Mechanical Vibrations (ME-502) - Unit 5 Short Notes

UNIT 5: MECHANICAL VIBRATIONS

I. FUNDAMENTALS & SINGLE DEGREE OF FREEDOM (SDOF) SYSTEMS

Degrees of Freedom (DOF)

• Definition: Minimum number of independent coordinates required to define the system's configuration.

• Types:

  • Translational DOF: Linear motion (e.g., mass on a spring).

  • Rotational DOF: Angular motion (e.g., pendulum, torsional system).

• Examples:

  • 1-DOF: Simple spring-mass system, simple pendulum.

  • 2-DOF: Two masses connected by springs, vehicle model (heave & pitch).

  • Multi-DOF: Multi-story building, complex machinery.

Equation of Motion (EOM) for SDOF

• Standard Form:

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

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

• Derivation via Newton's Second Law (FBD):

  1. Draw Free Body Diagram (FBD) of mass.

  2. Apply $$\displaystyle \sum F = m\ddot{x} $$: $$\displaystyle -k x - c\dot{x} + F(t) = m\ddot{x} $$.

  3. Rearrange to standard form.

• Derivation via Rayleigh's Energy Method (Frequently asked):

  • Total Energy: $$\displaystyle T + V = \text{constant} $$ (for conservative, undamped systems).

  • Kinetic Energy: $$\displaystyle T = \frac{1}{2}m\dot{x}^2 $$.

  • Potential Energy: $$\displaystyle V = \frac{1}{2}kx^2 $$.

  • Differentiate: $$\displaystyle \frac{d}{dt}(T+V) = 0 \Rightarrow m\dot{x}\ddot{x} + kx\dot{x} = 0 $$.

  • Divide by $\dot{x}$ (assuming $\dot{x} \neq 0$): $$\displaystyle m\ddot{x} + kx = 0 $$.

  [!TIP] Rayleigh's method is energy-based and avoids drawing FBDs. It's valid for conservative systems without non-conservative forces like damping or explicit forcing.

Free Vibration of Undamped SDOF

• EOM: $$\displaystyle m\ddot{x} + kx = 0 $$.

• Solution: $$\displaystyle x(t) = A\sin(\omega_n t) + B\cos(\omega_n t) $$.

• Natural Frequency:

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

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

• Natural Frequency of Systems with Multiple Springs:

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

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

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

Free Vibration of Damped SDOF

• Viscous Damping:

  • Model: Damping force $$\displaystyle F_d = c\dot{x} $$, proportional to velocity.

  • EOM: $$\displaystyle m\ddot{x} + c\dot{x} + kx = 0 $$.

• Coulomb (Dry Friction) Damping:

  • Model: Constant magnitude force $$\displaystyle F_d = \mu N \cdot \text{sign}(\dot{x}) $$, opposes motion.

  • Characteristics: Amplitude decreases linearly, frequency slightly less than $$\displaystyle \omega_n $$, motion is non-sinusoidal (saw-tooth).

• Damping Ratio ($\zeta$):

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

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

*   $$\displaystyle \zeta = 1 $$: Critically damped (fastest return to equilibrium without oscillation).

*   $$\displaystyle \zeta > 1 $$: Overdamped (slow return, no oscillation).

• Critical Damping Constant ($$\displaystyle c_c $$):

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

*   **Importance**: Boundary between oscillatory and non-oscillatory motion. Used as reference for damping ratio.

• Damped Natural Frequency ($$\displaystyle \omega_d $$): For underdamped ($$\displaystyle \zeta < 1 $$):

$$\omega_d = \omega_n \sqrt{1 - \zeta^2}$$

• Logarithmic Decrement ($\delta$):

  • Definition: $$\displaystyle \delta = \ln\left(\frac{x(t)}{x(t+T_d)}\right) $$, ratio of successive amplitudes.

  • Relation to $\zeta$:

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

*   **Limitation**: Formula assumes underdamped motion ($$\displaystyle \zeta < 1 $$). For $\zeta \geq 1$, no oscillations occur, so $\delta$ is undefined.

