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.

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.

Transient Thermomechanical Simulation of Laser Hammering in Optoelectronic Package Manufacturing

2004 ◽  
Vol 127 (3) ◽  
pp. 299-305 ◽  
Author(s):  
Ben Ting ◽  
Vincent P. Manno
Keyword(s):  
Steady State ◽  
Transient Response ◽  
Laser Energy ◽  
Steady State Solution ◽  
Heat Fluxes ◽  
State Solution ◽  
Asymptotic State ◽  
Element Analysis ◽  
Finite Element Analysis Simulation ◽  
Focus Spot

Laser hammering (LH) is a process used in the manufacturing of butterfly optoelectronic packages to correct laser-to-fiber misalignment that occurs when the semiconductor lasers are welded in place. High-power, precisely positioned pulsed lasers are used in LH to induce deformation of the fiber support housing to, in turn, induce realignment. A thermomechanical modeling study of LH is reported in this paper, which focuses on the degree to which a steady-state model can predict the asymptotic state of a transient response subjected to a periodic laser excitation. A baseline, two-dimensional fiber mounting/ferrule geometry is employed in a finite element analysis simulation case study. Various laser wave forms are applied to focus spot location sizes of 50 and 200μm over a range of applied heat fluxes (10-1000W∕mm2). Effects of laser energy deposition location, as well as the use of multiple lasers, are also studied. The results show that the steady-state solution is in good agreement with the asymptotic transient response for horizontal fiber displacement and fiber temperature. The laser focus spot surface temperature predictions are also found to be in reasonable agreement. However, the vertical fiber displacement tends to be overpredicted by the steady-state solution, sometimes by as much as an order of magnitude. The causes, both physical and computational, of this disagreement are discussed.

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.

Comparison of Steady State and Asymptotic Transient Thermal-Mechanical Simulations of Optoelectronic Laser Processing

2003 ◽  
Author(s):  
Ben Ting ◽  
Vincent P. Manno
Keyword(s):  
Steady State ◽  
Transient Response ◽  
Laser Energy ◽  
Steady State Solution ◽  
Heat Fluxes ◽  
State Solution ◽  
Final State ◽  
Element Analysis ◽  
Order Of Magnitude ◽  
Focus Spot

Traditional optoelectronic manufacturing of butterfly packages involves laser welding of a fiber mount followed by a tedious realignment procedure to reverse thermally induced distortions commonly referred to as Post Weld Shift (PWS). An alternative PWS compensation technique, Laser Hammering, entails manipulation of the fiber to light alignment through deformation of the fiber housing with high precision laser beams. The goal of this study is to predict and understand fiber displacements for butterfly packages subjected to the laser hammering process using finite element analysis. A standardized, two-dimensional fiber mounting/ferrule geometry is employed in a simulation case study. Various laser waveforms are applied to focus spot diameters of 50 and 200 μm over a range of applied heat fluxes (10 to 1000 W/mm2). The primary investigation focused on the degree to which a steady state (SS) model can predict the final state of a transient response (asymptotic steady state) subjected to a periodic laser excitation. Effects of laser energy deposition location and resolution, as well as the use of multiple lasers were also studied. The results obtained to date show that the steady state solution is in good agreement with the asymptotic transient response (ATR) for the center horizontal fiber displacement and the center fiber temperature. The focus spot region surface temperature predictions of steady state and asymptotic transient simulations were also found to be in reasonable agreement. However, the vertical fiber displacement tends to be over predicted by the steady state solution, sometimes by as much as an order of magnitude. The causes, both physical and computational, of this disagreement are discussed in the paper.

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.

Response of a Submerged Cylindrical Shell to an Axially Propagating Step Wave

1965 ◽  
Vol 32 (4) ◽  
pp. 788-792 ◽  
Author(s):  
M. J. Forrestal ◽  
G. Herrmann
Keyword(s):  
Cylindrical Shell ◽  
Steady State ◽  
Wave Front ◽  
Transverse Shear ◽  
Shell Theory ◽  
Steady State Solution ◽  
State Solution ◽  
Rotatory Inertia ◽  
Shear Deformations ◽  
Axial Coordinate

An infinitely long, circular, cylindrical shell is submerged in an acoustic medium and subjected to a plane, axially propagating step wave. The fluid-shell interaction is approximated by neglecting fluid motions in the axial direction, thereby assuming that cylindrical waves radiate away from the shell independently of the axial coordinate. Rotatory inertia and transverse shear deformations are included in the shell equations of motion, and a steady-state solution is obtained by combining the independent variables, time and the axial coordinate, through a transformation that measures the shell response from the advancing wave front. Results from the steady-state solution for the case of steel shells submerged in water are presented using both the Timoshenko-type shell theory and the bending shell theory. It is shown that previous solutions, which assumed plane waves radiated away from the vibrating shell, overestimated the dumping effect of the fluid, and that the inclusion of transverse shear deformations and rotatory inertia have an effect on the response ahead of the wave front.

scholarly journals Comparison of Solvers Performance for Load Flow Analysis

2019 ◽  
Vol 3 (1) ◽  
pp. 26 ◽  
Author(s):  
Vishnu Sidaarth Suresh
Keyword(s):  
Steady State ◽  
Power System ◽  
Reactive Power ◽  
Flow Analysis ◽  
Steady State Solution ◽  
Load Flow ◽  
State Solution ◽  
Computational Time ◽  
Load Flow Analysis ◽  
Active And Reactive Power

Load flow studies are carried out in order to find a steady state solution of a power system network. It is done to continuously monitor the system and decide upon future expansion of the system. The parameters of the system monitored are voltage magnitude, voltage angle, active and reactive power. This paper presents techniques used in order to obtain such parameters for a standard IEEE – 30 bus and IEEE-57 bus network and makes a comparison into the differences with regard to computational time and effectiveness of each solver

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