UNIT 5: STRUCTURAL DYNAMICS - EXAM-FOCUSED SHORT NOTES

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

Equation of Motion (EOM) Formulation

• Newton's Second Law: Sum of forces = mass × acceleration.

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

where $m$ = mass, $c$ = damping coefficient, $k$ = stiffness, $u$ = displacement.

• D'Alembert's Principle: Incorporates inertia force $-m\ddot{u}$ as a fictitious force. Applied to complex systems (e.g., rotating masses, pulleys) by drawing a Free Body Diagram (FBD) and summing all forces (including $-m\ddot{u}$) to zero.

  [!TIP] For figure-based questions (like Fig 2, 3 from Nov 2022), first identify the generalized coordinate (displacement $u$), then write expressions for spring force ($$\displaystyle k u_{eq} $$), damping force ($c \dot{u}$), and inertia force ($-m\ddot{u}$).

Free Vibration Analysis

• Undamped ($$\displaystyle c=0 $$): $$\displaystyle m\ddot{u} + ku = 0 $$

  • Natural Frequency: $$\displaystyle \omega_n = \sqrt{\frac{k}{m}} $$ (rad/s)

  • Period: $$\displaystyle T = \frac{2\pi}{\omega_n} = 2\pi\sqrt{\frac{m}{k}} $$

• Damped ($$\displaystyle c>0 $$): $$\displaystyle m\ddot{u} + c\dot{u} + ku = 0 $$

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

  • Damped Natural Frequency: $$\displaystyle \omega_d = \omega_n\sqrt{1-\zeta^2} $$ (for $$\displaystyle \zeta < 1 $$).

  • Logarithmic Decrement ($\delta$): Ratio of successive amplitudes in underdamped vibration.

$$\delta = \ln\left(\frac{u_i}{u_{i+1}}\right) = \frac{2\pi\zeta}{\sqrt{1-\zeta^2}}$$

    > [!IMPORTANT] $\delta$ is used to **experimentally determine $\zeta$**. Sketch shows exponential decay; $\delta$ increases with $\zeta$.

*   **Critically Damped ($$\displaystyle \zeta = 1 $$):** System returns to equilibrium fastest without oscillation. $$\displaystyle c = c_c = 2\sqrt{km} $$.

Forced Vibration Analysis

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

  • Steady-State Response: $$\displaystyle u(t) = U\sin(\omega t - \phi) $$

  • Amplitude: $$\displaystyle U = \frac{F_0/k}{\sqrt{(1-r^2)^2 + (2\zeta r)^2}} $$, where $$\displaystyle r = \omega/\omega_n $$ (frequency ratio).

  • Phase Angle: $$\displaystyle \tan\phi = \frac{2\zeta r}{1-r^2} $$.

  • Resonance: Maximum $U$ occurs at $$\displaystyle r = \sqrt{1-2\zeta^2} $$ (for $$\displaystyle \zeta < 0.707 $$). At resonance, $$\displaystyle \phi = 90^\circ $$.

• Response to Arbitrary Force: Use principle of superposition.

Numerical Evaluation Methods

• Newmark's Method (β-method): Implicit time-stepping scheme for numerical integration.

  • Assumptions: $$\displaystyle \dot{u}_{t+\Delta t} = \dot{u}_t + \Delta t[(1-\gamma)\ddot{u}_t + \gamma\ddot{u}_{t+\Delta t}] $$

  $$\displaystyle u_{t+\Delta t} = u_t + \Delta t\dot{u}_t + \frac{\Delta t^2}{2}[(1-2\beta)\ddot{u}_t + 2\beta\ddot{u}_{t+\Delta t}] $$

  • Common Choice: $$\displaystyle \gamma = 1/2 $$, $$\displaystyle \beta = 1/4 $$ → Average Acceleration Method (unconditionally stable).

  • Procedure: At each step, solve $$\displaystyle [M]\ddot{u}_{t+\Delta t} + [C]\dot{u}_{t+\Delta t} + [K]u_{t+\Delta t} = F_{t+\Delta t} $$ for $$\displaystyle \ddot{u}_{t+\Delta t} $$, then update $\dot{u}$ and $u$.

• Duhamel's Integral (Convolution Integral) for Undamped Systems:

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

*   **Unit Impulse Response ($$\displaystyle F(t) = I\delta(t) $$):** $$\displaystyle h(t) = \frac{I}{m\omega_n}\sin(\omega_n t) $$.

