UNIT 3: STRUCTURAL DYNAMICS – EXAM-FOCUSED SHORT NOTES

I. FUNDAMENTALS OF SINGLE DEGREE OF FREEDOM (SDOF) SYSTEMS

Equation of Motion (EOM)

Derived using Newton’s Second Law for a spring-mass-damper system:

• Free body diagram: inertial force $-m\ddot{u}$, damping force $-c\dot{u}$, spring force $-ku$, external force $F(t)$.

• Standard form:

$$m\ddot{u} + c\dot{u} + ku = F(t)$$

• Earthquake excitation (base acceleration $$\displaystyle \ddot{u}_g $$):

  Absolute displacement $$\displaystyle u_{abs} = u + u_g $$, inertial force $$\displaystyle -m(\ddot{u} + \ddot{u}_g) $$.

$$m\ddot{u} + c\dot{u} + ku = -m\ddot{u}_g$$

• D’Alembert’s Principle: Include inertial force as a dynamic force, sum of all forces = 0:

  $$\displaystyle -ku - c\dot{u} + F(t) - m\ddot{u} = 0 $$ → same EOM.

[!TIP]

Common Pitfall: For earthquake loading, the RHS is $$\displaystyle -m\ddot{u}_g $$, not $$\displaystyle +m\ddot{u}_g $$. Always consider relative displacement $u$.

Free Vibration Analysis

• Undamped ($$\displaystyle c=0 $$):

  $$\displaystyle m\ddot{u} + ku = 0 $$ → $$\displaystyle \ddot{u} + \omega_n^2 u = 0 $$, where $$\displaystyle \omega_n = \sqrt{k/m} $$.

  Solution: $$\displaystyle u(t) = A\sin(\omega_n t) + B\cos(\omega_n t) $$.

• Damped ($c \neq 0$):

  Characteristic equation: $$\displaystyle m\lambda^2 + c\lambda + k = 0 $$.

  Damping ratio: $$\displaystyle \zeta = \frac{c}{2\sqrt{mk}} = \frac{c}{c_c} $$, where $$\displaystyle c_c = 2\sqrt{mk} $$ (critical damping).

  • Underdamped ($$\displaystyle \zeta < 1 $$): $$\displaystyle \lambda = -\zeta\omega_n \pm i\omega_d $$, $$\displaystyle \omega_d = \omega_n\sqrt{1-\zeta^2} $$.

    Solution: $$\displaystyle u(t) = e^{-\zeta\omega_n t}(A\sin\omega_d t + B\cos\omega_d t) $$.

  • Critically damped ($$\displaystyle \zeta = 1 $$): $$\displaystyle \lambda = -\omega_n $$ (repeated). Fastest return to equilibrium without oscillation.

  • Overdamped ($$\displaystyle \zeta > 1 $$): two real roots, slower return.

[!TIP]

Critical Damping Example: Door closers, shock absorbers. $$\displaystyle c_c $$ is the minimum damping to prevent oscillation.

Logarithmic Decrement ($\delta$)

• Definition: $$\displaystyle \delta = \ln\left(\frac{u(t)}{u(t+T_d)}\right) $$ for underdamped systems, where $$\displaystyle T_d = 2\pi/\omega_d $$ (damped period).

• Derivation:

  $$\displaystyle u(t) = A e^{-\zeta\omega_n t} \sin(\omega_d t + \phi) $$

  $$\displaystyle u(t+T_d) = A e^{-\zeta\omega_n (t+T_d)} \sin(\omega_d t + \phi + 2\pi) = u(t) e^{-\zeta\omega_n T_d} $$

  $$\displaystyle \therefore \delta = \zeta\omega_n T_d = \frac{2\pi\zeta}{\sqrt{1-\zeta^2}} $$

• Variation with $\zeta$: $\delta$ increases with $\zeta$. For small $\zeta$, $\delta \approx 2\pi\zeta$.

• Use: Experimental determination of $\zeta$ from decay of amplitudes.

[!TIP]

$\delta$ is defined only for underdamped systems. For $\zeta \geq 1$, no oscillation → $\delta$ undefined.

