A Substructure Technique for Dynamics of Flexible Mechanical Systems With Contact-Impact

1987 ◽  
Author(s):  
S.-C. Wu ◽  
E. J. Haug
Keyword(s):  
Lagrange Multipliers ◽  
Synthesis Method ◽  
Mechanical Systems ◽  
Normal Modes ◽  
Local Deformation ◽  
Constraint Forces ◽  
Transverse Impact ◽  
Contact Impact ◽  
Fixed Interface ◽  
Deformation Modes

Abstract A substructure synthesis method is proposed to account for contact-impact effects in flexible components of mechanical systems. Components that may come into contact is divided into substructures, on each of which local deformation modes are defined to described deformation fields of components. Constraint modes and fixed interface normal modes are used to account for elastic deformation within each substructure. A constraint addition-deletion technique is used to account for contact between impacting bodies. Lagrange multipliers associated with the constraints, which represent constraint forces, are used to determine separation of contacting nodes. Use of the method is illustrated on problems of longitudinal and transverse impact of bodies.

A Substructure Technique for Dynamics of Flexible Mechanical Systems With Contact-Impact

1990 ◽  
Vol 112 (3) ◽  
pp. 390-398 ◽  
Author(s):  
S.-C. Wu ◽  
E. J. Haug
Keyword(s):  
Lagrange Multipliers ◽  
Synthesis Method ◽  
Mechanical Systems ◽  
Normal Modes ◽  
Local Deformation ◽  
Contact Forces ◽  
Transverse Impact ◽  
Contact Impact ◽  
Fixed Interface ◽  
Deformation Modes

A substructure synthesis method is proposed to account for contact-impact effects in flexible components of mechanical systems. Components that may come into contact are divided into substructures, on each of which local deformation modes are defined to describe deformation fields of components. Constraint modes and fixed interface normal modes are used to account for elastic deformation within each substructure. A constraint addition-deletion technqiue is used to determine when contact occurs, account for the effect of contact constraints during the period of contact, and delete constraints after the completion of the contact event. Lagrange multipliers associated with the contact constraints, which represent contact forces, are used to determine the time of separation of contacting nodes. Use of the method is illustrated for longitudinal and transverse impact of elastic bars.

Modeling of Flexible Bodies for Multibody Dynamic Systems Using Ritz Vectors

1994 ◽  
Vol 116 (2) ◽  
pp. 437-444 ◽  
Author(s):  
H. T. Wu ◽  
N. K. Mani
Keyword(s):  
Dynamic Analysis ◽  
Mechanical Systems ◽  
Normal Modes ◽  
Body Motion ◽  
Constraint Forces ◽  
Flexible Bodies ◽  
Static Correction ◽  
Ritz Vectors ◽  
Regeneration Procedure ◽  
Special Distribution

Vibration normal modes and static correction modes have been previously used to model flexible bodies for dynamic analysis of mechanical systems. The efficiency and accuracy of using these modes to model a system depends on both the flexibility of each body and the applied loads. This paper develops a generalized method for the generation of a set of Ritz vectors to be used in addition to vibration normal modes to form the modal basis to model flexible bodies for dynamic analysis of multibody mechanical systems. The Ritz vectors are generated using special distribution of the D’Alembert force and the kinematic constraint forces due to gross-body motion of a flexible body. Combined with vibration normal modes, they form more efficient vector bases for the modeling of flexible bodies comparing to using vibration normal modes alone or using the combination of static correction modes and vibration normal modes. Ritz vectors can be regenerated when the system undergoes significant changes of its configuration and the regeneration procedure is inexpensive. The effectiveness of using the combination of vibration normal modes and the proposed Ritz vectors is demonstrated using a planar slider-crank mechanism.

scholarly journals On the Interpretation of the Lagrange Multipliers in the Constraint Formulation of Contact Problems; or Why Are Some Multipliers Always Zero?

2014 ◽  
Author(s):  
A. L. Schwab
Keyword(s):  
Mechanical System ◽  
Lagrange Multipliers ◽  
Mechanical Systems ◽  
Contact Problems ◽  
Constraint Forces ◽  
Basic Notion ◽  
Constraint Equations ◽  
Linear Dependency ◽  
Constraint Approach ◽  
Two Bodies

One method for modeling idealized contact between two bodies in mechanical system is based on the constraint approach, where Lagrange multipiers are introduced, which serve as constraint forces. In the usage of this formulation, there exists a linear dependancy between the Lagrange multipliers. Moreover, it has been observed that some Lagrange multipliers are always identical to zero. This sort of contradicts the basic notion that Lagrange multipliers in mechanical systems act as constraint forces which, when constraints are violated, push the system back in the desired configuration. In this paper it will be shown, by theory and example, that the above-mentioned linear dependency of the Lagrange multipliers, together with specific entries in the Jacobian of the constraint equations, results in some Lagrange multipliers being identical to zero.