*   **Rectangular Pulse (Height $A$, Duration $$\displaystyle T_p $$):** $$\displaystyle F(t) = A[1(t) - 1(t-T_p)] $$.

$$u(t) = \frac{A}{k} \left[ 1 - \frac{\sin(\omega_n t - \phi)}{\sin\phi} \right] \text{ for } t > T_p, \text{ where } \phi = \omega_n T_p.$$

Laplace Transform Methods

• Rectangular Pulse: $$\displaystyle F(t) = A[1(t) - 1(t-T_p)] \xrightarrow{\mathcal{L}} F(s) = \frac{A}{s}(1 - e^{-sT_p}) $$.

• Unit Impulse (Dirac Delta): $\delta(t) \xrightarrow{\mathcal{L}} 1$.

  • Deduced as limit of rectangular pulse as $A \to \infty$, $$\displaystyle T_p \to 0 $$, with area = 1.

• Solving EOM: Take Laplace of $$\displaystyle m\ddot{u} + c\dot{u} + ku = F(t) $$ with zero ICs:

$$U(s) = \frac{F(s)}{ms^2 + cs + k}. \quad u(t) = \mathcal{L}^{-1}\{U(s)\}.$$

2. DAMPING IN STRUCTURAL SYSTEMS

• Equivalent Viscous Damping: Non-viscous damping is represented by an equivalent viscous damper that dissipates same energy per cycle.

$$c_{eq} = \frac{\text{Energy dissipated per cycle}}{2\pi \omega_n \times (\text{max strain energy})}$$

• Damping Ratio from Experiment: $$\displaystyle \zeta = \frac{\delta}{\sqrt{4\pi^2 + \delta^2}} $$ (from logarithmic decrement $\delta$).

3. MULTI-DEGREE OF FREEDOM (MDOF) SYSTEMS

Matrix Formulation of EOM

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

• $[M]$: Mass matrix (symmetric, positive definite).

• $[K]$: Stiffness matrix (symmetric, positive semi-definite).

• $[C]$: Damping matrix (often assumed proportional: $$\displaystyle [C] = \alpha[M] + \beta[K] $$).

Eigenvalue Problem (Free Vibration)

$$([K] - \omega^2[M])\{\phi\} = \{0\}$$

• Characteristic Equation: $$\displaystyle \det([K] - \omega^2[M]) = 0 $$ → gives $n$ natural frequencies $$\displaystyle \omega_1, \omega_2, ..., \omega_n $$.

• Mode Shapes (Eigenvectors): $$\displaystyle \{\phi^{(1)}\}, \{\phi^{(2)}\}, ... $$ corresponding to each $$\displaystyle \omega_i $$.

Methods for Solving Eigenvalue Problems

• Matrix Iteration Method (Power Method):

  1. Guess initial vector $$\displaystyle \{x_0\} $$.

  2. Solve $$\displaystyle [K]\{x_{i+1}\} = [M]\{x_i\} $$ (or vice versa).

  3. Normalize $$\displaystyle \{x_{i+1}\} $$ (e.g., max component = 1).

  4. Compute Rayleigh quotient: $$\displaystyle \omega^2 = \frac{\{x_{i+1}\}^T[K]\{x_{i+1}\}}{\{x_{i+1}\}^T[M]\{x_{i+1}\}} $$.

  5. Iterates to fundamental (lowest) frequency and mode.

• Holzer Method: Hand-calculation method for torsional systems or systems with known $[J]$ (mass moment of inertia) and $[K]$ (torsional stiffness). Based on trial-and-error for $\omega$ until all amplitudes are consistent.

  [!TIP] Matrix Iteration gives fundamental mode. To find higher modes, use orthogonality to deflate the matrix or use inverse iteration.

Mode Shapes & Normalization

• Physical Significance: Each mode is a standalone independent vibration pattern at its natural frequency.

• Normalization:

  1. Mass Normalization: $$\displaystyle \{\phi_i\}^T[M]\{\phi_i\} = 1 $$.

  2. Unity at a Point: Set a specific displacement component = 1.

  3. Stiffness Normalization: $$\displaystyle \{\phi_i\}^T[K]\{\phi_i\} = \omega_i^2 $$.

• Orthogonality Properties:

$$\{\phi_i\}^T[M]\{\phi_j\} = 0 \quad \text{and} \quad \{\phi_i\}^T[K]\{\phi_j\} = 0 \quad \text{for } i \neq j.$$

This is key for **modal superposition**.

