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Engineering · Civil Engineering
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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 (kz,c2). At both ends, the girder to is supported by bearings that can be modeled by two spring-dashpot systems (ki,ks and c,c). The pier with mass mi has the stiffness ki and damping ci. The pier and girder are assumed to move in-plane in longitudinal direction only, denoted as x: and x, respectively; as shown in Figure 1. Falt) ks k4 Girder X2tt) Pier F-(0) m,, Xt) Ci 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 IL T-V where R is the dissipation function and X is external force. Prove that the equation of motion is : where: F(T) R(o) PI, k, k, +k, +k Х,(r) 1.2. Assuming the following values: mi:m尸1000 kg, k,-10000 N/m, k,-k,-k-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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