17 Modal Analysis of a Time-Variable Ropeway System: Model Reduction and Vibration Instability Detection 135 Cable Grip & Vehicle Horizontal De ection Bullwheel Vertical De ection Support Fig. 17.2 Interactions between each mechanical part of a ropeway n t e z e z e x e y Fig. 17.3 Cable model with local frame. (t,n,ez) . d dS T(S, t) d dS X(S, t) +f(S, t) =μ ∂ 2 X(S, t) ∂t2 , (17.2 ) where.T(S, t) =T(S, t) t is the tension vector and. f the external linear load along the cable. In the specific case of the static (time t independent) and uniform (space S independent) external load due to gravity .f =−μg e y , the resolution of Eq. (17.2) leads to catenary analytical solution for the cable shape .y(x) and the tension. T(x) depending on.x ∈ [xA,xB] Cartesian parameter given in the global frame.(e x,ey,ez) , .y(x) =τ cosh x τ +K1 +K2, (17.3 ) .T(x) =μgτ cosh x τ +K1 , (17.4 ) wher e . τ , . K1 , an d. K2 depend on the boundary conditions at the ends of the cable. An additional equation accounts for the effect of the behavior law on the current cable length L .L=L0 + xB xA ˆ ε(x) dx =L0 + xB xA T(x) EA dx. (17.5 ) The linearization of Eq. (17.2) in the local frame.(t,n,e z) , see Fig. 17.3, around an in-plane static configuration. Xs(S) = x ex + y ey calculated above is given as a dynamic displacement .U(S, t) =u(S, t) t + v(S, t) n + w(S, t) e z give n i n the framework of small dynamic displacement with a linear . ε strain measure, . −μ¨u(S, t) − 2 κ[v(S, t) +κu(S, t)] =0, (17.6)
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