Stability of Motion of Elastic Planar Linkages With Application to Slider Crank Mechanism

1982 ◽  
Vol 104 (4) ◽  
pp. 698-703 ◽  
Author(s):  
I. G. Tadjbakhsh
Keyword(s):  
Steady State ◽  
Equations Of Motion ◽  
Steady State Solution ◽  
Critical Values ◽  
Length Ratio ◽  
Viscous Damping ◽  
Stability Of Motion ◽  
Start Up ◽  
Planar Linkages ◽  
Crank Mechanism

The problem of stability of motion of elastic planar linkages is considered in the context of the classical Euler-Bernoulli equations of motion. The case of slider-crank mechanism is considered in detail and the critical values of the dimensionless parameters measuring slenderness, speed, and length ratio which may cause instability are determined. The start-up and the steady-state solution of the mechanism without viscous damping and the effects of flexibility on piston force and efficiency is evaluated.

Member Initial Curvature Effects on the Elastic Slider-Crank Mechanism Response

1982 ◽  
Vol 104 (1) ◽  
pp. 159-167 ◽  
Author(s):  
M. Badlani ◽  
A. Midha
Keyword(s):  
Steady State ◽  
Natural Frequency ◽  
Equations Of Motion ◽  
Excitation Frequency ◽  
Beam Theory ◽  
Connecting Rod ◽  
Initial Curvature ◽  
Curvature Effects ◽  
Transient Solutions ◽  
Crank Mechanism

Parametric vibration of initially curved columns loaded by axial-periodic loads has received considerable attention, concluding that regions of instability exist and that excitation frequencies less than the natural frequency of the principal resonance may occur. Recent publications have cautioned against the use of curved members in machines designed for precise operation, suggesting a detrimental coupling of the longitudinal and transverse deformations. In this work, the dynamic behavior of a slider-crank mechanism with an initially curved connecting rod is investigated. Governing equations of motion are developed using the Euler-Bernoulli beam theory. Both steady-state and transient solutions are determined, and compared with those obtained for the mechanism possessing a geometrically perfect (straight) connecting rod. A very small initial curvature is shown to cause a significantly greater steady-state response. The magnification in its transient response is shown to be even greater than that due to a straight connecting rod. Additionally, an excitation frequency less than the natural frequency is also shown to occur.

A Variational Principle for the Elastodynamic Motion of Planar Linkages

1976 ◽  
Vol 98 (4) ◽  
pp. 1306-1312 ◽  
Author(s):  
B. S. Thompson ◽  
A. D. S. Barr
Keyword(s):  
Variational Principle ◽  
Equations Of Motion ◽  
Problem Definition ◽  
Compatibility Conditions ◽  
Elastic Deformations ◽  
Rigid Body Motions ◽  
Planar Linkages ◽  
Crank Mechanism ◽  
Approximate Equations ◽  
Motion Boundary

A variational principle is presented that may be used for setting up the equations describing the elastodynamic motion of planar linkages in which all the members are considered to be flexible. These systems are modeled as a set of continua in which elastic deformations are superimposed on gross rigid-body motions. Displacement continuity at pin joints, or any other special constraints that are peculiar to the linkage being analyzed, are incorporated by the use of Lagrange multipliers. By permitting independent variations of the stress, strain, displacement, and velocity parameters for each link approximate equations of motion, boundary and compatibility conditions for the complete mechanism may be systematically constructed. As an illustrative example, the derivation of the problem definition for a flexible slider-crank mechanism is given.

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.

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.

A Computational Investigation of the Steady State Vibrations of Unbalanced Flexibly Supported Rigid Rotors Damped by Short Magnetorheological Squeeze Film Dampers

2013 ◽  
Vol 135 (6) ◽  
Author(s):  
Jaroslav Zapoměl ◽  
Petr Ferfecki ◽  
Paola Forte
Keyword(s):  
Steady State ◽  
Equations Of Motion ◽  
Dynamical Behavior ◽  
Steady State Solution ◽  
Squeeze Film ◽  
Damping Force ◽  
Lateral Vibrations ◽  
Lubricating Layer ◽  
Squeeze Film Dampers ◽  
Rigid Rotors

Unbalance is the principal cause of excitation of lateral vibrations of rotors and generation of the forces transmitted through the rotor supports to the foundations. These effects can be significantly reduced if damping devices are added to the constraint elements. To achieve their optimum performance, their damping effect must be controllable. The possibility of controlling the damping force is offered by magnetorheological squeeze film dampers. This article presents an original investigation of the dynamical behavior of a rigid flexibly supported rotor loaded by its unbalance and equipped with two short magnetorheological squeeze film dampers. In the computational model, the rotor is considered as absolutely rigid and the dampers are represented by force couplings. The pressure distribution in the lubricating layer is governed by a modified Reynolds equation adapted for Bingham material, which is used to model the magnetorheological fluid. To obtain the steady state solution of the equations of motion, a collocation method is employed. Stability of the periodic vibrations is evaluated by means of the Floquet theory. The proposed approach to study the behavior of rigid rotors damped by semi-active squeeze film magnetorheological dampers and the developed efficient computational methods to calculate the system steady state response and to evaluate its stability represent new contributions of this article.

