For the system below, the mass, m is attached to the end of a cantilever beam of length, l, elasticity, E, and moment of inertia, I. Derive the equation of motions and find the natural frequencies in two cases by lagrange energy method    a) k1=k2=0 b) k1≠k2≠0

A simply supported beam of 1m length and 50 mm diameter is carrying a mass 100 kg at a distance of 0.25 m from one end. Find the natural frequency of transverse vibrations. Assume Young's modulus of beam material as 250 GN/m.

Q-3 mechanical forces (Shear type). Find the propagation speed (vs) (for both longitudinal and transverse vibrations) of the elastic waves propagating along [111] direction to dissipate the gained elastic energy through the sample. The necessary equations can be obtained from the equation given below for the displacement vector A continuum bcc structure is deformed elastically by x, y, and z directional R = R,& + Ryỹ + R22 p(ô²R/@t) = C11(8²R$ /ôx²)+ C44 (a²R¾/@y²+ ô²Rx /ô z²)+(C12+C4)(ô²Ry/ Ô x ô y+ ô²Rz/@xð z) Here, C11, C44, C12 and p are the moduli of elasticity (Stiffness constants) and the mass density of the bcc structure, respectively.

An aluminum sphere with a radius of 8 in. is attached to one end of a 20 ft long aluminum rod of 1 in. diameter. For aluminum, modulus of elasticity, G = 3.44 x 10 Ib/in while its density is 0.097 Ib/in The other end of the rod is fixed as shown. Find the period of this pendulum