• Comparison: Underdamped vs. Coulomb Damped Free Vibration:

Forced Vibration of SDOF (Harmonic Force)

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

• Steady-State Response: $$\displaystyle x_{ss}(t) = X \sin(\omega t - \phi) $$.

• Magnification Factor (Dynamic Magnification, $M$):

  • Definition: Ratio of dynamic amplitude to static deflection under $$\displaystyle F_0 $$.

$$M = \frac{X}{X_{st}} = \frac{X}{F_0/k}$$

• Derivation & Expression:

  Substitute $$\displaystyle x_{ss} = X\sin(\omega t - \phi) $$ into EOM $$\displaystyle m\ddot{x}+c\dot{x}+kx=F_0\sin\omega t $$.

  After algebra:

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

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

• Resonance:

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

  • Amplitude at Resonance ($$\displaystyle r=1 $$):

$$M_{res} = \frac{1}{2\zeta}$$

*   **Phase at Resonance**: $$\displaystyle \phi = \pi/2 $$.

• Effect of Damping:

  • Resonance Peak: Decreases as $\zeta$ increases.

  • Bandwidth: Increases as $\zeta$ increases. Bandwidth $\Delta r \approx 2\zeta$ for small $\zeta$.

Special Cases & Applications of SDOF

• Base Excitation / Vibration Isolation:

  • Setup: Base motion $$\displaystyle y(t) = Y \sin(\omega t) $$.

  • Relative Displacement: $$\displaystyle z = x - y $$.

  • EOM (in terms of $z$): $$\displaystyle m\ddot{z} + c\dot{z} + kz = -m\ddot{y} $$.

  • Transmissibility (Displacement):

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

*   **Transmissibility (Force)**: $$\displaystyle T_f = \frac{F_{trans}}{F_0} = \sqrt{1 + (2\zeta r)^2} \cdot T_d $$.

*   **Isolator Design**:

    *   For **isolation** ($$\displaystyle \omega > \omega_n $$), need $$\displaystyle r > \sqrt{2} $$.

    *   **Stiffness** from transmissibility requirement: $$\displaystyle k = m\omega_n^2 $$.

    *   **Maximum Isolator Deformation**: $$\displaystyle \delta_{max} = X_{max} + Y $$ (if base and mass move in phase).

• Rotating Unbalance:

  • Equivalent Force: $$\displaystyle F_0 = m_e e \omega^2 $$, where $$\displaystyle m_e $$ = eccentric mass, $e$ = eccentricity.

  • Steady-State Amplitude:

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

*   **Critical Speed**: Speed at which $$\displaystyle \omega = \omega_n $$ (resonance). Must be avoided in rotating machinery.

• Vehicle/Trailer Vibration on Sinusoidal Road (Jun 2025):

  • Model: Trailer as SDOF with base excitation from road profile $$\displaystyle y = Y \sin(\frac{2\pi}{\lambda} v t) $$, where $\lambda$ = wavelength, $v$ = speed.

  • Critical Speed: Occurs when forcing frequency $$\displaystyle \omega = \frac{2\pi v}{\lambda} = \omega_n $$.

$$v_{crit} = \frac{\lambda \omega_n}{2\pi}$$

*   **Amplitude at Speed $v$**: Use $$\displaystyle r = \omega/\omega_n $$ in transmissibility formula $$\displaystyle X = T_d Y $$.

• Recoil Mechanisms (Jun 2025):

  • Model: Critically damped SDOF ($$\displaystyle \zeta=1 $$).

  • EOM: $$\displaystyle m\ddot{x} + c_c\dot{x} + kx = 0 $$, with $$\displaystyle c_c = 2\sqrt{km} $$.

  • Design for Maximum Recoil Distance $$\displaystyle x_{max} $$:

    Given initial recoil velocity $$\displaystyle v_0 $$, initial displacement $$\displaystyle x_0=0 $$.

    For critical damping, solution: $$\displaystyle x(t) = (A + Bt)e^{-\omega_n t} $$.

