Structural Dynamics (CE-702 (C)) - Unit 2 Short Notes

UNIT 2: STRUCTURAL DYNAMICS

I. SINGLE DEGREE OF FREEDOM (SDOF) SYSTEMS

A. Fundamentals & Equation of Motion

• D'Alembert's Principle: Inertia forces are introduced to convert a dynamic problem into a static equilibrium problem. For a mass \( m \) with displacement \( u(t) \), the principle states:

  \[ F_{\text{applied}} - F_{\text{inertia}} - F_{\text{damping}} - F_{\text{stiffness}} = 0 \]

  where \( F_{\text{inertia}} = m\ddot{u} \).

• Newton's Second Law: Directly gives the Equation of Motion (EoM) for an SDOF system with viscous damping and external force \( F(t) \):

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

*   \( m \): Mass

*   \( c \): Damping coefficient

*   \( k \): Stiffness

*   \( u \): Displacement from equilibrium

B. 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 \( \omega_n = \sqrt{k/m} \) is the **natural circular frequency** (rad/s).

**General Solution:** \( u(t) = A \sin(\omega_n t) + B \cos(\omega_n t) \)

*   \( A, B \) are constants determined by initial conditions.

*   **Natural Frequency (Hz):** \( f_n = \omega_n / (2\pi) \).

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

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

where \( \zeta = c / (2\sqrt{mk}) \) is the **damping ratio**.

*   **Characteristic Equation:** \( s^2 + 2\zeta\omega_n s + \omega_n^2 = 0 \)

*   **Roots:** \( s_{1,2} = -\zeta\omega_n \pm \omega_n\sqrt{\zeta^2 - 1} \)

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

• Logarithmic Decrement (\(\delta\)):

  • Definition: The natural logarithm of the ratio of any two successive amplitudes of an underdamped system.

  \[ \delta = \ln\left(\frac{u(t)}{u(t+T_d)}\right) \]

  where \( T_d = 2\pi / \omega_d \) is the damped period.

  • Derivation: For \( u(t) = U_0 e^{-\zeta\omega_n t} \sin(\omega_d t + \phi) \), amplitudes at \( t \) and \( t+T_d \) are \( U_0 e^{-\zeta\omega_n t} \) and \( U_0 e^{-\zeta\omega_n (t+T_d)} \). Thus:

  \[ \delta = \zeta\omega_n T_d = \frac{2\pi\zeta}{\sqrt{1-\zeta^2}} \]

  \boxed{\delta = \frac{2\pi\zeta}{\sqrt{1-\zeta^2}}}

  • Relationship: For small \(\zeta\), \( \zeta \approx \delta / (2\pi) \).

  [!TIP] Exam Tip: Sketch of \(\delta\) vs. \(\zeta\) shows it increases slowly at first, then sharply as \(\zeta \to 1\). For \(\zeta > 0.2\), use exact formula; for \(\zeta < 0.2\), approximate \(\zeta = \delta/2\pi\).

C. Forced Vibration Analysis

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

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

  • Amplitude Equation:

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

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

  \boxed{U = \frac{F_0/k}{\sqrt{(1 - r^2)^2 + (2\zeta r)^2}}}

  • Phase Angle:

  \[ \tan \phi = \frac{2\zeta r}{1 - r^2} \]

  • Resonance: Maximum amplitude occurs at \( r_r = \sqrt{1 - 2\zeta^2} \) (for \(\zeta < 1/\sqrt{2}\)). At resonance:

  \[ U_{\text{max}} = \frac{F_0/k}{2\zeta\sqrt{1-\zeta^2}} \]

  For light damping (\(\zeta \ll 1\)), \( U_{\text{max}} \approx F_0/(2\zeta k) \).

• Equation of Motion for Damped Forced Vibration:

  Derived directly from Newton's Law as given above.

• Response to Arbitrary Forcing: Duhamel's Integral (Convolution Integral)

  • Concept: The response of an undamped SDOF system to any force \( F(t) \) is the superposition (integral) of its responses to a series of infinitesimal impulses.

  • For Undamped System (\(\zeta=0\)):

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

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

  • Application for Unit Impulse: A unit impulse \( F(\tau) = \delta(\tau) \) (Dirac delta) applied at \( \tau=0 \) gives the impulse response function:

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

  • Application for Rectangular Pulse: \( F(t) = F_0 \) for \( 0 \leq t \leq t_d \), 0 otherwise.

