State Space Form of Linear Constrained Systems

1 State Space Form of Linear Constrained Systems 

The transformation to state space form reduces the number of diflFerential equations to its minimum by incorporating the constraint equations. In the resulting equations the generalized constraint force no longer appears and an additional equation must be provided if its value is required. 

In this section the numerical generation of the state space form for linear time invariant systems will be discussed. We saw in Sec. 1.5 how the equations of motion (1.5.2) can be linearized to 

Herein, G is the n x rip constraint matrix, K,D M are the stiffness-, damping-and mass matrices, respectively, and z are time dependent kinematical excitations. To keep formulas short we assume u i) = 0. Furthermore, we want to exclude redundant constraints by assuming rank(G) = n.  

We define an Up x riy matrix V such that its columns span a basis of this space. The general form of all solutions of the homogeneous system is then obtained as 

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