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MULTI-DEGREE OF FREEDOM LINEAR STRUCTURES
8.4 FREE, UNDAMPED MOTION
The values of a.’n and xn are computed from the condition of free, undamped
structural motion. That is, C = p = 0, for which the governing équation (8.1)
becomes
M £ +K £ = 0
(8.57)
Frequencies
Assume a harmonie solution to équation (8.57) in the form
« = !•**
(8.58)
in which £ is the time-independent amplitude vector and lü is the frequency
parameter. When £ and £ from the last équation are substituted into équation
(8.57), the resuit is
(—w2M£+K£p
=0
(8.59)
Since
is arbitrary, then the bracket term in the last équation is zéro, which
can be expressed as
(K - cu2M)£ = 0
(8.60)
This last resuit represents a set of N linear, simultaneous, algebraic, homogeneous équations in the vector components £„, n = 1, 2,... , N. By Cramer’s
rule, nontrivial solutions for £ exist only if the déterminant of the bracket term
vanishes, or
det(K - u2M) = 0
(8.61)
W hen this déterminant is expanded, the resuit is an AT h order polynominal
in lu2, for which the N consecutive roots or eigenvalues are positive numbers,
designated as eu2, eu2, • • • , eu2,... ,eu^. The numbers obtained from the positive
square root of each eigenvalue are the free vibration frequencies of the structural
System. For convenience, these frequencies are arranged from the smallest to
the largest in order of the ascending subscripts: iui,lu2,... , eun,... ,Wjv- F°r
A
3, the only practical way to détermine these frequencies is to employ a
computer package, Mathematica® (1999), for instance.
Modal Vectors and Normalization
For the nth frequency cun there is a corresponding modal vector £n which
can be computed from équation (8.60), rewritten as
(K-o£M)ên = 0
(8-62)
The modal vector in component form is
t. = [£in, ê2n, ... ,£Nn]T = [i, £2„,...
(8-63)
MULTI-DEGREE OF FREEDOM LINEAR STRUCTURES
8.4 FREE, UNDAMPED MOTION
The values of a.’n and xn are computed from the condition of free, undamped
structural motion. That is, C = p = 0, for which the governing équation (8.1)
becomes
M £ +K £ = 0
(8.57)
Frequencies
Assume a harmonie solution to équation (8.57) in the form
« = !•**
(8.58)
in which £ is the time-independent amplitude vector and lü is the frequency
parameter. When £ and £ from the last équation are substituted into équation
(8.57), the resuit is
(—w2M£+K£p
=0
(8.59)
Since
is arbitrary, then the bracket term in the last équation is zéro, which
can be expressed as
(K - cu2M)£ = 0
(8.60)
This last resuit represents a set of N linear, simultaneous, algebraic, homogeneous équations in the vector components £„, n = 1, 2,... , N. By Cramer’s
rule, nontrivial solutions for £ exist only if the déterminant of the bracket term
vanishes, or
det(K - u2M) = 0
(8.61)
W hen this déterminant is expanded, the resuit is an AT h order polynominal
in lu2, for which the N consecutive roots or eigenvalues are positive numbers,
designated as eu2, eu2, • • • , eu2,... ,eu^. The numbers obtained from the positive
square root of each eigenvalue are the free vibration frequencies of the structural
System. For convenience, these frequencies are arranged from the smallest to
the largest in order of the ascending subscripts: iui,lu2,... , eun,... ,Wjv- F°r
A
3, the only practical way to détermine these frequencies is to employ a
computer package, Mathematica® (1999), for instance.
Modal Vectors and Normalization
For the nth frequency cun there is a corresponding modal vector £n which
can be computed from équation (8.60), rewritten as
(K-o£M)ên = 0
(8-62)
The modal vector in component form is
t. = [£in, ê2n, ... ,£Nn]T = [i, £2„,...
(8-63)