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ME-502 · Mechanical Vibrations/Unsolved PYQ Paper

ME-502 Mechanical Vibration - Nov 2022 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
    aA vibratory system consists of a mass $12.5\ \text{kg}$, a spring of stiffness $1000\ \text{N/m}$, and a dashpot with damping coefficient of $15\ \text{Ns/m}$. Find the value of critical damping of the system and the value of logarithmic decrement of the vibrating system.
    How to answer
    • Define critical damping Cc = 2*m*wn = 2*sqrt(k*m)
    • Compute wn = sqrt(k/m), Cc value
    • Compute damping ratio zeta = c/Cc
    • Logarithmic decrement delta = 2*pi*zeta/sqrt(1-zeta^2)
  2. 7 Marks
    bDefine the following terms?
    1. i)Vector representation of the displacement, velocity, and acceleration of SHM
    2. ii)Fundamental mode of vibration
    3. iii)Mathematical definition of Simple Harmonic Motion (SHM)
    How to answer
    • Define rotating vector for SHM displacement
    • Show velocity leads displacement by 90 deg and acceleration by 180 deg
    • Define fundamental mode / lowest natural frequency mode shape
    • Give mathematical condition d2x/dt2 + omega_n^2 x = 0 for SHM
  3. 7 Marks
    a2. a)
    1. i)Draw the FBD of the system and derived the differential equation governing by vibrating system is (Figure 1)
    2. ii)Why we use Viscous damping for modelling the vibrating system. Explain in brief.
    Given Figure 1: mass m on horizontal surface with damper c on left connected to fixed point with coordinate y, and spring k on right connected to fixed wall with coordinate x on mass. Damper symbol C, spring k, mass m.
    How to answer
    • Draw FBD of mass m with spring force kx and damper force c*dx/dt
    • Apply Newton's second law in x-direction
    • Obtain m*x_ddot + c*x_dot + k*x = 0
    • Explain linearization and mathematical convenience of viscous damping model
  4. 7 Marks
    bA system is shown in the following figure 2. The bar AB is assumed to be rigid and weightless. The natural frequency of vibration of the system is given by
    Given Figure 2: horizontal bar AB pivoted at A on left wall. Spring k1 attached at distance a from wall to ceiling. At end B length l from A, spring k2 attached vertically with mass m below. Dimensions a and l marked.
    How to answer
    • Identify rigid bar AB pivoted at A with springs k1 at distance a and k2-mass at length l
    • Write moment equation about hinge A for small angle
    • Obtain equivalent stiffness and natural frequency expression
    • State assumptions of small oscillations and weightless bar
  5. 7 Marks
    aProve that the critical speed of whirling speed for a rotating shaft is same as the frequency of natural transverse vibration.
    How to answer
    • Write equation of rotating shaft with eccentricity
    • Show deflection grows when rotation equals transverse natural frequency
    • Define critical / whirling speed condition
    • Conclude critical speed equals natural transverse frequency
  6. 7 Marks
    bA uniform rigid slender bar of mass $10\ \text{kg}$, hinged at the left end is suspended with the help of spring and damper arrangement as shown in the figure where $K = 2\ \text{kN/m}$, $C = 500\ \text{Ns/m}$ and the stiffness of the torsional spring $k_{\theta}$ is $1\ \text{kN/m/rad}$. Ignore the hinge dimensions.
    1. i)Find the un-damped natural frequency of oscillations of the bar about the hinge point.
    2. ii)The damping coefficient in the vibration equation.
    Given figure for Q3 b): horizontal bar hinged at left end, spring K and damper C in parallel attached 400 mm from hinge, total bar length 500 mm. Hinge has torsional spring k_theta.
    How to answer
    • Take moment about hinge, include torsional spring k_theta, spring K and damper C
    • Find mass moment of inertia of bar about hinge (1/3)mL^2
    • Obtain undamped natural frequency sqrt(k_eq/J)
    • Extract equivalent viscous damping coefficient from moment equation
  7. 7 Marks
    aWhat are the principles on which a Vibrometer and an accelerometer are based? Derive the expression.