Component Mode Synthesis Method Using Partial Interface Modes: Application to Tuned and Mistuned Bladed Disk With Local Non-Linearity

2009 ◽  
Author(s):  
Duc-Minh Tran
Keyword(s):  
Synthesis Method ◽  
Normal Modes ◽  
Component Mode Synthesis ◽  
New Method ◽  
Reduced Order Models ◽  
Reduced Order ◽  
Generalized Coordinates ◽  
Bladed Disk ◽  
Static Condensation ◽  
Fixed Interface

A new fixed interface component mode synthesis method using partial interface modes is presented. Partial interface modes are the structure normal modes which result from the static condensation of the structure to the interface between the substructures and which are clamped at a part of this interface. This method is the generalization of the classical component mode synthesis method which keeps all the interface physical displacements in the assembled reduced system and the method using interface modes which eliminates all of them. It allows one to reduce the number of the interface generalized coordinates and at the same time to keep some of the physical displacements at the interface. This latter capability is very useful to build reduced order models in which the presence of physical displacements are essential, for example in order to impose prescribed motions or to take into account local non-linearities. The new method is applied to a bladed disk in both tuned and mistuned cases.

Contact Modeling Technique of a Flexible Piston and Cylinder Using Modal Synthesis Method

2005 ◽  
Author(s):  
W. K. Kim ◽  
S. H. Sohn ◽  
H. J. Cho ◽  
D. S. Bae ◽  
J. H. Choi
Keyword(s):  
Contact Force ◽  
Synthesis Method ◽  
Normal Modes ◽  
Modeling Technique ◽  
Mode Shapes ◽  
Dynamics Analysis ◽  
Contact Modeling ◽  
Static Correction ◽  
Modal Synthesis ◽  
Contact Surfaces

In this paper, contact modeling technique and dynamics analysis of piston and cylinder system are presented by using modal synthesis method. It is very important to select mode shapes representing a global or local behavior of a flexible body due to a specified loading condition. This paper proposes a technique to generate the static correction modes which are nicely representing a motion by a contact force between a piston and cylinder. First normal modes of piston and cylinder under a boundary condition are computed, and then static correction modes due to a contact force applied at contacted nodes are added to the normal modes. Also, this paper proposes an efficient dynamics analysis process while changing the shape of the piston and cylinder. In optimization process or design study, their geometric data can be changed a bit. The slight changes of their contact surfaces make a high variation of the magnitude of a contact force, and it can yield the different dynamic behavior of an engine system. But, since the variations of the normal and correction modes are very small, the re-computation of their normal and correction modes due to the change of contact surfaces can be useless. Until now, whenever their contact surfaces are changed at a design cycle, the modes have been recomputed. Thus, most engineers in industries have been spent many times in very tedious and inefficient design process. In this paper, the normal and correction modes from the basic geometry of the piston and cylinder are computed. If the geometry shape is changed, nodal positions of the original modal model are newly calculated from an interpolation method and changed geometry data. And then the updated nodes are used to compute a precise contact force. The proposed methods illustrated in this investigation have good agreement with results of a nodal synthesis technique and proved that it is very efficient design method.

Impact on Beams

1936 ◽  
Vol 3 (2) ◽  
pp. A55-A61
Author(s):  
H. L. Mason
Keyword(s):  
Historical Development ◽  
Contact Region ◽  
Local Deformation ◽  
Lateral Vibration ◽  
Transverse Impact ◽  
Concentrated Mass ◽  
Impact Theory ◽  
Inelastic Impact ◽  
Maximum Contact

Abstract This paper deals with transverse impact on beams the mass of which is of importance. Experimental results are presented for comparison with theory. Impacts which appear single to the eye are shown to consist in reality of several blows in quick succession. Section 1 of the paper traces the historical development of this subject by discussing the investigations of Young, Hodgkinson, Cox, Saint Venant, and Timoshenko. Section 2 treats a simplified system in which a concentrated mass strikes a smaller concentrated mass having a “soft” spring restraint. For elastic impact, theory predicts for the struck mass a path composed of sinusoidal elements separated by instantaneous blows. For inelastic impact it predicts a joint harmonic motion. Records of the paths of both masses were obtained experimentally. Section 3 of the paper uses Timoshenko’s method of combining local deformation of the contact region with lateral vibration of the beam. An experimental investigation of maximum contact pressure and of blow duration gives what is believed to be the first confirmation of this theory. Section 4 describes an experimental determination of flexural stresses in elastic and inelastic impact on a 3-in. I-beam by the use of a Westinghouse magnetic strain gage. The indication is that stresses may be higher than those calculated by the usual approximations.