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

    (Derived by direct integration of Duhamel's or using Laplace transform).

• Vibration Isolation:

  • Theory: A system (mass-spring-damper) is mounted on a vibrating base (e.g., machine on foundation). The goal is to minimize force transmission (\( F_T \)) to the base or displacement of the mass.

  • Transmissibility (Force): \( T_F = \frac{F_T}{F_0} = \sqrt{\frac{1 + (2\zeta r)^2}{(1 - r^2)^2 + (2\zeta r)^2}} \)

  • Condition for Isolation: \( r > \sqrt{2} \). For effective isolation, operating frequency should be > 1.414 times the natural frequency.

  [!TIP] Common Pitfall: Transmissibility < 1 only when \( r > \sqrt{2} \). At \( r=1 \), \( T_F = 1/(2\zeta) \), which is large for small \(\zeta\).

D. Special Topics & Methods

• Laplace Transform Methods:

  • Transform of Rectangular Pulse (height A, duration \( t_d \)):

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

  • Transform of Unit Impulse: \( \mathcal{L}\{\delta(t)\} = 1 \).

  • Application to SDOF EoM: Take Laplace transform of \( m\ddot{u} + c\dot{u} + ku = F(t) \), solve algebraic equation for \( U(s) \), then inverse transform to get \( u(t) \). Particularly useful for pulses and impulses.

• Fourier Transform Methods (Basic Concept):

  Represents any arbitrary force \( F(t) \) as a continuous spectrum of harmonic components. The response is the sum of responses to each harmonic component. The Frequency Response Function \( H(\omega) = 1/(k - m\omega^2 + i c \omega) \) is central.

II. MULTI-DEGREE OF FREEDOM (MDOF) SYSTEMS

A. Formulation & Matrix Methods

• Equation of Motion in Matrix Form:

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

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

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

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

  • \( \{u\} \): Displacement vector

• Assembly: Using direct stiffness method. For each element, derive its local stiffness/mass matrices, transform to global coordinates, and assemble into global \( [K] \) and \( [M] \).

• D'Alembert's Principle for MDOF: \( \{F_{\text{applied}}\} - [M]\{\ddot{u}\} - [C]\{\dot{u}\} - [K]\{u\} = \{0\} \).

B. Eigenvalue Problem (Free Vibration)

• Concept: Solve for natural frequencies (\(\omega\)) and corresponding mode shapes (\(\Phi\)) by setting \( \{F(t)\}=0 \) and assuming harmonic solution \( \{u\} = \{\Phi\} \sin \omega t \).

• General Eigenvalue Problem:

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

  For non-trivial solution (\(\{\Phi\} \neq \{0\}\)), the determinant must be zero:

  \[ \det([K] - \omega^2 [M]) = 0 \]

  This is the frequency (characteristic) equation, a polynomial of order \( n \) (number of DOFs), yielding \( n \) roots \( \omega_i^2 \) (natural frequencies).

• Methods for Solving Eigenvalue Problems:

  • Matrix Iteration Method (Power Method): Iteratively solves \( [K]\{\Phi^{(k+1)}\} = \lambda [M]\{\Phi^{(k)}\} \) where \( \lambda = \omega^2 \). Finds the fundamental (lowest) frequency and mode first. Subsequent modes found by orthogonalization (e.g., using \( [M] \)-orthogonality).

  • Holzer's Method: Primarily for torsional systems (rotational DOFs). Uses an iterative procedure based on the determinant condition, often tabular. Suitable for systems with lumped inertias and torsional springs.

• Mode Orthogonality & Normalization:

  • Orthogonality: Distinct mode shapes are orthogonal with respect to both \( [M] \) and \( [K] \):

    \[ \{\Phi_i\}^T [M] \{\Phi_j\} = 0, \quad \{\Phi_i\}^T [K] \{\Phi_j\} = 0 \quad \text{for } i \neq j \]

  • Normalization: Scale mode shapes for convenience. Common choices:

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

    2. Unit Displacement at a Point: Set a specific component of \(\Phi_i\) to 1.

C. Forced Vibration Analysis

• Mode Superposition Method (Decoupling):

  1. Assume solution as a sum of modal contributions: \( \{u(t)\} = \sum_{i=1}^n \{\Phi_i\} q_i(t) \), where \( q_i(t) \) are generalized coordinates.