    How to answer
    • Explain seismic instrument with mass-spring-damper base excitation
    • Derive transmissibility for displacement pickup (vibrometer)
    • Derive for acceleration pickup (accelerometer) with high natural frequency
    • State frequency ratio conditions for each instrument
  8. 7 Marks
    bFind the natural frequencies and first two normal mode shapes of the system shown in Figure 3. Assume $k_{1} = k_{2} = k$ and $m_{1} = m_{2} = m$
    Given Figure 3: two masses m1, m2 on rollers on horizontal surface. m1 connected to left wall by k1, m1 connected to m2 by K2 (k2). Displacements x1, x2 marked upward arrows.
    How to answer
    • Write equations of motion for m1, m2 with k1, k2
    • Form mass and stiffness matrices, solve determinant |K - omega^2 M|=0
    • Obtain two natural frequencies omega1, omega2
    • Find amplitude ratios for mode shapes
  9. 14 Marks
    Find the lowest natural frequency of vibration of system shown in Figure 4. Assume $E = 1.96 \times 10^{11}\ \text{N/m}^{2}, I = 4 \times 10^{-7}\ \text{m}^{4}$
    Given Figure 4: simply supported shaft EI=CONST total length 4.5 m with 1.5 m segments. 100 kg disc 1.5 m from left, 50 kg disc 3.0 m from left.
    How to answer
    • Model shaft as simply supported beam with two concentrated masses
    • Use Dunkerley or Rayleigh method for transverse vibration
    • Insert E, I, spans 1.5 m each, masses 100 kg and 50 kg
    • Compute lowest natural frequency
  10. 7 Marks
    aDerive the expression for frequency of free damped vibrations.
    How to answer
    • Write equation m*x_ddot + c*x_dot + k*x = 0
    • Assume solution x = e^(st), find roots with damping ratio
    • Obtain damped frequency omega_d = omega_n*sqrt(1-zeta^2)
    • Discuss under, critical and overdamped cases
  11. 7 Marks
    bIn a spring mass vibrating system, the natural frequency of vibration is $3.56 \text{Hz}$. When the amount of the suspended mass is increased by $5 \text{kg}$, the natural frequency is lowered to $2.9 \text{Hz}$. Determine the original unknown mass and the spring constant.
    How to answer
    • Write fn = (1/2pi)*sqrt(k/m) for two cases
    • Form ratio to eliminate k, solve for unknown mass m
    • Compute k from either frequency
    • Check units Hz to rad/s
  12. 14 Marks
    A right cantilever steel shaft of $0.3\ \text{m}$ effective length has a heavy rotor fixed at its end. The mass of the rotor is $10\ \text{kg}$ and has a radius of gyration of $12\ \text{cm}$ about its axis. The thickness of the rotor is $6\ \text{cm}$. The moment of inertia of the section of the shaft about its neutral axis is $10\ \text{cm}^{4}$. This shaft is to run at $10,000\ \text{rpm}$. Check if this operating speed is safe.
    How to answer
    • Compute mass moment of inertia of rotor and shaft stiffness 3EI/L^3
    • Find transverse natural frequency / critical speed
    • Include gyroscopic effect of disc thickness if required
    • Compare operating speed 10000 rpm with critical, comment on safety
  13. 7 Marks
    aWrite the Logical Explanation with neat sketch if needed:
    1. i)What is SPL in noise?
    2. ii)Does 0db SPL mean no sound pressure?
    3. iii)How loud is 100dB SPL?
    How to answer
    • Define SPL = 20 log10(p/p_ref) with reference 20 microPa
    • Explain 0 dB is threshold of hearing, not zero pressure
    • Give scale for 100 dB loudness example
    • Add neat sketch of dB scale if needed
  14. 7 Marks
    bWhat is octave band analysis of sound?
    How to answer
    • Define octave band with factor 2 frequency ratio
    • Explain filters and centre frequencies
    • State use for noise analysis and control
    • Give example octave bands
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