UNIT 3: Mechanical Vibrations - Comprehensive Short Notes

I. FUNDAMENTAL CONCEPTS & SINGLE DEGREE OF FREEDOM (SDOF) SYSTEMS

A. Introduction & Modeling

• Causes of Vibration: Imbalance, misalignment, irregular friction, external excitation, wind/earthquake loads.

• Degree of Freedom (DoF): Minimum number of independent coordinates required to define system configuration.

  • Types: SDOF (1 coordinate), 2-DOF, MDOF (n coordinates).

• Simple Harmonic Motion (SHM):

  • Mathematical Definition: $$\displaystyle x(t) = A \sin(\omega t + \phi) $$, where $A$=amplitude, $\omega$=circular frequency, $\phi$=phase.

  • Vector Representation: Displacement, velocity ($$\displaystyle \dot{x} = A\omega \cos(\omega t + \phi) $$), acceleration ($$\displaystyle \ddot{x} = -A\omega^2 \sin(\omega t + \phi) $$) represented as rotating vectors.

• Modeling: Physical system → idealized spring-mass-damper model.

B. Equation of Motion & Natural Frequency

• Newton's Second Law (for mass-spring-damper on smooth surface):

$$m\ddot{x} + c\dot{x} + kx = F(t)$$

> [!TIP] Always draw **Free Body Diagram (FBD)** before writing EoM. Identify forces: $-k x$ (spring), $-c \dot{x}$ (damper), $F(t)$ (external).

• Rayleigh's Energy Method (for undamped systems):

$$T_{max} = V_{max} \implies \frac{1}{2}m\dot{x}_{max}^2 = \frac{1}{2}k x_{max}^2$$

For assumed mode shape $$\displaystyle x(t) = u(t) \cdot f(x) $$, leads to $$\displaystyle \omega_n^2 = \frac{\int_0^L EI (f'')^2 dx}{\int_0^L \rho A f^2 dx} $$.

• Natural Frequency (Undamped):

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

$$f_n = \frac{\omega_n}{2\pi} \quad \text{(Hz)}$$

$$T = 2\pi\sqrt{\frac{m}{k}} \quad \text{(Time period)}$$

• Spring Combinations:

  • Series: $$\displaystyle \frac{1}{k_{eq}} = \frac{1}{k_1} + \frac{1}{k_2} + ... $$

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

  [!CAUTION] For complex systems, use equivalent stiffness at the mass point. Consider displacement compatibility.

C. Damping Models & Free Vibration Response

• Viscous Damping:

  • Force $$\displaystyle F_d = -c\dot{x} $$ (proportional to velocity).

  • Rationale: Represents fluid resistance, internal material friction; mathematically convenient (linear).

• Coulomb (Dry Friction) Damping:

  • Force $$\displaystyle F_d = \mu N \cdot \text{sgn}(\dot{x}) $$ (constant magnitude, opposes motion).

  • Characteristics: Non-linear, causes linear decay of amplitude per cycle, frequency independent of amplitude.

• Comparison: Underdamped vs. Coulomb Damping

  | Feature | Viscous (Underdamped) | Coulomb | | :--- | :--- | :--- | | Force Law | $F \propto \dot{x}$ | $$\displaystyle F = \text{constant} $$ | | Amplitude Decay | Exponential ($$\displaystyle e^{-\zeta\omega_n t} $$) | Linear ($$\displaystyle \Delta x = \frac{4\mu N}{k} $$ per cycle) | | Frequency | $$\displaystyle \omega_d = \omega_n\sqrt{1-\zeta^2} $$ (amplitude dependent) | $$\displaystyle \omega_n $$ (amplitude independent) | | EoM | Linear | Non-linear | | Energy Loss | Proportional to $$\displaystyle \dot{x}^2 $$ | Constant per half-cycle |

• Critical Damping Constant:

$$c_c = 2m\omega_n = 2\sqrt{km}$$

*   **Importance**: Minimum damping to prevent oscillation; fastest return to equilibrium without overshoot.