Reduction of Hamiltonian Mechanical Systems With Affine Constraints: A Geometric Unification

2016 ◽  
Vol 12 (2) ◽  
Author(s):  
Robin Chhabra ◽  
M. Reza Emami ◽  
Yael Karshon
Keyword(s):  
Symmetry Group ◽  
Mechanical Systems ◽  
Hamiltonian Formalism ◽  
Geometrical Approach ◽  
Constraint Forces ◽  
Dynamical Equations ◽  
Relative Configuration ◽  
D’Alembert Equation ◽  
Dynamical Reduction ◽  
Configuration Manifold

This paper presents a geometrical approach to the dynamical reduction of a class of constrained mechanical systems. The mechanical systems considered are with affine nonholonomic constraints plus a symmetry group. The dynamical equations are formulated in a Hamiltonian formalism using the Hamilton–d'Alembert equation, and constraint forces determine an affine distribution on the configuration manifold. The proposed reduction approach consists of three main steps: (1) restricting to the constrained submanifold of the phase space, (2) quotienting the constrained submanifold, and (3) identifying the quotient manifold with a cotangent bundle. Finally, as a case study, the dynamical reduction of a two-wheeled rover on a rotating disk is detailed. The symmetry group for this example is the relative configuration manifold of the rover with respect to the inertial space. The proposed approach in this paper unifies the existing reduction procedures for symmetric Hamiltonian systems with conserved momentum, and for Chaplygin systems, which are normally treated separately in the literature. Another characteristic of this approach is that although it tracks the structure of the equations in each reduction step, it does not insist on preserving the properties of the system. For example, the resulting dynamical equations may no longer correspond to a Hamiltonian system. As a result, the invariance condition of the Hamiltonian under a group action that lies at the heart of almost every reduction procedure is relaxed.

Modeling of Flexible Bodies for Dynamic Analysis of Multi-Body Dynamic Systems Using Ritz Vectors

1991 ◽  
Author(s):  
Henry T. Wu ◽  
Neel K. Mani
Keyword(s):  
Dynamic Analysis ◽  
Mechanical Systems ◽  
Normal Modes ◽  
Flexible Bodies ◽  
Static Correction ◽  
Ritz Vectors ◽  
Regeneration Procedure ◽  
Efficient Vector ◽  
Multi Body ◽  
Body Dynamic

Abstract Vibration normal modes and static correction modes have been previously used to model flexible bodies for dynamic analysis of mechanical systems. The efficiency and accuracy of using these modes to model a system depends on both the flexibility of each body and the applied loads. This paper develops a generalized method for the generation of a set of Ritz vectors to model flexible bodies for dynamic analysis of multi-body mechanical systems. The Ritz vectors are generated using the distribution of dynamic loading on a flexible body. Therefore they form the most efficient vector basis for the spatial distribution of the loadings. The Ritz vectors can be re-generated when the system undergoes significant changes of its configuration and the regeneration procedure is inexpensive. The combinations of vibration normal modes and the proposed Ritz vectors thus form more efficient and accurate vector bases for the modeling of flexible bodies for dynamic analysis.

Influence of Reference Conditions in the Analysis of the Impact of Flexible Bodies

2001 ◽  
Author(s):  
José L. Escalona ◽  
Juana Mayo ◽  
Jaime Domínguez
Keyword(s):  
Rigid Body ◽  
Reference Conditions ◽  
Normal Modes ◽  
Frame Of Reference ◽  
The Body ◽  
Natural Boundary Condition ◽  
Flexible Bodies ◽  
Free Interface ◽  
Fixed Interface ◽  
The Impact

Abstract In this paper, the floating frame of reference approach is applied to the dynamics of the impact of flexible bodies, while component mode synthesis is used to describe deformation. The influence of the reference conditions, that indicate the type of attachment between the body fixed frame of reference and the flexible bodies, is investigated. Rigid and free attachments allow the use of fixed interface and free interface normal modes, respectively. A finite number of fixed interface modes does not fulfil the natural boundary condition at the attachment point. Free interface normal modes cannot describe the compressive forces at the contact surface. However, it is shown that both set of modes are able to describe the impact-induced elastic waves. In the evaluation of the kinematic coefficient of restitution, these two approaches differ significantly. When free attachment is considered, the derivatives of the reference co-ordinates coincide with the equivalent rigid body velocities of the flexible bodies, remaining constant after the impact. However, if the body frame of reference is rigidly attached, the equivalent rigid body velocities of the flexible body have to be evaluated as a linear combination of the derivative of reference and elastic co-ordinates. The axial impact of a rigid body on a flexible rod and the transverse impact of a flexible pendulum with a fixed stop are simulated to illustrate these facts. Hertzian contact forces are assumed to occur during impact.

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