    Apply ICs: $$\displaystyle x(0)=0 \Rightarrow A=0 $$; $$\displaystyle \dot{x}(0)=v_0 \Rightarrow B=v_0 $$.

    Max displacement occurs at $$\displaystyle t=0 $$? Actually for $$\displaystyle \zeta=1 $$, $$\displaystyle x(t) = v_0 t e^{-\omega_n t} $$. Max at $$\displaystyle t=1/\omega_n $$.

$$x_{max} = \frac{v_0}{e \omega_n} \approx \frac{v_0}{2.718 \omega_n}$$

    Given $$\displaystyle x_{max} $$ and $$\displaystyle v_0 $$, solve for $$\displaystyle \omega_n $$, then $$\displaystyle k = m\omega_n^2 $$.

Total Response of SDOF

• Principle of Superposition:

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

*   $$\displaystyle x_h(t) $$: **Complementary (Transient) Solution** to homogeneous EOM ($$\displaystyle F(t)=0 $$). Depends on damping.

*   $$\displaystyle x_p(t) $$: **Particular (Steady-State) Solution** to forced EOM. Independent of ICs.

• Initial Conditions: $$\displaystyle x(0) = x_0 $$, $$\displaystyle \dot{x}(0) = v_0 $$.

• Complete Solution Forms:

  • Underdamped ($$\displaystyle \zeta<1 $$):

$$x(t) = e^{-\zeta\omega_n t} \left[ C_1 \sin(\omega_d t) + C_2 \cos(\omega_d t) \right] + X\sin(\omega t - \phi)$$

    Determine $$\displaystyle C_1, C_2 $$ from ICs.

*   **Critically Damped** ($$\displaystyle \zeta=1 $$):

$$x(t) = (C_1 + C_2 t) e^{-\omega_n t} + x_p(t)$$

*   **Overdamped** ($$\displaystyle \zeta>1 $$):

$$x(t) = C_1 e^{s_1 t} + C_2 e^{s_2 t} + x_p(t)$$

    where $$\displaystyle s_{1,2} = -\zeta\omega_n \pm \omega_n\sqrt{\zeta^2-1} $$.

II. MULTI DEGREE OF FREEDOM (MDOF) SYSTEMS

Modeling & Equations of Motion

• Matrix Form:

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

where $[M]$, $[C]$, $[K]$ are mass, damping, stiffness matrices; $\mathbf{x}$ is displacement vector.

• Writing EOM: Use Newton's method or D'Alembert's principle. Draw FBD for each mass, write $$\displaystyle \sum F = m_i\ddot{x}_i $$, include internal forces from springs/dampers.

Free Vibration Analysis (Undamped)

• Assumption: Harmonic motion $$\displaystyle \mathbf{x}(t) = \boldsymbol{\phi} \sin(\omega t + \phi) $$.

• Eigenvalue Problem:

  Substitute into undamped EOM $$\displaystyle [M]\ddot{\mathbf{x}} + [K]\mathbf{x} = \mathbf{0} $$:

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

• Natural Frequencies ($$\displaystyle \omega_i $$):

  • Roots of characteristic equation: $$\displaystyle \det([K] - \omega^2 [M]) = 0 $$.

  • For $n$-DOF system, $n$ eigenvalues $$\displaystyle \omega_1^2, \omega_2^2, \dots, \omega_n^2 $$ (ordered $$\displaystyle \omega_1 < \omega_2 < \dots $$).

• Normal Modes / Mode Shapes ($$\displaystyle \boldsymbol{\phi}_i $$):

  • Definition: Deflected shape at which all points move sinusoidally at frequency $$\displaystyle \omega_i $$.

  • Amplitude Ratio: For a given mode $i$, ratio $$\displaystyle \phi_{j,i} / \phi_{k,i} $$ is constant (relative amplitudes of coordinates $j,k$).