• Damping Ratio:

$$\zeta = \frac{c}{c_c}$$

*   **Classification**:

    *   $$\displaystyle \zeta < 1 $$: Underdamped (oscillatory)

    *   $$\displaystyle \zeta = 1 $$: Critically damped

    *   $$\displaystyle \zeta > 1 $$: Overdamped (non-oscillatory, slow)

• Underdamped Free Vibration Solution:

$$x(t) = e^{-\zeta\omega_n t} \left[ A \cos(\omega_d t) + B \sin(\omega_d t) \right]$$

$$\omega_d = \omega_n \sqrt{1 - \zeta^2} \quad \text{(damped natural frequency)}$$

*   $A, B$ determined from $x(0)$, $\dot{x}(0)$.

• Logarithmic Decrement:

$$\delta = \frac{1}{n} \ln \frac{x(t)}{x(t+nT_d)} = \frac{2\pi\zeta}{\sqrt{1-\zeta^2}}$$

*   **Use**: Experimental measurement of $\zeta$ from decay of successive peaks.

> [!CAUTION] Valid only for **$$\displaystyle \zeta < 1 $$** (underdamped). For $\zeta \ge 1$, no oscillations exist.

• Hysteresis & Structural Damping:

  • Hysteresis: Stress-strain curve forms a loop; area = energy dissipated per cycle.

  • Structural Damping: Modeled as complex stiffness $k(1 + i\eta)$, where $\eta$ = loss factor. Energy loss proportional to $$\displaystyle x^2 $$.

II. FORCED VIBRATION & RESPONSE ANALYSIS

A. Harmonic Forcing

• Equation: $$\displaystyle m\ddot{x} + c\dot{x} + kx = F_0 \sin \omega t $$

• Steady-State Response:

$$x(t) = X \sin(\omega t - \phi)$$

• Magnification Factor (Dynamic Magnifier):

$$M.F. = \frac{X}{X_{st}} = \frac{1}{\sqrt{(1 - r^2)^2 + (2\zeta r)^2}}$$

where $$\displaystyle r = \omega / \omega_n $$ (frequency ratio), $$\displaystyle X_{st} = F_0/k $$ (static deflection).

> [!TIP] **Derivation Key**: Assume $$\displaystyle x_p = X \sin(\omega t - \phi) $$, substitute into EoM, equate coefficients. Get:

> 

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

• Resonance:

  • Condition: $r \approx 1$ (for low $\zeta$).

  • Amplitude at Resonance (for $$\displaystyle \zeta < 1/\sqrt{2} $$):

$$X_{res} = \frac{F_0/k}{2\zeta\sqrt{1-\zeta^2}} \approx \frac{F_0}{2\zeta k}$$

*   **Phase at Resonance**: $$\displaystyle \phi = 90^\circ $$ (for $$\displaystyle \zeta < 1/\sqrt{2} $$).

B. Forced Response with Initial Conditions

• Total Response:

$$x(t) = x_c(t) + x_p(t)$$

*   $$\displaystyle x_c(t) $$: Complementary (transient) solution → homogeneous EoM, decays with time ($$\displaystyle \zeta > 0 $$).

*   $$\displaystyle x_p(t) $$: Particular (steady-state) solution → persists for $t \to \infty$.

• Procedure:

  1. Find $$\displaystyle x_p(t) = X \sin(\omega t - \phi) $$.

  2. Find $$\displaystyle x_c(t) $$ from homogeneous EoM (depends on $\zeta$).

  3. Apply $x(0)$, $\dot{x}(0)$ to determine constants in $$\displaystyle x_c(t) $$.

C. Base Excitation & Vibration Isolation

• Setup: $$\displaystyle y = Y \sin \omega t $$ (base motion). Relative displacement $$\displaystyle z = x - y $$.

