Vibration Analysis Questions -

Q1. For the given system, determine the equilibrium position and its equation of vibration about it. Spring force = 0 when θ = 0.

Q2. A cord of length l and mass per unit length ρ is under tension T with the left end fixed and the right end attached to a spring-mass system, as shown in the figure below. Determine the equation for the natural frequencies.

Q3. A uniform rod of length 1 cross sectional area A is fixed at the upper end and is loaded with a weight W on the other end. Show that the natural frequencies are determined from the equation ---

ωl√(ρ/E)tan(ωl√(ρ/E)) = Aρlg/W

Q4. A uniform bar has these specifications: length l, mass density per unit volume ρ, torsional rigidity I_pG where I_p is polar moment of inertia and G is the shear modulus. The end x = 0 is fastened to a torsional spring of stiffness K Nm/rad and the end x = l is fixed. Determine the transcendental equation from which the natural frequencies can be established. Verify the correctness of this equation by considering the cases K = 0 and K = ∞.

Q5. Determine the natural frequencies of a uniform beam of length l clamped at both ends.