Structural Dynamics (CE-702 (C)) - Unit 4 Short Notes

1.0 SINGLE DEGREE OF FREEDOM (SDOF) SYSTEMS: FUNDAMENTALS & GOVERNING EQUATIONS

1.1 Equation of Motion for SDOF Systems

The fundamental equation describing the dynamic behavior of an SDOF system is derived from equilibrium.

• Using Newton's Second Law:

  \[ \sum F = m\ddot{u} \]

  Applying to the mass \(m\) displaced by \(u(t)\):

  \[ F(t) - F_s - F_d = m\ddot{u} \]

  where \(F_s = ku\) (spring force) and \(F_d = c\dot{u}\) (damping force for viscous damping).

• Using D'Alembert's Principle:

  Introduces inertial force \(-m\ddot{u}\). The system is in "dynamic equilibrium" under applied force \(F(t)\), spring force \(F_s\), damping force \(F_d\), and inertial force.

  \[ F(t) - F_s - F_d - m\ddot{u} = 0 \]

  This is powerful for systems with accelerating coordinates.

Standard Form (Damped Forced Vibration):

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

1.2 Free Vibration Analysis

• Undamped Free Vibration (\(c=0, F=0\)):

  \[ m\ddot{u} + ku = 0 \quad \Rightarrow \quad \ddot{u} + \omega_n^2 u = 0 \]

  where Natural Frequency \(\omega_n = \sqrt{\frac{k}{m}}\) (rad/s). Solution: \(u(t) = A\sin(\omega_n t) + B\cos(\omega_n t)\).

• Damped Free Vibration (\(F=0\)):

  \[ \ddot{u} + 2\zeta\omega_n\dot{u} + \omega_n^2 u = 0 \]

  where Damping Ratio \(\zeta = \frac{c}{c_c}\) and Critical Damping Coefficient \(c_c = 2m\omega_n = 2\sqrt{km}\).

  Types of Damping:

  | Damping Type | Force Model | Common in | | :--- | :--- | :--- | | Viscous | \(F_d = c\dot{u}\) | Hydraulic dampers, air dashpots | | Coulomb (Friction) | \(F_d = \mu N \cdot \text{sgn}(\dot{u})\) | Sliding interfaces, joints | | Hysteretic | \(F_d\) proportional to displacement amplitude | Material internal friction (steel, concrete) |

  Logarithmic Decrement (\(\delta\)):

  Ratio of successive amplitudes in underdamped vibration (\(\zeta < 1\)).

  \[ \delta = \ln\left(\frac{u(t)}{u(t+T_d)}\right) = \frac{2\pi\zeta}{\sqrt{1-\zeta^2}} \]

  where \(T_d = \frac{2\pi}{\omega_d}\) is the damped period, \(\omega_d = \omega_n\sqrt{1-\zeta^2}\).

  [!TIP] Exam Focus: Derivation of \(\delta\) formula and sketch of \(\delta\) vs. \(\zeta\) (monotonically increasing, \(\delta \to \infty\) as \(\zeta \to 1\)).

1.3 Forced Vibration Analysis

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

  Steady-State Response: \(u_{ss}(t) = U\sin(\omega t - \phi)\)

  \[ U = \frac{F_0/k}{\sqrt{(1 - r^2)^2 + (2\zeta r)^2}}, \quad \tan\phi = \frac{2\zeta r}{1 - r^2} \]

  where \(r = \omega / \omega_n\) (frequency ratio).

  Resonance: For light damping (\(\zeta < 0.2\)), peak response near \(r = \sqrt{1 - 2\zeta^2} \approx 1\). Amplitude at resonance: \(U_{res} \approx \frac{F_0}{2\zeta k}\).

• Response to Arbitrary Force: Concept of convolution integral. The response at time \(t\) depends on the history of loading \(F(\tau)\) for \(\tau < t\).

• Response to Unit Impulse: Impulse \(I = \int F(t)dt\). A unit impulse (\(I=1\)) applied at \(t=0\) gives the impulse response function \(h(t)\). For an undamped system, \(h(t) = \frac{1}{m\omega_n}\sin(\omega_n t)\).

• Viscous Dampers: Devices providing force \(F_d = c\dot{u}\). They dissipate energy as heat, reducing vibration amplitudes, especially near resonance. Their coefficient \(c\) directly defines \(\zeta\).

2.0 ANALYTICAL METHODS FOR DETERMINING RESPONSE

2.1 Duhamel's Integral (Principle of Superposition)

Integrates the effect of a continuous force history \(F(\tau)\) using the impulse response function.