  2. Pre-multiply EoM by \( \{\Phi_j\}^T \) and use orthogonality properties to decouple equations:

     \[ \ddot{q}_j + 2\zeta_j \omega_j \dot{q}_j + \omega_j^2 q_j = \frac{\{\Phi_j\}^T \{F(t)\}}{\{\Phi_j\}^T [M] \{\Phi_j\}} \]

     Each equation is an independent SDOF equation.

  3. Solve each SDOF equation (e.g., for harmonic force) to get \( q_j(t) \).

  4. Total response: \( \{u(t)\} = \sum \{\Phi_j\} q_j(t) \).

• Practical Considerations:

  • Modal Damping: Often assume proportional damping, so each mode has its own \(\zeta_j\).

  • Truncation: Only the first few modes (which capture most mass/energy) are used in the summation. Higher modes are often neglected.

D. Specific System Analysis

• Analysis of Discrete Spring-Mass Systems: Standard procedure: (1) Identify DOFs, (2) Assemble \( [M] \) and \( [K] \) using direct stiffness, (3) Solve eigenvalue problem for \(\omega_i, \Phi_i\).

• Torsional Vibration of Shafts with Mounted Discs:

  • Derivation of EoM: For a shaft of torsional stiffness \( GJ/L \) (G: modulus, J: polar moment, L: length) with discs of mass moment of inertia \( I_1, I_2, ... \) at various points.

  • Using D'Alembert's principle for rotational motion (\( \sum M = I\alpha \)):

    For disc 1: \( -k_{12}(\theta_1 - \theta_2) = I_1 \ddot{\theta}_1 \)

    For disc 2: \( k_{12}(\theta_1 - \theta_2) - k_{23}(\theta_2 - \theta_3) = I_2 \ddot{\theta}_2 \), etc.

    In matrix form: \( [I]\{\ddot{\theta}\} + [K_\theta]\{\theta\} = \{0\} \), where \( [I] \) is diagonal (inertias), \( [K_\theta] \) is torsional stiffness matrix.

III. NUMERICAL METHODS FOR DYNAMIC RESPONSE

A. Newmark's Beta Method

• Detailed Discussion: A direct time-integration method for solving the equation \( [M]\{\ddot{u}\}_{t+\Delta t} + [C]\{\dot{u}\}_{t+\Delta t} + [K]\{u\}_{t+\Delta t} = \{F\}_{t+\Delta t} \).

• Algorithm (Key Steps):

  1. Assume acceleration and velocity variation over \(\Delta t\):

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

     where \(\gamma\) and \(\beta\) are parameters.

  2. Effective stiffness matrix: \( [\hat{K}] = [K] + \gamma/(\beta\Delta t)[C] + 1/(\beta\Delta t^2)[M] \).

  3. Effective load vector:

     \[ \{\hat{F}\}_{t+\Delta t} = \{F\}_{t+\Delta t} + [M]\left( \frac{1}{\beta\Delta t^2}\{u\}_t + \frac{1}{\beta\Delta t}\{\dot{u}\}_t + \left(\frac{1}{2\beta}-1\right)\{\ddot{u}\}_t \right) + [C]\left( \frac{\gamma}{\beta\Delta t}\{u\}_t + \left(\frac{\gamma}{\beta}-1\right)\{\dot{u}\}_t + \Delta t\left(\frac{\gamma}{2\beta}-1\right)\{\ddot{u}\}_t \right) \]

  4. Solve \( [\hat{K}]\{\Delta u\} = \{\hat{F}\}_{t+\Delta t} \) for displacement increment \(\{\Delta u\}\).

  5. Update \( \{u\}_{t+\Delta t}, \{\dot{u}\}_{t+\Delta t}, \{\ddot{u}\}_{t+\Delta t} \).

• Stability Conditions:

  • Unconditional Stability: Requires \( \beta \geq \gamma/2 \), \( \gamma \geq 1/2 \). Common choice: Average Acceleration Method (\(\beta=1/4, \gamma=1/2\)) which is unconditionally stable and second-order accurate.

  • Conditional Stability: For \(\beta=0\) (Central Difference Method), stable only if \(\Delta t \leq 2/\omega_{\text{max}}\).

• Application: Used for both SDOF and MDOF systems. For MDOF, matrices are large but sparse; efficient solvers (like Cholesky) are used.