Forced Vibration Analysis

• Harmonic Force $$\displaystyle F(t) = F_0\sin\omega t $$:

  Steady-state response: $$\displaystyle u(t) = U\sin(\omega t - \phi) $$

  Amplitude:

$$U = \frac{F_0}{\sqrt{(k - m\omega^2)^2 + (c\omega)^2}}$$

Phase: $$\displaystyle \phi = \tan^{-1}\left(\frac{c\omega}{k - m\omega^2}\right) $$

Resonance: For light damping ($$\displaystyle \zeta < 1/\sqrt{2} $$), peak amplitude at $$\displaystyle \omega_r = \omega_n\sqrt{1-2\zeta^2} $$.

• Solution Methods for Arbitrary Forcing:

  • Duhamel’s Integral (Principle of Superposition) for undamped systems:

$$u(t) = \frac{1}{m\omega_n} \int_0^t F(\tau) \sin[\omega_n(t-\tau)] d\tau$$

- **Unit Impulse** $$\displaystyle F(t) = I\delta(t) $$:  

  Impulse response $$\displaystyle h(t) = \frac{1}{m\omega_n}\sin\omega_n t $$ for $$\displaystyle t>0 $$.

- **Rectangular Pulse** $$\displaystyle F(t) = F_0 $$ for $$\displaystyle 0<t<\tau $$, else 0:  

  Apply Duhamel or Laplace (see below).

• Laplace Transform Method:

  Take LT of EOM: $$\displaystyle (ms^2 + cs + k)U(s) = F(s) $$.

  Solve $$\displaystyle U(s) = H(s)F(s) $$, where $$\displaystyle H(s) = \frac{1}{ms^2+cs+k} $$ is transfer function.

  For rectangular pulse: $$\displaystyle F(s) = F_0\frac{1-e^{-s\tau}}{s} $$.

  Inverse LT via partial fractions.

  Deduce unit impulse: As $\tau\to0$, $$\displaystyle F_0\tau = I $$, $F(s)\to I$, so $$\displaystyle U(s) = I \cdot H(s) $$.

Stiffness & Properties

• Effective stiffness $$\displaystyle k_{eff} $$:

  • Springs in series: $$\displaystyle \frac{1}{k_{eff}} = \sum \frac{1}{k_i} $$

  • Springs in parallel: $$\displaystyle k_{eff} = \sum k_i $$

• Viscous damper: Force $$\displaystyle F_d = c\dot{u} $$, damping coefficient $c$ (N·s/m). Energy dissipated per cycle: $$\displaystyle E_d = \pi c \omega U^2 $$.

II. NUMERICAL METHODS FOR SDOF SYSTEMS

Newmark’s Beta Method

• Procedure: Assume acceleration variation over $\Delta t$:

$$ \begin{aligned} u_{t+\Delta t} &= u_t + \Delta t \dot{u}_t + \Delta t^2\left[\left(\frac{1}{2}-\beta\right)\ddot{u}_t + \beta \ddot{u}_{t+\Delta t}\right] \\ \dot{u}_{t+\Delta t} &= \dot{u}_t + \Delta t\left[(1-\gamma)\ddot{u}_t + \gamma \ddot{u}_{t+\Delta t}\right] \end{aligned} $$

• Solve for $$\displaystyle \ddot{u}_{t+\Delta t} $$ from effective stiffness:

$$\hat{k} u_{t+\Delta t} = \hat{F}_{t+\Delta t}$$

where $$\displaystyle \hat{k} = m + \gamma \Delta t c + \beta \Delta t^2 k $$, $$\displaystyle \hat{F}_{t+\Delta t} = F_{t+\Delta t} + m\left(\frac{u_t}{\Delta t^2} + \frac{\dot{u}_t}{\Delta t} + \left(\frac{1}{2}-\beta\right)\ddot{u}_t\right) + c\left(\dot{u}_t + \Delta t(1-\gamma)\ddot{u}_t\right) $$.

• Stability:

  • $$\displaystyle \beta = 1/4 $$ (average acceleration): unconditionally stable.

  • $$\displaystyle \beta = 1/6 $$ (linear acceleration): conditionally stable if $$\displaystyle \Delta t \leq \frac{2}{\omega_n} $$.

• Common choice: $$\displaystyle \beta=1/4 $$, $$\displaystyle \gamma=1/2 $$.

[!TIP]

Newmark is implicit for $$\displaystyle \beta>0 $$. For explicit ($$\displaystyle \beta=0 $$), conditionally stable with very small $\Delta t$.

