UNIT 1: FUNDAMENTALS OF STRUCTURAL DYNAMICS

1.0 SINGLE-DEGREE-OF-FREEDOM (SDOF) SYSTEMS

1.1 Fundamental Concepts

• Degree of Freedom (DOF): Minimum number of independent coordinates required to define the system's configuration.

  • Discrete systems: Finite DOF (e.g., mass-spring).

  • Continuous systems: Infinite DOF (e.g., beams, bars).

• Idealization: Real structures approximated as SDOF by lumping mass at critical points and assuming deformation shape.

1.2 Equation of Motion (EOM) Formulation

• Newton's Second Law:

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

where $m$ = mass, $c$ = damping coefficient, $k$ = stiffness, $u$ = displacement, $F(t)$ = external force.

• D'Alembert's Principle:

  $$\displaystyle F_{\text{applied}} - m\ddot{u} - c\dot{u} - ku = 0 $$ (inertial force included).

• Earthquake Ground Acceleration ($$\displaystyle u_g $$):

  Relative displacement $$\displaystyle u_r = u - u_g $$ leads to:

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

[!TIP]

Common Pitfall: Forgetting the negative sign when substituting base excitation.

1.3 Free Vibration Analysis

• Undamped Free Vibration:

  $$\displaystyle m\ddot{u} + ku = 0 $$ → Natural frequency:

$$ \omega_n = \sqrt{\frac{k}{m}} \quad \text{(rad/s)} $$

Period: $$\displaystyle T = \frac{2\pi}{\omega_n} $$.

• Damped Free Vibration:

  $$\displaystyle m\ddot{u} + c\dot{u} + ku = 0 $$ → Damping ratio:

$$ \xi = \frac{c}{2\sqrt{km}} = \frac{c}{c_c} $$

where $$\displaystyle c_c = 2\sqrt{km} $$ (critical damping).

• Types of Damping:

  | Type | Force-Velocity Relation | Typical Use | |------|------------------------|-------------| | Viscous | $$\displaystyle F = c\dot{u} $$ | Common in analysis | | Coulomb (Friction) | $$\displaystyle F = \mu N \cdot \text{sgn}(\dot{u}) $$ | Sliding interfaces | | Structural (Hysteretic) | $F \propto \dot{u}$ with phase lag | Material damping |

• Critical Damping:

  $$\displaystyle c = c_c $$ → System returns to equilibrium fastest without oscillation.

  Example: Shock absorbers in vehicles.

• Logarithmic Decrement ($\delta$):

$$ \delta = \ln\left(\frac{x_1}{x_2}\right) = \frac{2\pi\xi}{\sqrt{1-\xi^2}} $$

[!TIP]

For small $\xi$, $\delta \approx 2\pi\xi$.

1.4 Forced Vibration Analysis

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

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

$$ U = \frac{F_0/k}{\sqrt{(1-r^2)^2 + (2\xi r)^2}}, \quad \phi = \tan^{-1}\left(\frac{2\xi r}{1-r^2}\right) $$

$$\displaystyle r = \omega/\omega_n $$ (frequency ratio).

Resonance: $$\displaystyle \omega \approx \omega_n $$ for $\xi \approx 0$ → $U \to \infty$ (theoretical).

• Base Excitation (Seismic):

  Transmissibility for displacement:

$$ T_d = \frac{U}{u_g} = \frac{r^2}{\sqrt{(1-r^2)^2 + (2\xi r)^2}} $$

For $$\displaystyle r > \sqrt{2} $$, $$\displaystyle T_d < 1 $$ (isolation).

• Arbitrary Forcing:

  Principle of superposition: Response = sum of responses to individual force components.

1.5 Key Parameters & Properties

• Stiffness ($k$): Force per unit displacement.

• Effective Stiffness for spring systems:

  • Series: $$\displaystyle 1/k_{\text{eq}} = 1/k_1 + 1/k_2 $$

  • Parallel: $$\displaystyle k_{\text{eq}} = k_1 + k_2 $$

• Natural Frequency for Simple Systems:

  • Mass-spring: $$\displaystyle \omega_n = \sqrt{k/m} $$

  • Simple pendulum: $$\displaystyle \omega_n = \sqrt{g/L} $$

  • Compound pendulum: $$\displaystyle \omega_n = \sqrt{\frac{mgh}{I}} $$

2.0 ANALYTICAL METHODS FOR FORCED VIBRATION RESPONSE

2.1 Duhamel's Integral (Principle of Superposition)

• Concept: Response to arbitrary force $F(t)$ as integral of impulse responses.

