## Structures with Rigid Body Modes

Structures with rigid-body modes have poles at zero, therefore they behave like systems with integrators. The corresponding H2, Hm, and Hankel norms for systems with poles at zero do not exist as their values tend to infinity. However, the infinite values of the norms of some modes should not be an obstacle in the reduction process. These values indicate that the corresponding states should be retained in the reduced model, regardless of the norms of other modes. The reduction problem can be solved by determining the inverses of grammians, as in [58]. Here we use two simple approaches for the reduction of systems with integrators.

In the first approach, the system is represented in modal coordinates by the following system triple (see Section 4.3):

A =

"0m 0 "

, B =

' Br '

_ 0 A0 _

where 0m is an m x m zero matrix. The triple (A0, B0, C0) has no poles at zero. It is itself in modal coordinates. The vector of the corresponding modal Hm norms is denoted h0 . This vector is arranged in descending order, and the remaining infinite norms are added h = {inf, h0}

to obtain the vector of Hm norms of the (A, B, C) representation, where inf = {<», <», ..., <»} contains m values at infinity. The system is reduced by truncation, as described at the beginning of this chapter.

The second approach is based on the approximate evaluation of the Hm norms. From (5.22) we find

and coi = 0 for the poles at zero; thus, ||G:i|^ <». For nonzero poles we determine the finite norms from the above equation, and order in a descending order. The corresponding state-space representation is reduced by truncation.

Example 6.8. Consider a simple system from Fig. 1.1 with the following parameters: m1 = m2 = 1, m3 = 2, k1 = k4 = 0, k2 = 0.3, and k3 = 1. Damping is proportional to the stiffnesses, dt = 0.03ki, i = 1, 2, 3, the input force is applied at mass m2 , and the output rate is measured at the same location. This system has two poles at zero. Find its Hm norms and reduce the system.

The modal representation of the system without rigid-body modes is as follows; see (6.13):

-0.0264

1.3260

0

0

-1.3260

-0.0264

0

0

0

0

-0.0051

0.5840

0

0

-0.5840

6.0883 x10' 35.8240

with the following H norms ¡0^ = 13.9202 and ||02||m = 1.3247. Therefore, the vector of the norms of the modes of the (ao, bo, co) representation is ho = {13.9202, 13.9202, 1.3247, 1.3247}, and the vector of the Hm norms of the full system (A, B, C) is h = {<», 13.9202,13.9202,1.3247,1.3247}.

By deleting the last two states in the state-space representation, related to the smallest norms, one obtains the reduced-order model as follows:

 0 0 0 0 " 0 0 0 0 0 0 -0.0264 -1.3260 0 0 1.326 6.0883 x10' 35.8240 -1.1608 1062x!0~ The plots of the impulse response and the magnitude of the transfer function of the full and reduced models are shown in Fig. 6.8(a),(b). The plots show that the error of the reduction is small. In fact, for the impulse response, the error was less than 1%, namely, In the above, y denotes the impulse response of the full model and yr denotes the impulse response of the reduced model. Example 6.9. Consider the Deep Space Network antenna azimuth model that has a pole at zero. The identified state-space representation of the open-loop model has n = 36 states, including states with a pole at zero. Reduce this model in modal coordinates, by determining the Hankel norms (or Hankel singular values) for states with nonzero poles. The plot of the Hankel singular values is shown in Fig. 6.9. By deleting the states with Hankel singular values smaller than 0.003 we obtain the 18-state model. The reduced model preserves properties of the full model, as is shown by the magnitude and phase of the transfer function in Fig. 6.10(a),(b). The state-space representation of the reduced antenna model is given in Appendix C.3. frequency, rad/s Figure 6.8. Full model (solid line) and reduced model (dashed line) of a simple structure with poles at zero: (a) Impulse responses; and (b) magnitudes of the transfer function. The figure shows good coincidence between the responses of the reduced and full models. frequency, rad/s Figure 6.8. Full model (solid line) and reduced model (dashed line) of a simple structure with poles at zero: (a) Impulse responses; and (b) magnitudes of the transfer function. The figure shows good coincidence between the responses of the reduced and full models. io 10 io 10 15 20 state number Figure 6.9. Hankel singular values of the DSS26 antenna rate-loop model: 17 states out of 35 states are retained in the reduced model. 15 20 state number Figure 6.9. Hankel singular values of the DSS26 antenna rate-loop model: 17 states out of 35 states are retained in the reduced model. frequency, Hz frequency, Hz frequency, Hz Figure 6.10. Transfer function of the full- (solid line) and reduced-order (dashed line) models of the DSS26 antenna shows good coincidence in (a) magnitude; and (b) phase. frequency, Hz Figure 6.10. Transfer function of the full- (solid line) and reduced-order (dashed line) models of the DSS26 antenna shows good coincidence in (a) magnitude; and (b) phase.
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