Answer Happy

Figure below is a diagrammatic representation of a propped bridge structure suspended on springs as shown. Upon applying the free body diagram method to the figure, the equations of motion of the system can be given as (assuming small angle of rotation, θ ) mx+2kx+41​kLθ=0 e. Assume that an inital displacement x=1.8 mm is applied to the bar from rest. lθ+41​kLx+163​kL2θ=M(t) Determine the expression for the time history of the resulting motion. (ii) Hint: The general solution is given by x(t)=AX1​cosω1​t+BX1​sinω1​t+CX2​cosω2​t+ DX2​sinα2​t. where A,B,C, and D are constants of integraton. The inital conditions are {x}= (0.00180​)m;{x}=(00​)m a. Re-write the above 6.0 m. in matrix format. (3) b. Assume that the kinetic energy, T and the potential energy. V of the system at an arbitrary instant are given by T=21​mx˙2+21​lθ˙2 and V=21​k(x−41​Lθ)2+21​k(x+21​Lθ)2. Thus, the Lagrangian, L can be written as L=T−V=21​mx˙2+21​Iθ˙2−21​k(x−41​Lθ)2− 21​k(x+21​Lθ)2. Use the Lagrange's equations to derive the system's equations of motion from the formula dtΔ​(∂x∂x​)−∂x∂t​=Q where x and Q represent the generalised displacements and forces. (7) c. Calculate the eigenvalues and natural frequencies (Hz) of the system if m= 5 kg,l=0.5 kg.m2, L=0.8 km and k=2×10∘N/m. d. Determine the mode shape vectors for the system. (4)