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134) . ½Qn ŠN ŠT . 136) The general stiffness matrix is 2 ½Kn ŠÃ1 Knà ¼ 4 « 0 Considering node coordinates overlapping between neighborhood elements when assembled, the general coordinate is introduced as Qz ¼ ½ q q1 w1 q2 w2 q3 w3 . 137) Then there exists the relationship between the local coordinate and the general coordinate QÃn ¼ b Qz , where the transformation matrix b is a (5N)  (2N þ 1) sparse matrix. 140) While taking boundary conditions into account, the beam is fixed at one end and it holds that q1 ¼ w1 h 0.

Based on these considerations, it is necessary to investigate the influences of rotary inertia and shear deformation on the vibration of flexible manipulators. Another factor that obviously also affects the vibration of flexible manipulators is the load carried by end-effectors (the tip load). If the influence of the tip load on the vibration of flexible manipulators is shown to be significant, non-adaptive control approaches should be considered inappropriate or at least inefficient for the control of flexible manipulators.

2 EulereBernoulli Model after Linearization Using Eq. 56) Note that in the dynamic models presented in Ref. e. S ¼ 0 and ac ¼ 0 were assumed). 60) Note that when no tip load is present, the equations in this case are the same as those given in Ref. [7]. 56), we find that Eq. 56) can be used to calculate hub rotation q and thus to eliminate q from dynamic equations. To compensate for the lost boundary condition, Eq. 58) can be considered as the new condition. In fact, 36 Chapter 3 Eq. 58) represents the dynamic torque balance equation of the hub according to the EulereBernoulli or Timoshenko beam model.

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