# A recursive algorithm for generating the equations of motion of spatial mechanical systems with application to the five-point suspension

## Abstract

In this paper, a recursive formulation for generating the equations of motion of spatial mechanical systems is presented. The rigid bodies are replaced by a dynamically equivalent constrained system of particles which avoids introducing any rotational coordinates. For the open-chain system, the equations of motion are generated recursively along the serial chains using the concepts of linear and angular momenta. Closed-chain systems are transformed to open-chain systems by cutting suitable kinematic joints and introducing cut-joint constraints. The formulation is used to carry out the dynamic analysis of multi-link five-point suspension. The results of the simulation demonstrate the generality and simplicity of the proposed dynamic formulation.

## Key Words

Multibody System Dynamics Equations of Motion System of Rigid Bodies Mechanisms Machine Theory## Nomenclature

*d*_{i,j}Distance between points

*i*and*j**G*_{1}Vector sum of the moments of the external forces and force couples acting on the body with respect to particle 1

*I*_{ξξ},*I*_{ηη},*I*_{ζζ}Moments of inertia of the body with respect to the body attached coordinate frame

*I*_{ξη},*I*_{ξζ},*I*_{ζη}Products of inertia of the body with respect to the body attached coordinate frame

*m*Mass of the body

*m*_{i}Mass of particle

*i**m*_{i,j}Mass of the secondary particle that is located between the primary particles

*i*and*j*- \(\bar r_G \)
The position vector of the centre of mass of the body with respect to the body attached coordinate frame

- \(\bar r_i \)
Position vector of particle i with respect to the body attached coordinate frame

- \(r_i ,\dot r,\ddot r_i \)
Position, velocity, and acceleration vectors of particle i with respect to an inertial reference frame

- \(r_{i,j} ,\dot r_{i,j} ,\ddot r_{i,j} \)
Relative position, velocity, and acceleration vectors between particles

*i*and*j**r*_{i}^{T}r_{j}Algebraic notation denotes the dot product operation (r

_{i},_{r})- \(\tilde r_i r_j \)
Algebraic notation denotes the cross product operation (r

_{i}xr_{j})- R
Vector sum of the external forces acting on the rigid body

*ξ*_{i}, η_{i}, ζ_{i}Coordinates of particle i with respect to the body attached coordinate frame

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