• EoM (relative coordinate):

$$m\ddot{z} + c\dot{z} + kz = -m\ddot{y}$$

• Transmissibility (TR):

$$\frac{X}{Y} = \frac{\sqrt{1 + (2\zeta r)^2}}{\sqrt{(1 - r^2)^2 + (2\zeta r)^2}}$$

*   **Force Transmissibility** (force transmitted to base):

$$F_T = kX + c\dot{X} \implies \frac{F_T}{F_0} = \frac{r^2 \sqrt{1 + (2\zeta / r)^2}}{\sqrt{(1 - r^2)^2 + (2\zeta r)^2}}$$

• Isolation vs. Amplification:

  • $$\displaystyle r < \sqrt{2} $$: Amplification ($$\displaystyle X/Y > 1 $$). Avoid operating here.

  • $$\displaystyle r > \sqrt{2} $$: Isolation ($$\displaystyle X/Y < 1 $$). Effective for $$\displaystyle r > \sqrt{2} $$ and $\zeta$ small.

  • Rule of Thumb: For >90% isolation, need $$\displaystyle r > 3 $$.

• Applications:

  • Vehicle on Sinusoidal Road:

    • Critical speed: $$\displaystyle \omega_c = \omega_n \sqrt{g/Y} $$ (when $X$ max).

    • Amplitude at speed $v$: $$\displaystyle r = \frac{2\pi v}{\lambda \omega_n} $$, use TR formula.

  • Machine on Resilient Foundation:

    • Given static deflection $$\displaystyle \delta_{st} = mg/k \implies \omega_n = \sqrt{g/\delta_{st}} $$.

    • Use $X/Y$ formula to find $\zeta$ or other parameters.

  • Isolator Design:

    • Given max acceleration $$\displaystyle a_{max} $$, find $k$: $$\displaystyle X_{max} = a_{max}/\omega^2 $$, use TR.

D. Rotating Unbalance

• Model: $$\displaystyle m_e e \omega^2 \sin \omega t $$ (harmonic force), where $$\displaystyle m_e $$ = eccentric mass, $e$ = eccentricity.

• Steady-State Amplitude:

$$X = \frac{m_e e \omega^2 / k}{\sqrt{(1 - r^2)^2 + (2\zeta r)^2}} = \frac{m_e e}{m} \cdot \frac{r^2}{\sqrt{(1 - r^2)^2 + (2\zeta r)^2}}$$

• Force Transmitted to Base: $$\displaystyle F_T = m \omega^2 X $$ (in-phase with unbalance force for $$\displaystyle r<1 $$).

E. General Forcing Functions

• Fourier Series Expansion (for periodic force $F(t)$ with period $T$):

$$F(t) = a_0 + \sum_{n=1}^{\infty} \left( a_n \cos n\omega_0 t + b_n \sin n\omega_0 t \right)$$

where $$\displaystyle \omega_0 = 2\pi/T $$.

*   **Response**: $$\displaystyle x(t) = \sum_{n=1}^{\infty} X_n \sin(n\omega_0 t - \phi_n) $$, with $$\displaystyle X_n $$ for each harmonic.

• Convolution Integral (for arbitrary $F(t)$):

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

III. MULTI-DEGREE OF FREEDOM (MDOF) SYSTEMS

A. Modeling & Equations of Motion

• DoF Determination: Count independent coordinates required to describe configuration (consider constraints).

• Matrix EoM (undamped, no forcing):

$$\mathbf{M}\ddot{\mathbf{x}} + \mathbf{K}\mathbf{x} = \mathbf{0}$$

where $$\displaystyle \mathbf{x} = [x_1, x_2, ..., x_n]^T $$.

• Assumption: Proportional damping ($$\displaystyle \mathbf{C} = \alpha\mathbf{M} + \beta\mathbf{K} $$) allows modal decoupling.

B. Undamped Free Vibration

• Eigenvalue Problem:

  Assume $$\displaystyle \mathbf{x}(t) = \boldsymbol{\phi} e^{i\omega t} \implies (\mathbf{K} - \omega^2 \mathbf{M})\boldsymbol{\phi} = \mathbf{0} $$

• Natural Frequencies:

$$\det(\mathbf{K} - \omega^2 \mathbf{M}) = 0$$

Solve for $$\displaystyle \omega_1^2, \omega_2^2, ..., \omega_n^2 $$ (eigenvalues).