Model and Method for a Time-Efficient Analysis of Lateral Drillstring Dynamics

2015 ◽  
Author(s):  
Ilja Gorelik ◽  
Marcus Neubauer ◽  
Jörg Wallaschek ◽  
Oliver Höhn
Keyword(s):  
Steady State ◽  
Equations Of Motion ◽  
Element Model ◽  
Steady State Solution ◽  
Contact Forces ◽  
Lumped Parameter Model ◽  
Computational Time ◽  
Operational Parameters ◽  
Real Time Analysis ◽  
Inclined Boreholes

The lumped parameter model and numerical method proposed in this paper aim at gaining a better understanding of the mechanisms leading to harmful lateral drillstring vibrations in inclined boreholes. The shooting method is applied to the equations of motion in order to skip transients and to arrive quickly at a steady state solution. In combination with a sequential continuation technique, parameter maps are generated that show regions where harmful vibrations can be avoided. Comparisons to a finite element model show that the steady state is predicted accurately, while consuming only a fraction of the computational time. The proposed model is experimentally validated on a test rig. Special emphasis is put upon the evaluation of contact forces and the frequency content of the signals. The presented investigations create the basis for real-time analysis of drillstring dynamics and can be used to give recommendations to adjust operational parameters.

Effect of Internal Material Damping on the Dynamics of a Slider-Crank Mechanism

1983 ◽  
Vol 105 (3) ◽  
pp. 452-459 ◽  
Author(s):  
M. Badlani ◽  
A. Midha
Keyword(s):  
Steady State ◽  
Transverse Vibration ◽  
Equations Of Motion ◽  
Viscoelastic Model ◽  
Mathieu Equation ◽  
Perturbation Approach ◽  
Material Damping ◽  
Connecting Rod ◽  
The Stability ◽  
Crank Mechanism

A study of the effect of internal material damping on the dynamic response behavior of a slider-crank mechanism is presented in this paper. In developing the governing equations of motion, an assumption of a linear viscoelastic model for the connecting rod is made. A perturbation approach is utilized for reducing these coupled axial and transverse nonlinear equations to a nonhomogeneous damped Mathieu equation, describing the transverse vibration of the connecting rod. Both steady-state and transient solutions are determined and compared to those obtained from the use of an undamped connecting rod. It is demonstrated that the viscoelastic material damping can have significant influence, both favorable and adverse, in attempting to attenuate the steady-state and transient response of the connecting rod. The response is computed for several combinations of the excitation parameter and the frequency ratio. The stability of the transverse vibration of the connecting rod is also investigated in this paper.

Suspended Decoupler Design of Hydraulic Engine Mount

2011 ◽  
Author(s):  
Reza N. Jazar ◽  
M. Mahinfalah ◽  
J. Christopherson
Keyword(s):  
Steady State ◽  
Transient Response ◽  
Equations Of Motion ◽  
State Behavior ◽  
State Response ◽  
Nonlinear Finite Elements ◽  
Lumped Parameter ◽  
Start Up ◽  
Engine Mount ◽  
Static Conditions

A known problem in classical hydraulic engine mount is that because of the density mismatch between the decoupler and surrounding fluid, the decoupler might float, or stick to the cage bounds, assuming static conditions. The problem appears in the transient response of a bottomed up floating decoupler hydraulic engine mount. To overcome the bottomed up problem, a suspended decoupler design for improved decoupler control is introduced. The new design does not noticeably effect the mechanisms steady state behavior, but improves start up and transient response. Additionally, the decoupler mechanism is incorporated into a smaller, lighter, yet more tunable and hence more effective hydraulic mount design. Ususally the elastomechanical components in a hydraulic engine mount are assumed lumped and linear. To have a more realistic modeling, utilizing nonlinear finite elements in conjunction with a lumped parameter modeling approach, we evaluate the resorting characteristics of the components and implement them in the equations of motion. The steady state response of a dimensionless model of the mount is examined utilizing the averaging perturbation method applied to a set of second order nonlinear ordinary differential equations. It is shown that the frequency responses of the floating and suspended decoupled designs are similar and functional.

scholarly journals Suspended Decoupler: A New Design of Hydraulic Engine Mount

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
J. Christopherson ◽  
M. Mahinfalah ◽  
Reza N. Jazar
Keyword(s):  
Steady State ◽  
Transient Response ◽  
Equations Of Motion ◽  
State Behavior ◽  
Engine Mounts ◽  
Nonlinear Finite Elements ◽  
Lumped Parameter ◽  
Start Up ◽  
Engine Mount ◽  
Static Conditions

Because of the density mismatch between the decoupler and surrounding fluid, the decoupler of all hydraulic engine mounts (HEM) might float, sink, or stick to the cage bounds, assuming static conditions. The problem appears in the transient response of a bottomed-up floating decoupler hydraulic engine mount. To overcome the bottomed-up problem, a suspended decoupler design for improved decoupler control is introduced. The new design does not noticeably affect the mechanism's steady-state behavior, but improves start-up and transient response. Additionally, the decoupler mechanism is incorporated into a smaller, lighter, yet more tunable and hence more effective hydraulic mount design. The steady-state response of a dimensionless model of the mount is examined utilizing the averaging perturbation method applied to a set of second-order nonlinear ordinary differential equations. It is shown that the frequency responses of the floating and suspended decoupled designs are similar and functional. To have a more realistic modeling, utilizing nonlinear finite elements in conjunction with a lumped parameter modeling approach, we evaluate the nonlinear resorting characteristics of the components and implement them in the equations of motion.

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.

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