Mechanical Systems: Equations of Motion and Stability
The Chapter ‘Mechanical Systems: Equations of Motion and Stability’ corresponds to the material presented in five lectures given at the CISM Course no. 418. The first parts deal with the form of the equations of motion of mechanical systems, in particular
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. Spelsberg-Korspeter (Eds.), Active and Passive Vibration Control of Structures, CISM International Centre for Mechanical Sciences DOI 10.1007/ 978-3-7091-1821-4_1 © CISM Udine 2014
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P. Hagedorn
1 Equations of Motion of Discrete Mechanical Systems In this lecture we will shortly recapitulate the form of the equations of motion of discrete mechanical systems (which may of course be an approximation of continuous systems). We will highlight certain aspects which, although elementary, are not always stressed in basic vibration courses. For a holonomic system of n degrees of freedom and generalized coordinates q = (q1 , q2 , · · · , qn )T
(1)
the equations of motion can be obtained from the Lagrange equations of the second type, based on the Lagrangian L = T − U,
(2)
where T is the kinetic energy and U the potential energy function (which we assume may also depend on the time t). Lagrange’s equations then read ∂L ∂ ∂L − = Qs , ∂t q˙s ∂qs
(3)
where the Qs are the generalized forces not represented by the potential U . For a large class of systems these equations can be written as ˙ t). M q¨ + Gq˙ + Kq = f (q, q,
(4)
˙ t) contains for example the damping and other nonconserThe term f (q, q, vative terms, as well as for example control forces. The linearized equations (linearized about an equilibrium of the unforced autonomous system) can then be written as M q¨ + (D + G)q˙ + (K + N )q = f (t).
(5)
This linearized form of the equations of motion is usually employed to develop appropriate active or passive vibration control. The control strategies based on these linear models may then later be tested for the nonlinear model. In many cases, setting up the equations of motion using Lagrange’s equations is not a practical approach and other methods may be more efficient. The form of the equations will however be the same as above. Unless
Mechanical Systems: Equations of Motion and Stability
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stated otherwise, we will assume the following properties for the matrices: Mass matrix:
M = M T,
M > 0 (symmetric, positive definite)
Damping matrix:
D = DT ,
D≥0
(symmetric, positive semidef.)
T
K≥0
(symmetric, positive semidef.)
Stiffness matrix: Gyroscopic matrix:
K=K , T
G = −G , T
Circulatory matrix: N = −N ,
(skew symmetric) (skew symmetric)
In this lecture and in the next one, we will discuss in some more detail the significance of the different matrices for the behavior of the mechanical systems. We will first consider the free vibrations, i.e. the case f (t) = 0: M q¨ + (D + G)q˙ + (K + N )q = 0.
1.1
(6)
The Eigenvalue Problem
Since (5) is a system of ode’s with constant coefficients, the exponential ansatz q(t) = reλt
(7)
is successful, leading to λ2 M + λ(D + G) + K + N reλt = 0.
(8)
In order for (8) to be valid for all times, the condition λ2 M + λ(D + G) + K + N r = 0
(9)
must be fulfilled. Equation (9) is the eigenvalue problem, and the sought values of λ and r are respectively the eigenvalues and the eigenvectors. The eigenvalue problem is a linear homogeneous algebraic system in t
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