Control beam

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Simply-supported beam F m, EJ, l xs xa l = 0:55 m t = 0:002 m w = 0:05 m  = 7700 Kg=m3 E = 200 GPa J = 1 12wt3 m4 xa = 0:7 l xs = 0:4 l – p. 1 Model PLV : Z l 0 v 00T EJv 00 dx = Z l 0 vT mvdx + Z l 0 vT F(t)(x xa)dx Solution : v(x; t) = 1X i=1 i(x)qi(t) = h 1(x) 2(x) ::: i 8 > > < > > : q 1(t) q2(t) ::: 9 > > = > > ; = (x)q(t) i(x) = r 2 ml sin ix l  q(t)T Z l 0  00 (x)T EJ 00 (x)dx  q(t)+q(t)T Z l 0 (x)T m(x)dx  q(t) = q(t)T (xa)T F(t)  00 (x) = h 00 1 (x) 00 2 (x) ::: ::: 00 i (x) ::: ::: i – p. 2 Model M q(t) + Kq(t) = (xa)T F(t) Mij = Z l 0 m i(x) j(x)dx = 2 l Z l 0 sin ix l  sin jx l  dx = 8 < : 1 i = j 0 i 6= j K ij = Z l 0 EJ 00 i (x) 00 j (x)dx = 2EJ ml i l 2 j l 2 Z l 0 sin ix l  sin jx l  dx = = 8 < : EJ m i l
4 i = j 0 i 6= j !2 i = EJ m i l 4 – p. 3 Design model qi(t) + !2 i qi(t) = i(xa)F (t) i = 1; 2;:::;np np = number of modes in the design model qi(t) + 2i!i _qi(t) + !2 i qi(t) = i(xa)F (t) i = 1; 2;:::;np i = damping factor of mode i ( = 0:01 ) – p. 4 Design model (state-space) 8 < : _x(t) = A px(t) + Bpu(t) y(t) = Cpx(t) + Dpu(t) x(t) = 8 > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > : q 1(t) _q1(t) q2(t) _q2(t) ::: ::: q np (t) _qnp (t) 9 > > > > > > > > > > > > > > > > = > > > > > > > > > > > > > > > > ; = n ::: ::: q i(t) _qi(t) ::: ::: oT u(t) = F(t) y(t) = v(xs; t) = (xs)q(t) = npX i=1 i(xs)qi(t) – p. 5