B. Other Direct Integration Methods (Conceptual)

• Wilson's θ-method: A modification of Newmark's method with \(\theta \geq 1.0\) (usually 1.37). It is unconditionally stable and introduces numerical damping to suppress high-frequency oscillations. Uses an extra step at \( t+\theta\Delta t \) (where \(\theta>1\)).

IV. CONTINUOUS SYSTEMS (Distributed Parameter Systems)

A. Derivation of EoM & Natural Frequencies

• General Approach: Apply Newton's Law to an infinitesimal element, derive partial differential equation (PDE) of motion.

• Axial Vibration of a Bar:

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

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

  \[ \frac{\partial^2 u}{\partial x^2} = \frac{1}{c_a^2} \frac{\partial^2 u}{\partial t^2}, \quad c_a = \sqrt{E/\rho} \]

  Boundary Conditions: e.g., fixed (\(u=0\)), free (\(EA u_x=0\)).

  Frequency Equation: From separation of variables \(u(x,t)=\Phi(x)T(t)\), get \(\Phi'' + (\omega^2/c_a^2)\Phi = 0\). Apply BCs to get transcendental equation for \(\omega\).

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

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

  For uniform beam:

  \[ \frac{\partial^4 u}{\partial x^4} = \frac{1}{c_b^4} \frac{\partial^2 u}{\partial t^2}, \quad c_b = \sqrt[4]{EI/(\rho A)} \]

  Boundary Conditions: Require continuity of displacement, slope, bending moment (\(EI u_{xx}\)), and shear force (\(EI u_{xxx}\)).

• Torsional Vibration of a Shaft:

  \[ \frac{\partial}{\partial x}\left( GJ \frac{\partial \theta}{\partial x} \right) = \rho J_p \frac{\partial^2 \theta}{\partial t^2} \]

  For uniform shaft:

  \[ \frac{\partial^2 \theta}{\partial x^2} = \frac{1}{c_t^2} \frac{\partial^2 \theta}{\partial t^2}, \quad c_t = \sqrt{G/\rho} \]

  Boundary Conditions: e.g., fixed (\(\theta=0\)), free (\(GJ \theta_x=0\)).

B. Mode Shapes & Frequency Calculation

• Standard Boundary Conditions:

  | System | BC at x=0 | BC at x=L | Frequency Equation | First \(\omega_1\) (approx) | | :--- | :--- | :--- | :--- | :--- | | Axial Bar | Fixed (\(u=0\)) | Fixed (\(u=0\)) | \(\sin \beta L = 0\) | \(\beta L = \pi\) | | | Free (\(u_x=0\)) | Free (\(u_x=0\)) | \(\cos \beta L = 0\) | \(\beta L = \pi/2\) | | Simply Supported Beam | \(u=0, M=0\) | \(u=0, M=0\) | \(\cos \beta L \cosh \beta L = 1\) | \(\beta L \approx 4.73\) | | Cantilever Beam | \(u=0, u_x=0\) | \(M=0, V=0\) | \(\cos \beta L \cosh \beta L = -1\) | \(\beta L \approx 1.875\) | | Clamped-Free (Cantilever) Torsional | \(\theta=0\) | \(M_t=0\) | \(\sin \beta L = 0\) | \(\beta L = \pi/2\) |

  • \(\beta^4 = \rho A \omega^2 / EI\) (beam), \(\beta^2 = \rho \omega^2 / G\) (torsion), etc.

• Determining Frequencies & Sketching Mode Shapes:

  1. Solve frequency equation for \(\beta_n L\).

  2. Compute \(\omega_n = c \cdot (\beta_n)^n\) (n=1 for axial/torsion, n=2 for beam).

  3. Mode shape \(\Phi_n(x)\) from spatial solution (e.g., \(\Phi_n(x) = \sin(\beta_n x)\) for fixed-fixed bar, \(\Phi_n(x) = \cosh(\beta_n x) - \cos(\beta_n x) + \eta(\sin(\beta_n x) - \sinh(\beta_n x))\) for cantilever beam).

  4. Sketch: Ensure correct number of nodes (zero displacement points, excluding ends) = \(n-1\) for the \(n^{th}\) mode. Ends satisfy BCs.

• Example: Uniform Beam Clamped at One End and Free at the Other (Cantilever)

  • Frequency Equation: \( \cos \beta L \cosh \beta L = -1 \)

  • First three \(\beta L\): 1.875, 4.694, 7.855

  • Mode Shapes:

    • 1st Mode: No internal nodes. Shape like a tilted "S" (maximum deflection at free end).

