On the Use of Momentum Balance and the Assumed Modes Method in Transverse Impact Problems

1992 ◽  
Vol 114 (3) ◽  
pp. 364-373 ◽  
Author(s):  
H. Palas ◽  
W. C. Hsu ◽  
A. A. Shabana
Keyword(s):  
Degrees Of Freedom ◽  
Mass Matrix ◽  
Coefficient Of Restitution ◽  
Body Motion ◽  
Jump Discontinuity ◽  
Momentum Balance ◽  
Transverse Impact ◽  
Orthogonal Functions ◽  
Reaction Forces ◽  
Impact Problems

The objective of this investigation is to examine the validity of applying the assumed modes method and the generalized impulse momentum approach that involves the coefficient of restitution in the analysis of transverse impact in constrained elastic systems. A simple impact model that consists of a rotating beam which is subjected to a transverse impact by a mass (impact hammer) moving with a constant velocity is used. For the purpose of comparison and in order to check the validity of using the proposed technique, the transverse deformation of the beam with respect to the beam coordinate system is described using three different assumed sets of orthogonal functions. The different sets of modes are the clamped-free modes, pin-free modes, and a set of assumed harmonic functions. The system mass matrix that accounts for the coupling between the rigid body motion and the elastic deformation is identified and used with the Jacobian matrix of the kinematic constraints and the coefficient of restitution to define the algebraic generalized impulse momentum equations that describe the transverse impact. The series solution obtained using the generalized impulse momentum equations is used to define the generalized impulse, the jump discontinuity in the system reference and modal velocities, and the jump discontinuities in the generalized joint reaction forces. It is shown in this investigation, that by increasing the number of elastic degrees of freedom, the jump discontinuity in the angular velocity of the rod as well as the generalized impulse converge to zero regardless of the assumed complete set of modes used. The effect of the coefficient of restitution and the mass ratio on the jump in the system velocities and the generalized reaction forces is also examined.

On the Use of the Momentum Balance in the Impact Analysis of Constrained Elastic Systems

1990 ◽  
Vol 112 (1) ◽  
pp. 119-126 ◽  
Author(s):  
J. Rismantab-Sany ◽  
A. A. Shabana
Keyword(s):  
Impact Analysis ◽  
Deformable Body ◽  
Coefficient Of Restitution ◽  
Jump Discontinuity ◽  
Momentum Balance ◽  
Constrained Motion ◽  
Elastic Systems ◽  
Deformable Bodies ◽  
Impact Problems ◽  
The Impact

In elastic systems, impulsive forces that act at a point on a deformable body produce stress waves that travel with finite speeds. This paper examines, both theoretically and numerically, the validity of using the generalized impulse momentum approach in modeling impact or collisions in the constrained motion of deformable bodies. The generalized impulse momentum equations that involve the coefficient of restitution and the kinematic constraint Jacobian matrix are used to predict the jump discontinuity in the velocity vector as well as the joint reaction forces. The series solutions obtained by solving these algebraic equations are used to establish a closed form relationship between the jump discontinuity in velocities and joint reactions due to impact and the number of elastic degrees of freedom. It is shown that by increasing the number of elastic coordinates these series converge to their limits. The convergence of these series is used to prove that the generalized impulse momentum equations with the coefficient of restitution can be used with confidence to study impact problems in constrained multibody systems consisting of interconnected rigid and deformable bodies. The results obtained are compared with the classical treatment of the impact problems in the theory of elasticity wherein the case of perfectly plastic impact is assumed.

