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2 Finite Modal Model of EulereBernoulli Beam The dynamics of a flexible manipulator system are described by an infinite-dimensional mathematical model, since the model consists of partial differential equations. 7: The Fifth Modal Shape. 8: The Sixth Modal Shape. design a finite-dimensional controller, a finite-dimensional system model is needed. To achieve this goal, a finite-dimensional approximation needs to be used to model the flexible manipulator, to retain a finite number of modes and to truncate the other, less significant modes based on the requirements of the controller.

1, if we set A ¼ 1, then C ¼ À1. 80) is redundant, and for each frequency it becomes ! ! 81) b2 b4 D b3 À b1 By solving this, B and D are obtained. 2. In fact, the number of modal shapes is infinite, because of the distributed parameter beam system. Only the first 10 modal shapes and their frequencies are listed in the calculation of the state-space equation in the next chapter. Generally, the more modal shapes involved in the system, the more complicated the system model will be. According to Hastings and Book [35,36] and Hughes [37], the first two modes have a greater influence on the model accuracy than the higher-order modes.

111) into Eq. ; 0; 0; 0ŠTnþ1 . M and K denote the mass and rigidity matrices of the system respectively, and are expressed as follows: 3 2I 6 6 6 6 6 6 M¼6 6 6 6 6 4 H rS1 1 rSi 0 7 7 7 7 7 7 7   7 Jp Jp 1 7 þ a þ rS À M c p nÀ1 7 3 2 Dx Dx Dx   7  5 Jp Jp 2ac Mp 1 1 À M þ þ a þ M c p p nþ1;nþ1 Dx Dx2 Dx Dx Dx2 0 1 K ¼ ½K1 ; K2 ; K3 ; . Ki ; . ; n À 1Þ), row of matrix K, all the elements are zero, except the (i À 2)th to (i þ 2)th ones and kn;nÀ2 ¼ bnÀ2 ; kn;nÀ1 ¼ À2ðbnÀ1 þ bnÀ2 Þ; kn;n ¼ ð4bnÀ1 þ bnÀ2 Þ; knþ1;nÀ1 ¼ ÀbnÀ1 ; knþ1;n ¼ 2bnÀ1 ; knþ1;nþ1 ¼ ÀbnÀ1 kn;nþ1 ¼ À2bnÀ1 On this basis, first-order differential equations and then the state-space equations can be easily obtained for dynamic analysis and the control scheme design.

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