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Equations of motion in the state and confiruration spaces

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Equations of motion in the state and
confiruration spaces

Appendix A
EQUATIONS OF MOTION
IN THE STATE
AND CONFIGURATION SPACES

A.1 EQUATIONS OF MOTION OF DISCRETE
LINEAR SYSTEMS
A.1.1 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
m¨
x + cx˙ + kx = f (t) .

(A.1)

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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
¨ + A2 x˙ + A3 x = f (t) ,
A1 x

(A.2)

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;
• A1 , A2 and A3 are matrices, whose order is n × 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 A1 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
M¨
x + (C + G) x˙ + (K + H) x = f (t) ,
(A.3)
where:
• M, the mass matrix of the system, is a symmetrical matrix of order n × n
(coincides with A1 ). Usually it is not singular.
• C is the real symmetric viscous damping matrix (the symmetric part of
A2 ).
• K is the real symmetric stiffness matrix (the symmetric part of A3 ).
• G is the real skew-symmetric gyroscopic matrix (the skew-symmetric part
of A2 ).
• H is the real skew-symmetric circulatory matrix (the skew-symmetric part
of A3 ).
Remark A.1 Actually it is possible to write the set of li...
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Equations of motion in the state and
confiruration spaces
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