◽  
Degree Of Freedom ◽  
Buckling Loads ◽  
Total Potential ◽  
Total Energy Balance ◽  
Highly Nonlinear ◽  
Planar Systems ◽  
Two Degree Of Freedom

Nonlinear dynamic buckling of autonomous potential two-degree-of-freedom nondissipative systems with static unstable critical points lying on nonlinear primary equilibrium paths is studied via a geometric approach. This is based on certain salient properties of the zero level total potential energy “surface” which in conjunction with the total energy-balance equation allow establishment of new dynamic buckling criteria for planar systems. These criteria yield readily obtained “exact” dynamic buckling loads without solving the highly nonlinear initial-value problem. The simplicity, reliability, and efficiency of the proposed technique is illustrated with the aid of various dynamic buckling analyses of two two-degree-of-freedom models which are also compared with those obtained by the Verner-Runge-Kutta scheme.

Dynamic Buckling of 1-DOF Autonomous Potential Systems Under Tilted Cusp Catastrophe

2001 ◽  
Author(s):  
Anthony N. Kounadis
Keyword(s):  
Autonomous Systems ◽  
Total Potential Energy ◽  
Dynamic Buckling ◽  
Local Analysis ◽  
Cusp Catastrophe ◽  
Imperfection Sensitivity ◽  
Buckling Loads ◽  
Total Potential ◽  
Constant Direction ◽  
Step Loading

Abstract Nonlinear dynamic buckling of one-degree-of freedom (1-DOF) undamped systems under step loading (autonomous systems) of constant direction and infinite duration is discussed in detail using Catastrophe Theory. Attention is focused on the relation of static cuspoind catastrophes to the corresponding dynamic catastrophes for 1-DOF autonomous undamped systems by determining properly the dynamic singularity and bifurcational sets for such systems. Using local analysis one has to classify first the total potential energy (TPE) function of the system into one of the elementary Thom’s catastrophes by defining the corresponding control (unfolding) parameters. Subsequently, using global analyses one can readily obtain exact results for the dynamic buckling loads (DBLs) and their imperfection sensitivity of systems subjected to dynamic dual cusp and tilted cusp catastrophes. It was found that the maximum DBL of the dynamic tilted cusp catastrophe corresponds to a limit point lying in the vicinity of the hysteresis point (related to the static tilted cusp catastrophe). Numerical examples illustrate the methodology proposed herein.

The Probabilistic Solutions to Nonlinear Random Vibrations of Multi-Degree-of-Freedom Systems

1999 ◽  
Vol 67 (2) ◽  
pp. 355-359 ◽  
Author(s):  
G.-K. Er
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Degree Of Freedom ◽  
State Variables ◽  
Algebraic Equations ◽  
Nonlinear Random Vibration ◽  
Highly Nonlinear ◽  
Fpk Equation ◽  
Two Degree Of Freedom

The probability density function of the responses of nonlinear random vibration of a multi-degree-of-freedom system is formulated in the defined domain as an exponential function of polynomials in state variables. The probability density function is assumed to be governed by Fokker-Planck-Kolmogorov (FPK) equation. Special measure is taken to satisfy the FPK equation in the average sense of integration with the assumed function and quadratic algebraic equations are obtained for determining the unknown probability density function. Two-degree-of-freedom systems are analyzed with the proposed method to validate the method for nonlinear multi-degree-of-freedom systems. The probability density functions obtained with the proposed method are compared with the obtainable exact and simulated ones. Numerical results showed that the probability density function solutions obtained with the presented method are much closer to the exact and simulated solutions even for highly nonlinear systems with both external and parametric excitations. [S0021-8936(00)01602-0]

ENERGY APPROACH FOR DYNAMIC BUCKLING OF AN UNDAMPED ARCH MODEL UNDER STEP LOADING WITH INFINITE DURATION

2010 ◽  
Vol 10 (03) ◽  
pp. 411-439 ◽  
Author(s):  
YONG-LIN PI ◽  
MARK ANDREW BRADFORD ◽  
SHUGUO LIANG
Keyword(s):  
Equations Of Motion ◽  
Equation Of Motion ◽  
Buckling Analysis ◽  
Dynamic Buckling ◽  
Energy Approach ◽  
Equilibrium Path ◽  
Arch Model ◽  
Buckling Loads ◽  
Step Loading ◽  
Two Degree Of Freedom

Performing a dynamic buckling analysis of structures is more difficult than carrying out its static buckling analysis counterpart. Some structures have a nonlinear primary equilibrium path including limit points and an unstable equilibrium path. They may also have bifurcation points at which equilibrium bifurcates from the primary equilibrium path to an unstable secondary equilibrium path. When such a structure is subjected to a load that is applied suddenly, the oscillation of the structure may reach the unstable primary or secondary equilibrium path and the structure experiences an escaping-motion type of buckling. For these structures, complete solutions of the equations of motion are usually not needed for a dynamic buckling analysis, and what is really sought are the critical states for buckling. Nonlinear dynamic buckling of an undamped two degree-of-freedom arch model is investigated herein using an energy approach. The conditions for the upper and lower dynamic buckling loads are presented. The merit of the energy approach for dynamic buckling is that it allows the dynamic buckling load to be determined without the need to solve the equations of motion. The solutions are compared with those obtained by an equation of motion approach.

Comparison of Open Chain and Closed Chain Planar Two Degree of Freedom Manipulator for Positional Error

2014 ◽  
Vol 6 (2) ◽  
Author(s):  
H. P. Jawale ◽  
H. T. Thorat
Keyword(s):  
Geometric Approach ◽  
Degree Of Freedom ◽  
The Other ◽  
Dimensionless Number ◽  
Positional Error ◽  
Positional Accuracy ◽  
Open Chain ◽  
Closed Chain ◽  
Error Index ◽  
Two Degree Of Freedom

Open chain and closed chain manipulators are designed for specific objectives. Closed chain five bar manipulator is possible to be configured as a substitute to an open chain two degree of freedom (DoF) manipulator. Positional accuracy is one of the factors for performance evaluation, characterizing suitability of a configuration over the other. Present paper attempts comparative analysis of positional inaccuracy of closed chain five bar manipulator and serial chain configuration. Both manipulators are modeled for positional deviations under identical specifications considering randomness due to joint clearances and backlash in drive. The maximum positional inaccuracy is expressed in terms of dimensionless number as error index (EI) to estimate the comparative behavior of the manipulators. Positional error under influence of backlash and clearances is quantified. Comparison of two configurations is presented and conditional superiority of a configuration over the other is commented using geometric approach.

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