• For Undamped System:

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

• For Damped System:

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

Applications:

• Unit Impulse: Set \(F(\tau) = \delta(\tau)\) (Dirac delta). Integral yields \(h(t)\) as above.

• Rectangular Pulse (height \(A\), duration \(T\)):

  \[ F(t) = \begin{cases} A & 0 \leq t \leq T \\ 0 & t > T \end{cases} \]

  Response for \(t > T\):

  \[ u(t) = \frac{A}{k} \left[ 1 - \frac{e^{-\zeta\omega_n (t-T)}}{\sqrt{1-\zeta^2}} \cos\left( \omega_d (t-T) - \phi \right) \right] \]

  where \(\tan\phi = \frac{2\zeta r}{\sqrt{1-\zeta^2}}\) for \(r=1\) (pulse frequency = \(\omega_n\)).

2.2 Laplace Transform Methods

Transforms differential equations into algebraic equations in \(s\)-domain.

• Laplace Transform of Rectangular Pulse:

  \[ \mathcal{L}\{F(t)\} = \mathcal{L}\{A[1(t) - 1(t-T)]\} = \frac{A}{s}(1 - e^{-sT}) \]

• Laplace Transform of Unit Impulse (Dirac Delta):

  \[ \mathcal{L}\{\delta(t)\} = 1 \]

  Deduced as the derivative of the unit step function \(1(t)\): \(\mathcal{L}\{1(t)\} = \frac{1}{s}\), so \(\mathcal{L}\{\delta(t)\} = s \cdot \frac{1}{s} = 1\).

• Solving EOM: Transform \(m\ddot{u} + c\dot{u} + ku = F(t)\) using \(\mathcal{L}\{\ddot{u}\} = s^2U(s) - su(0) - \dot{u}(0)\), etc. Solve for \(U(s)\), then inverse transform.

2.3 Fourier Transform Methods (Brief)

Decomposes arbitrary force \(F(t)\) into harmonic components via frequency domain. Response \(U(\omega)\) is \(H(\omega)F(\omega)\), where \(H(\omega)\) is the frequency response function. Inverse transform gives time response. Useful for random vibrations.

3.0 NUMERICAL METHODS FOR DYNAMIC RESPONSE

3.1 Newmark's Method

A step-by-step integration scheme for solving \(m\ddot{u}_{t+\Delta t} + c\dot{u}_{t+\Delta t} + ku_{t+\Delta t} = F_{t+\Delta t}\).

• Assumptions (for velocity & displacement):

  \[ \dot{u}_{t+\Delta t} = \dot{u}_t + \Delta t[(1-\gamma)\ddot{u}_t + \gamma\ddot{u}_{t+\Delta t}] \]

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

  Common choices:

  • Average Acceleration: \(\gamma = \frac{1}{2}, \beta = \frac{1}{4}\) → Unconditionally stable, 2nd order accurate.

  • Linear Acceleration: \(\gamma = \frac{1}{2}, \beta = \frac{1}{6}\) → Conditionally stable (\(\Delta t < \frac{2}{\omega_n}\)).

• Algorithm (for step \(t \to t+\Delta t\)):

  1. Predict: \(\tilde{u} = u_t + \Delta t\dot{u}_t + \frac{\Delta t^2}{2}(1-2\beta)\ddot{u}_t\), \(\tilde{\dot{u}} = \dot{u}_t + (1-\gamma)\Delta t\ddot{u}_t\).

  2. Solve for effective stiffness: \(K_{eff} = m + \gamma c\Delta t + \beta k\Delta t^2\).

  3. Compute effective load: \(\tilde{F} = F_{t+\Delta t} + m(\frac{u_t}{\beta\Delta t^2} + \frac{\dot{u}_t}{\beta\Delta t}) + c(\frac{\gamma u_t}{\beta\Delta t} + (\frac{\gamma}{\beta}-1)\dot{u}_t + \frac{\Delta t}{2}(\frac{\gamma}{\beta}-2)\ddot{u}_t)\).

  4. Solve: \(\ddot{u}_{t+\Delta t} = \frac{1}{K_{eff}}(\tilde{F} - c\tilde{\dot{u}} - k\tilde{u})\).

  5. Correct: \(u_{t+\Delta t} = \tilde{u} + \beta\Delta t^2\ddot{u}_{t+\Delta t}\), \(\dot{u}_{t+\Delta t} = \tilde{\dot{u}} + \gamma\Delta t\ddot{u}_{t+\Delta t}\).

[!TIP] Exam Focus: State assumptions, write effective stiffness formula, and mention stability conditions.