• Undamped SDOF:

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

• Unit Impulse Response ($$\displaystyle F(t) = \delta(t) $$):

$$ h(t) = \frac{1}{m\omega_n} \sin\omega_n t \quad (t \geq 0) $$

• Rectangular Pulse (height $$\displaystyle F_0 $$, duration $T$):

$$ u(t) = \begin{cases} \frac{F_0}{k}(1 - \cos\omega_n t) & 0 \leq t \leq T \\ \frac{F_0}{k}(\cos\omega_n(t-T) - \cos\omega_n t) & t > T \end{cases} $$

2.2 Laplace Transform Methods

• Laplace Transform of $F(t)$:

$$ F(s) = \mathcal{L}\{F(t)\} = \int_0^\infty F(t)e^{-st} dt $$

• Rectangular Pulse:

$$ F(t) = F_0[1(t) - 1(t-T)] \quad \Rightarrow \quad F(s) = \frac{F_0(1-e^{-sT})}{s} $$

• Unit Impulse (Dirac Delta):

$$ \mathcal{L}\{\delta(t)\} = 1 $$

• Solving EOM:

  Transform EOM: $$\displaystyle (ms^2 + cs + k)U(s) = F(s) + \text{initial conditions} $$.

  Inverse transform to get $u(t)$.

2.3 Fourier Transform Methods

• Fourier Transform of $F(t)$:

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

• Frequency Response Function (FRF):

$$ H(\omega) = \frac{1}{k - m\omega^2 + i c \omega} $$

• Response: $$\displaystyle u(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} H(\omega) F(\omega) e^{i\omega t} d\omega $$.

• Application: Efficient for periodic/aperiodic forces, spectral analysis.

3.0 MULTI-DEGREE-OF-FREEDOM (MDOF) SYSTEMS

3.1 Matrix Formulation of Equations of Motion

• General form:

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

where $[M]$, $[C]$, $[K]$ are $n \times n$ mass, damping, stiffness matrices.

3.2 Eigenvalue Problem & Normal Modes

• Free Vibration: $$\displaystyle [M]\{\ddot{u}\} + [K]\{u\} = 0 $$ (assume $$\displaystyle \{u\} = \{\phi\}e^{i\omega t} $$).

  Eigenvalue problem:

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

• Methods:

  | Method | Principle | Best For | |--------|-----------|----------| | Matrix Iteration (Power) | Iterate $$\displaystyle [K]^{-1}[M]\{\phi\} $$ | Largest eigenvalue | | Holzer | Trial $\omega$, compute displacements, check equilibrium | Lumped mass systems | | Rayleigh-Ritz | Assume mode shape $\phi(x)$, compute $$\displaystyle \omega^2 = \frac{\{\phi\}^T[K]\{\phi\}}{\{\phi\}^T[M]\{\phi\}} $$ | Approximate first mode |

• Orthogonality:

$$ \{\phi_i\}^T[M]\{\phi_j\} = 0, \quad \{\phi_i\}^T[K]\{\phi_j\} = 0 \quad (i \neq j) $$

• Normalization:

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

  • Unity at a point: $$\displaystyle \phi_i(x_{\text{ref}}) = 1 $$

3.3 Solution of Forced Vibration

• Modal Superposition:

$$ \{u\} = [\Phi]\{q\}, \quad [\Phi] = [\phi_1\ \phi_2\ \cdots\ \phi_n] $$

• Uncoupling: Premultiply by $$\displaystyle [\Phi]^T $$:

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

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

• Combination of Modal Responses:

  | Method | Formula | Applicability | |--------|---------|---------------| | SRSS | $$\displaystyle R = \sqrt{\sum R_i^2} $$ | Well-separated modes ($$\displaystyle \omega_i/\omega_j > 1.5 $$) | | CQC | $$\displaystyle R = \sqrt{\sum\sum R_i R_j \rho_{ij}} $$ | Closely spaced modes |

  where $$\displaystyle \rho_{ij} = \frac{8\xi^2 r^{1+r}}{(1+r^2)^2 + 4\xi^2 r(1+r)^2} $$, $$\displaystyle r = \omega_i/\omega_j \leq 1 $$.

  [!TIP]

  IS 1893: Use SRSS for ≤ 3 modes; CQC for more or close modes.

