V

Assume that the test object can be represented by an n-degree-of-freedom lumped-parameter model

(1)

H<M « .7

H

H

(5)

The nxn matrices M. D, and K_ are mass matrix. damplng matrix. and stiffness matrix respectively. The forcing input vector is fjt). and y is the corre-sponding response. Notę that, during testing. only one element of the vector fjt) is nonzero. It is cus-tomary to assume that the energy dissipation can be modeled by viscous proportional damping. so that normal (real) modes exist. This assumption can be relaxed eventually by incorporating complex modes. Proportionally damped systems, in the absence of forcing excitations, can be excited by a suitable initial condition so that all degrees of freedom move in proportion at a fixed frequency. This fixed pro-portion, which can be represented by a vector of arbitrary scalę, describes a modę shape. and the associated frequency is the natural frequency of that modę.

Assume that each degree of freedom has an associated inertia (mass) element. This means that the system does not possess static modes (or residual flexibilities). If static modes are indeed present, they can be accounted for by assigning rełatively large natural frequencies to the corresponding modes in the present formulation. Rigid body modes (or zero-natural-frequency modes) can be directly in-corporated. An n-degree-of-freedom system has n modes. The corresponding modę shape vectors are linearly independent in the present formulation. The matrix of modę shape vectors, or modal matrix, normal ized with respect to the mass matrix is given by

nj

Each diagonal term Hj is a simple-oscillator transfer function given by

w w*

¥ * tJ^2^# • •*

Y^m*Q

V(jw) “ H(io» FJjw)

with j denoting the imaginary unity \/-T and

H/j.)

cj)

(6)

Notę that Wj is the i th undamped natural frequency (in the time domain) and is approximately equal to the frequency at the i th resonance (in the frequency domain) for Iow damping. The modal damping ratio is denoted by f j.

Equation (5) can be written in the scalar form as

H>>

n

I

r«1

r r

(7)

*

s

(2)

The modal matrix is nonsingular in this case. Be-cause the transformation

(3)

diagonalizes equation (1) into a set of simple oscilla-tor equations, it follows that the input-output transfer function matrix JH(jcj) can be expressed as

in which and the are the i th and k th elements of the modę shape vector in the r th modę.

Symmetry and reciprocłty. Notę from equations (5) or (7) that the transfer function matrix issymmetric. This can be interpreted as Maxwell's principle of reciprocity. Specifically, in the frequency domain, the response at the k th degree of freedom when an excitation is applied at the i th degree of freedom is equal to the response at the i th degree of freedom when the same excitation is applied at the k th degree of freedom.

Extractlon of mass, ttiffrmt, and damping matrices. After the modal data have been experimentally deter-mined. the time domain model, given by equation (1), is extracted (identified) by using the following relationships; they result from the transformation shown in equation (3).

V

V

. •

(4)

The mass matrix is given by

M • W V^J)~'

(8)

k\

I

V

&

r

:s

_kv.

AJ