Forced Vibration: Modal Superposition

1. Assume solution: $$\displaystyle \{u\} = [\Phi]\{q(t)\} $$, where $$\displaystyle [\Phi] = [\{\phi^{(1)}\} \{\phi^{(2)}\} ...] $$ (mode shape matrix).

2. Pre-multiply by $$\displaystyle [\Phi]^T $$: $$\displaystyle [\Phi]^T[M][\Phi]\{\ddot{q}\} + [\Phi]^T[C][\Phi]\{\dot{q}\} + [\Phi]^T[K][\Phi]\{q\} = [\Phi]^T\{F(t)\} $$.

3. Using orthogonality, equations uncouple:

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

where $$\displaystyle m_i = \{\phi_i\}^T[M]\{\phi_i\} $$, $$\displaystyle k_i = \{\phi_i\}^T[K]\{\phi_i\} = m_i\omega_i^2 $$, $$\displaystyle Q_i = \{\phi_i\}^T\{F(t)\} $$.

4. Solve each SDOF equation independently for $$\displaystyle q_i(t) $$.

5. Total response: $$\displaystyle \{u(t)\} = \sum_{i=1}^n \{\phi_i\} q_i(t) $$.

Modal Combination (IS 1893)

• SRSS (Square Root of Sum of Squares):

$$R_{max} = \sqrt{\sum_{i=1}^n (R_i)^2}$$

*   **Applicable when:** Natural frequencies are well separated ($$\displaystyle \omega_i/\omega_{i-1} > 1.5 $$).

• CQC (Complete Quadratic Combination):

$$R_{max} = \sqrt{\sum_{i=1}^n \sum_{j=1}^n \delta_{ij} R_i R_j}$$

where $$\displaystyle \delta_{ij} = \frac{8\zeta^2 (r_i r_j)^{1.5}}{(1-r_i^2)(1-r_j^2) + 4\zeta^2 r_i r_j (1+r_i r_j)} $$, $$\displaystyle r_i = \omega_i/\omega_j $$.

*   **Applicable for:** Closely spaced modes. Accounts for **mode coupling**.

> [!IMPORTANT] **IS 1893:2002** uses **CQC** for regular structures and **SRSS** for irregular ones. For base shear, use **SRSS** of modal base shears.

4. CONTINUOUS SYSTEMS (Distributed Parameter)

Axial Vibration of a Bar

• EOM (PDE): $$\displaystyle \frac{\partial}{\partial x}\left(EA\frac{\partial u}{\partial x}\right) = \rho A \frac{\partial^2 u}{\partial t^2} $$

  For uniform bar: $$\displaystyle \frac{\partial^2 u}{\partial x^2} = \frac{1}{c_a^2} \frac{\partial^2 u}{\partial t^2} $$, where $$\displaystyle c_a = \sqrt{E/\rho} $$ (wave speed).

• BCs: Fixed ($$\displaystyle u=0 $$), Free ($$\displaystyle \partial u/\partial x = 0 $$).

• Natural Frequencies & Modes: $$\displaystyle \omega_n = \frac{n\pi c_a}{L} $$ for fixed-fixed; $$\displaystyle \phi_n(x) = \sin(n\pi x/L) $$.

Bending Vibration of a Beam (Euler-Bernoulli)

• EOM (PDE): $$\displaystyle EI \frac{\partial^4 u}{\partial x^4} + \rho A \frac{\partial^2 u}{\partial t^2} = 0 $$

  or $$\displaystyle \frac{\partial^4 u}{\partial x^4} = \frac{1}{c_b^2} \frac{\partial^2 u}{\partial t^2} $$, $$\displaystyle c_b = \sqrt{EI/(\rho A)} $$.

• BCs (for transverse vibration):

  • Simply Supported: $$\displaystyle u=0 $$, $$\displaystyle M=0 $$ ($$\displaystyle \frac{\partial^2 u}{\partial x^2}=0 $$).

  • Cantilever (Clamped-Free): $$\displaystyle u=0 $$, $$\displaystyle \theta=0 $$ ($$\displaystyle \frac{\partial u}{\partial x}=0 $$) at fixed; $$\displaystyle M=0 $$, $$\displaystyle V=0 $$ ($$\displaystyle \frac{\partial^3 u}{\partial x^3}=0 $$) at free.