III. MULTI-DEGREE OF FREEDOM (MDOF) SYSTEMS – THEORY & FORMULATION

Matrix Formulation of Equations of Motion

• General form:

$$[M]\{\ddot{u}\} + [C]\{\dot{u}\} + [K]\{u\} = \{F(t)\}$$

where $[M]$, $[C]$, $[K]$ are mass, damping, stiffness matrices (symmetric positive definite for stable systems).

• Assembly: Element matrices assembled using connectivity (stiffness method) or consistent mass formulation.

Eigenvalue Problem (Free Vibration)

• Assume $$\displaystyle \{u\} = \{\phi\} e^{i\omega t} $$ → $$\displaystyle ([K] - \omega^2[M])\{\phi\} = \{0\} $$.

• Natural frequencies $$\displaystyle \omega_i $$: roots of $$\displaystyle \det([K] - \omega^2[M]) = 0 $$.

• Mode shapes $$\displaystyle \{\phi_i\} $$: eigenvectors corresponding to $$\displaystyle \omega_i $$.

• Orthogonality Properties (for distinct $$\displaystyle \omega_i $$):

$$ \begin{aligned} \{\phi_i\}^T[M]\{\phi_j\} &= 0 \quad (i \neq j) \quad \text{(mass-orthogonality)} \\ \{\phi_i\}^T[K]\{\phi_j\} &= 0 \quad (i \neq j) \quad \text{(stiffness-orthogonality)} \end{aligned} $$

• Mode Normalization:

  • Mass-normalization: $$\displaystyle \{\phi_i\}^T[M]\{\phi_i\} = 1 $$

  • Max displacement = 1: $$\displaystyle \max(\phi_{ij}) = 1 $$

Solution Methods for Eigenvalue Problem

• Matrix Iteration Methods:

  • Power method: For fundamental (largest) frequency. Iterate $$\displaystyle \{\phi^{(k+1)}\} = [K]^{-1}[M]\{\phi^{(k)}\} $$, normalize. Converges to $$\displaystyle \omega_1^2 = 1/\lambda_1 $$ where $$\displaystyle \lambda_1 $$ is largest eigenvalue of $$\displaystyle [K]^{-1}[M] $$.

  • Inverse iteration: For smallest frequency (highest mode).

• Holzer Method (for torsional systems or general MDOF):

  1. Assume a trial frequency $\omega$.

  2. Starting from one end, compute torque in each shaft using $$\displaystyle T_i = J_i \omega^2 \theta_i - T_{i-1} $$.

  3. Check if torque at other end matches boundary condition (e.g., zero at free end).

  4. Adjust $\omega$ until convergence.

• Standard Eigenvalue Solvers: Jacobi method for symmetric $[K] - \lambda[M]$, or reduce to standard form $$\displaystyle [M]^{-1}[K]\{\phi\} = \omega^2 \{\phi\} $$ then use QR algorithm.

[!TIP]

Holzer is hand-calculation friendly for systems with many DOF but sparse matrices (torsional). Matrix iteration is computational.

IV. RESPONSE ANALYSIS OF MDOF SYSTEMS

Modal Superposition / Decoupling

• Concept: Express displacement as sum of modal contributions:

$$\{u(t)\} = \sum_{i=1}^{N} \{\phi_i\} q_i(t)$$

where $$\displaystyle q_i(t) $$ are generalized coordinates.

• Substitute into EOM, pre-multiply by $$\displaystyle \{\phi_j\}^T $$, use orthogonality:

$$m_i^* \ddot{q}_i + c_i^* \dot{q}_i + k_i^* q_i = Q_i^*(t)$$

with:

$$ \begin{aligned} m_i^* &= \{\phi_i\}^T[M]\{\phi_i\} \quad \text{(generalized mass)} \\ k_i^* &= \{\phi_i\}^T[K]\{\phi_i\} = \omega_i^2 m_i^* \\ c_i^* &= \{\phi_i\}^T[C]\{\phi_i\} \quad \text{(modal damping)} \\ Q_i^*(t) &= \{\phi_i\}^T\{F(t)\} \quad \text{(generalized force)} \end{aligned} $$

• Uncoupling achieved if damping is proportional (Rayleigh damping: $$\displaystyle [C] = \alpha[M] + \beta[K] $$), then $$\displaystyle c_i^* = \alpha m_i^* + \beta k_i^* $$, diagonal.

