(Solved): 3. Question 3 [25 marks] Consider the mass-spring system of Figure 3 where the masses of \( 2 m \) ...

3. Question 3 [25 marks] Consider the mass-spring system of Figure 3 where the masses of \( 2 m \) and \( m \) are bound to each other via a spring of stiffness \( k \) and connected to rigid walls via springs of stiffness \( 2 k \) and \( k \) as shown. The coordinates expressing the displacements of each mass are \( x_{1} \) and \( x_{2} \). Figure 3: Mass-spring system a) Calculate the Lagrangian of the system. \( [5 \) marks] b) Derive the equations of motion of the system. [5 marks] c) Calculate the natural frequencies of the system. [5 marks] d) Calculate the mode shape vectors of the system. [5 marks] e) The masses are released at \( t=0 \) from positions \( x_{1}(0)=0.5 \) meter and \( x_{2}(0)=1 \) meter with zero initial velocities, that is, \( \dot{x}_{1}(0)=0 \mathrm{~m} / \mathrm{s} \) and \( \dot{x}_{2}(0)=0 \mathrm{~m} / \mathrm{s} \). Find the expressions for the motion of each mass, \( x_{1}(t) \) and \( x_{2}(t) \), for \( t \geq 0 \). [5 marks] 4. Question 4 [25 marks] Consider the three-bar assembly, loaded with force \( P \), and constrained at the two ends, as shown in Figure 4. The bars have identical length, \( L \), and are made of the same material of Young's modulus \( E \). The bars have cross-sectional areas \( 3 A, 2 A \) and \( A \) respectively. Figure 4: Three-bar assembly a) Sketch the appropriate finite element model for the three-bar assembly showing the degrees of freedom. [5 marks] b) Find the stiffness matrix of the three-bar assembly. [5 marks] c) Find the displacements \( u_{1} \) and \( u_{2} \). [5 marks] d) Calculate the support reaction forces at the two ends of the tree-bar assembly. [5 marks] e) Calculate the stresses in the system's bars. [5 marks]