a. Draw the free body diagrams. b. Calculate the stiffness matrix, the mass matrix and the inverse mass matrix. c. Use the matrix iteration method to find the highest natural frequency and the associated mode shape. k1 m1 Χ1 k2 m2 x2 k3 W m3 m₁ = 3m m2 = m k₁ = k k₂ = 2k m3 = 2m k3 = 3k X3

Q5. For a point at distance 50 m and angle 450 to the axis which of the following statements are correct? Consider an infinite baffled piston of radius 5 cm driven at 2 kHz in air with velocity 10 m/s. You may choose multiple options. a. The constant term is given by 515.03 b. The distance dependent term is given by e-jk50/50 c. The directivity term is given by j1(Ka sin 450)/sin 450 d. There is no time dependent term.

INSTRUCTIONS: Solve the following problem by manually applying the finite element method, following these steps: Pre-processing: Geometry identification. Material properties identification. Load identification. Boundary condition identification. Development of the connectivity matrix for nodes and elements. Processing: Calculation of local stiffness matrices. Assembly of the global stiffness matrix. Assembly of the global force matrix (if required). Application of boundary conditions. Resolution of the system of equations. Derivation of the complete displacement vector (u). Post-processing: Reaction forces calculation. Stress analysis. Problem Statement: the truss nodes and elements (in parentheses) are already numbered. The areas (are in cm^2 ) are underlined. members are made of structural steel, modulus of elasticity is (E) of 20×10^6 N/cm^2 . the lengths are given in cm.

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3. Use the second method of frame assignment, and find the Jacobian of the following robot using the velocity propagation method (Ignore all theta related offsets): 03 02 L2 01 Please answer the question precisely and completely to use the second method of frame assignment to find the D-H table [alpha (i-1) , a (i-1) , d (i) , theta (i)], and find the jacobian matrix using the velocity propagation method

1. For the linkage system show, Link 1 moves along the vertical direction via a linear motor. Link 2 and 3,,dependent on the motion of Link 1, will move accordingly. Perform position analysis using Chace Mehod: Draw the loop closure vectors for the problem. (a) (b) Determine the unknowns. (c) Determine which kinematic case it is (d) Determine equations for the unknowns as function of known variables. (e) Assume link 1 is located 3 inches above the x-y coordinates, and link 2 has a length of 5 inches. Determine the numerical values of the two unknowns for this specific configuration. Link 1 X Link 2 Link 3

Given the vibrating system below: K4 Y(t) =Ysin30t where for = 30 and Y=20mm Find the following K1 K2 m C3 H C2 C1 C5 C4 1. Frequency Ratio 2. Displacement Transmissibility Ratio 3. Absolute displacement of the mass 4. Type of Damping 5. Equation of motion x(t). Assume Initial conditions for displacement and velocity 6. Graph 2 cycles of the vibrating system. You can use third party app for this. M = 10 kg K1=100 N/m K2= 80 N/m K3=75 N/m K4= 120 N/m C1 = 20Ns/m C2=40 Ns/m C3= 35Ns/m C4= 15 Ns/m C5= 10 Ns/m

2. Consider the following matrix 10 0 140 A1 = 0 3 A2 = 1 1 1 , A3 = 1 0 -1 3 (a) Find the eigenvalues and eigenvectors of each matrix in (2). 1 10 010 1 2 3 (2) (b) Find the modal matrix M and diagonal each matrix (2) through similarity transforma- tion. Which matrix in (2) will lead to the Jordan form?

Assigned necessary coordinate frames and obtained the total transformation matrix using the D-H method.

A9