A Parametric and Experimental Study of a Power Construction Tool

1993 ◽  
Author(s):  
Dragomir C. Marinkovich ◽  
John M. Kremer ◽  
A. A. Shabana
Keyword(s):  
Computer Model ◽  
Combustion Engine ◽  
Coefficient Of Restitution ◽  
Absorption Capacity ◽  
Jump Discontinuity ◽  
Momentum Balance ◽  
Premature Failure ◽  
Power Construction ◽  
Alternative Material ◽  
Circular Saws

Abstract Power tools play a large role in the construction industry. From power nailers to circular saws, these tools provide an enormous and relatively untapped opportunity in the field of dynamic analysis. The intent of this paper is to analyze a unique power construction tool and then develop a computer model to simulate its dynamics. The study model used in this investigation is a nailer which utilizes a linear internal combustion engine to drive nails into wood and is completely cordless and portable. Analytical expressions for the forces acting on the system are developed and the obtained analytical results are verified experimentally. The computer model is then used to parametrically study the performance of this nailer under varying conditions. The parameters varied are the piston mass, by choosing materials with different densities, and the bumper material on which the piston impacts, to vary the coefficient of restitution between the bumper and piston. The jump discontinuity in the system velocities is predicted using a momentum balance and the restitution conditions. The coefficient of restitutions used in this study are determined experimentally using a simple impact test. It is found that there is an ideal piston mass that will provide the highest kinetic energy at the end of the stroke. It is also shown that the lower the coefficient of restitution between the piston and bumper, the lower the piston rebound height and velocity. The material chosen as having the lowest coefficient of restitution was not acceptable as an alternative material due to its low durometer, or hardness. In addition, the high elastic-energy-absorption capacity of this material may also result in increased heating of the bumper resulting in premature failure.

The Efficacy of the Momentum Balance Method in Transverse Impact Problems

1998 ◽  
Vol 120 (1) ◽  
pp. 47-53 ◽  
Author(s):  
A. S. Yigit ◽  
A. P. Christoforou
Keyword(s):  
Permanent Deformation ◽  
Local Deformation ◽  
Balance Method ◽  
Impact Behavior ◽  
Momentum Balance ◽  
Transverse Impact ◽  
Balance Solution ◽  
Post Impact ◽  
The Difference ◽  
Impact Problems

An elastic-plastic contact law that incorporates local permanent deformation in the contact zone has been used to investigate the efficacy of the momentum balance method in transverse impact problems. The momentum balance solution is compared to the numerical solution of the same equations using the contact law. It is shown that the momentum balance method gives very good results for the post impact behavior of both the impacter and the beam. In certain cases, it is also shown to yield satisfactory results for the dynamic behavior during contact. It is demonstrated that the momentum balance method with an appropriate coefficient of restitution obtained from the contact law, is physically the same as using the contact law itself, with the difference that the local deformation is neglected.

Dynamics of a Radially Rotating Beam With Impact, Part 1: Theoretical and Computational Model

1990 ◽  
Vol 112 (1) ◽  
pp. 65-70 ◽  
Author(s):  
A. S. Yigit ◽  
A. G. Ulsoy ◽  
R. A. Scott
Keyword(s):  
Rigid Body ◽  
Computational Model ◽  
Computational Algorithm ◽  
Coefficient Of Restitution ◽  
Experimental Results ◽  
Body Motion ◽  
Momentum Balance ◽  
Rotating Beam ◽  
Before And After ◽  
Simulation Results

A model is presented for the dynamics of a radially rotating beam with impact. The model uses a momentum balance method and a coefficient of restitution, and enables one to predict the rigid body motion as well as the elastic motion before and after impact. A computational algorithm is also developed to implement the model. In Part 2 simulation results will be compared with experimental results to investigate the validity of the model.

The Efficacy of the Momentum Balance Method in Transverse Impact Problems

1995 ◽  
Author(s):  
Ahmet S. Yigit ◽  
Andreas P. Christoforou
Keyword(s):  
Permanent Deformation ◽  
Local Deformation ◽  
Balance Method ◽  
Impact Behavior ◽  
Momentum Balance ◽  
Transverse Impact ◽  
Balance Solution ◽  
Post Impact ◽  
The Difference ◽  
Impact Problems

Abstract An elastic-plastic contact law that incorporates local permanent deformation in the contact zone has been used to investigate the efficacy of the momentum balance method in transverse impact problems. The momentum balance solution is compared to the numerical solution of the same equations using the contact law. It is shown that the momentum balance method gives very good results for the post impact behavior of both the impacter and the beam. In certain cases, it is also shown to yield satisfactory results for the dynamic behavior during contact. It is demonstrated that the momentum balance method with an appropriate coefficient of restitution obtained from the contact law, is physically the same as using the contact law itself, with the difference that the local deformation is neglected.

