Dynamic Analysis of Mechanical Systems With Elastic Telescopic Members

1994 ◽  
Author(s):  
B. O. Al-Bedoor ◽  
Y. A. Khulief
Keyword(s):  
Dynamic Model ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Elastic Beam ◽  
Vibrational Motion ◽  
Dynamic Coupling ◽  
Gravitational Effects ◽  
Coupling Terms ◽  
Tip Mass ◽  
Stiffening Effect

Abstract A dynamic model for the vibrational motion of an elastic beam-like telescopic member is presented. In addition to translation, the elastic member is allowed to execute large reference rotation. The Lagrangian approach in conjunction with the assumed modes technique are employed in deriving the equations of motion. The developed model accounts for all the dynamic coupling terms, as well as the stiffening effect due to the beam reference rotation. The tip mass dynamics is included together with the associated dynamic coupling between the modal degrees of freedom. In addition, the devised dynamic model takes into account the gravitational effects, thus permitting motions in either vertical or horizontal planes. Numerical simulation of a mechanical system with an elastic telescopic member is presented.

Toward a Minimal Dynamic Model of a 6-DOF Parallel Robot

1997 ◽  
Author(s):  
L. Beji ◽  
M. Pascal ◽  
P. Joli
Keyword(s):  
Dynamic Model ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Computational Cost ◽  
Closed Form Solution ◽  
Parallel Robot ◽  
Form Solution ◽  
Lagrangian Formalism ◽  
Inverse Kinematic Problem ◽  
Geometrical Properties

Abstract In this paper, an architecture of a six degrees of freedom (dof) parallel robot and three limbs is described. The robot is called Space Manipulator (SM). In a first step, the inverse kinematic problem for the robot is solved in closed form solution. Further, we need to inverse only a 3 × 3 passive jacobian matrix to solve the direct kinematic problem. In a second step, the dynamic equations are derived by using the Lagrangian formalism where the coordinates are the passive and active joint coordinates. Based on geometrical properties of the robot, the equations of motion are derived in terms of only 9 coordinates related by 3 kinematic constraints. The computational cost of the obtained dynamic model is reduced by using a minimum set of base inertial parameters.

Nonlinear Dynamic Modeling of Elastic Beam Fixed on a Moving Cart and Carrying Lumped Tip Mass Subjected to External Periodic Force

2009 ◽  
Author(s):  
Fadi A. Ghaith
Keyword(s):  
Linear Model ◽  
Equations Of Motion ◽  
Elastic Beam ◽  
Open Loop ◽  
Beam Deflection ◽  
Mode Shapes ◽  
Euler Beam ◽  
Periodic Force ◽  
Assumed Mode Method ◽  
Tip Mass

In the present work, a Bernoulli – Euler beam fixed on a moving cart and carrying lumped tip mass subjected to external periodic force is considered. Such a model could describe the motion of structures like forklift vehicles or ladder cars that carry heavy loads and military airplane wings with storage loads on their span. The nonlinear equations of motion which describe the global motion as well as the vibration motion were derived using Lagrangian approach under the inextensibility condition. In order to investigate the influence of the axial movement of the cart on the response of the system, unconstrained modal analysis has been carried out, and accurate mode shapes of the beam deflection were obtained. The assumed mode method was utilized for approximating the beam elastic deformation based on the single unconstrained mode shapes. Numerical simulation has been carried out to estimate the open-loop response of the nonlinear beam-mass-cart model as well as for the simplified linear model under the influence of the periodic excitation force. Also a comparison study between the responses of the linear and nonlinear models was established. It was shown that the maximum values of the beam tip deflection estimated from the nonlinear model are lower than the corresponding values obtained via the linear model, which reveals the importance of considering nonlinear hardening term in formulating the equations of motion for such system in order to come with more accurate and reliable model.

