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 Poisson-Based Formulation for Frictional Impact Analysis of Open and Closed-Loop Multibody Mechanical Systems

1999 ◽  
Author(s):  
Hamid M. Lankarani ◽  
Murthy Ayyagari
Keyword(s):  
Impact Analysis ◽  
Closed Loop ◽  
Mechanical Systems ◽  
Friction Model ◽  
Momentum Balance ◽  
Contact Period ◽  
Double Pendulum ◽  
Canonical Momentum ◽  
Impact Problems ◽  
The Impact

Abstract Frictional impact analysis requires a friction model capable of correct detection of all possible impact modes such as sliding, sticking, and reverse sliding. Conventional methods for frictional impact analysis have either shown energy gain or not developed for jointed mechanical system, and especially not for closed-chain multibody systems. This paper presents a general formulation for the analysis of impact problems with friction in both open- and closed-loop multibody mechanical systems. The Poisson’s hypothesis is used for the definition of the coefficient of restitution, and thus the energy gains inherent with the use of the Newton’s hypothesis are avoided. A canonical form of the system equations of motion using Cartesian coordinates and Cartesian momenta is utilized. The canonical momentum-balance equations are formulated and solved for the change in the system Cartesian momenta using an extension of Routh’s graphical method for the normal and tangential impulses. The velocity jumps are calculated by balancing the accumulated system momenta during the contact period. The formulation is shown to recognize all modes of impact; i.e., sliding, sticking, and reverse sliding. The impact problems are classified into seven cases, and based on the pre-impact system configuration and velocities, expressions for the normal and tangential impulses are derived for each impact case. Examples including the impact of a falling rod on the ground, the tip of a double pendulum impacting the ground, and the impact of the rear wheel and suspension system of an automobile executing a very stiff bump are analyzed with the developed formulation.

Frictional Impact Analysis in Open-Loop Multibody Mechanical Systems

1999 ◽  
Vol 121 (1) ◽  
pp. 119-127 ◽  
Author(s):  
S. Ahmed ◽  
H. M. Lankarani ◽  
M. F. O. S. Pereira
Keyword(s):  
Impact Analysis ◽  
Mechanical Systems ◽  
Classical Problem ◽  
Open Loop ◽  
Tangential Component ◽  
Momentum Balance ◽  
Double Pendulum ◽  
Canonical Momentum ◽  
Impact Problems ◽  
The Impact

Analysis of impact problems in the presence of any tangential component of impact velocity requires a friction model capable of correct detection of the impact modes. This paper presents a formulation for the analysis of impact problems with friction in open-loop multibody mechanical systems. The formulation recognizes the correct mode of impact; i.e., sliding, sticking, and reverse sliding. Poisson’s hypothesis is used for the definition of the coefficient of restitution, and thus the energy gains inherent with the use of the Newton’s hypothesis are avoided. The formulation is developed by using a canonical form of the system equations of motion using joint coordinates and joint momenta. The canonical momentum-balance equations are solved for the change in joint momenta using Routh’s graphical method. The velocity jumps are calculated balancing the accumulated momenta of the system during the impact process. The impact cases are classified based on the pre-impact positions and velocities, and inertia properties of the impacting systems, and expressions for the normal and tangential impulse are derived for each impact case. The classical problem of impact of a falling rod with the ground (a single object impact) is solved with the developed formulation and verified. Another classical problem of a double pendulum striking the ground (a multibody system impact) is also presented. The results obtained for the double pendulum problem confirms that the energy gain in impact analysis can be avoided by considering the correct mode of impact and using the Poisson’s instead of the Newton’s hypothesis.

A Poisson-Based Formulation for Frictional Impact Analysis of Multibody Mechanical Systems With Open or Closed Kinematic Chains

1999 ◽  
Vol 122 (4) ◽  
pp. 489-497 ◽  
Author(s):  
Hamid M. Lankarani
Keyword(s):  
Mechanical System ◽  
Impact Analysis ◽  
Mechanical Systems ◽  
Friction Model ◽  
Momentum Balance ◽  
Contact Period ◽  
Double Pendulum ◽  
Canonical Momentum ◽  
Impact Problems ◽  
The Impact