Numerical Simulation of Multiple Floating Structures With Nonlinear Constraints

2002 ◽  
Vol 124 (2) ◽  
pp. 104-109 ◽  
Author(s):  
Subrata K. Chakrabarti
Keyword(s):  
Degrees Of Freedom ◽  
Order Theory ◽  
Frequency Domain Analysis ◽  
Body Motion ◽  
Single Frequency ◽  
Integration Scheme ◽  
Mooring Lines ◽  
Floating Structures ◽  
Irregular Wave ◽  
Exciting Forces

A versatile and efficient numerical analysis is developed to compute the responses of a moored floating system composed of multiple floating structures. Structures such as tankers, semisubmersibles, FPSOs, SPARs, TLPs, and SPMs connected by mooring lines, connectors or fenders may be analyzed individually or collectively including multiple interaction. The analysis is carried out in the time domain assuming rigid body motion for the structures, and the solution is generated by a forward integration scheme. The analysis includes the nonlinearities in the excitation, damping, and restoring terms encountered in a typical mooring system configuration. It also allows for instabilities in the tower oscillation as well as slack mooring lines. Certain simplifications in the analysis have been made, which are discussed. The exciting forces in the analysis are wind, current, and waves (including a steady and an oscillating drift force), which are not necessarily collinear. The waves can be single frequency or composed of multiple frequency components. For regular waves either linear, stretched linear or fifth order theory may be used. The irregular wave may be included as a given spectral model (e.g., PM or JONSWAP). The vessels are free to respond to the exciting forces in six degrees of freedom—surge, sway, heave, roll, pitch, and yaw. The tower, when present, is free to respond in two degrees of freedom—oscillation and precession. The loads in the mooring lines are determined from prescribed tension-strain tables for the lines. Rigid mooring arms can be analyzed by allowing for compression in the load-strain table. Fenders may be input similarly through load compression tables. In order to establish the stability and accuracy of the solution, comparison of the results with linearized frequency domain analysis was made. The analysis is verified by several different model test results for different structure configurations in regular and random seas. Some of the interesting aspects of nonlinear system are shown with a few examples.

Human Carrying Simulation With Symmetric and Asymmetric Loads

2012 ◽  
Author(s):  
Yujiang Xiang ◽  
Jasbir S. Arora ◽  
Salam Rahmatalla ◽  
Hyun-Joon Chung ◽  
Rajan Bhatt ◽  
...  
Keyword(s):  
Degrees Of Freedom ◽  
Human Motion ◽  
Human Model ◽  
Physical Constraints ◽  
Digital Human Model ◽  
Load Carrying ◽  
Reaction Forces ◽  
Asymmetric Load ◽  
Optimization Schemes ◽  
Digital Human

Human carrying is simulated in this work by using a skeletal digital human model with 55 degrees of freedom (DOFs). Predictive dynamics approach is used to predict the carrying motion with symmetric and asymmetric loads. In this process, the model predicts joints dynamics using optimization schemes and task-based physical constraints. The results indicated that the model can realistically match human motion and ground reaction forces data during symmetric and asymmetric load carrying task. With such prediction capability the model could be used for biomedical and ergonomic studies.