Response to Dynamic Loading

• For arbitrary $\{F(t)\}$, solve each uncoupled SDOF equation for $$\displaystyle q_i(t) $$ (e.g., Duhamel, numerical methods), then sum.

• Response Spectrum Method (for seismic loads):

  • Construction: For a given ground motion $$\displaystyle \ddot{u}_g(t) $$, compute peak response (displacement $$\displaystyle S_d $$, velocity $$\displaystyle S_v $$, acceleration $$\displaystyle S_a $$) of SDOF systems over range of $$\displaystyle \omega_n $$ and $\zeta$.

  • MDOF approximation: Modal displacement peak:

    $$\displaystyle u_i^{max} = \Gamma_i \cdot S_d(\omega_i, \zeta_i) $$

    where modal participation factor $$\displaystyle \Gamma_i = \frac{\{\phi_i\}^T[M]\mathbf{1}}{\{\phi_i\}^T[M]\{\phi_i\}} $$ ($\mathbf{1}$ = vector of ones for uniform acceleration).

  • Combination of modal responses:

    • SRSS (Square Root of Sum of Squares):

      $$\displaystyle R^{max} = \sqrt{\sum_{i=1}^{N} (R_i^{max})^2} $$

      Applicable when modes are well-separated ($$\displaystyle \omega_i/\omega_j > 1.5 $$).

    • CQC (Complete Quadratic Combination):

      $$\displaystyle R^{max} = \sqrt{\sum_{i=1}^{N} \sum_{j=1}^{N} \rho_{ij} R_i^{max} R_j^{max}} $$

      with mode correlation coefficient:

$$ \rho_{ij} = \frac{8\zeta^2 (r^{1+\zeta})}{(1-r^2)^2 + 4\zeta^2 r^2}, \quad r = \omega_i/\omega_j \text{ or } \omega_j/\omega_i \text{ (take } r \geq 1\text{)} $$

  Used for **closely spaced modes**.

[!TIP]

Static vs Dynamic Analysis: Dynamic analysis essential when fundamental period $$\displaystyle T_1 > 0.1 $$ s or when higher mode effects significant (tall buildings).

V. EARTHQUAKE ENGINEERING & SEISMIC ANALYSIS (IS 1893:2002)

Dynamic Analysis Procedure as per IS 1893:2002

1. Lumped Mass Model: Concentrate masses at floor levels (including live load participation factor).

2. Seismic Force Distribution:

   • Base shear: $$\displaystyle V_b = A_h \cdot W $$, where $$\displaystyle A_h $$ = horizontal seismic coefficient from Table 3 (based on seismic zone, importance factor, and $$\displaystyle T_1 $$), $W$ = total seismic weight.

   • Lateral force at floor $i$:

     $$\displaystyle F_i = \frac{w_i h_i^k}{\sum_{j=1}^{N} w_j h_j^k} \cdot V_b $$

     where $$\displaystyle w_i $$ = weight at floor $i$, $$\displaystyle h_i $$ = height from base, $k$ = exponent:

     • For moment-resisting frames: $$\displaystyle k = 1 $$ if $$\displaystyle T_1 \leq 0.75 $$ s, $$\displaystyle k = 2 $$ if $$\displaystyle T_1 \geq 2.0 $$ s, linear interpolation in between.

3. Fundamental Time Period $$\displaystyle T_1 $$:

   • For moment-resisting frames (RC/steel):

     $$\displaystyle T_1 = 0.075 h^{0.75} $$ (for RC, $h$ in meters)

     or $$\displaystyle T_1 = 0.09 h / \sqrt{d} $$ (steel, $d$ = base dimension in direction of vibration).

   • For shear walls: $$\displaystyle T_1 = \frac{2\pi}{\sqrt{g}} \sqrt{\frac{\sum_{i=1}^{N} (m_i \delta_i)}{\sum_{i=1}^{N} (f_i \delta_i)}} $$ (exact method).

Combination of Modal Responses

• SRSS: Simple, conservative for separated modes. May overestimate for close modes.

• CQC: Accounts for modal coupling, more accurate for closely spaced modes. $$\displaystyle \rho_{ij} \to 0 $$ when $r \to \infty$ or $r \to 0$, reduces to SRSS for well-separated modes.

[!TIP]

In IS 1893, CQC is recommended when $$\displaystyle T_1 $$ and $$\displaystyle T_2 $$ are close (difference < 10%). Always check mode spacing.