4.0 CONTINUOUS SYSTEMS (BEAMS, BARS, SHAFTS)

4.1 Derivation of EOM & Natural Frequencies

• Axial Vibration of Bar:

$$ \rho A \frac{\partial^2 u}{\partial t^2} = EA \frac{\partial^2 u}{\partial x^2} $$

Boundary conditions: fixed ($$\displaystyle u=0 $$), free ($$\displaystyle EA \frac{\partial u}{\partial x}=0 $$).

Natural frequencies: $$\displaystyle \omega_n = \frac{n\pi}{L}\sqrt{\frac{E}{\rho}} $$ (fixed-fixed).

• Bending Vibration of Beam (Euler-Bernoulli):

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

Boundary conditions depend on end conditions (clamped, simply-supported, free).

• Torsional Vibration of Shaft-Disc:

  For uniform shaft with disc at end:

$$ GJ \frac{\partial^2 \theta}{\partial x^2} = \rho J \frac{\partial^2 \theta}{\partial t^2} $$

Natural frequency: $$\displaystyle \omega_n = \frac{n\pi}{L}\sqrt{\frac{G}{\rho}} $$ (fixed-free).

4.2 Mode Shapes & Frequency Calculation

• Uniform Beam, Clamped-Free (Cantilever):

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

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

  Mode shape:

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

• Sketch Mode Shapes:

  • First mode: one half-sine, max at free end.

  • Second mode: one full sine + half, node near 0.78L.

  • Third mode: 1.5 sine, nodes at ~0.55L and ~0.87L.

  [!TIP]

  Always verify boundary conditions: clamped ($$\displaystyle u=0 $$, $$\displaystyle \frac{du}{dx}=0 $$), free ($$\displaystyle \frac{d^2u}{dx^2}=0 $$, $$\displaystyle \frac{d^3u}{dx^3}=0 $$).

5.0 DAMPING DEVICES & VIBRATION CONTROL

5.1 Viscous Dampers

• Principle: Force proportional to velocity: $$\displaystyle F = c\dot{u} $$.

• Applications: Bridges (seismic retrofit), buildings (reduce wind/earthquake response).

• Types: Fluid viscous, elastomeric, friction dampers.

5.2 Vibration Isolation

• Transmissibility ($T$): Ratio of transmitted force/motion to applied.

• Force Transmissibility:

$$ T_F = \frac{\sqrt{1 + (2\xi r)^2}}{\sqrt{(1-r^2)^2 + (2\xi r)^2}} $$

• Motion Transmissibility:

$$ T_d = \frac{r^2}{\sqrt{(1-r^2)^2 + (2\xi r)^2}} $$

• Design: For isolation, $$\displaystyle r > \sqrt{2} $$ and $\xi$ small (typically 0.05–0.1).

  DiagramCANVAS: Plot of transmissibility vs frequency ratio showing isolation region

6.0 SEISMIC ANALYSIS & DESIGN (PER IS 1893:2002/2016)

6.1 Dynamic Analysis Procedure for Buildings

• Lumped Mass Model: Mass at each floor level:

  $$\displaystyle W_i = \text{floor area} \times \text{height} \times \text{density} + \text{live load fraction} $$.

• Seismic Force Distribution (Mode $i$):

$$ F_{ij} = \frac{W_j \phi_{ij}}{\sum_{k=1}^n W_k \phi_{ik}} \cdot F_i $$

where $$\displaystyle F_i = \frac{Z I S_a(T_i) W_i \phi_{i1}}{\sum_{k=1}^n W_k \phi_{ik}^2} $$ (for fundamental mode).

• Total Base Shear: $$\displaystyle V_b = \sum_{i=1}^n F_i $$.

6.2 Response Spectrum

• Definition: Plot of peak response (acceleration $$\displaystyle S_a $$, velocity $$\displaystyle S_v $$, displacement $$\displaystyle S_d $$) vs natural period $T$ for a given damping ratio $\xi$.

• Construction: From accelerograms, compute response of SDOF with varying $T$, take maxima.

• Use: Directly read peak response for each mode in MDOF systems.

