On Efficient Computation of the Steady-State Response of Linear Systems With Periodic Coefficients

1996 ◽  
Vol 118 (3) ◽  
pp. 522-526 ◽  
Author(s):  
T. J. Selstad ◽  
K. Farhang
Keyword(s):  
Steady State ◽  
Linear Systems ◽  
Steady State Solution ◽  
Time Varying ◽  
Steady State Response ◽  
Efficient Computation ◽  
Periodicity Condition ◽  
Varying Coefficients ◽  
State Response ◽  
Time Varying Coefficients

An efficient method for obtaining the steady-state response of linear systems with periodically time varying coefficients is developed. The steady-state solution is obtained by dividing the fundamental period into a number of intervals and establishing, based on a fourth-order Rung-Kutta formulation, the relation between the response at the start and end of the period. Imposition of periodicity condition upon the response facilitates computation of the initial condition that yields the steady-state values in a single pass; i.e., integration over only one period. Through a practical example, the method is shown to be more accurate and computationally more efficient than other known methods for computing the steady-state response.

An Efficient Algorithm for Computing the Steady-State Dynamic Response of High-Speed Mechanisms

1994 ◽  
Author(s):  
Tyler J. Selstad ◽  
Kambiz Farhang
Keyword(s):  
Steady State ◽  
High Speed ◽  
Steady State Solution ◽  
Time Varying ◽  
Initial Condition ◽  
Steady State Response ◽  
Periodicity Condition ◽  
Varying Coefficients ◽  
State Response ◽  
Time Varying Coefficients

Abstract An efficient method for obtaining the steady-state response of linear systems with periodically time varying coefficients is developed. The steady-state solution is obtained by dividing the fundamental period into a number of intervals and establishing, based on a fourth-order Rung-Kutta formulation, the relation between the response at the start and end of the period. Imposition of periodicity condition upon the response facilitates computation of the initial condition that yields the steady-state values in a single pass; i.e. integration over only one period. Through a practical example, the method is shown to be more accurate and computationally more efficient than other known methods for computing the steady-state response.

Steady-State Response of Periodically Time-Varying Linear Systems, With Application to an Elastic Mechanism

1995 ◽  
Vol 117 (4) ◽  
pp. 633-639 ◽  
Author(s):  
K. Farhang ◽  
A. Midha
Keyword(s):  
Steady State ◽  
Differential Equations ◽  
Closed Form ◽  
Linear Systems ◽  
Direct Method ◽  
Time Varying ◽  
Steady State Response ◽  
Varying Coefficients ◽  
State Response ◽  
Elastic Mechanism

This paper presents the development of an efficient and direct method for evaluating the steady-state response of periodically time-varying linear systems. The method is general, and its efficacy is demonstrated in its application to a high-speed elastic mechanism. The dynamics of a mechanism comprised of elastic members may be described by a system of coupled, inhomogeneous, nonlinear, second-order partial differential equations with periodically time-varying coefficients. More often than not, these governing equations may be linearized and, facilitated by separation of time and space variables, reduced to a system of linear ordinary differential equations with variable coefficients. Closed-form, numerical expressions for response are derived by dividing the fundamental time period of solution into subintervals, and establishing an equal number of continuity constraints at the intermediate time nodes, and a single periodicity constraint at the end time nodes of the period. The symbolic solution of these constraint equations yields the closed-form numerical expression for the response. The method is exemplified by its application to problems involving a slider-crank mechanism with an elastic coupler link.

An Efficient Method for Evaluating Steady-State Response of Periodically Time-Varying Linear Systems, With Application to an Elastic Slider-Crank Mechanism

1994 ◽  
Author(s):  
Kambiz Farhang ◽  
Ashok Midha
Keyword(s):  
Steady State ◽  
Differential Equations ◽  
Closed Form ◽  
Linear Systems ◽  
Direct Method ◽  
Time Varying ◽  
Steady State Response ◽  
Varying Coefficients ◽  
State Response ◽  
Crank Mechanism

Abstract This paper presents the development of an efficient and direct method for evaluating the steady-state response of periodically time-varying linear systems. The method is general, and its efficacy is demonstrated in its application to a high-speed elastic mechanism. The dynamics of a mechanism comprised of elastic members may be described by a system of coupled, inhomogeneous, nonlinear, second-order partial differential equations with periodically time-varying coefficients. More often than not, these governing equations may be linearized and, facilitated by separation of time and space variables, reduced to a system of linear ordinary differential equations with variable coefficients. Closed-form, numerical expressions for response are derived by dividing the fundamental time period of solution into subintervals, and establishing an equal number of continuity constraints at the intermediate time nodes, and a single periodicity constraint at the end time nodes of the period. The symbolic solution of these constraint equations yields the closed-form numerical expression for the response. The method is exemplified by its application to problems involving a slider-crank mechanism with an elastic coupler link.

A SIMPLIFIED CIRCUIT OF THE SEISMIC ELECTRIC METHOD AND ITS STEADY‐STATE SOLUTION

Geophysics ◽  
1936 ◽  
Vol 1 (3) ◽  
pp. 336-339 ◽  
Author(s):  
M. M. Slotnick
Keyword(s):  
Steady State ◽  
Steady State Solution ◽  
State Solution ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
Steady State Response ◽  
State Response ◽  
In Series ◽  
The One ◽  
Electric Effect

The Seismic Electric Effect gives rise to the problem of finding the steady state response of a circuit consisting of an inductance and a response of a circuit consisting of an inductance and a resistance of the form R+A cos cot (R>A) in series with a D.C. input. In this paper a solution is given, other than the one usually obtained by the method of successive approximations.

Closed-Form, Steady-State Solution for the Unbalance Response of a Rigid Rotor in Squeeze Film Damper

1983 ◽  
Vol 105 (3) ◽  
pp. 551-556 ◽  
Author(s):  
D. L. Taylor ◽  
B. R. K. Kumar
Keyword(s):  
Steady State ◽  
Closed Form ◽  
Closed Form Solution ◽  
Steady State Solution ◽  
State Solution ◽  
Squeeze Film ◽  
Steady State Response ◽  
Rigid Rotor ◽  
Squeeze Film Damper ◽  
State Response

This paper considers the steady-state response due to unbalance of a planar rigid rotor carried in a short squeeze film damper with linear centering spring. The damper fluid forces are determined from the short bearing, cavitated (π film) solution of Reynold’s equation. Assuming a circular centered orbit, a change of coordinates yields equations whose steady-state response are constant eccentricity and phase angle. Focusing on this steady-state solution results in reducing the problem to solutions of two simultaneous algebraic equations. A method for finding the closed-form solution is presented. The system is nondimensionalized, yielding response in terms of an unbalance parameter, a damper parameter, and a speed parameter. Several families of eccentricity-speed curves are presented. Additionally, transmissibility and power consumption solutions are present.

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