The Timoshenko Beam on an Elastic Foundation and Subject to a Moving Step Load, Part 1: Steady-State Response

1996 ◽  
Vol 118 (3) ◽  
pp. 277-284 ◽  
Author(s):  
S. F. Felszeghy
Keyword(s):  
Steady State ◽  
Boundary Conditions ◽  
Elastic Foundation ◽  
Timoshenko Beam ◽  
Equations Of Motion ◽  
Steady State Solution ◽  
State Solution ◽  
Actual Problem ◽  
State Response ◽  
Particular Solution

The response of a simply supported semi-infinite Timoshenko beam on an elastic foundation to a moving step load is determined. The response is found from summing the solutions to two mutually complementary sets of governing equations. The first solution is a particular solution to the forced equations of motion. The second solution is a solution to a set of homogeneous equations of motion and nonhomogeneous boundary conditions so formulated as to satisfy the initial and boundary conditions of the actual problem when the two solutions are summed. As a particular solution, the steady-state solution is used which is the motion that would appear stationary to an observer traveling with the load. Steady-state solutions are developed in Part 1 of this article for all load speeds greater than zero. It is shown that a steady-state solution which is identically zero ahead of the load front exists at every load speed, in the sense of generalized functions, including the critical speeds when the load travels at the minimum phase velocity of propagating harmonic waves and the sonic speeds. The solution to the homogeneous equations of motion is developed in Part 2 where the two solutions in question are summed and numerical results are presented as well.

The Timoshenko Beam on an Elastic Foundation and Subject to a Moving Step Load, Part 2: Transient Response

1996 ◽  
Vol 118 (3) ◽  
pp. 285-291 ◽  
Author(s):  
S. F. Felszeghy
Keyword(s):  
Steady State ◽  
Boundary Conditions ◽  
Elastic Foundation ◽  
Transient Response ◽  
Timoshenko Beam ◽  
Equations Of Motion ◽  
Steady State Solution ◽  
State Solution ◽  
Critical Speeds ◽  
Particular Solution

The transient response of a simply supported semi-infinite Timoshenko beam on an elastic foundation to a moving step load is determined. The response is found from summing the solutions to two mutually complementary sets of governing equations. The first solution is a particular solution to the forced equations of motion. The second solution is a solution to a set of homogeneous equations of motion and nonhomogeneous boundary conditions so formulated as to satisfy the initial and boundary conditions of the actual problem when the two solutions are summed. As a particular solution, the steady-state solution is used which is the motion that would appear stationary to an observer traveling with the load. Steady-state solutions were developed in Part 1 of this article for all load speeds greater than zero. The solution to the homogeneous equations of motion is developed here in Part 2. It is shown that the latter solution can be obtained by numerical integration using the method of characteristics. Particular attention is given to the cases when the load travels at the critical speeds consisting of the minimum phase velocity of propagating harmonic waves and the sonic speeds. It is shown that the solution to the homogeneous equations combines with the steady-state solution in such a manner that the beam displacements are continuous and bounded for all finite times at all load speeds including the critical speeds. Numerical results are presented for the critical load speed cases.

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.

Comparison of Transient and Steady-State Behaviors in Unwinding Mechanism

2011 ◽  
Author(s):  
Wan-Suk Yoo ◽  
Kun-Woo Kim ◽  
Deuk-Man An ◽  
Jae-Wook Lee
Keyword(s):  
Steady State ◽  
Tensile Force ◽  
Equations Of Motion ◽  
Finite Difference Methods ◽  
Nonlinear Partial Differential Equations ◽  
Inertial Force ◽  
Steady State Solution ◽  
State Solution ◽  
Resistance Force ◽  
Transient Solution

In this study, the transient analysis of a cable unwinding from a cylindrical spool package is first studied and compared to experiment. Then, a steady-state solution is also compared to transient solution. Cables are assumed to be withdrawn with a constant velocity through a fixed point which is located along the axis of the package. When the cable is flown out of the package, several dynamic forces, such as inertial force, Coriolis force, centrifugal force, tensile force, and fluid-resistance force are acting on the cable. Consequently, the cable becomes to undergo very nonlinear and complex unwinding behavior which is called unwinding balloon. In this paper, to prevent the problems during unwinding such as tangling or cutting, unwinding behaviors of cables in transient state were derived and analyzed. First of all, the governing equations of motion of cables unwinding from a cylindrical spool package were systematically derived using the extended Hamilton’s principles of an open system in which mass is transported at each boundary. And the modified finite difference methods are suggested to solve the derived nonlinear partial differential equations. Time responses of unwinding cables are calculated using Newmark time integration methods. The transient solution is compared to physical experiment, and then the steady-state solution is compared to transient solution.

