Here’s an Assessment for 1.58: Structural Dynamics & Vibrations, structured to test
conceptual understanding, problem-solving ability, and applied analysis across the main topics
of the course. This assessment includes 10 questions: multiple choice, short answer, and
quantitative problems.
📝 Structural Dynamics & Vibrations – Assessment
Section A: Conceptual Understanding (2 pts each)
Q1. Which of the following does not affect the natural frequency of a single-degree-of-freedom
(SDOF) system?​
A. Mass of the system​
B. Stiffness of the spring​
C. Damping coefficient​
D. Boundary conditions
Q2. In a damped system, increasing the damping ratio will:​
A. Increase natural frequency​
B. Decrease response amplitude​
C. Increase the number of vibrations​
D. Have no effect on phase lag
Q3. The primary reason for using modal analysis in multi-degree-of-freedom (MDOF) systems is
to:​
A. Avoid using matrix algebra​
B. Convert the system to a single DOF model​
C. Decouple the equations of motion​
D. Increase the damping ratio
Section B: Short Answer (4 pts each)
Q4. Define critical damping and explain its physical significance in engineering applications.
Q5. Write down the equation of motion for a SDOF system subjected to a harmonic force and
identify each term.
Section C: Analytical Problems (8 pts each)
Q6. A spring–mass–damper system has:
●​ Mass ( m = 2 ) kg,​
●​ Stiffness ( k = 500 ) N/m,​
●​ Damping ( c = 10 ) Ns/m.​
Find:​
a) The damping ratio​
b) The damped natural frequency​
c) The general solution for free vibration if ( x(0) = 0.1 ) m, ( \dot x(0) = 0 )
Q7. A system with ( m = 5 ) kg, ( k = 200 ) N/m is subjected to a harmonic force ( F(t) =
50\cos(3t) ) N.​
Assuming negligible damping, compute:​
a) Natural frequency​
b) Steady-state amplitude of vibration​
c) Comment on resonance condition
Q8. Consider a 2-DOF system with mass and stiffness matrices:​
[​
[\mathbf M] = \begin{bmatrix} 2 & 0 \ 0 & 1 \end{bmatrix},\quad​
[\mathbf K] = \begin{bmatrix} 4 & -2 \ -2 & 4 \end{bmatrix}​
]
a) Write the equations of motion in matrix form​
b) Find the natural frequencies of the system
Section D: Advanced Application (10 pts)
Q9. A simply supported beam (length ( L = 1.5 ) m, ( EI = 2 \times 10^4 ) Nm², ( \rho A = 15
) kg/m) is undergoing transverse vibrations.​
Find the first two natural frequencies using the beam vibration theory.
Q10. A sensitive device (mass = 15 kg) is mounted on a vibration isolator. The base vibrates as
( y(t) = 0.01\sin(40t) ). The isolator stiffness is 6000 N/m, and damping is negligible.
a) Derive the governing equation for relative displacement​
b) Compute the transmissibility​
c) Is the system effectively isolating vibrations?
Total: 80 Points
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