Analysis of frictional impact in a multibody mechanical system requires a friction model capable of correct detection of all possible impact modes such as sliding, sticking, and reverse sliding. Conventional methods for frictional impact analysis have either shown energy gain or not developed for jointed mechanical system, and especially not for closed-chain multibody systems. This paper presents a general formulation for the analysis of impact problems with friction in both open- and closed-loop multibody mechanical systems. Poisson’s hypothesis is used for the definition of the coefficient of restitution, and thus the energy gains inherent with the use of Newton’s hypothesis are avoided. A canonical form of the system equations of motion using Cartesian coordinates and Cartesian momenta is utilized. The canonical momentum-balance equations are formulated and solved for the change in the system Cartesian momenta using an extension of Routh’s graphical method for the normal and tangential impulses. The velocity jumps are calculated by balancing the accumulated system momenta during the contact period. The formulation is shown to recognize all modes of impact; i.e., sliding, sticking, and reverse sliding. The impact problems are classified into seven types, and based on the pre-impact system configuration and velocities, expressions for the normal and tangential impulses are derived for each impact type. Examples including the tip of a double pendulum impacting the ground with some experimental verification, and the impact of the rear wheel and suspension system of an automobile executing a very stiff bump are analyzed with the developed formulation. [S1050-0472(00)02304-7]

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.

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.

A New Contact Force Model for Low Coefficient of Restitution Impact

2012 ◽  
Vol 79 (6) ◽  
Author(s):  
Mohamed Gharib ◽  
Yildirim Hurmuzlu
Keyword(s):  
Contact Force ◽  
Contact Stiffness ◽  
Coefficient Of Restitution ◽  
Force Model ◽  
Reference Case ◽  
Practical Applications ◽  
Contact Force Model ◽  
Post Impact ◽  
Impact Problems ◽  
The Impact

Impact problems arise in many practical applications. The need for obtaining an accurate model for the inelastic impact is a challenging problem. In general, two approaches are common in solving the impact problems: the impulse-momentum and the compliance based methods. The former approach included the coefficient of restitution which provides a mechanism to solve the problem explicitly. While the compliance methods are generally tailored to solve elastic problems, researchers in the field have proposed several mechanisms to include inelastic losses. In this paper, we present correlations between the coefficient of restitution in the impulse-momentum based method and the contact stiffness in the compliance methods. We conducted numerical analysis to show that the resulting solutions are indeed identical for a specific range of impact conditions. The impulse-momentum based model is considered as a reference case to compare the post impact velocities. The numerical results showed that, the impulse-momentum and the compliance based methods can produce similar outcomes for specific range of coefficient of restitution if they satisfied a set of end conditions. The correlations lead to introduce a new contact force model with hysteresis damping for low coefficient of restitution impact.

Interference Impact Analysis of Multibody Systems

1999 ◽  
Vol 121 (1) ◽  
pp. 128-135 ◽  
Author(s):  
D. Wang ◽  
C. Conti ◽  
D. Beale
Keyword(s):  
Impact Analysis ◽  
Initial Conditions ◽  
The Body ◽  
Contact Joint ◽  
Joint Coordinates ◽  
Computer Aided Analysis ◽  
Constrained Multibody System ◽  
Impact Problems ◽  
The Impact ◽  
Two Bodies

A new computer aided analysis method for frictionless impact problems due to interference between two bodies in a constrained multibody system is presented in this paper. A virtual contact joint concept is used to detect interference between two bodies and calculate the jump in the body momenta, velocity discontinuities and rebounds. The interference surfaces can be described by the joint coordinates of the virtual contact joint, which are very useful for determining the impact time, the types and positions of two impact surfaces and impact initial conditions when an interference happens between two bodies.

Defining the Coefficient of Restitution for the Impact Between Free and Constrained Bodies

1999 ◽  
Author(s):  
J. L. Escalona ◽  
J. M. Mayo ◽  
J. Domínguez
Keyword(s):  
Mechanical Energy ◽  
Coefficient Of Restitution ◽  
Rigid Bodies ◽  
Momentum Balance ◽  
Balance Equations ◽  
Flexible Bodies ◽  
Local Loss ◽  
Contact Points ◽  
Before And After ◽  
The Impact

Abstract This paper revisits the coefficient of restitution involved in the impulse-momentum balance equations for colliding rigid bodies and examines its extension to impacts between flexible bodies. The analytical solution to axial impact on a flexible rod is used to demonstrate that the coefficient of restitution is not inherent in the underlying physical process. In fact, the type of coefficient to be used in each case depends on the particular model employed by the analyst to describe flexibility in the bodies concerned. It is demonstrated that the coefficient of restitution used in the generalized impulse-momentum balance for flexible bodies does not represent a physical magnitude. In any case, as shown in this paper, the ratio between the relative velocities at the contact points or surfaces of the flexible bodies before and after impact is no measure of the local loss of mechanical energy during the process.