• Mode Shapes (Eigenvectors) $$\displaystyle \boldsymbol{\phi}^{(i)} $$:

  • Solve $$\displaystyle (\mathbf{K} - \omega_i^2 \mathbf{M})\boldsymbol{\phi}^{(i)} = \mathbf{0} $$ for each $$\displaystyle \omega_i $$.

  • Normalization:

    1. Mass-normalized: $$\displaystyle \boldsymbol{\phi}^{(i)T}\mathbf{M}\boldsymbol{\phi}^{(i)} = 1 $$

    2. Max=1: Set largest component = 1.

• Orthogonality Properties (for distinct $$\displaystyle \omega_i^2 $$):

$$\boldsymbol{\phi}^{(i)T}\mathbf{M}\boldsymbol{\phi}^{(j)} = 0 \quad (i \neq j)$$

$$\boldsymbol{\phi}^{(i)T}\mathbf{K}\boldsymbol{\phi}^{(j)} = 0 \quad (i \neq j)$$

> [!TIP] **Proof**: Use $$\displaystyle (\mathbf{K} - \omega_i^2\mathbf{M})\boldsymbol{\phi}^{(i)}=0 $$ and $$\displaystyle (\mathbf{K} - \omega_j^2\mathbf{M})\boldsymbol{\phi}^{(j)}=0 $$, pre-multiply and subtract.

C. Solution of Standard MDOF Problems

• Two-DOF System (masses $$\displaystyle m_1, m_2 $$, springs $$\displaystyle k_1, k_2, k_3 $$):

  1. Write EoM: $$\displaystyle m_1\ddot{x}_1 + (k_1+k_2)x_1 - k_2 x_2 = 0 $$, $$\displaystyle m_2\ddot{x}_2 - k_2 x_1 + (k_2+k_3)x_2 = 0 $$.

  2. Assume $$\displaystyle x_i = \phi_i e^{i\omega t} $$.

  3. Get characteristic equation: $$\displaystyle \det\begin{bmatrix} k_1+k_2 - m_1\omega^2 & -k_2 \\ -k_2 & k_2+k_3 - m_2\omega^2 \end{bmatrix} = 0 $$.

  4. Solve quadratic for $$\displaystyle \omega_1^2, \omega_2^2 $$.

  5. For each $$\displaystyle \omega_i $$, find mode shape ratio $$\displaystyle \phi_1/\phi_2 $$ from one equation.

• Torsional System:

$$J_1\ddot{\theta}_1 + (k_{t1}+k_{t2})\theta_1 - k_{t2}\theta_2 = 0, \quad J_2\ddot{\theta}_2 - k_{t2}\theta_1 + k_{t2}\theta_2 = 0$$

Similar procedure, replace $m \to J$, $$\displaystyle k \to k_t $$, $x \to \theta$.

• Systems with Rigid Bars: Use compatibility conditions to relate displacements of connected masses. Often reduces to effective mass/spring.

• Beam/Foundation Systems: Model as lumped mass on elastic support. $$\displaystyle k = 3EI/L^3 $$ for tip load on cantilever.

D. Principal (Modal) Coordinates

• Transformation: $$\displaystyle \mathbf{x} = \boldsymbol{\Phi} \mathbf{q} $$, where $$\displaystyle \boldsymbol{\Phi} = [\boldsymbol{\phi}^{(1)} \ \boldsymbol{\phi}^{(2)} \ ...] $$ (modal matrix).

• Principal Coordinates $\mathbf{q}(t)$: New coordinates in which equations decouple.

• Decoupling (for undamped, no forcing):

$$\ddot{q}_i + \omega_i^2 q_i = 0 \quad \text{(independent SDOF equations)}$$

• Determination: Given $\mathbf{x}(t)$, compute $$\displaystyle q_i = \boldsymbol{\phi}^{(i)T}\mathbf{M}\mathbf{x} $$ (if mass-normalized).