  • Orthogonality Properties:

$$\boldsymbol{\phi}_i^T [M] \boldsymbol{\phi}_j = 0 \quad (i \neq j)$$

$$\boldsymbol{\phi}_i^T [K] \boldsymbol{\phi}_j = 0 \quad (i \neq j)$$

    Modes are orthogonal with respect to mass and stiffness matrices.

• Modal Analysis Procedure:

  1. Form $[M]$, $[K]$.

  2. Solve $$\displaystyle \det([K] - \omega^2 [M]) = 0 $$ for $$\displaystyle \omega_i^2 $$.

  3. For each $$\displaystyle \omega_i $$, solve $$\displaystyle ([K] - \omega_i^2[M])\boldsymbol{\phi}_i = \mathbf{0} $$ for $$\displaystyle \boldsymbol{\phi}_i $$ (up to scale factor).

  4. Normalize mode shapes (e.g., $$\displaystyle \boldsymbol{\phi}_i^T[M]\boldsymbol{\phi}_i = 1 $$).

• Example Systems: Two-mass, three-mass spring-damper systems. Use matrix formulation.

Principal Coordinates

• Concept: Transform physical coordinates $\mathbf{x}$ to principal coordinates $\mathbf{q}$ such that EOMs decouple.

$$\mathbf{x} = [\Phi] \mathbf{q}$$

where $$\displaystyle [\Phi] = [\boldsymbol{\phi}_1, \boldsymbol{\phi}_2, \dots, \boldsymbol{\phi}_n] $$ is the modal matrix.

• Decoupling: Pre-multiply by $$\displaystyle [\Phi]^T $$:

$$[\Phi]^T[M][\Phi]\ddot{\mathbf{q}} + [\Phi]^T[K][\Phi]\mathbf{q} = [\Phi]^T\mathbf{F}(t)$$

Due to orthogonality, $$\displaystyle [\Phi]^T[M][\Phi] $$ and $$\displaystyle [\Phi]^T[K][\Phi] $$ are diagonal matrices.

Result: $n$ independent SDOF equations:

$$m_i^* \ddot{q}_i + k_i^* q_i = f_i(t)$$

where $$\displaystyle m_i^* = \boldsymbol{\phi}_i^T[M]\boldsymbol{\phi}_i $$, $$\displaystyle k_i^* = \boldsymbol{\phi}_i^T[K]\boldsymbol{\phi}_i = \omega_i^2 m_i^* $$.

• Determination: For a given spring-mass system, find $$\displaystyle \omega_i $$, $$\displaystyle \boldsymbol{\phi}_i $$ as above. Then $$\displaystyle \mathbf{q} = [\Phi]^{-1}\mathbf{x} $$. Since $[\Phi]$ is not necessarily orthogonal (only mass-orthogonal), use $$\displaystyle [\Phi]^T[M][\Phi] = [I] $$ if mass-normalized.

Torsional Systems

• Modeling: Replace linear springs with torsional springs ($$\displaystyle k_t $$ in N·m/rad), masses with rotary inertias ($J$ in kg·m²).

• EOM: $$\displaystyle J_i \ddot{\theta}_i + \sum k_t(\theta_i - \theta_j) = T_i(t) $$.

• Natural Frequencies: Solve $$\displaystyle \det([K_t] - \omega^2 [J]) = 0 $$, where $$\displaystyle [K_t] $$ is torsional stiffness matrix, $[J]$ is diagonal inertia matrix.

• Given Relationships: If $$\displaystyle k_0=2k_1 $$, $$\displaystyle J_2=2J_1 $$, substitute into matrices before solving eigenvalue problem.

III. SPECIAL SYSTEMS & APPLICATIONS

Continuous Systems (Approximate Methods)

• Beams in Bending:

  • Natural Frequency of a beam with concentrated mass $m$ at a point:

    Use Rayleigh's method: Assume mode shape (e.g., static deflection curve under $m$). Then:

$$\omega_n^2 = \frac{k_{eq}}{m} = \frac{\text{Strain Energy}}{\text{Max Kinetic Energy}} = \frac{\int_0^L EI (y'')^2 dx}{m \cdot (\text{max deflection})^2}$$

    For a simply supported beam with mass at midspan, $$\displaystyle y(x) = \frac{m g}{2EI} \left( \frac{L}{2}x - x^2 \right) $$ for $0 \le x \le L/2$.