3.2 Other Numerical Procedures

• Central Difference Method: Explicit, conditionally stable (\(\Delta t < \frac{2}{\omega_n}\)), 2nd order.

• Wilson's θ Method: Implicit, unconditionally stable for \(\theta \geq 1.37\).

• Runge-Kutta Methods: General ODE solvers (e.g., 4th order).

4.0 MULTI-DEGREE OF FREEDOM (MDOF) SYSTEMS

4.1 Matrix Formulation of Equations of Motion

For an \(n\)-DOF system:

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

where \([M]\) = mass matrix, \([C]\) = damping matrix, \([K]\) = stiffness matrix, \(\{u\}\) = displacement vector.

4.2 Eigenvalue Problem

For undamped free vibration (\(\{F\}=0, [C]=0\)):

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

• Natural Frequencies (\(\omega_i\)): Roots of the characteristic equation \(\det([K] - \omega^2[M]) = 0\). For \(n\)-DOF, there are \(n\) frequencies \(\omega_1 \leq \omega_2 \leq ... \leq \omega_n\).

• Mode Shapes (\(\{\phi_i\}\)): Non-trivial displacement vectors corresponding to each \(\omega_i\).

• Normalization of Modes:

  1. Mass Normalization: \(\{\phi_i\}^T[M]\{\phi_i\} = 1\).

  2. Unit Displacement Normalization: Set one component of \(\{\phi_i\}\) to 1 (often the top floor for buildings).

4.3 Matrix Iteration Methods

To find eigenvalues/vectors without full determinant.

• Power Method: Finds fundamental (\(\omega_1\)) mode and frequency. Iterate: \(\{\phi^{(k+1)}\} = [K]^{-1}[M]\{\phi^{(k)}\}\), normalize. Converges to mode with highest frequency.

• Inverse Iteration (with Shift): Finds highest mode or mode near a guessed frequency \(\lambda\). Solve \(([K] - \lambda[M])\{\phi^{(k+1)}\} = [M]\{\phi^{(k)}\}\).

• Subspace Iteration: Finds multiple modes simultaneously. More efficient for several modes.

4.4 Holzer Method (Short Note)

A determinant-based tabular method primarily for torsional systems. For each trial frequency \(\omega\), compute torques and rotations at each inertia, starting from one end. The method converges when the sum of torques at the other end is zero. Useful for systems with lumped inertias and shafts.

4.5 Uncoupling of Equations of Motion

• Modal Analysis: Transform physical coordinates \(\{u\}\) to modal coordinates \(\{q\}\):

  \[ \{u\} = [\Phi]\{q\} \]

  where \([\Phi] = [\{\phi_1\} \{\phi_2\} ... \{\phi_n\}]\) is the matrix of mode shapes.

• Conditions for Uncoupling:

  1. Proportional (Rayleigh) Damping: \([C] = \alpha[M] + \beta[K]\). Then \([\Phi]^T[C][\Phi] = \text{diagonal}\).

  2. Orthogonality of Modes: \([\Phi]^T[M][\Phi] = [I]\) (if mass-normalized) and \([\Phi]^T[K][\Phi] = \text{diag}(\omega_i^2)\).

• Principal Modes: The set of mode shapes \(\{\phi_i\}\) that satisfy the orthogonality conditions. They form a complete set for representing any motion.

5.0 CONTINUOUS SYSTEMS (AXIAL & BENDING VIBRATION)

5.1 Derivation of Equation of Motion

• Axial Vibration of a Bar:

  Consider an infinitesimal element \(dx\). Force equilibrium:

  \[ (A\sigma)_{x} - (A\sigma)_{x+dx} = \rho A dx \cdot \frac{\partial^2 u}{\partial t^2} \]

  Using \(\sigma = E\varepsilon = E\frac{\partial u}{\partial x}\):

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

  For uniform bar (\(EA\) constant):

  \[ \boxed{\frac{\partial^2 u}{\partial x^2} = \frac{1}{c_a^2} \frac{\partial^2 u}{\partial t^2}} \quad \text{where} \quad c_a = \sqrt{\frac{E}{\rho}} \quad \text{(wave speed)} \]

• Bending Vibration of a Beam (Euler-Bernoulli):

  Moment-curvature: \(M = -EI\frac{\partial^2 u}{\partial x^2}\). Shear force equilibrium on element:

  \[ \frac{\partial^2 M}{\partial x^2} = \rho A \frac{\partial^2 u}{\partial t^2} \]

  Substituting \(M\):

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

5.2 Natural Frequencies & Mode Shapes

Assume separation of variables: \(u(x,t) = \phi(x) \cdot q(t)\).

