EQUATIONS OF MOTION: LAGRANGE’S FORMULATION
213
Second, it is not difficult to show that the Lagrange équations (8.44) can
be équivalent to the matrix form of (8.1), provided that the dynamic System is
simple, that the motion is small, and that K and V are expressed in quadratic
form. To show this équivalence, let K. V, and 6W hâve the forms of équations (8.45)-(8.47), except let Vg - 0. When the terms of équations (8.44) are
evaluated using équations (8.37)-(8.39), then the resuit is
M£+K£=g
(8.54)
Since the nonconservative force vector is g = p - qD. in which qD = C£
from équation (8.22), then the last équation becomes identical to the linear
form, équation (8.1), or
M£+C£+K£ = p
(8.55)
The last comment is that, once the équations of motions are formulated,
with ail time-varying environmental loads and the constant coefficients identified, then those équations can be solved directly to détermine the time-varying
responses = £,(t), i = 1, 2,... , N. To do this, the analyst has a wide choice of
a computer software packages, including PSI-Plot (1999) and Mathematica®
(1999). For such computations, 2N initial conditions must be specified, or
C(0) = [€r(0),€2(0).........^-(0)]T
(8.56a)
€(0) =[êi(0),é2(0),... ,G,(0)]T
(8.56b)
In general, steady State solutions £f(t) with light damping are sought, and
those responses occur in the numerical solutions if the run time t is sufficiently
long. In such solutions, the damping eventually éliminâtes the initial transient
responses so that the choice of initial conditions expressed by équations (8.56)
is of no conséquence. Recall that for a single degree of freedom System, light
structural damping was based on the parameter £ = Ci/2\/k\m in which C was
in the measured range of 0.05 to 0.1. See équations (5.58) and (5.64). Using
this latter information, rough estimâtes of <., = Cji can be made to obtain the
steady State numerical solutions. A more exact way to relate to c,j is discussed
later in this chapter.
Although numerical solutions can generally be computed by what is sometimes called the brute force method, it is quite appropriate and often more
physically meaningful to obtain closed form solutions to the équations of motion by the classical normal mode method. The remainder of this chapter is
devoted to this latter method: the computation of the System s characteristic
frequencies in free vibration u>n and their corresponding modal vectors xn, and
the superposition of these modal vectors to obtain the structure s shape in terms
of the steady State response vectors = Ç,(t)-
213
Second, it is not difficult to show that the Lagrange équations (8.44) can
be équivalent to the matrix form of (8.1), provided that the dynamic System is
simple, that the motion is small, and that K and V are expressed in quadratic
form. To show this équivalence, let K. V, and 6W hâve the forms of équations (8.45)-(8.47), except let Vg - 0. When the terms of équations (8.44) are
evaluated using équations (8.37)-(8.39), then the resuit is
M£+K£=g
(8.54)
Since the nonconservative force vector is g = p - qD. in which qD = C£
from équation (8.22), then the last équation becomes identical to the linear
form, équation (8.1), or
M£+C£+K£ = p
(8.55)
The last comment is that, once the équations of motions are formulated,
with ail time-varying environmental loads and the constant coefficients identified, then those équations can be solved directly to détermine the time-varying
responses = £,(t), i = 1, 2,... , N. To do this, the analyst has a wide choice of
a computer software packages, including PSI-Plot (1999) and Mathematica®
(1999). For such computations, 2N initial conditions must be specified, or
C(0) = [€r(0),€2(0).........^-(0)]T
(8.56a)
€(0) =[êi(0),é2(0),... ,G,(0)]T
(8.56b)
In general, steady State solutions £f(t) with light damping are sought, and
those responses occur in the numerical solutions if the run time t is sufficiently
long. In such solutions, the damping eventually éliminâtes the initial transient
responses so that the choice of initial conditions expressed by équations (8.56)
is of no conséquence. Recall that for a single degree of freedom System, light
structural damping was based on the parameter £ = Ci/2\/k\m in which C was
in the measured range of 0.05 to 0.1. See équations (5.58) and (5.64). Using
this latter information, rough estimâtes of <., = Cji can be made to obtain the
steady State numerical solutions. A more exact way to relate to c,j is discussed
later in this chapter.
Although numerical solutions can generally be computed by what is sometimes called the brute force method, it is quite appropriate and often more
physically meaningful to obtain closed form solutions to the équations of motion by the classical normal mode method. The remainder of this chapter is
devoted to this latter method: the computation of the System s characteristic
frequencies in free vibration u>n and their corresponding modal vectors xn, and
the superposition of these modal vectors to obtain the structure s shape in terms
of the steady State response vectors = Ç,(t)-