VI. CONTINUOUS SYSTEMS – NATURAL VIBRATIONS

Axial Vibration of a Bar

• EOM (1D wave equation):

$$\frac{\partial}{\partial x}\left(EA \frac{\partial u}{\partial x}\right) = \rho A \frac{\partial^2 u}{\partial t^2}$$

• For uniform bar ($EA$, $\rho A$ constant):

$$\frac{\partial^2 u}{\partial x^2} = \frac{1}{c^2} \frac{\partial^2 u}{\partial t^2}, \quad c = \sqrt{E/\rho}$$

• Boundary Conditions:

  • Fixed: $$\displaystyle u=0 $$

  • Free: $$\displaystyle \frac{\partial u}{\partial x}=0 $$ (stress-free)

• Solution (separation of variables $$\displaystyle u(x,t)=\phi(x)\sin\omega t $$):

  $$\displaystyle \phi(x) = A\sin(kx) + B\cos(kx) $$, $$\displaystyle k = \omega/c $$.

• Natural frequencies:

$$\omega_n = \frac{n\pi}{L}\sqrt{\frac{E}{\rho}} \quad \text{(fixed-fixed)}$$

For clamped-free (fixed at $$\displaystyle x=0 $$, free at $$\displaystyle x=L $$):

$$\displaystyle u(0)=0 $$, $$\displaystyle \frac{du}{dx}(L)=0 $$ → $$\displaystyle \cos(kL)=0 $$ → $$\displaystyle kL = (2n-1)\frac{\pi}{2} $$

$$\omega_n = \frac{(2n-1)\pi}{2L}\sqrt{\frac{E}{\rho}}$$

Bending Vibration of Euler-Bernoulli Beam

• EOM:

$$\frac{\partial^2}{\partial x^2}\left(EI \frac{\partial^2 u}{\partial x^2}\right) + \rho A \frac{\partial^2 u}{\partial t^2} = 0$$

• For uniform beam:

$$EI \frac{\partial^4 u}{\partial x^4} + \rho A \frac{\partial^2 u}{\partial t^2} = 0$$

• Assume $$\displaystyle u(x,t) = \phi(x)\sin\omega t $$ →

  $$\displaystyle EI \phi'''' - \rho A \omega^2 \phi = 0 $$.

  Let $$\displaystyle \beta^4 = \frac{\rho A \omega^2}{EI} $$, so $$\displaystyle \phi(x) = A\sin\beta x + B\cos\beta x + C\sinh\beta x + D\cosh\beta x $$.

• Boundary Conditions (examples):

  • Clamped (fixed): $$\displaystyle u=0 $$, $$\displaystyle \frac{du}{dx}=0 $$

  • Simply supported (pinned): $$\displaystyle u=0 $$, $$\displaystyle \frac{d^2u}{dx^2}=0 $$

  • Free: $$\displaystyle \frac{d^2u}{dx^2}=0 $$, $$\displaystyle \frac{d^3u}{dx^3}=0 $$

• Characteristic equations:

  • Cantilever (clamped-free): $$\displaystyle \cos\beta L \cosh\beta L = -1 $$

    First three $\beta L$: 1.875, 4.694, 7.855

$$\omega_n = \beta_n^2 \sqrt{\frac{EI}{\rho A L^4}}$$

• Fixed-fixed: $$\displaystyle \cos\beta L \cosh\beta L = 1 $$ → $\beta L$: 4.730, 7.853, 10.996, ...

• Pinned-pinned: $$\displaystyle \sin\beta L = 0 $$ → $$\displaystyle \beta L = n\pi $$ → $$\displaystyle \omega_n = \frac{n^2\pi^2}{L^2}\sqrt{\frac{EI}{\rho A}} $$

Example: Cantilever Beam (Clamped-Free)

• First three frequencies:

  $$\displaystyle \omega_1 = (1.875)^2 \sqrt{\frac{EI}{\rho A L^4}} $$,

  $$\displaystyle \omega_2 = (4.694)^2 \sqrt{\frac{EI}{\rho A L^4}} $$,

  $$\displaystyle \omega_3 = (7.855)^2 \sqrt{\frac{EI}{\rho A L^4}} $$.

• Mode shapes (mass-normalized):

$$\phi_n(x) = \cos\beta_n x - \cosh\beta_n x - \frac{\cos\beta_n L + \cosh\beta_n L}{\sin\beta_n L + \sinh\beta_n L}(\sin\beta_n x - \sinh\beta_n x)$$

• Sketch: First mode: one half-sine, no nodes except clamped end. Second: one node, third: two nodes.

