tailieunhanh - Equations of motion in the state and confiruration spaces

Consider a system with a single degree of freedom and assume that the equation expressing its dynamic equilibrium is a second order ordinary differential equation (ODE) in the generalized coordinate x. Assume as well that the forces entering the dynamic equilibrium equation are • a force depending on acceleration (inertial force), • a force depending on velocity (damping force), • a force depending on displacement (restoring force), • a force, usually applied from outside the system, that depends neither on coordinate x nor on its derivatives, but is a generic function of time (external forcing function). If the dependence of. | Appendix A EQUATIONS OF MOTION IN THE STATE AND CONFIGURATION SPACES EQUATIONS OF MOTION OF DISCRETE LINEAR SYSTEMS Configuration space Consider a system with a single degree of freedom and assume that the equation expressing its dynamic equilibrium is a second order ordinary differential equation ODE in the generalized coordinate x. Assume as well that the forces entering the dynamic equilibrium equation are a force depending on acceleration inertial force a force depending on velocity damping force a force depending on displacement restoring force a force usually applied from outside the system that depends neither on coordinate x nor on its derivatives but is a generic function of time external forcing function . If the dependence of the first three forces on acceleration velocity and displacement respectively is linear the system is linear. Moreover if the constants of such a linear combination usually referred to as mass m damping coefficient c and stuffiness k do not depend on time the system is time-invariant. The dynamic equilibrium equation is then mX cx kx f t . 666 Appendix A. EQUATIONS OF MOTION If the system has a number n of degrees of freedom the most general form for a linear time invariant set of second order ordinary differential equations is A1x A2X A3x f t where x is a vector of order n n is the number of degrees of freedom of the system where the generalized coordinates are listed Ai A2 and A3 are matrices whose order is n x n they contain the characteristics independent of time of the system f is a vector function of time containing the forcing functions acting on the system. Matrix Ai is usually symmetrical. The other two matrices in general are not. They can be written as the sum of a symmetrical and a skew-symmetrical matrices MX C G X K H x f t where M the mass matrix of the system is a symmetrical matrix of order n x n coincides with A1 . Usually it is not singular. C is the real symmetric viscous damping matrix the