IV. APPLICATIONS & SPECIALIZED TOPICS

A. Whirling of Shafts & Rotor Dynamics

• Critical Speed (Whirling Speed): Rotational speed at which rotor deflection becomes infinite (resonance).

• Proof: For a simple rotor on massless shaft, $$\displaystyle \omega_n = \sqrt{k_{eq}/m} $$, where $$\displaystyle k_{eq} $$ is shaft lateral stiffness. Critical speed $$\displaystyle N_c = \frac{60}{2\pi} \omega_n $$ (rpm).

• Safety Check: Operating speed should be > 1.3–1.5 × $$\displaystyle N_c $$ (avoid resonance zone) or < 0.7 × $$\displaystyle N_c $$.

• Calculation:

  1. Find lateral stiffness $k$ of shaft (for given boundary conditions).

  2. $$\displaystyle \omega_n = \sqrt{k/m} $$ (or use $$\displaystyle \omega_n^2 = \frac{EI}{\rho A L^4} $$ for distributed mass).

  3. $$\displaystyle N_c = \frac{60\omega_n}{2\pi} $$.

B. Vibration Measurement & Instruments

• Seismic Instruments (mass-spring-damper fixed to vibrating body):

  • Relative Motion: $$\displaystyle z = x - y $$ (where $y$ = base motion).

  • Vibrometer (displacement): Measures $z \approx y$ for $$\displaystyle \omega \ll \omega_n $$.

  • Accelerometer: Measures $\ddot{z} \approx -\ddot{y}$ for $$\displaystyle \omega \gg \omega_n $$.

  [!TIP] Derivation: For $$\displaystyle \omega \ll \omega_n $$, $z \approx y$; for $$\displaystyle \omega \gg \omega_n $$, $$\displaystyle z \approx -\frac{\ddot{y}}{\omega_n^2} $$.

C. Dynamic Vibration Absorber (Tuned Mass Damper)

• Concept: Attach secondary mass-spring system ($$\displaystyle m_a, k_a $$) to primary system.

• Principle: Tune $$\displaystyle k_a $$ so $$\displaystyle \omega_a = \omega $$ (forcing freq). Absorbs energy, reduces primary response to zero at tuned freq (if undamped).

• With Damping: Broadens effective frequency range.

D. Human Response to Vibration & Noise

• Sound Pressure Level (SPL):

$$L_p = 10 \log_{10}\left(\frac{p^2}{p_{ref}^2}\right) = 20 \log_{10}\left(\frac{p}{p_{ref}}\right) \ \text{dB}$$

where $$\displaystyle p_{ref} = 20 \ \mu\text{Pa} $$ (threshold of hearing).

*   **0 dB SPL** ≠ no sound; it's the **reference pressure** (threshold).

• Sound Power Level (SWL):

$$L_W = 10 \log_{10}\left(\frac{W}{W_{ref}}\right) \ \text{dB}$$

where $$\displaystyle W_{ref} = 10^{-12} \ \text{W} $$.

*   **Relation**: $$\displaystyle L_W = L_p + 10 \log_{10}(4\pi r^2) $$ (for point source in free field).

• Octave Band Analysis:

  • Purpose: Break noise into frequency bands (octaves: $f, 2f$; 1/3 octaves: $$\displaystyle f, 2^{1/3}f, 2^{2/3}f, 2f $$).

  • Procedure: Use filters, measure SPL in each band → identify dominant frequencies.

• Hearing Conservation & Damage Risk Criteria:

  • Exposure Limits (OSHA/NIOSH): 90 dBA (OSHA) / 85 dBA (NIOSH) for 8 hrs.

  • Exchange Rate: 5 dB (OSHA) or 3 dB (NIOSH) – every 5/3 dB increase halves safe exposure time.

  • Precautions: Ear plugs/muffs, engineering controls (enclosure, damping), administrative controls (rotation).