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.

The Steady-State Response of the Double Bilinear Hysteretic Model

1965 ◽  
Vol 32 (4) ◽  
pp. 921-925 ◽  
Author(s):  
W. D. Iwan
Keyword(s):  
Steady State ◽  
Nonlinear Systems ◽  
Steady State Solution ◽  
Degree Of Freedom ◽  
State Solution ◽  
Steady State Response ◽  
Hysteretic Model ◽  
Jump Phenomenon ◽  
State Response ◽  
The Stability

The steady-state response of a one-degree-of-freedom double bilinear hysteretic model is investigated and it is shown that this model gives rise to the jump phenomenon which is associated with certain nonlinear systems. The stability of the steady-state solution is discussed and it is shown that the model predicts an unbounded resonance for finite excitation.

The Timoshenko Beam With a Moving Load

1968 ◽  
Vol 35 (3) ◽  
pp. 481-488 ◽  
Author(s):  
C. R. Steele
Keyword(s):  
Steady State ◽  
Timoshenko Beam ◽  
Moving Load ◽  
Beam Theory ◽  
Steady State Solution ◽  
State Solution ◽  
Asymptotic Results ◽  
Concentrated Load ◽  
Steady State Condition ◽  
Bar Velocity

The problem of a semi-infinite Timoshenko beam of an elastic foundation with a step load moving from the supported end at a constant velocity is discussed. Asymptotic solutions are obtained for all ranges of load speed. The solution is shown to approach the “steady-state” solution, except for three speeds at which the steady state does not exist. Previous investigators have considered only the steady-state solution for the moving concentrated load and have indicated that the three speeds are “critical.” It is shown, however, that only the lowest speed is truly critical in that the response increases with time. For the load speed equal to either the shear or bar velocity, the transients due to the end condition never leave the vicinity of the load discontinuities, so a steady-state condition is never attained. However, the response is shown to be bounded in time for a distributed load. Thus the nonexistence of a steady state does not necessarily indicate a critical condition. Furthermore, the concentrated load solution is shown to have validity at speeds the magnitude of the sonic speeds only for loads of a concentration beyond the limitations of beam theory. Asymptotic results have also been obtained for the beam without a foundation. Since the procedure is similar for beams with, and without, a foundation, only the results are included to show a comparison with the numerical results previously obtained by Florence.

Repetitive Control System Under Actuator Saturation and Windup Prevention

2009 ◽  
Vol 131 (4) ◽  
Author(s):  
D. Sbarbaro ◽  
M. Tomizuka ◽  
B. León de la Barra
Keyword(s):  
Steady State ◽  
Closed Loop ◽  
Actuator Saturation ◽  
Steady State Solution ◽  
State Solution ◽  
Valuable Insight ◽  
State Response ◽  
Saturating Actuators ◽  
Steady State Responses ◽  
Repetitive Controller

This work provides an analysis of the steady state response of a prototype repetitive controller applied to a class of nonlinear systems, i.e., systems with actuator saturation. First, it is shown that the steady state solution of the closed loop nonlinear system can be obtained by an iterative Picard process, which establishes the periodic nature of the steady state solution. Second, the conditions for obtaining bounded steady state responses are analyzed for a saturating nonlinearity commonly found in mechatronic applications. Valuable insight is provided into the effects of input signals and saturating actuators on the closed loop performance of a prototype repetitive controller. In order to improve the transient closed loop response, a simple antiwindup strategy tailored to repetitive controllers is proposed.

Periodic Response of a Link Coupling With Clearance

1989 ◽  
Vol 111 (2) ◽  
pp. 253-259 ◽  
Author(s):  
Y. S. Choi ◽  
S. T. Noah
Keyword(s):  
Steady State ◽  
Boundary Conditions ◽  
Solution Procedure ◽  
Contact Forces ◽  
Steady State Response ◽  
Algebraic Equations ◽  
State Response ◽  
Contact Points ◽  
Integration Schemes ◽  
Nonlinear Algebraic Equations

The nonlinear, steady-state response of a displacement-forced link coupling with clearance with finite stiffness is determined. The solution procedure is derived from satisfying the boundary conditions at the contact points and then solving the resulting nonlinear algebraic equations by setting the duration of contact as a parameter. This direct approach to determining periodic solutions for systems with clearances with finite stiffness is substantially more efficient than numerical integration schemes. Results in terms of contact forces and durations of contact are pertinent to fatigue and wear studies. Parametric relations are presented for effects of the variation of damping, stiffness, exciting displacement, and gap length on the dynamic behavior of the link pair.

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