The Natural Frequencies of Free and Constrained Non-Uniform Beams

1960 ◽  
Vol 64 (599) ◽  
pp. 697-699 ◽  
Author(s):  
R. P. N. Jones ◽  
S. Mahalingam
Keyword(s):  
Degrees Of Freedom ◽  
Natural Frequencies ◽  
Ritz Method ◽  
Normal Modes ◽  
Iteration Process ◽  
Conservative System ◽  
Basic System ◽  
Orthogonal Functions ◽  
Added Masses ◽  
The Given

The Rayleigh-Ritz method is well known as an approximate method of determining the natural frequencies of a conservative system, using a constrained deflection form. On the other hand, if a general deflection form (i.e. an unconstrained form) is used, the method provides a theoretically exact solution. An unconstrained form may be obtained by expressing the deflection as an expansion in terms of a suitable set of orthogonal functions, and in selecting such a set, it is convenient to use the known normal modes of a suitably chosen “ basic system.” The given system, whose vibration properties are to be determined, can then be regarded as a “ modified system,” which is derived from the basic system by a variation of mass and elasticity. A similar procedure has been applied to systems with a finite number of degrees of freedom. In the present note the method is applied to simple non-uniform beams, and to beams with added masses and constraints. A concise general solution is obtained, and an iteration process of obtaining a numerical solution is described.

Rolling Condition and Gyroscopic Moments in Curve Negotiations

2012 ◽  
Author(s):  
Ahmed A. Shabana ◽  
Martin B. Hamper ◽  
James J. O’Shea
Keyword(s):  
Coordinate System ◽  
Degrees Of Freedom ◽  
The Body ◽  
Contact Forces ◽  
Body Motion ◽  
Global Coordinate System ◽  
Gyroscopic Moment ◽  
Absolute Angular Velocity ◽  
Pure Rolling ◽  
The Stability

In vehicle system dynamics, the effect of the gyroscopic moments can be significant during curve negotiations. The absolute angular velocity of the body can be expressed as the sum of two vectors; one vector is due to the curvature of the curve, while the second vector is due to the rate of changes of the angles that define the orientation of the body with respect to a coordinate system that follows the body motion. In this paper, the configuration of the body in the global coordinate system is defined using the trajectory coordinates in order to examine the effect of the gyroscopic moments in the case of curve negotiations. These coordinates consist of arc length, two relative translations and three relative angles. The relative translations and relative angles are defined with respect to a trajectory coordinate system that follows the motion of the body on the curve. It is shown that when the yaw and roll angles relative to the trajectory coordinate system are constrained and the motion is predominantly rolling, the effect of the gyroscopic moment on the motion becomes negligible, and in the case of pure rolling and zero yaw and roll angles, the generalized gyroscopic moment associated with the system degrees of freedom becomes identically zero. The analysis presented in this investigation sheds light on the danger of using derailment criteria that are not obtained using laws of motion, and therefore, such criteria should not be used in judging the stability of railroad vehicle systems. Furthermore, The analysis presented in this paper shows that the roll moment which can have a significant effect on the wheel/rail contact forces depends on the forward velocity in the case of curve negotiations. For this reason, roller rigs that do not allow for the wheelset forward velocity cannot capture these moment components, and therefore, cannot be used in the analysis of curve negotiations. A model of a suspended railroad wheelset is used in this investigation to study the gyroscopic effect during curve negotiations.

Modeling of Multibody Flexible Articulated Structures With Mutually Coupled Motions: Part II — Application and Results

1992 ◽  
Author(s):  
Jiechi Xu ◽  
Joseph R. Baumgarten
Keyword(s):  
Experimental Data ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Open Loop ◽  
Body Motion ◽  
Universal Joint ◽  
Open Loop System ◽  
Numerical Examples ◽  
Kinematic Properties ◽  
Good Agreement

Abstract The application of the systematic procedures in the derivation of the equations of motion proposed in Part I of this work is demonstrated and implemented in detail. The equations of motion for each subsystem are derived individually and are assembled under the concept of compatibility between the local kinematic properties of the elastic degrees of freedom of those connected elastic members. The specific structure under consideration is characterized as an open loop system with spherical unconstrained chains being capable of rotating about a Hooke’s or universal joint. The rigid body motion, due to two unknown rotations, and the elastic degrees of freedom are mutually coupled and influence each other. The traditional motion superposition approach is no longer applicable herein. Numerical examples for several cases are presented. These simulations are compared with the experimental data and good agreement is indicated.

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