[!TIP]

For beam with discrete masses/springs (e.g., Fig. 6 from Nov 2022):

1. Model as lumped-mass system with equivalent stiffnesses.

2. Write EOM in terms of displacements $$\displaystyle u_1, u_2 $$.

3. Form $[M]$, $[K]$, solve eigenvalue problem $$\displaystyle ([K] - \omega^2[M])\{\phi\}=0 $$.

VII. VARIATIONAL PRINCIPLES & ADVANCED TOPICS

Hamilton’s Principle

• Statement: For conservative systems,

$$\delta \int_{t_1}^{t_2} (T - V) dt = 0$$

where $T$ = kinetic energy, $V$ = potential energy.

• Application to derive EOM (SDOF example):

  $$\displaystyle T = \frac{1}{2}m\dot{u}^2 $$, $$\displaystyle V = \frac{1}{2}ku^2 $$

  $$\displaystyle \int_{t_1}^{t_2} \left( m\dot{u}\delta\dot{u} - ku\delta u \right) dt = 0 $$

  Integrate by parts, apply $$\displaystyle \delta u(t_1)=\delta u(t_2)=0 $$ →

  $$\displaystyle m\ddot{u} + ku = 0 $$.

Vibration Isolation

• Concept: Reduce force transmission from a vibrating machine to foundation.

• Transmissibility $T$ = ratio of transmitted force to applied force.

• For force excitation $$\displaystyle F_0\sin\omega t $$ on mass:

$$T = \frac{\sqrt{1 + (2\zeta r)^2}}{\sqrt{(1-r^2)^2 + (2\zeta r)^2}}, \quad r = \frac{\omega}{\omega_n}$$

• Isolation region: $$\displaystyle r > \sqrt{2} $$ and $\zeta$ small → $$\displaystyle T < 1 $$.

• Design: Choose $$\displaystyle \omega_n $$ such that operating $\omega$ is in isolation region.

[!TIP]

For base excitation (earthquake), transmissibility of absolute acceleration is same formula. Isolation effective when $$\displaystyle r > \sqrt{2} $$.

Fourier Transform Methods

• Application: Solve vibration problems with arbitrary forcing by transforming to frequency domain.

• Take Fourier transform (FT) of EOM:

  $$\displaystyle [M](-\omega^2)\tilde{U}(\omega) + [C](i\omega)\tilde{U}(\omega) + [K]\tilde{U}(\omega) = \tilde{F}(\omega) $$

• Solve: $$\displaystyle \tilde{U}(\omega) = [H(\omega)]\tilde{F}(\omega) $$, where $[H(\omega)]$ is frequency response function matrix.

• Inverse FT gives $u(t)$.

• Advantage: Handles non-periodic forces, efficient for broadband excitation.

Uncoupling of Equations of Motion

• Mathematical procedure: Use modal matrix $$\displaystyle \Phi = [\{\phi_1\} \{\phi_2\} \cdots \{\phi_N\}] $$.

• Pre-multiply EOM by $$\displaystyle \Phi^T $$:

  $$\displaystyle \Phi^T[M]\Phi \ddot{q} + \Phi^T[C]\Phi \dot{q} + \Phi^T[K]\Phi q = \Phi^T F(t) $$

• If $\Phi$ contains mass-orthonormal modes ($$\displaystyle \Phi^T[M]\Phi = I $$) and damping is proportional ($$\displaystyle \Phi^T[C]\Phi = \text{diag}(2\zeta_i\omega_i) $$), then:

$$\ddot{q}_i + 2\zeta_i\omega_i \dot{q}_i + \omega_i^2 q_i = Q_i^*(t)$$

Decoupled into independent SDOF equations.

VIII. SPECIAL FOCUS: SHORT NOTES TOPICS (From Past Exams)

Fourier Transform Methods (Sec. VII)

• Transform EOM to algebraic equation in frequency domain.

• Solve for $\tilde{U}(\omega)$, inverse FT.

• Useful for non-periodic forces (e.g., impulses, random vibrations).

Response to Unit Impulse (Sec. I)

• Unit impulse: $$\displaystyle F(t) = \delta(t) $$ (Dirac delta).