A Three-Dimensional Dynamic Model for Determining the Equilibrium Configurations of Buoyantly Unstable Icebergs

1990 ◽  
Vol 112 (3) ◽  
pp. 253-262
Author(s):  
R. G. Jessup ◽  
S. Venkatesh
Keyword(s):  
Dynamic Model ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Static Potential ◽  
Initial State ◽  
Model Simulations ◽  
Six Degrees Of Freedom ◽  
Equilibrium Configurations ◽  
Variation Range

This paper describes a dynamic model developed for the purpose of determining the final equilibrium configurations of buoyantly unstable icebergs. The model places no restrictions on the size, shape, or dimensionality of the iceberg, or on the variation range of the configuration coordinates. Furthermore, it includes all six degrees of freedom and is based on a Lagrangian formulation of the dynamic equations of motion. It can be used to advantage in those situations in which the iceberg has a complicated potential function and can acquire enough momentum and kinetic energy in the initial phase of its motion to make its final configuration uncertain on the basis of a static potential analysis. The behavior of the model is examined through several model simulations. The sensitivity of the final equilibrium position to the initial orientation and shape of the iceberg is clearly evident in the model simulations. Model simulations also show that when an iceberg is released from a nonequilibrium initial state, the time taken for it to settle down varies from about 40 s for a growler to nearly 400 s for a large iceberg. While these absolute times may change with better parameterization of the forces, the relative variations with iceberg size are likely to be preserved.

Effect of an Original Gear Mesh Modeling on the Gearbox Dynamic Behavior

2007 ◽  
Author(s):  
Emmanuel Rigaud ◽  
Joe¨l Perret-Liaudet ◽  
Mohamed-Salah Mecibah
Keyword(s):  
Load Distribution ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Elastic Coupling ◽  
Gear Mesh ◽  
Meshing Stiffness ◽  
Face Width ◽  
Building Process ◽  
Coupling Terms

Prediction of the vibratory and acoustical behavior of gearboxes is generally based on characterization of the excitation sources, overall modeling of the gearbox, modal analysis and solving of the parametric equations of motion generated by these models. On the building process of such large degrees of freedom models, the elastic coupling induced by the gear mesh is generally described by the parametric meshing stiffness k(t) along the line of action. This kind of model is not able to take into account the load distribution along the tooth face width, even though the resulting moment can constrain rotational angles associated to wheels tilting and flexural deformation of shafts. The scope of this study is to introduce the coupling terms between wheels associated to these phenomena. Some examples show how they can influence the gear modal characteristics and dynamic response and, consequently, the vibratory and acoustical response of the gearbox.

Finite Element Dynamic Modeling of Elastic Beam With Prismatic Joint

1995 ◽  
Author(s):  
B. O. Al-Bedoor ◽  
Y. A. Khulief
Keyword(s):  
Finite Element ◽  
Dynamic Model ◽  
Transition Element ◽  
Flexible Multibody Dynamics ◽  
Elastic Beam ◽  
Fixed Number ◽  
Body Motion ◽  
Dynamic Coupling ◽  
Elastic Deformations ◽  
Prismatic Joint

Abstract A finite element dynamic model of a translating beam through a prismatic joint is presented. The method adopts a fixed number of elements. The element length, on the contrary to the previously reported models, is constant. The time dependent nature of the boundary conditions is utilized to impose the prismatic joint constraints by varying the stiffness of the transition element. This method preserves all the dynamic coupling terms between the axial rigid body motion and the elastic deformations. The end mass dynamics is conveniently considered in the formulation. Furthermore, the introduced dynamic model offers a convenient formulation that can be incorporated into a general flexible multibody dynamics code and lends itself for control applications. The developed model is evaluated through comparisons with previously reported results of other models.