*   **Parameters**: $E$ = Young's modulus, $I$ = area moment of inertia, $l$ = length.

• Shaft Whirling:

  • Critical Speed of Rotating Shaft: Rotational speed at which shaft deflection becomes very large (resonance).

  • Proof that Critical Whirling Speed = Natural Transverse Frequency:

    The equation for lateral vibration of a shaft with rotating disc is identical to that of a stationary beam with a concentrated mass, except the force is centrifugal $$\displaystyle m e \omega^2 $$. The critical condition occurs when forcing frequency $\omega$ equals the natural frequency $$\displaystyle \omega_n $$ of the shaft-disc system. Thus, critical whirling speed $$\displaystyle \omega_{crit} = \omega_n $$.

  • Safety Check (Nov 2022 - cantilever shaft):

    Given: shaft length $L$, rotor mass $m$, radius of gyration $k$, shaft $I$, $E$.

    1. Calculate $$\displaystyle \omega_n = \sqrt{\frac{EI}{m L^3}} $$ for cantilever with tip mass (approximate).

    2. Convert operating speed (rpm) to rad/s: $$\displaystyle \omega_{op} = \frac{2\pi N}{60} $$.

    3. Safe if $$\displaystyle \omega_{op} < 0.7 \omega_n $$ (typical margin). Or check if $$\displaystyle \omega_{op} \neq \omega_n $$.

  • Gyroscopic Effects: Briefly, for high-speed rotors, gyroscopic moments couple bending modes, splitting critical speeds.

Vibration Measurement & Instruments

• Vibrometer / Seismometer:

  • Principle: Measures displacement. Natural frequency $$\displaystyle \omega_n $$ is very low ($\ll$ frequency of vibration). Mass is relatively large, spring very soft.

  • Frequency Response: For $$\displaystyle \omega \ll \omega_n $$, $$\displaystyle X_{mass} \approx Y $$ (base displacement). Output proportional to relative displacement $x - y \approx -y$.

• Accelerometer:

  • Principle: Measures acceleration. Natural frequency $$\displaystyle \omega_n $$ is very high ($\gg$ frequency of vibration). Spring is very stiff, mass small.

  • Frequency Response: For $$\displaystyle \omega \gg \omega_n $$, $$\displaystyle m\ddot{x} \approx F(t) = -m\ddot{y} $$. So $\ddot{x} \approx \ddot{y}$. Output proportional to relative force $c\dot{z} \approx m\ddot{y}$.

• Derivation (General):

  Relative displacement $$\displaystyle z = x - y $$.

  EOM: $$\displaystyle m\ddot{z} + c\dot{z} + kz = -m\ddot{y} $$.

  Steady-state: $$\displaystyle Z = \frac{m\omega^2 Y}{\sqrt{(k - m\omega^2)^2 + (c\omega)^2}} $$.

  • Seismometer ($$\displaystyle \omega \ll \omega_n $$): $Z \approx Y$ → $x \approx 0$, $z \approx -y$.

  • Accelerometer ($$\displaystyle \omega \gg \omega_n $$): $$\displaystyle Z \approx \frac{m}{k}\omega^2 Y $$ → $\ddot{z} \approx \ddot{y}$.

Dynamic Vibration Absorber (Tuned Mass Damper - TMD)

• Concept: Attach a secondary mass-spring-damper ($$\displaystyle m_a, k_a, c_a $$) to primary system to reduce vibration at specific frequency $\omega$.

• Principle: Tune $$\displaystyle k_a $$ so that $$\displaystyle \omega_{na} = \omega $$ (the disturbing frequency). The absorber creates a force that cancels the force on the primary mass.

• Force Transmissibility Optimization (May 2024 - air compressor):

  • For primary system $m,k$ with unbalance force $$\displaystyle F_0 \sin\omega t $$.