• Human Response to Noise:

  • Effects: Hearing loss, stress, annoyance, communication interference.

  • Annoyance Threshold: Typically > 60–65 dBA for continuous noise.

V. PROBLEM-SOLVING METHODOLOGIES & FORMULAS

A. Key Formulas

B. Step-by-Step Solution Strategies

1. Natural Frequency of Complex Spring Systems:

   • Identify point where mass is attached.

   • Apply virtual displacement at that point.

   • Compute total strain energy in all springs: $$\displaystyle V = \frac{1}{2} \sum k_i (\Delta l_i)^2 $$.

   • Compute equivalent stiffness: $$\displaystyle k_{eq} = \frac{\partial^2 V}{\partial (\delta)^2} $$ where $\delta$ is virtual displacement.

   • $$\displaystyle \omega_n = \sqrt{k_{eq}/m} $$.

   Alternative: Use Rayleigh's method with assumed mode shape $f(x)$.

2. Damped System Parameters from Data:

   • Given $\delta$, $\zeta$, $$\displaystyle c_c $$, $m$, $k$, etc. → use $$\displaystyle \delta = 2\pi\zeta/\sqrt{1-\zeta^2} $$ to find $\zeta$.

   • Given $$\displaystyle \omega_d $$ and $$\displaystyle \omega_n $$ → $$\displaystyle \zeta = \sqrt{1 - (\omega_d/\omega_n)^2} $$.

   • Given $$\displaystyle c_c $$ and $m$ → $$\displaystyle \omega_n = c_c/(2m) $$, then $$\displaystyle k = m\omega_n^2 $$.

3. Base Excitation with Damping:

   • Identify $Y$, $\omega$, $m$, $k$, $c$ (or $\zeta$, $$\displaystyle \omega_n $$).

   • Compute $$\displaystyle r = \omega/\omega_n $$.

   • Use $X/Y$ formula directly.

   • For relative amplitude $$\displaystyle Z = |X - Y| $$, compute separately.

4. MDOF System Solution (2-DOF):

   • Step 1: Write two coupled EoMs from FBDs.

   • Step 2: Assume $$\displaystyle x_i = \phi_i e^{i\omega t} $$.

   • Step 3: Write in matrix form $$\displaystyle (\mathbf{K} - \omega^2\mathbf{M})\boldsymbol{\phi} = \mathbf{0} $$.

   • Step 4: Set determinant = 0 → quadratic in $$\displaystyle \omega^2 $$ → solve for $$\displaystyle \omega_1^2, \omega_2^2 $$.

   • Step 5: For each $$\displaystyle \omega_i $$, substitute into one equation to find ratio $$\displaystyle \phi_1/\phi_2 $$.

   • Step 6: Normalize mode shape (e.g., set $$\displaystyle \phi_2 = 1 $$ or mass-normalize).

C. Common Pitfalls & Conceptual Clarifications

• Logarithmic Decrement: Only for underdamped ($$\displaystyle \zeta < 1 $$). For $\zeta \ge 1$, no oscillations → $\delta$ undefined.

• Resonance in Damped Systems: Peak amplitude occurs at $$\displaystyle r = \sqrt{1 - 2\zeta^2} $$ (for $$\displaystyle \zeta < 1/\sqrt{2} $$), not exactly $$\displaystyle r=1 $$.

• Base Excitation: Forcing frequency = base frequency $\omega$, not $$\displaystyle \omega_n $$. Natural frequency $$\displaystyle \omega_n $$ is property of mass-spring.

• 0 dB SPL: Means sound pressure = reference pressure (20 µPa), not zero pressure. It's the quietest sound audible to average human.

• Mode Shape Normalization: Any scalar multiple is valid, but mass-normalization ($$\displaystyle \boldsymbol{\phi}^T\mathbf{M}\boldsymbol{\phi}=1 $$) simplifies modal analysis.

• Whirling Speed vs. Natural Frequency: For a rotor-shaft system, critical speed = natural frequency of transverse vibration (proved by equating centrifugal force to restoring force).