On the Dynamics and Multiple Equilibria of an Inverted Flexible Pendulum With Tip Mass on a Cart

2014 ◽  
Vol 136 (4) ◽  
Author(s):  
Ojas Patil ◽  
Prasanna Gandhi
Keyword(s):  
Dynamic Model ◽  
Multiple Equilibria ◽  
Chaotic Behavior ◽  
Elastic Beam ◽  
Superior Performance ◽  
Flexible Link ◽  
Equilibrium Solutions ◽  
Lagrange Formulation ◽  
Compact Design ◽  
Tip Mass

Flexible link systems are increasingly becoming popular for advantages like superior performance in micro/nanopositioning, less weight, compact design, lower power requirements, and so on. The dynamics of distributed and lumped parameter flexible link systems, especially those in vertical planes are difficult to capture with ordinary differential equations (ODEs) and pose a challenge to control. A representative case, an inverted flexible pendulum with tip mass on a cart system, is considered in this paper. A dynamic model for this system from a control perspective is developed using an Euler Lagrange formulation. The major difference between the proposed method and several previous attempts is the use of length constraint, large deformations, and tip mass considered together. The proposed dynamic equations are demonstrated to display an odd number of multiple equilibria based on nondimensional quantity dependent on tip mass. Furthermore, the equilibrium solutions thus obtained are shown to compare fairly with static solutions obtained using elastica theory. The system is demonstrated to exhibit chaotic behavior similar to that previously observed for vibrating elastic beam without tip mass. Finally, the dynamic model is validated with experimental data for a couple of cases of beam excitation.

A New Approach to Modeling Surface Defects in Bearing Dynamics Simulations

2008 ◽  
Vol 130 (4) ◽  
Author(s):  
Ankur Ashtekar ◽  
Farshid Sadeghi ◽  
Lars-Erik Stacke
Keyword(s):  
Dynamic Model ◽  
High Speed ◽  
Degrees Of Freedom ◽  
Surface Defects ◽  
Equations Of Motion ◽  
Infinite Domain ◽  
Step Size ◽  
Deep Groove ◽  
Dynamics Simulations ◽  
Bearing Dynamics

A dynamic model for deep groove and angular contact ball bearings was developed to investigate the influence of race defects on the motions of bearing components (i.e., inner and outer races, cage, and balls). In order to determine the effects of dents on the bearing dynamics, a model was developed to determine the force-deflection relationship between an ellipsoid and a dented semi-infinite domain. The force-deflection relationship for dented surfaces was then incorporated in the bearing dynamic model by replacing the well-known Hertzian force-deflection relationship whenever a ball/dent interaction occurs. In this investigation, all bearing components have six degrees-of-freedom. Newton’s laws are used to determine the motions of all bearing elements, and an explicit fourth-order Runge–Kutta algorithm with a variable or constant step size was used to integrate the equations of motion. A model was used to study the effect of dent size, dent location, and inner race speed on bearing components. The results indicate that surface defects and irregularities like dent have a severe effect on bearing motion and forces. Furthermore, these effects are even more severe for high-speed applications. The results also demonstrate that a single dent can affect the forces and motion throughout the entire bearing and on all bearing components. However, the location of the dent dictates the magnitude of its influence on each bearing component.

Planar Motion of a Flexible Beam With a Tip Mass Driven by Two Kinematic Rotational Degrees of Freedom

1998 ◽  
Vol 120 (1) ◽  
pp. 206-213
Author(s):  
D. C. Winfield ◽  
B. C. Soriano
Keyword(s):  
Finite Element Method ◽  
Degrees Of Freedom ◽  
Natural Frequencies ◽  
Equations Of Motion ◽  
Ritz Method ◽  
Planar Motion ◽  
Flexible Beam ◽  
The Finite Element Method ◽  
Rotational Degrees Of Freedom ◽  
Tip Mass

The objective was to model planar motion of a flexible beam with a tip mass that is driven by two kinematic rotational degrees of freedom which are (1) at the center of the hub and (2) at the point the beam is attached to the hub. The equations of motion were derived using Lagrange’s equations and were solved using the finite element method. The results for the natural frequencies of the beam especially at high tip masses and high rotational velocities of the hub were calculated and compared to results obtained using the Raleigh-Ritz method. The dynamic response of the beam due to a specified hub rotation was calculated for two cases.