  • With absorber, transmitted force to foundation is:

$$F_{trans} = F_0 \cdot \frac{r^2 \left[ (1 - r_a^2)^2 + (2\zeta_a r_a)^2 \right]}{\left[ (1 - r^2)(1 - r_a^2) - r^2 r_a^2 \right]^2 + \left[ 2\zeta_a r_a (1 - r^2) - 2\zeta r r_a^2 \right]^2}$$

    where $$\displaystyle r = \omega/\omega_n $$, $$\displaystyle r_a = \omega_a/\omega_n $$.

*   **Design for Minimum Transmission**: Set $$\displaystyle r_a = 1 $$ (tune to $\omega$). Then optimize $$\displaystyle \zeta_a $$ for minimum $$\displaystyle F_{trans} $$ at $$\displaystyle r=1 $$:

$$\zeta_{a,opt} = \sqrt{\frac{3}{8} \frac{m}{m_a}}$$

*   **Result**: At $$\displaystyle r=1 $$, $$\displaystyle F_{trans} = F_0 \cdot \frac{2\zeta_a}{\zeta_{a,opt}} $$? Actually, with optimal tuning, transmission can be made very small if $$\displaystyle m_a $$ is significant fraction of $m$.

*   **For Given % Transmission**: Rearrange formula to find required $$\displaystyle m_a/m $$ and $$\displaystyle \zeta_a $$.

IV. DAMPING MECHANISMS & MATERIALS

Types of Damping

• Viscous Damping:

  • Force: $$\displaystyle F_d = c\dot{x} $$.

  • Energy dissipated per cycle: $$\displaystyle E_D = \int_0^T c\dot{x}^2 dt = \pi c \omega X^2 $$ (for harmonic $$\displaystyle x=X\sin\omega t $$).

• Coulomb (Dry Friction) Damping:

  • Force: $$\displaystyle F_d = \mu N \cdot \text{sign}(\dot{x}) $$.

  • Energy dissipated per half-cycle: $$\displaystyle E_D = 4\mu N A $$ (constant, independent of frequency).

• Hysteresis / Solid Damping / Structural Damping:

  • Concept: Energy loss due to internal friction in material during cyclic loading. Stress-strain curve forms a hysteresis loop.

  • Area under Hysteresis Curve: Represents energy dissipated per unit volume per cycle.

  • Complex Stiffness Representation: Damping force in phase with velocity but modeled as imaginary stiffness:

$$F_d = i k^* x \quad \text{or} \quad k_{complex} = k(1 + i\eta)$$

    where $\eta$ = loss factor (damping ratio for structural damping: $$\displaystyle \zeta = \eta/2 $$ for small $\eta$).

Equivalent Damping

• Concept: Express energy dissipation of non-viscous damper in terms of an equivalent viscous damping constant $$\displaystyle c_{eq} $$ that dissipates same energy per cycle.

$$\int_0^T c_{eq} \dot{x}^2 dt = E_{D,actual}$$

For harmonic $$\displaystyle x = X\sin\omega t $$, LHS = $$\displaystyle \pi c_{eq} \omega X^2 $$.

*   **For Coulomb Damping**: $$\displaystyle E_D = 4\mu N A \approx 8\mu N X $$ (since $A \approx X$). So:

$$c_{eq} = \frac{4\mu N}{\pi \omega X} \quad \text{(amplitude-dependent!)}$$

*   **For Structural Damping**: $$\displaystyle E_D = \pi \eta k X^2 $$. So:

$$c_{eq} = \frac{\eta k}{\omega}$$

V. VIBRATION ISOLATION & CONTROL

• Isolation vs. Absorption:

  • Isolation: Prevent vibration from source to foundation (use isolators under machine).

  • Absorption: Add TMD to reduce vibration of structure.

• Design of Vibration Isolators:

  1. Determine Requirements: Limit transmitted acceleration/force or displacement.

  2. Calculate $$\displaystyle \omega_n $$: From transmissibility formula. For isolation ($$\displaystyle r > \sqrt{2} $$), $$\displaystyle T_d \approx 1/r^2 $$ for high $r$.