  • Fixed-Fixed: $$\displaystyle u=0 $$, $$\displaystyle \theta=0 $$ at both ends.

• Example: Clamped-Free Beam (Cantilever)

  • Frequency Equation: $$\displaystyle \cos\beta L \cosh\beta L = -1 $$.

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

  • Natural Frequencies: $$\displaystyle \omega_n = (\beta_n)^2 \sqrt{EI/(\rho A L^4)} $$.

  • Mode Shapes: $$\displaystyle \phi_n(x) = \cosh(\beta_n x) - \cos(\beta_n x) - \frac{\cosh\beta_n L + \cos\beta_n L}{\sinh\beta_n L + \sin\beta_n L} (\sinh(\beta_n x) - \sin(\beta_n x)) $$.

  [!DIAGRAM] Sketch: First mode (1.875) - one anti-node at tip; Second mode (4.694) - one node; Third mode (7.855) - two nodes.

Torsional Vibration of a Shaft

• System: Shaft (torsional stiffness $$\displaystyle K_t = GJ/L $$) with disc (mass moment of inertia $J$) at end. Neglect shaft mass.

• EOM: $$\displaystyle J \ddot{\theta} + K_t \theta = 0 $$.

• Natural Frequency: $$\displaystyle \omega_n = \sqrt{\frac{GJ}{JL}} = \sqrt{\frac{G}{\rho L^2}} $$ (if disc is uniform cylinder, $$\displaystyle J = \frac{1}{2}mR^2 $$).

  [!TIP] For Fig 1 (Nov 2022), treat as SDOF torsional system. $$\displaystyle K_t = \sum \frac{GJ_i}{L_i} $$ for series shafts.

5. EARTHQUAKE ENGINEERING & SEISMIC ANALYSIS

Equation of Motion for Base Excitation

• Absolute Displacement: $u(t)$ (relative to fixed ground).

• Base Motion: $$\displaystyle u_g(t) $$ (ground displacement).

• Relative Displacement: $$\displaystyle x(t) = u(t) - u_g(t) $$.

• Modified EOM:

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

or in absolute terms: $$\displaystyle m\ddot{u} + c(\dot{u}-\dot{u}_g) + k(u-u_g) = 0 $$.

• Effective Force Concept: The right-hand side $$\displaystyle -m\ddot{u}_g $$ acts as a pseudo force on the structure.

Dynamic Analysis as per IS 1893:2002 (Lumped Mass Model)

1. Lumped Mass: Mass of each story $$\displaystyle m_i $$ is lumped at floor level.

2. Stiffness: Story stiffness $$\displaystyle k_i $$ derived from lateral stiffness of columns/shear walls.

3. Base Shear ($$\displaystyle V_B $$): Total seismic force at base.

$$V_B = A_h \times W$$

where $$\displaystyle A_h = \frac{Z I S_a}{R g} $$ (seismic coefficient), $W$ = total seismic weight.

4. Distribution of Forces: Force at floor $i$:

$$F_i = \frac{W_i h_i}{\sum W_j h_j} \times V_B$$

where $$\displaystyle h_i $$ = height of floor $i$ from base.

5. Procedure: Apply $$\displaystyle F_i $$ at each floor, perform static analysis (like a lateral load case) to get story shears and displacements.

Response Spectrum

• Definition: Plot of maximum response (displacement, velocity, acceleration) of a SDOF system vs. its natural period ($$\displaystyle T_n $$) for a given damping ratio ($\zeta$) and a specific ground motion.

• Construction: For a given $$\displaystyle \ddot{u}_g(t) $$, solve many SDOF systems ($$\displaystyle \omega_n $$ varies) and record peak $$\displaystyle u_{max} $$, $$\displaystyle \dot{u}_{max} $$, or $$\displaystyle \ddot{u}_{max} + \ddot{u}_g $$.

• Use in Seismic Design (Response Spectrum Method):

  1. Obtain design response spectrum (from code, e.g., IS 1893) for site zone, damping, and importance factor.

  2. For MDOF building, compute natural periods ($$\displaystyle T_i $$) and mode shapes ($$\displaystyle \phi_i $$).

  3. Read spectral acceleration $$\displaystyle S_a(T_i) $$ for each mode.

  4. Compute modal participation factors $$\displaystyle \Gamma_i = \frac{\{\phi_i\}^T[M]\{1\}}{\{\phi_i\}^T[M]\{\phi_i\}} $$.