    • 2nd Mode: One internal node near 0.78L from fixed end. Deflection at free end in opposite direction to 1st mode.

    • 3rd Mode: Two internal nodes. More complex curvature.

  DiagramCANVAS: Sketch a cantilever beam. Draw first mode: smooth curve from fixed end (zero disp & slope) to max upward at free end. Second mode: fixed end zero, curve down then up crossing zero near 0.78L, max down at free end. Third mode: fixed end zero, curve up-down-up crossing zero twice, max up at free end.

V. SEISMIC ANALYSIS & RESPONSE SPECTRA

A. Earthquake Loading & IS 1893 Procedure

• Effect of Horizontal Ground Acceleration (\( \ddot{u}_g \)): Modifies EoM for base excitation. Relative displacement \( u \) (mass relative to ground) satisfies:

  \[ m(\ddot{u} + \ddot{u}_g) + c\dot{u} + ku = 0 \quad \Rightarrow \quad m\ddot{u} + c\dot{u} + ku = -m\ddot{u}_g \]

  The ground motion acts as a negative forcing function \(-m\ddot{u}_g\).

• Dynamic Analysis Procedure as per IS 1893:2002 (for multi-storey buildings):

  1. Lumped Mass Model: Masses lumped at floor levels. Total seismic mass \( W/g \) (where \( W \) = total seismic weight).

  2. Fundamental Time Period (\( T_1 \)): Estimated by empirical formulas based on height \( h \) and material:

     • For RC frames: \( T_1 = 0.075 h^{0.75} \) (h in meters)

     • For steel frames: \( T_1 = 0.085 h^{0.75} \)

  3. Seismic Force Distribution: Base shear \( V_b = A_h \cdot W \), where \( A_h \) = horizontal seismic coefficient from design spectrum based on \( T_1 \), zone, and damping.

  4. Distribution along Height: Force at level \( i \):

     \[ Q_i = \frac{W_i h_i}{\sum_{j=1}^n W_j h_j} \cdot V_b \]

     where \( W_i \) = weight at level \( i \), \( h_i \) = height from base.

B. Response Spectrum

• Concept: A plot of the maximum response (e.g., displacement, velocity, acceleration) of a family of SDOF systems (with varying \(\omega_n\) or \(T_n\)) subjected to the same ground motion \(\ddot{u}_g(t)\), versus their natural period \(T\) or frequency.

• Construction:

  1. Choose a set of periods \( T_i \) and damping ratios \(\zeta\) (usually 5%).

  2. For each \( T_i \), solve the EoM \( \ddot{u} + 2\zeta\omega_n\dot{u} + \omega_n^2 u = -\ddot{u}_g(t) \) numerically (e.g., Newmark).

  3. Find peak values \( u_{\text{max}}(T_i), \dot{u}_{\text{max}}(T_i), \ddot{u}_{\text{max}}(T_i) \).

  4. Plot these maxima vs. \( T_i \).

• Use in Seismic Design:

  • Design Spectrum: Smooth, conservative spectrum derived from many records for a site. Used in codes (like IS 1893) to compute base shear.

  • Modal Response Spectrum Method: For MDOF, compute natural frequencies/modes. For each mode \(i\), read its maximum modal response \( q_i^{\text{max}} \) from the spectrum (using \( T_i \)). Then combine modal maxima using SRSS or CQC to get total response.

C. Modal Combination Methods

• SRSS (Square Root of Sum of Squares):

  \[ R_{\text{max}} = \sqrt{\sum_{i=1}^N (R_i^{\text{max}})^2} \]

  • Assumption: Modes are statistically independent (well-separated frequencies, i.e., \( \omega_j / \omega_i > 1.5 \) or \( T_j/T_i > 1.2 \)).

  • When to use: For structures with regular mass/stiffness distribution and light damping (\(\zeta < 5\%\)).