Linear Dynamic Coupling in Geared Rotor Systems

1986 ◽  
Vol 108 (2) ◽  
pp. 171-176 ◽  
Author(s):  
J. W. David ◽  
L. D. Mitchell
Keyword(s):  
Gear Tooth ◽  
Equations Of Motion ◽  
Dynamic Coupling ◽  
Rotating Shaft ◽  
Rotor Systems ◽  
Linear Dynamic ◽  
Heavy Lift ◽  
Reaction Forces ◽  
Coupling Terms ◽  
Nonlinear Equations Of Motion

The ability to analyze accurately the torsional-axial-lateral coupled response of geared systems is the key to the prediction of dynamic gear forces, shaft moments and torques, dynamic reaction forces, and moments at all bearing points. These predictions can, in turn, be used to estimate gear-tooth lives, shaft lives, housing vibrational response, and noise generation. Typical applications would be the design and analysis of gear drives in heavy-lift helicopters, industrial speed reducers, Naval propulsion systems, and heavy, off-road equipment. In this paper, the importance of certain linear dynamic coupling terms on the predicted response of geared rotor systems is addressed. The coupling terms investigated are associated with those components of a geared system that can be modeled as rigid disks. First, the coupled, nonlinear equations of motion for a disk attached to a rotating shaft are presented. The conventional argument for ignoring these dynamic coupling terms is presented and the error in this argument is revealed. It is shown that in a geared system containing gears with more than about 50 teeth, the magnitude of some of the dynamic-coupling terms is potentially as large as the magnitude of the linear terms that are included in most rotor analyses. In addition, it is shown that the dynamic coupling terms produce the multi-frequency responses seen in geared systems. To quantitatively determine the effects of the linear dynamic-coupling terms on the predicted response of geared rotor systems, a trial problem is formulated in which these effects are included. The results of this trial problem shows that the inclusion of the linear dynamic-coupling terms changed the predicted response up to eight orders of magnitude, depending on the response frequency. In addition, these terms are shown to produce sideband responses greater than the unbalanced response of the system.

Coupling of In-Plane Flexural, Tangential, and Shear Wave Modes of a Curved Beam

2011 ◽  
Vol 134 (1) ◽  
Author(s):  
B. Kang ◽  
C. H. Riedel
Keyword(s):  
Shear Wave ◽  
Equations Of Motion ◽  
High Order ◽  
Curved Beam ◽  
Beam Model ◽  
Elastic Coupling ◽  
Dynamic Coupling ◽  
Frequency Spectra ◽  
Coupling Terms ◽  
Wave Modes

In this paper, the coupling effects among three elastic wave modes, flexural, tangential, and radial shear, on the dynamics of a planar curved beam are assessed. Two sets of equations of motion governing the in-plane motion of a curved beam are derived, in a consistent manner, based on the theory of elasticity and Hamilton’s principle. The first set of equations retains all resulting linear coupling terms that includes both static and dynamic coupling among the three wave modes. In the derivation of the second set of equations, the effects of Coriolis acceleration and high-order elastic coupling terms are neglected, which leads to a set of equations without dynamic coupling terms between the tangential and shear wave modes. This second set of equations of motion is the one most commonly used in studies on thick curved beams that include the effects of centerline extensibility, rotary inertia, and shear deformation. The assessment is carried out by comparing the dynamic behavior predicted by each curved beam model in terms of the dispersion relations, frequency spectra, cutoff frequencies, natural frequencies and mode shapes, and frequency responses. In order to ensure the comparison is based on accurate results, the wave propagation technique is applied to obtain exact wave solutions. The results suggest that the contributions of the dynamic and high-order elastic coupling terms to the response of a thick curved beam are quite significant and that these coupling terms should not be neglected for an accurate analysis of a thick curved beam with a large curvature parameter.

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