14.6 Derivatives of Transient Response 749
Table 14.3 Derivatives of Eigenvalues [14.4]
i Eigenvalue,λi 10 −^9
∂λi
∂x 1
10 −^9
∂λi
∂x 3
10 −^2
∂Y 5 i
∂x 1
10 −^2
∂Y 6 i
∂x 1
1 24.66 0.3209 –0.1582 1.478 –2.298
2 974.7 3.86 –0.4144 0.057 –3.046
3 7782.0 23.5 21.67 0.335 –5.307
For illustration, a cylindrical cantilever beam is considered [14.4]. The beam is
modeled with three finite elements with six degrees of freedom as indicated in Fig. 14.4.
The diameters of the beam are considered as the design variables,xi, i = 1 , 2, 3. The
first three eigenvalues and their derivatives are shown in Table 14.3 [14.4].
14.6 Derivatives of Transient Response
The equations of motion of ann-degree-of-freedom mechanical/structural system with
viscous damping can be expressed as [14.10]
[M]Y ̈+[C]Y ̇+[K]Y=F(t) (14.52)
where [M],[C], and [K] are then×nmass, damping, and stiffness matrices, respec-
tively,F(t)is then-component force vector,Yis then-component displacement vector,
and a dot over a symbol indicates differentiation with respect to time. Equations
(14.52) denote a set ofncoupled second-order differential equations. In most practical
cases,nwill be very large and Eqs. (14.52) are stiff; hence the numerical solution of
Eqs. (14.52) will be tedious and produces an accurate solution only for low-frequency
components. To reduce the size of the problem, the displacement solution, Y, is
expressed in terms ofrbasis functions 1 , 2 ,... ,andr(withr≪n)as
Y=[]q or yj=
∑r
k= 1
j kqk(t), j= 1 , 2 ,... , n (14.53)
where
[]=[ 1 2 · · ·r]
is the matrix of basis functions,j kthe element in rowjand columnkof the matrix
[],qanr-component vector of reduced coordinates, andqk(t) het kth component
of the vectorq. By substituting Eq. (14.53) into Eq. (14.52) and premultiplying the
resulting equation by []T, we obtain a system ofrdifferential equations:
[M]q ̈+[C]q ̇+[K]q=F(t) (14.54)
where
[M]=[]T[ [M]] (14.55)
[C]=[]T[ [C]] (14.56)
[K]=[]T[ [K]] (14.57)
F(t)=[]TF t)( (14.58)
