Issue Section:
Discussion
Keywords:
finite element analysis, modelling, vibration control, damping
Topics:
Condensation, Damping, Finite element analysis, Modeling, Vibration control

The dynamic condensation method (1) was successfully extended by Rao 2 to handle the unsymmetric systems with damping. This method is very interesting and useful in the finite element modeling, vibration control, etc. However, one misunderstanding occurred when this approach was utilized in substructure synthesis.

As stated by the author in Sec. 4, the reduced order matrices [MR] and [KR] of each substructure in Eqs. (16) and (17) have the form
(1)
in which [MmmR],[CmmR], and [KmmR] are the reduced order mass, damping, and stiffness matrices of order m×m.

Actually, if the reduced order matrices [MR] and [KR] are computed from Eqs. (16) and (17) as indicated by the author, these two matrices are generally fully populated and do not have the forms shown in Eq. (1). This will be explained in detail later. Hence one cannot simply convert these two matrices into the displacement space with the explicit forms of the reduced order matrices [MmmR],[CmmR] and [KmmR]. If the matrices on the right-hand sides of Eq. (1) are known and those on the left-hand sides are unknowns, the relations shown in this equation are right. However, the problem is how we get the reduced matrices [MmmR],[CmmR], and [KmmR] before we have [MR] and [KR].

To simplify the discussion, consider a symmetric problem. After the simplification, the full order matrices [M]¯ and [K]¯ in Ref. 2 become
(2)
if the eigenproblem in Sec. 3 rather than the dynamic equations of equilibrium in Sec. 2 is considered. The transformation matrices [R] and [S] are the same and indicated by [R]. The corresponding governing equation for the transformation matrix is given by
(3)
and the initial approximation is
(4)
A very simple numerical example is given to show the form of reduced order matrices [MR] and [KR]. In this example, the mass, damping, and stiffness matrices are
(5)
Two cases that the first and the third degrees of freedom are, respectively, selected as the master degrees of freedom are considered. The resulted reduced order matrices [MR] and [KR] from the initial approximation and the first three iterations are listed in Table 1. The results show that reduced order matrices [MR] and [KR] obtained from the initial approximation, that is, Guyan condensation, have the forms given in Eq. (2). This conclusion can be proven simply. After two iterations, both reduced order matrices are fully populated. The further discussion on the dynamic condensation of viscously damped, symmetric models may be found in Refs. 3,4,5 6.
Table
Reduced order matrices [MR] and [KR] during iteration
Iteration Case 1 Case 2
[MR][KR][MR][KR]
0 0 −1 −1 0 0 −1 −1 0
−1 −1 0 1 −1 −0.1111 0 0.3333
1 0 −3 −3 0 0.2469 −1.4815 −1.4815 0
−3 −1 0 1 −1.4815 −0.1111 0 0.3333
2 −3.3333 −4.6667 −2.4444 0.5556 0.1963 −1.5716 −1.5042 −0.0311
−4.6667 −1 0.5556 1.5556 −1.5716 −0.1962 −0.0311 0.3512
3 −3.0183 −5.1174 −3.0159 0.6003 0.2750 −1.6296 −1.6196 −0.04670
−5.1174 −0.6281 0.6003 1.6537 −1.6296 −0.2148 −0.04670 0.3507
Reduced order matrices [MR] and [KR] during iteration
Iteration Case 1 Case 2
[MR][KR][MR][KR]
0 0 −1 −1 0 0 −1 −1 0
−1 −1 0 1 −1 −0.1111 0 0.3333
1 0 −3 −3 0 0.2469 −1.4815 −1.4815 0
−3 −1 0 1 −1.4815 −0.1111 0 0.3333
2 −3.3333 −4.6667 −2.4444 0.5556 0.1963 −1.5716 −1.5042 −0.0311
−4.6667 −1 0.5556 1.5556 −1.5716 −0.1962 −0.0311 0.3512
3 −3.0183 −5.1174 −3.0159 0.6003 0.2750 −1.6296 −1.6196 −0.04670
−5.1174 −0.6281 0.6003 1.6537 −1.6296 −0.2148 −0.04670 0.3507

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