6.3 Combination of Modal Responses

7.0 NUMERICAL METHODS FOR DYNAMIC RESPONSE

7.1 Newmark's Method

• Algorithm (Average Acceleration: $$\displaystyle \beta = 1/4 $$, $$\displaystyle \gamma = 1/2 $$ – unconditionally stable):

  1. Predict:

$$ \tilde{u}_{i+1} = u_i + \Delta t \, v_i + \frac{\Delta t^2}{2}(1-2\beta) a_i $$

$$ \tilde{v}_{i+1} = v_i + \Delta t (1-\gamma) a_i $$

2. Solve for acceleration:

$$ a_{i+1} = \frac{1}{m + \gamma c \Delta t + \beta k \Delta t^2} \left[ F_{i+1} - c \tilde{v}_{i+1} - k \tilde{u}_{i+1} \right] $$

3. Correct:

$$ u_{i+1} = \tilde{u}_{i+1} + \beta \Delta t^2 a_{i+1} $$

$$ v_{i+1} = \tilde{v}_{i+1} + \gamma \Delta t \, a_{i+1} $$

• Parameters:

  • $$\displaystyle \beta = 1/4 $$, $$\displaystyle \gamma = 1/2 $$: Unconditionally stable, second-order accurate.

  • $$\displaystyle \beta = 1/6 $$, $$\displaystyle \gamma = 1/2 $$: Linear acceleration (conditionally stable).

• Application: Step-by-step integration for SDOF/MDOF (assemble global matrices).

8.0 SPECIAL TOPICS & ADVANCED METHODS (SHORT NOTE SYLLABUS)

8.1 Hamilton's Principle

• Variational Formulation:

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

for conservative systems, where $T$ = kinetic energy, $V$ = potential energy.

• Application: Derives EOM without considering internal forces explicitly.

8.2 Holzer Method

• For MDOF Free Vibration:

  Assume trial frequency $\omega$, compute displacements from equilibrium, adjust $\omega$ until compatibility satisfied.

• Procedure: Start from one end, use $$\displaystyle k_i \phi_i = \omega^2 m_i \phi_i $$ to propagate.

8.3 Response Spectrum

• As in 6.2. Key: Provides peak response for any SDOF with given $T$, $\xi$.

8.4 Uncoupling of Equations of Motion

• Use modal matrix $[\Phi]$ such that:

$$ [\Phi]^T[M][\Phi] = [I], \quad [\Phi]^T[K][\Phi] = [\omega^2] $$

Then EOM decouples into independent SDOF equations.

8.5 Fourier Transform Methods

• Convert time-domain EOM to frequency-domain:

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

Solve for $\{U(\omega)\}$, inverse Fourier transform to get $\{u(t)\}$.

8.6 Response to Unit Impulse

• Impulse Response Function $h(t)$: Response to $\delta(t)$.

• Undamped: $$\displaystyle h(t) = \frac{1}{m\omega_n} \sin\omega_n t $$.

• Damped: $$\displaystyle h(t) = \frac{1}{m\omega_d} e^{-\xi\omega_n t} \sin\omega_d t $$, $$\displaystyle \omega_d = \omega_n\sqrt{1-\xi^2} $$.

• Use: Convolution with $F(t)$ gives response (Duhamel's integral).

8.7 Vibration Isolation

• As in 5.2. Design goal: $$\displaystyle T_d < 1 $$ for $$\displaystyle r > \sqrt{2} $$.

8.8 Arbitrary Force

• Treatment:

  • Duhamel's integral (time domain).

  • Fourier transform (frequency domain).

  • Laplace transform (if initial conditions matter).

8.9 Equation of Motion

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

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

• Base Excitation: Replace $F(t)$ with $$\displaystyle -m\ddot{u}_g $$ and use relative displacement.

8.10 Matrix Formulation

• Assembly: Element matrices $$\displaystyle [m^e] $$, $$\displaystyle [c^e] $$, $$\displaystyle [k^e] $$ → global $[M]$, $[C]$, $[K]$.

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

• Modal Analysis: $$\displaystyle [\Phi]^T[M][\Phi] = [I] $$, $$\displaystyle [\Phi]^T[K][\Phi] = [\omega^2] $$.

Exam Focus:

• Derive EOM for SDOF (Newton/D'Alembert).

• Compute $$\displaystyle \omega_n $$, $\xi$, $\delta$ for given $m,c,k$.

• Steady-state response for harmonic force (amplitude, phase, resonance).

• Duhamel integral for rectangular pulse.

• Eigenvalue problem for 2-DOF systems (matrix iteration).

• Mode shapes for uniform beam (clamped-free, simply-supported).

• IS 1893 dynamic analysis steps and SRSS/CQC.

• Newmark's algorithm steps.

• Short notes on Hamilton, Holzer, response spectrum, uncoupling.