Chapter 5 Robust Performance Tailoring with Tuning - SSL - MIT

percent change in the performance due to a percent change in the parameter:
¯
dσz
dp
= x0
σz0
∂σz
∂p
where σz0 is the RMS performance of the nominal design and ∂σz
∂x
(2.27)
is the derivative of
the RMS performance with respect to the parameter, x as defined in Equation 2.17.
In the case of the design masses, the nominal values are zero, so the average of the
structural mass in the consistent mass matrix in the x and y degrees of freedom
at the appropriate node is used in the sensitivity calculation. Note from the table
that increasing the cross-sectional area of the truss segments in the nominal design
decreases the variance of the OPD thereby improving the performance of the system.
2.3.4 Finite Element Gradients
In order to compute the performance gradients it is necessary to obtain the gradients
of the stiffness and mass matrices with respect to the design variables. The stiff-
ness and mass matrices depend on the cross-sectional diameter of the truss segments
through the area and inertia properties of the beam elements. Therefore, the chain
rule is employed to find the derivatives of the global stiffness and mass matrices with
respect to di:
∂K
∂di
∂M
∂di
= ∂K
∂Ii
∂Ii
∂di
= ∂M ∂Ai
∂Ai ∂di
+ ∂K ∂Ai
∂Ai ∂di
(2.28)
(2.29)
It is assumed that the bar elements have a circular cross-section, so that the inertia
and area of the elements in the i th truss segment are defined as:
Ii = π
64 d4 i
Ai = π
4 d2 i
52
(2.30)
(2.31)