• CQC (Complete Quadratic Combination):

  \[ R_{\text{max}} = \sqrt{\sum_{i=1}^N \sum_{j=1}^N R_i^{\text{max}} R_j^{\text{max}} \rho_{ij}} \]

  where \( \rho_{ij} \) is the modal cross-correlation coefficient:

  \[ \rho_{ij} = \frac{8\zeta^2 (r/r_0)(1+r/r_0)^{1.5}}{(1 - r^2)^2 + 4\zeta^2 r^2 (1 + r/r_0)^{1.5}}, \quad r = \omega_j/\omega_i \leq 1, \quad r_0 = \frac{\omega_j}{\omega_i} \text{ for } i=j \]

  • Advantage: Accounts for closely spaced modes and their interaction. More accurate for irregular, high-rise buildings.

  • When to use: For structures with closely spaced frequencies or significant damping.

• Comparison:

  | Feature | SRSS | CQC | | :--- | :--- | :--- | | Basis | Statistical independence | Full correlation | | Computational Cost | Low | Higher (double sum) | | Accuracy for Close Modes | Poor (may underestimate) | Good | | Code Recommendation | For regular structures | For irregular/close modes |

VI. ADVANCED ANALYTICAL TOOLS & CONCEPTS

A. Variational Principles

• Hamilton's Principle (Brief): The actual motion of a conservative system between times \( t_1 \) and \( t_2 \) makes the action integral stationary (usually minimum):

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

  where \( T \) = kinetic energy, \( V \) = potential energy (strain energy - work of external forces). Can derive EoM for complex systems (including continuous) without considering individual forces.

B. Specialized Topics (from Short Notes)

• Viscous Dampers:

  • Types: Dashpots (fluid orifice), viscoelastic, tuned mass dampers (TMD).

  • Modeling: Represented by force \( F_d = c\dot{u} \). In MDOF, added to \( [C] \) matrix. Often assumed proportional damping (\( [C] = \alpha[M] + \beta[K] \)) to preserve mode orthogonality.

• Equation of Motion (General Form):

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

  Specific forms for SDOF, MDOF, continuous systems (PDEs).

• Response to Unit Impulse (Impulse Response Function):

  • For SDOF undamped: \( h(t) = \frac{1}{m\omega_n} \sin(\omega_n t) \).

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

  • For MDOF: Matrix of impulse response functions \( [h(t)] \), where \( h_{ij}(t) \) is displacement at \( i \) due to unit impulse at \( j \). Total response: \( \{u(t)\} = \int_0^t [h(t-\tau)] \{F(\tau)\} d\tau \).

• Arbitrary Force (General Solution Approach):

  For linear systems, use principle of superposition. Solution = Complementary Function (homogeneous solution, free vibration) + Particular Integral (steady-state/forced response). For arbitrary \( F(t) \), Duhamel's integral (convolution) is the particular integral for undamped systems.

• Concordant Cable & Linear Transformation:

  • Concordant Cable: A tendon profile that, if made concordant (i.e., its eccentricity varies exactly to balance a given load), produces zero net moment at all sections. The moment due to prestress exactly balances external moment.

  • Linear Transformation: A method to find an equivalent tendon profile (with different eccentricities) that produces the same internal moment diagram as a given profile, by adding a linear transformation (a straight tendon with constant eccentricity). Used in prestressed concrete analysis.

C. Uncoupling of Equations of Motion

• Concept: Transform the coupled MDOF equations \( [M]\{\ddot{u}\} + [K]\{u\} = \{0\} \) (undamped free vibration) into a set of independent SDOF equations using the mode shapes as transformation coordinates.

• Procedure:

  1. Find natural frequencies \(\omega_i\) and normalized mode shapes \(\Phi_i\) (e.g., \(\Phi_i^T[M]\Phi_i = 1\)).

  2. Define transformation: \( \{u\} = [\Phi]\{q\} \), where \( [\Phi] = [\Phi_1 \Phi_2 ... \Phi_n] \).

  3. Pre-multiply EoM by \( [\Phi]^T \):

     \[ [\Phi]^T[M][\Phi]\{\ddot{q}\} + [\Phi]^T[K][\Phi]\{q\} = \{0\} \]

  4. Due to orthogonality, \( [\Phi]^T[M][\Phi] = [I] \) (identity) and \( [\Phi]^T[K][\Phi] = [\Omega^2] \) (diagonal matrix with \(\omega_i^2\)).

  5. Result: \( \ddot{q}_i + \omega_i^2 q_i = 0 \) for each mode \( i \). Decoupled!

• For Damped/Forced Systems: If damping is proportional (\( [C] = \alpha[M] + \beta[K] \)), the same transformation decouples the equations. Otherwise, equations remain coupled.