CE-702-C Structural Dynamics - Nov 2023 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
   aDiscuss Newmark's method for numerical evaluation of dynamic response of single degree of freedom system.
   How to answer

   • Definition of Newmark's method as step-by-step numerical integration for SDOF dynamic response
   • Governing equation $m\ddot{u}+c\dot{u}+ku=p(t)$ and Newmark assumptions with parameters $\gamma, \beta$
   • Recurrence / incremental algorithm steps for displacement, velocity, acceleration
   • Discussion of stability, accuracy and special cases: average acceleration and linear acceleration methods
2. 7 Marks
   bA single degree of system consists of a mass 20 kg, spring of stiffness 2200N/m and a dashpot with a damping coefficient of 60 N-s/m is subjected to a harmonic excitation of $F = 200\sin 5t$ Newton's. Determine the steady state of response and write the solution of equation of motion.
   How to answer

   • Compute natural frequency $\omega_n=\sqrt{k/m}$, damping ratio $\zeta$ and excitation frequency $\omega=5$ rad/s
   • Compute frequency ratio and dynamic magnification factor for harmonic excitation
   • Determine steady-state amplitude and phase lag
   • Write total solution as transient plus steady-state $u(t)$ and equation of motion $m\ddot{u}+c\dot{u}+ku=F_0\sin\omega t$
3. 7 Marks
   aWhat is damping? Discuss the critical damping with example.
   How to answer

   • Define damping as energy dissipation mechanism in vibrating system
   • Define critical damping $c_c=2m\omega_n$ and damping ratio $\zeta=c/c_c$
   • Discuss critically damped response $\zeta=1$ with non-oscillatory return to equilibrium
   • Give structural/mechanical example contrasting under, critical and over damping
4. 7 Marks
   bDerive equation of motion for single degree of freedom system damped forced vibration.
   How to answer

   • Draw SDOF mass-spring-damper with harmonic force and free-body diagram
   • Apply Newton's second law / D'Alembert's principle
   • Include inertia, damping $c\dot{u}$, stiffness $ku$ and forcing $p(t)$ terms
   • Obtain $m\ddot{u}+c\dot{u}+ku=p(t)$ for damped forced vibration
5. 14 Marks
   Find the natural frequency for given system in Fig. 1.

   

   How to answer

   • Reduce series and parallel spring combinations to equivalent stiffness $k_{eq}$
   • Identify mass $2m$ in the spring-mass system fixed at both ends
   • Use $\omega_n=\sqrt{k_{eq}/m_{eq}}$
   • Compute natural frequency and cyclic frequency $f_n$
6. 7 Marks
   aExplain 'Logarithmic decrement' and 'Duhamel's integral' method for undamped system.
   How to answer

   • Define logarithmic decrement $\delta=\ln(u_n/u_{n+1})$ and relation to damping ratio
   • Explain free-decay measurement for undamped/light-damped system
   • State Duhamel's integral for response to arbitrary loading via impulse superposition
   • Give formula $u(t)=\int_0^t p(\tau)h(t-\tau)d\tau$ for undamped system
7. 7 Marks
   bFind the Laplace transformation of a pulse height 'A' and duration '$\tau$' for rectangular pulse as given in Fig. 2. Deduce the Laplace transform of unit impulse.

   

   How to answer

   • Express rectangular pulse $f(t)=A$ for $0<t<\tau$, zero otherwise using unit step functions
   • Apply Laplace definition $F(s)=\int_0^{\infty}f(t)e^{-st}dt$
   • Obtain $F(s)=A(1-e^{-s\tau})/s$
   • Take limit for unit impulse to deduce $L\{\delta(t)\}=1$
8. 7 Marks
   aDiscuss in detail the Eigen value problem and the methods used for solving the problems.
   How to answer

   • Define eigenvalue problem $(K-\omega^2M)\phi=0$ and characteristic equation
   • Explain natural frequencies and mode shapes as eigenpairs
   • List solution methods: characteristic polynomial, matrix iteration, Stodola, Holzer, Jacobi, QR
   • Discuss orthogonality and expansion of response
9. 7 Marks
   bDiscuss the methods of matrix iteration.
   How to answer

   • State principle of matrix iteration for fundamental mode by repeated multiplication
   • Explain Stodola / power method with assumed trial vector and normalization
   • Discuss convergence, sweeping and higher mode iteration
   • Mention advantages and limitations for MDOF systems
10. 7 Marks
    aWrite notes on viscous dampers.
    How to answer

    • Define viscous damper and linear force-velocity relation $f_D=c\dot{u}$
    • Describe construction and energy dissipation mechanism
    • Give equation of motion inclusion and damping ratio
    • List structural applications
11. 7 Marks
    bExplain D'Alembert's principle and its applications.
    How to answer

    • State D'Alembert's principle: dynamic equilibrium with inertia force $-m\ddot{u}$
    • Convert dynamics to equivalent statics $\sum F - m\ddot{u}=0$
    • Derive SDOF/MDOF equations of motion using it
    • Give structural dynamics applications
12. 14 Marks
    Derive equation of natural frequency for axial vibration of bar and bending vibration of beam as continuous system.
    How to answer

    • Consider bar element with axial displacement $u(x,t)$ and beam in bending $v(x,t)$
    • Apply equilibrium on differential element to get wave equation $EA u''=m\ddot{u}$ and beam equation $EI v''''+m\ddot{v}=0$
    • Assume harmonic solution and apply boundary conditions
    • Obtain frequency equations and $\omega_n$ expressions for modes
13. 14 Marks
    Write note on any four of the followings :

    1. i)Fourier transform methods
    2. ii)Response to unit impulse
    3. iii)Vibration isolation
    4. iv)Arbitrary force
    5. v)Equation of motion
    6. vi)Matrix formulation
    How to answer

    • Select any four topics from the six listed items
    • Give definition, governing formula and physical significance for each
    • Include diagram or example where relevant
    • Summarize applications in structural dynamics