• For axial bar: \(\phi(x) = C_1\sin(kx) + C_2\cos(kx)\), \(k = \frac{\omega}{c_a}\).

• For beam: \(\phi(x) = C_1\sin(kx) + C_2\cos(kx) + C_3\sinh(kx) + C_4\cosh(kx)\), \(k^4 = \frac{\rho A \omega^2}{EI}\).

Boundary Conditions determine \(kL\) and thus \(\omega_n\).

• Clamped-Free (Cantilever) Beam:

  \[ \cos(kL)\cosh(kL) = -1 \]

  First three roots: \(k_1L \approx 1.875\), \(k_2L \approx 4.694\), \(k_3L \approx 7.855\).

  \[ \omega_i = (k_i L)^2 \sqrt{\frac{EI}{\rho A L^4}} \]

  Mode Shapes \(\phi_i(x)\) are orthogonal: \(\int_0^L \phi_i(x)\phi_j(x)\rho A dx = 0\) for \(i \neq j\).

[!TIP] Exam Focus: Derive wave equation for bar and beam equation. Solve for cantilever beam frequencies. Sketch first three mode shapes (1st: one inflection, 2nd: two inflections, etc.).

6.0 EARTHQUAKE ENGINEERING & CODE-BASED PROCEDURES

6.1 Earthquake Ground Motion

For base excitation \(u_g(t)\) (ground displacement), absolute displacement \(u(t)\):

\[ m(\ddot{u}_g + \ddot{u}_r) + c\dot{u}_r + ku_r = 0 \]

where \(u_r = u - u_g\) is relative displacement. Standard form:

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

The RHS is an effective force \(-m\ddot{u}_g\).

6.2 Dynamic Analysis Procedure as per IS 1893:2002

For multi-story buildings:

1. Lumped Mass Model: Total seismic mass \(W = \sum \text{(floor weight)}\). Mass lumped at floor levels.

2. Seismic Force Distribution:

   • Base shear: \(V_b = A_h \cdot W\) (where \(A_h\) = horizontal seismic coefficient from response spectrum).

   • Force at floor \(i\): \(F_i = \frac{W_i h_i}{\sum W_j h_j} \cdot V_b\) (where \(h_i\) = height from base).

   • DiagramSEARCH: is 1893 base shear distribution diagram

3. Response Spectrum Method:

   • Construct elastic response spectrum (max response vs. \(\omega_n\) or \(T_n\)) for given damping (usually 5%).

   • For each mode \(i\): \(u_{i,max} = S_d(T_i) \cdot \frac{\Gamma_i W}{k_i}\), where \(\Gamma_i = \frac{\{\phi_i\}^T[M]\{1\}}{\{\phi_i\}^T[M]\{\phi_i\}}\) (modal participation factor).

   • Combine modal maxima to get total response.

4. CQC vs. SRSS:

   | CQC (Complete Quadratic Combination) | SRSS (Square Root of Sum of Squares) | | :--- | :--- | | Used when mode frequencies are close (spacing \(\leq 0.2\omega_n\)). | Used when mode frequencies are well separated (> \(0.2\omega_n\)). | | Accounts for modal coupling via a cross-term coefficient \(\rho_{ij}\) (function of damping ratio and frequency ratio). | Assumes modal responses are statistically independent. | | \(R_{total} = \sqrt{\sum_i \sum_j R_i R_j \rho_{ij}}\) | \(R_{total} = \sqrt{\sum_i R_i^2}\) | | More accurate, computationally heavier. | Simpler, may be conservative or unconservative if modes are close. |

[!TIP] Exam Focus: Write step-by-step IS 1893 procedure. Clearly distinguish CQC (for close frequencies) and SRSS (for separated frequencies).

7.0 SPECIALIZED CONCEPTS & APPLICATIONS (SHORT NOTE SCOPE)

7.1 Vibration Isolation

• Concept: Reduce transmitted force/motion from a vibrating source to a foundation or sensitive equipment using a resilient mount (spring-damper).

• Transmissibility (TF): Ratio of transmitted force to applied force, or displacement ratio.

  \[ TF = \sqrt{ \frac{1 + (2\zeta r)^2}{(1 - r^2)^2 + (2\zeta r)^2} } \]

  • Force Isolation: \(r > \sqrt{2}\) (high frequency). TF < 1.

  • Motion Isolation: \(r > 1\) (above resonance). TF < 1 for \(r > \sqrt{2}\).

  • At resonance (\(r=1\)), TF = 1 (no isolation, amplification).

