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126) 3. 129) is expressed as 3 0 0 0 07 7 0 07 7 0 05 0 0 where IH ¼ Ih þ rl3/3. 132) where Ai and Ii are the cross-sectional area and rotary inertia of the ith segment of the beam. 134) . ½Qn N T . 136) The general stiffness matrix is 2 ½Kn Ã1 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.
0; 0; 0Tnþ1 . 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.
It is assumed that the initial (neutral) longitudinal axis of the beam coincides with the x-axis. For the sake of convenience, the following differential notations have been used: ( )0 ¼ v( )/vx and ð Þ ¼ vð Þ=vt. The beam is of length L with a transverse section with moment of inertia I and area A. Young’s modulus, shear modulus, and mass per unit length of the beam are denoted by E, G, and r respectively. The tip load is of mass Mp and inertia moment Jp, and its center of mass in AO0 B is denoted as (ac,bc).