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ME-502 · Mechanical Vibrations/Unsolved PYQ Paper

ME-502 Mechanical Vibration - Jun 2025 Question Paper

Under each question, How to answer lists the points an answer should cover, outlined by CampusPrep from the syllabus. Check the details against your textbook.

  1. 7 Marks
    aThe impact force created by a forging hammer can be modeled as shown in Fig.1. Determine the Fourier series expansion of the impact force.
    Plot of x(t) vs t with vertical axis x(t) marked A and 0. Three identical half-sine arches: first from 0 to tau/2? Actually 0 to ~tau? labelled tau/2 at peak, tau at end; second from tau to 2tau with peak at 3tau/2; third from 2tau to ~5tau/2 with peak at 5tau/2. Zero between pulses.
    How to answer
    • Identify periodic impact force $x(t)$ with period $\tau$ from Fig.1: half-sine pulses of amplitude $A$.
    • Write Fourier series $x(t)=a_0/2+\sum a_n\cos n\omega t + b_n\sin n\omega t$ with $\omega=2\pi/\tau$.
    • Evaluate coefficients by integration over 0 to $\tau$.
    • Present final expansion for the repeated half-sine.
  2. 7 Marks
    bDetermine the natural frequency of undamped single degree of freedom as shown in Fig.2. Assume the values of stiffness $k_1 = 1 N/m, k_2 = 2 N/m, k_3 = 1 N/m, k_4 = 4 N/m, k_5 = 6 N/m$.
    Spring-mass system Figure 2: top fixed support. Two k1 springs in parallel connected in series with k2 below, then with two k3 springs in parallel below k2. That branch connects via k5 to 3 kg mass. A second branch with k4 alone connects top support directly to the junction above k5. Mass 3 kg at bottom with downward arrow.
    How to answer
    • Reduce series-parallel spring combination to equivalent stiffness $k_{eq}$.
    • Identify $k_1$ in parallel, $k_3$ in parallel, then series-parallel steps with $k_2, k_4, k_5$.
    • Use $\omega_n = \sqrt{k_{eq}/m}$ with $m=3\,kg$.
    • Substitute $k_1=1, k_2=2, k_3=1, k_4=4, k_5=6\,N/m$ and compute frequency in rad/s and Hz.
  3. 7 Marks
    aThe maximum permissible recoil distance of a gun is specified as 0.5 m. If the initial recoil velocity is 10 m/s, find the spring stiffness of the recoil mechanism. Assume that a critically damped dashpot is used in the recoil mechanism and the mass of the gun has to be 500 kg.
    How to answer
    • Write equation for critically damped system $\zeta=1$ with initial velocity.
    • Use $x(t)=(x_0+(v_0+\omega_n x_0)t)e^{-\omega_n t}$ with $x_0=0$.
    • Maximize $x(t)$: set $\dot{x}=0$ to get $t_{max}=1/\omega_n$.
    • Solve $x_{max}=v_0/(e\omega_n)=0.5$ for $\omega_n$ then $k=m\omega_n^2$.
  4. 7 Marks
    bA viscously damped system has a stiffness of 5,000 N/m, critical damping constant of 0.2 Ns/mm, and a logarithmic decrement of 2.0. If the system is given an initial displacement of 0 and velocity of 1 m/s, determine the maximum displacement of the system.
    How to answer
    • Compute $\omega_n=\sqrt{k/m}$, $c_c$, damping ratio $\zeta$ from logarithmic decrement $\delta=2.0$.
    • Find damped natural frequency $\omega_d=\omega_n\sqrt{1-\zeta^2}$.
    • Write free response with $x_0=0, \dot{x}_0=1\,m/s$.
    • Differentiate and set to zero to obtain $x_{max}$.
  5. 7 Marks
    aDefine the term magnification factor. Derive the expression for magnification factor of spring mass damper system subjected to sinusoidal harmonic force $F = F_0 \sin \omega t$.
    How to answer
    • Define magnification factor as ratio of dynamic to static amplitude.
    • Write equation $m\ddot{x}+c\dot{x}+kx=F_0\sin\omega t$.
    • Assume steady-state harmonic solution and derive amplitude.
    • Express $M=1/\sqrt{(1-r^2)^2+(2\zeta r)^2}$ with $r=\omega/\omega_n$.
  6. 7 Marks
    bThe spring of an automobile trailer are compressed 0.1 m under its own weight. Find the critical speed when the trailer is travelling over a road with a profile approximated by a sine wave of amplitude 0.08 m and wave length of 14 meters. What will be the amplitude of vibration at 60 km/hour?