7.2 Hamilton's Principle (Variational Principle)

Statement: Among all possible kinematically admissible displacements \(\{u(t)\}\) satisfying boundary conditions, the actual motion makes the Lagrangian Action \(S = \int_{t_1}^{t_2} (T - V) dt\) stationary (\(\delta S = 0\)), where \(T\) = kinetic energy, \(V\) = potential energy.

• Application: Derive equations of motion for complex systems (e.g., beams, plates) without isolating elements. Yields same EOM as Newton/D'Alembert.

7.3 Response Spectrum

• Definition: Plot of maximum response (displacement, velocity, acceleration) of a single-degree-of-freedom system vs. its natural period \(T\) or frequency, for a given ground motion and damping ratio.

• Construction: For each \(T\), solve SDOF EOM with \(u_g(t)\), record peak \(u_{max}\), \(v_{max}\), \(a_{max}\).

• Use in MDOF: Approximate peak response of each mode using spectrum value at \(T_i\), then combine (SRSS/CQC).

7.4 Torsional Vibration

• Example: Rigid disc (mass \(m\), polar moment \(J\)) mounted on a flexible shaft (torsional stiffness \(K_t\), damping \(C_t\)).

• Equation of Motion:

  \[ J\ddot{\theta} + C_t\dot{\theta} + K_t\theta = T(t) \]

  Identical form to translational SDOF. Natural frequency: \(\omega_n = \sqrt{K_t/J}\).

7.5 Effective Stiffness Calculation

• Definition: The stiffness \(k\) relating force to displacement in a spring-mass system: \(F = k u\).

• For Parallel Springs: \(k_{eq} = k_1 + k_2 + ...\)

• For Series Springs: \(\frac{1}{k_{eq}} = \frac{1}{k_1} + \frac{1}{k_2} + ...\)

• For Distributed Systems: From strain energy \(U = \frac{1}{2}\int \sigma\varepsilon dV\). For axial bar, \(k = \frac{EA}{L}\); for cantilever beam tip load, \(k = \frac{3EI}{L^3}\).

8.0 FREQUENTLY ASKED SHORT NOTE TOPICS (SYNTHESIS)

8.1 Comparison & Contrast Topics

• CQC vs. SRSS: See 6.2. CQC for close modes, SRSS for separated.

• Pre-tensioning vs. Post-tensioning: (Context: Prestressed Concrete, not core Dynamics). Pre-tensioning: steel tensioned before concrete cast. Post-tensioning: steel tensioned after concrete hardens.

8.2 Conceptual Definitions & Explanations

• Logarithmic Decrement: \(\delta = \ln(u_t / u_{t+T_d}) = 2\pi\zeta / \sqrt{1-\zeta^2}\). Measures rate of decay.

• Duhamel's Integral: \(u(t) = \int_0^t F(\tau) h(t-\tau) d\tau\). Superposition of impulse responses.

• Eigenvalue Problem: \(([K] - \omega^2[M])\{\phi\}=0\). Solves for \(\omega_i, \{\phi_i\}\).

• Modal Analysis/Uncoupling: Transform to \(\{u\} = [\Phi]\{q\}\); uncouples EOM if \([C]\) proportional.

• Response Spectrum: Max SDOF response vs. \(T\) for given ground motion.

• Vibration Isolation: Use resilient mount; effective for \(r > \sqrt{2}\).

8.3 Method Discussions

• Newmark's Method: Step-by-step; assumptions \(\gamma, \beta\); average acceleration (\(\gamma=0.5, \beta=0.25\)) unconditionally stable.

• Matrix Iteration Methods: Power (fundamental mode), Inverse (highest/near guess), Subspace (multiple modes).

• Holzer Method: Determinant/tabular for torsional systems; torque sum zero at end.

• IS 1893 Dynamic Analysis: Lumped mass → base shear \(V_b = A_h W\) → distribute by \(W_i h_i\) → response spectrum → CQC/SRSS.

8.4 Derivations & Formulations

• SDOF Damped Forced EOM: \(m\ddot{u} + c\dot{u} + ku = F(t)\) from Newton/D'Alembert.

• Axial Bar Wave Eq: \(\frac{\partial^2 u}{\partial x^2} = \frac{1}{c_a^2}\frac{\partial^2 u}{\partial t^2}\), \(c_a = \sqrt{E/\rho}\).

• Beam Bending Eq: \(EI\frac{\partial^4 u}{\partial x^4} + \rho A\frac{\partial^2 u}{\partial t^2} = 0\).

• Laplace Transform of Rectangular Pulse: \(\frac{A}{s}(1 - e^{-sT})\). Unit impulse: \(\mathcal{L}\{\delta(t)\}=1\).