  5. Modal base shear: $$\displaystyle V_{Bi} = A_{hi} W \Gamma_i $$, where $$\displaystyle A_{hi} = \frac{S_a(T_i)}{g} $$.

  6. Combine modal shears/displacements using CQC (for regular) or SRSS (for irregular) to get design values.

6. SPECIALIZED TOPICS & METHODS (Short Notes)

• Fourier Transform Methods: Converts time-domain signal $f(t)$ to frequency-domain $F(\omega)$. Used to find response spectrum from ground motion by solving for each frequency component.

$$F(\omega) = \int_{-\infty}^{\infty} f(t) e^{-i\omega t} dt$$

• Vibration Isolation: Reduces transmitted force from a vibrating machine to foundation.

  • Transmissibility (TR): $$\displaystyle TR = \frac{F_{trans}}{F_{applied}} = \sqrt{\frac{1+(2\zeta r)^2}{(1-r^2)^2 + (2\zeta r)^2}} $$.

  • Isolation: Requires $$\displaystyle r > \sqrt{2} $$ (above resonant frequency). For $r \gg 1$, $$\displaystyle TR \approx 1/(r^2) $$.

• Hamilton's Principle: Variational principle. For conservative systems, $$\displaystyle \delta \int_{t_1}^{t_2} (T - V) dt = 0 $$, where $T$ = kinetic energy, $V$ = potential energy. Yields EOM.

• Gaussian Plume Model: Air pollution dispersion model (not core structural dynamics, but appears in papers). Predicts concentration $C(x,y,z)$ downwind from a continuous point source.

$$C(x,y,z) = \frac{Q}{2\pi \sigma_y \sigma_z U} \exp\left(-\frac{y^2}{2\sigma_y^2}\right) \left[ \exp\left(-\frac{(z-H)^2}{2\sigma_z^2}\right) + \exp\left(-\frac{(z+H)^2}{2\sigma_z^2}\right) \right]$$

where $Q$ = emission rate, $U$ = wind speed, $H$ = effective stack height, $$\displaystyle \sigma_y, \sigma_z $$ = dispersion parameters (function of downwind distance $x$ and stability class).

SYSTEM ANALYSIS FROM FIGURES (Key Strategy)

For any figure (like Fig 1, 3, 4, 5, 6 from Nov 2022/2023):

1. Identify Generalized Coordinate(s): Single $u$ for SDOF, $$\displaystyle u_1, u_2 $$ for 2-DOF.

2. Write Energy/Force Expressions:

   • Springs: $$\displaystyle F_s = k \times (\text{relative deformation}) $$. For series/parallel, find equivalent stiffness $$\displaystyle k_{eq} $$.

   • Dampers: $$\displaystyle F_d = c \times (\text{relative velocity}) $$.

   • Inertia: $-m\ddot{u}$.

3. Apply Newton's 2nd Law or D'Alembert's Principle: Sum forces on mass(es) = $m\ddot{u}$.

4. For MDOF (Fig 5, 6): Write EOM for each mass. Assemble $[M]$, $[K]$. For free vibration, solve eigenvalue problem $$\displaystyle ([K] - \omega^2[M])\{\phi\}=0 $$ using Matrix Iteration or direct solution for 2-DOF.

5. Normalize Modes: As instructed (e.g., "unit vertical deflection at free end").

Final Exam Checklist:

• [ ] Derive EOM for SDOF (Newton & D'Alembert).

• [ ] Define & sketch Logarithmic Decrement vs. $\zeta$.

• [ ] Write Duhamel's Integral for rectangular pulse.

• [ ] Explain Newmark's method steps (β=1/4, γ=1/2).

• [ ] Set up eigenvalue problem & describe Matrix Iteration.

• [ ] State orthogonality conditions.

• ] Write uncoupled modal equation.

• [ ] Distinguish CQC vs. SRSS.

• [ ] Derive beam EOM (Euler-Bernoulli) & list BCs.

• [ ] Write IS 1893 base shear & force distribution formulas.

• [ ] Sketch response spectrum.

• [ ] Define transmissibility & condition for isolation.