    How to answer
    • Find $\omega_n=\sqrt{g/\delta_{st}}$ from static deflection 0.1 m.
    • Critical speed corresponds to resonance: excitation frequency equals $\omega_n$.
    • Relate vehicle speed $V$ to excitation $\omega=2\pi V/\lambda$ with $\lambda=14\,m$.
    • Compute vibration amplitude at 60 km/hour using magnification factor for undamped/base excitation.
  7. 14 Marks
    A heavy machine, weighting 3000 N, is supported on a resilient foundation. The static deflection of the foundation due to the weight of the machine is found to be 7.5 cm. It is observed that the machine vibrates with an amplitude of 1 cm when the base of the foundation is subjected to harmonic oscillation at the undamped natural frequency of the system with an amplitude of 0.25 cm. Find:
    1. i)the damping constant of the foundation.
    2. ii)the dynamic force amplitude on the base, and
    3. iii)the amplitude of the displacement of the machine relative to the base.
    How to answer
    • Compute $\omega_n=\sqrt{g/\delta_{st}}$ from static deflection 7.5 cm.
    • Use base excitation transmissibility $X/Y$ at $r=1$ to solve for $\zeta$ and $c$.
    • Compute transmitted force amplitude $F_T$ using transmissibility.
    • Compute relative displacement $Z = X-Y$ phasor magnitude.
  8. 14 Marks
    Find the total response of a single-degree-of-freedom system with $m = 10 kg, c = 20 N-s/m, k = 4000 N/m, x_0 = 0.01 m,$ and $\dot{x}_0 = 0$ under the following conditions:
    1. a)An external force $F_t = F_0 \cos \omega t$ acts on the system with $F_0 = 100 N$ and $\omega = 10 rad/s$.
    2. b)Free vibration response with $F(t) = 0$.
    How to answer
    • Compute $\omega_n, \zeta, \omega_d$ from $m,c,k$.
    • Write total response = transient + steady-state for forced case.
    • Apply $x_0=0.01\,m, \dot{x}_0=0$ to find constants.
    • For $F(t)=0$, write underdamped free vibration decay expression.
  9. 14 Marks
    Find the natural frequencies and amplitude ratio of the system shown in Fig. 3. with $k_1 = 20 N/m, k_2 = 10 N/m, m_1 = 20 kg$ and $m_2 = 7 kg$.
    Figure 3: mass m1 on rollers connected to wall by k1 on left and to mass m2 on rollers by k2. Displacements x1(t), x2(t) and forces f1(t), f2(t) marked to the right.
    How to answer
    • Write equations of motion for $m_1, m_2$ coupled by $k_2$.
    • Assume harmonic solution to get frequency determinant.
    • Solve $|K-\omega^2 M|=0$ for $\omega_{n1}, \omega_{n2}$.
    • Find amplitude ratios $(X_1/X_2)$ for each mode.
  10. 14 Marks
    Determine the principal coordinates for the spring-mass system shown in Fig.4.
    Figure 4 Two Degree of freedom system: vertical chain fixed top and bottom. From top: spring k1=k, mass m1=m with displacement x1(t) downward, spring k2=nk, mass m2=m with displacement x2(t) downward, spring k3=k to ground.
    How to answer
    • Write kinetic and potential energies and equations of motion.
    • Form mass and stiffness matrices for $x_1, x_2$.
    • Find eigenvalues/eigenvectors to get natural frequencies and modal vectors.
    • Transform to principal/modal coordinates $q = P^{-1}x$.
  11. 14 Marks
    Write short notes on the following:
    1. a)Relationship between sound pressure level and sound power level.
    2. b)Octave Band Analysis
    3. c)Various hearing conservation and damage risk criterion.
    4. d)Distinguish between viscous damping and Coulomb damping.
    How to answer
    • Define SPL vs PWL with reference pressure/power and relation via intensity.
    • Explain octave band centre frequencies and filtering for noise analysis.
    • List hearing conservation criteria: OSHA, ISO, damage risk levels.
    • Contrast viscous (velocity proportional) vs Coulomb (constant friction) damping characteristics.
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