Rigid Spring Research Articles

Application of graph theory to the determination of potential energy of systems consisting of rigid bodies and springs

The process of determining the potential energy, as a function of generalized coordinates, of a system consisting of rigid bodies and springs is often extremely laborious. In this paper, a method is presented by means of which all the potential energy terms of a system are calculated in a systematic manner. It is assumed that the system of rigid bodies has a topological tree structure, and that linear springs are situated arbitrarily between the rigid bodies. The method is based on a graph-theoretical approach and is consistent with that of the calculation of kinetic energy presented in ( 2 , J. Franklin Inst., Vol. 324, No. 3, pp. 351–367, 1987). The paper contains the derivation of general formulae for potential energies of gravity and elastic forces. These formulae are implemented for the case of a planar system; an example is also provided.

25—ANALYSES OF SPRING-UNDERMOTION SUB-MECHANISMS

The spring parameters that affect the form of the undermotion-force-lift characteristic are discussed. The method of analysis previously described is then applied to specific undermotion submechanisms. The usual form of spring-side mechanism is analysed in terms of two geometrical parameters, and the ways in which both the theoretical working range and the form of the force-angular-position characteristic depend on these parameters are determined. For springs with initial tension, this requires that the spring index and material and the effective lengths of the rigid spring connectors be assumed. The general effects of changes in this index and connector length are indicated. The contributions to the characteristic of two common types of connexion from heald frame to undermotion are also determined and shown to be small. Finally, the results of a comparison of the proposed method of analysis with a direct analysis of a complete undermotion are given.

Dynamical Analysis of Suspension Systems

Suspension systems with spatial kinematics are modeled by multibody systems consisting of rigid bodies, springs, dashpots and joints. The nonlinear and linearized equations of motion are found by a computerized method in symbolical representation. The dynamical analysis of the vehicle's riding comfort, riding safety and joint forces is presented. An example shows the application of the method.

Three-dimensional mathematical model of the human spine structure

A method is presented for constructing discrete-parameter type three-dimensional mathematical models and governing equations of motion of the spine structure. The anatomic structure is represented by any combination of rigid bodies, springs, and dashpots in space. These are positioned, orientated, and connected in a manner to represent the true mechanical function of the structure. The rigid bodies are of any shape and have 6 degrees-of-freedom, allowing three-dimensional motion. The springs and dashpots may have up to twenty-one stiffness and damping coefficients respectively to precisely represent the three-dimensional coupled behavior. The method is straightforward and simple to apply. The governing equations are in the matrix form and are easily generated and solved by computer techniques.

Simulation of the Dynamics of Machinery

The dynamics of motion of general, two-dimensional machine systems incorporating linkage elements is studied by a procedure which eliminates equation writing. The method is similar to chemical engineering process analysis procedures utilizing an interconnection matrix to describe attachments between components of a system. A digital computer program based on the procedure, termed the Machine Dynamics-Universal System Analyzer (MEDUSA), has been written and tested. This program contains subroutines for simulating the behavior of the following standard components: rigid links, torsional and linear springs, torsional and linear dashpots, force sources, motion generators, and flexible beams. Gears, chain drives, hydraulic transmissions, electric motors, and other components may be added to the simulation without difficulty. To use the program, the engineer follows a standard method to develop a special subroutine calling various component subroutines in proper order. The program computes the accelerations at the centers of mass of the various rigid elements and the interconnection forces simultaneously. Numerical integration alternating with the acceleration computation determines the position and force history of the machine. Experience shows that the engineering effort needed to develop a dynamic simulation of a general machine system using MEDUSA is greatly reduced compared to deriving the equations of motion.

The Riding Qualities of Wheeled Vehicles

A wheeled vehicle traversing an irregular road surface is subjected to forced vibration. This vibration is analysed into heaving, pitching, and rolling modes, and means are described to compute or simulate the response of the vehicle both to continuous excitation at definite frequencies, and to isolated obstacles affecting one wheel at a time. It is shown that a simple model, comprising only rigid masses, linear springs, and velocity damping, suffices to represent the riding quality of the vehicle, and a procedure by which the model may be matched to the vehicle as closely as possible is described. The theory is illustrated by computation of the impact loads imposed on the wheels of a six-wheeled lorry when surmounting artificial obstacles, allowance being made for the effects both of the wheels leaving the road, and of interaction between the four wheels in the rear truck. Good agreement with experimental records of wheel load is shown, and it is suggested that similar computation or simulation applied to passenger-carrying vehicles would prove of value in the improvement of riding quality.