• Impulse response function $h(t)$: solution to $$\displaystyle m\ddot{h} + c\dot{h} + kh = \delta(t) $$, $$\displaystyle h(0^-)=0 $$, $$\displaystyle \dot{h}(0^-)=0 $$.

• For undamped: $$\displaystyle h(t) = \frac{1}{m\omega_n}\sin\omega_n t $$ ($$\displaystyle t>0 $$).

• For damped: $$\displaystyle h(t) = \frac{1}{m\omega_d} e^{-\zeta\omega_n t} \sin\omega_d t $$ ($$\displaystyle t>0 $$).

• Duhamel’s integral for arbitrary force: $$\displaystyle u(t) = \int_0^t F(\tau) h(t-\tau) d\tau $$.

Vibration Isolation (Sec. VII)

• Goal: reduce transmitted force/acceleration.

• Transmissibility $$\displaystyle T = \frac{\sqrt{1+(2\zeta r)^2}}{\sqrt{(1-r^2)^2+(2\zeta r)^2}} $$.

• Isolation when $$\displaystyle r > \sqrt{2} $$.

• Design: select $$\displaystyle \omega_n $$ low enough so that operating $r$ is in isolation region.

Arbitrary Force (Sec. I & IV)

• SDOF: Use Duhamel’s integral (undamped) or convolution with impulse response (damped).

• MDOF: Modal superposition → solve each modal SDOF with arbitrary $$\displaystyle Q_i^*(t) $$.

Equation of Motion (Sec. I & III)

• SDOF: $$\displaystyle m\ddot{u} + c\dot{u} + ku = F(t) $$ (relative coordinate).

• MDOF: $$\displaystyle [M]\{\ddot{u}\} + [C]\{\dot{u}\} + [K]\{u\} = \{F(t)\} $$.

• Derivation via Newton’s law or D’Alembert’s principle (SDOF), Lagrange’s equations or finite element assembly (MDOF).

Matrix Formulation (Sec. III)

• Assemble $[M]$, $[C]$, $[K]$ from element matrices.

• General EOM: $$\displaystyle [M]\{\ddot{u}\} + [C]\{\dot{u}\} + [K]\{u\} = \{F(t)\} $$.

• For free vibration: $$\displaystyle ([K] - \omega^2[M])\{\phi\} = \{0\} $$.

Hamilton Principle (Sec. VII)

• $$\displaystyle \delta \int_{t_1}^{t_2} (T - V) dt = 0 $$ for conservative systems.

• Derives EOM without considering forces directly.

Holzer Method (Sec. III)

• For torsional systems:

  1. Assume trial $\omega$.

  2. Starting from one end, compute torque in each shaft: $$\displaystyle T_i = J_i \omega^2 \theta_i - T_{i-1} $$.

  3. Adjust $\omega$ until torque at other end satisfies BC (e.g., $$\displaystyle T_N=0 $$ for free end).

• Advantage: Hand calculation for systems with many DOF.

Response Spectrum (Sec. IV)

• Plot of peak response (displacement $$\displaystyle S_d $$, velocity $$\displaystyle S_v $$, acceleration $$\displaystyle S_a $$) vs. natural period $T$ for a given ground motion and damping $\zeta$.

• Use in MDOF: Approximate modal peak responses $$\displaystyle u_i^{max} = \Gamma_i S_d(\omega_i, \zeta_i) $$.

• Construction: Solve SDOF for range of $$\displaystyle \omega_n $$, record maxima.

Uncoupling of Equations of Motion (Sec. IV)

• Use modal matrix $\Phi$ with mass-orthonormal modes.

• Pre-multiply EOM by $$\displaystyle \Phi^T $$ → diagonal matrices if damping proportional.

• Result: $N$ independent SDOF equations.

Key Formulas Summary

Final Exam Strategy:

1. Derivations: Practice EOM (Newton/D’Alembert), $$\displaystyle \omega_n $$, $\delta$, Duhamel, orthogonality, Hamilton’s principle.

2. Numerical Problems: SDOF forced vibration, Newmark’s method, eigenvalue for 2-DOF, IS 1893 base shear.

3. Conceptual Questions: Distinguish damping types, SRSS vs CQC, vibration isolation regions.

4. Diagrams: Sketch mode shapes for cantilever beam, explain Holzer method flowchart, response spectrum sketch.

5. Code Application: IS 1893 period formulas, force distribution, modal combination rules.