Impact Analysis of Plates Using Quasi-Static Approach

1994 ◽  
Author(s):  
Shashishekar Shivaswamy ◽  
Jianmin Li ◽  
Hamid M. Lankarani
Keyword(s):  
Impact Analysis ◽  
Steel Plates ◽  
Stress Wave Propagation ◽  
Deformation Model ◽  
Static Force ◽  
Non Linear ◽  
Static Approach ◽  
Linear Force ◽  
Impact Problems ◽  
The Impact

Abstract Impact calculations suffer from several practical limitations which limit their application to establishing the approximate magnitude of the various phenomena involved. The transient force deformation response of a body subjected to impact can be explained accurately using stress wave propagation theory. As this approach is very complicated, a simpler quasi-static approach with non-linear force deformation Hertz relations can be employed for impact analysis. However, these relations can not explain the energy absorption and permanent deformations encountered during the impact. This necessitates independent non-linear force-deformation relations for compression and restitution phases of impact. In the present paper, impact tests conducted on Aluminum and Steel plates have been reported. The impact response of the system was compared with the various theoretical quasi-static force models. Considering the assumptions made in the quasi-static force models, the experimental results matched very well with the theoretical results. Non-linear force-deformation model with independent relations for compression and restitution phases was found to be the best approach to analyze impact problems. The value of the index in the non-linear force-deformation relations was found to be approximately 1.71 and 1.78 for Aluminum and Steel respectively. The values of impact parameters for a given material were found to depend on impact velocity.

Frictional Impact Analysis in Constrained Multibody Systems

1996 ◽  
Author(s):  
Shakil Ahmed ◽  
Hamid M. Lankarani ◽  
Manual F. O. S. Pereira
Keyword(s):  
Impact Velocity ◽  
Canonical Form ◽  
Multibody Systems ◽  
Impact Analysis ◽  
Classical Problem ◽  
Double Pendulum ◽  
Joint Coordinates ◽  
Impact Problems ◽  
Energy Gains ◽  
The Impact

Abstract Analysis of impact problem in the presence of any tangential component of impact velocity requires a friction model capable of correct detection of the impact modes such as sliding, sticking, and reverse sliding. A survery of literature has shown that studies on the impact analysis of multibody systems have either been limited to the direct impact type with only a normal component of impact velocity (no frictional effect) or the ones that include friction have shown energy gains in the results due to the inherent problem in the use of Newton’s hypothesis. This paper presents a formulation for the analysis of impact problems with friction in constrained multibody mechanical systems. The formulation recognizes the correct mode of impact, i.e., sliding, sticking, and reverse sliding. The Poisson’s hypothesis is used for the definition of the coefficient of restitution, and thus the energy gains inherent with the use of Newton’s hypothesis are avoided. The formulation is developed by using a canonical form of the system equation of motion using joint coordinates and joint momenta. The use of canonical formulation is a natural way of balancing the momenta for impact problems. The joint coordinates reduces the equations of motion to a minimal set, and eliminate the complications arised from the kinematic constraint equations. The canonical form of equations are solved for the change in joint momenta using Routh’s graphical method. The velocity jumps are then calculated balancing the accumulated momenta of the system during the impact process. The impact cases are classified based on the pre-impact positions and velocities, and mass properties of the impacting systems. Analytical expressions for normal and tangential impulse are derived for each impact case. The classical problem of impact of a falling rod with the ground (a single object impact) is solved with the developed formulation, and the results are compared and verified by the solution from other studies. Another classical problem of a double pendulum striking the ground (a multibody impact) is also solved. The results obtained for the double pendulum problem confirms that the energy gain in impact analysis can be avoided by considering the correct mode of impact and using Poisson’s instead of Newton’s hypothesis.

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