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Engineering · Civil Engineering
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No.1 30 points) Consider a bridge system rests on single pier in the middle and two supports bearing at the ends. The bridge can be modeled as a two-degree of freedom system and only vibration in x-direction is considered. The girder moves as a rigid body with mass m; and is connected to pier by spring-dashpot system (k,c). At both ends, the girder to is supported by bearings that can be modeled by two spring-dashpot systems (ks, ks and csc). The pier with mass mi has the stiffness k and damping c1. The pier and girder are assumed to move in-plane in longitudinal direction only, denoted as xs and xa, respectively; as shown in Figure 1. Problem k4 x2lt) Girder C3 ky Pier Figure 1. 1.1. Obtain the kinetic (T), potential (V) and the Dissipation energy of the system (R), and derive the equation of motion of the system by Lagranges principle (L T-V): where R is the dissipation function and X, is external force. Prove that the equation of motion is: where -c, c, +c,+e,! L-k, k: + k,+ k,于にle(of..th(t) 1.2. Assuming the following values: mi-m,-1000 kg, k,-10000 N/m.k2zksk,-5000 N/m; and c,-c,-100 N s/m, c,-c,-50 N s/m, check the damping proportionality condition of the system, define the eigenvalue problem, obtain the natural frequencies, modal damping and mode-shapes using modal analysis procedure. Plot the mode-shapes.

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