  3. Select Stiffness $k$: $$\displaystyle k = m \omega_n^2 $$.

  4. Check Damping: Low $\zeta$ (0.05–0.1) gives better isolation at high $r$, but higher $\zeta$ reduces resonance peak. Trade-off.

• Isolator Stiffness for Acceleration Limit (May 2024 - flow monitor):

  • Given: base acceleration amplitude $$\displaystyle Y\omega^2 $$, max allowed mass acceleration $$\displaystyle a_{max} $$.

  • Transmitted acceleration amplitude $$\displaystyle A = \omega^2 X $$.

  • From transmissibility: $$\displaystyle X = T_d Y $$, so $$\displaystyle A = \omega^2 T_d Y $$.

  • Set $$\displaystyle A \le a_{max} $$ → solve for $r$, then $k$.

  • Example: $$\displaystyle a_{max} = 5g $$, $$\displaystyle Y\omega^2 = ? $$ Given $Y$ and $\omega$, compute $r$ from $$\displaystyle T_d = A/(\omega^2 Y) $$, then $$\displaystyle k = m \omega_n^2 = m (\omega/r)^2 $$.

• Maximum Displacement of Isolated System:

  $$\displaystyle X_{max} = T_d Y_{max} $$.

• Maximum Isolator Deformation:

  $$\displaystyle \delta_{max} = |x - y|_{max} = |X - Y| $$ if in phase, or $X + Y$ if out of phase. Generally, $$\displaystyle \delta_{max} \approx X_{max} + Y_{max} $$ (worst case).

VI. NOISE & VIBRATION – ACOUSTICS (Frequently Tested)

Sound Fundamentals

• Sound Pressure Level (SPL):

  • Definition: Logarithmic measure of RMS sound pressure $p$ relative to reference pressure $$\displaystyle p_{ref} = 20 \ \mu\text{Pa} $$ (threshold of hearing).

$$\boxed{SPL \ (\text{dB}) = 20 \log_{10}\left(\frac{p}{p_{ref}}\right)}$$

*   **0 dB SPL**: Means $$\displaystyle p = p_{ref} $$ (threshold of hearing), **not** zero sound pressure. It is a reference level.

• Sound Power Level (SWL):

  • Definition: Logarithmic measure of acoustic power $W$ radiated by source relative to reference power $$\displaystyle W_{ref} = 10^{-12} \ \text{W} $$.

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

*   **Relationship with SPL**: For a point source in free field, SPL decreases with distance $r$: $$\displaystyle SPL = SWL - 20\log_{10}(r) - 11 \ \text{dB} $$ (for $r$ in meters). SPL is **pressure-based**, SWL is **power-based** (source strength).

Sound Analysis

• Octave Band Analysis:

  • Purpose: Break down complex noise into frequency bands to identify dominant frequencies and assess human ear response (which is logarithmic).

  • Concept: Each band has upper frequency = 2 × lower frequency. Center frequency $$\displaystyle f_c $$ is geometric mean: $$\displaystyle f_c = \sqrt{f_1 f_2} $$.

    • Example: 31.5 Hz band: 22.4–45 Hz (center 31.5).

  • Use: Noise control (target dominant bands), compliance with standards (e.g., OSHA limits in octave bands), hearing protector selection.

Human Response & Conservation

• Response of Human to Noise:

  • Physiological: Hearing damage (temporary/permanent threshold shift), stress, hypertension.

  • Psychological: Annoyance, reduced concentration, sleep disturbance.

  • Frequency Sensitivity: Human ear most sensitive 2–5 kHz (speech frequencies).

• Hearing Conservation & Damage Risk Criteria:

  • Permissible Exposure Limits (PEL): Time-weighted average (TWA) noise exposure allowed.

    • OSHA: 90 dBA for 8 hours (5 dB exchange rate: 90 dBA/8h, 95 dBA/4h, 100 dBA/2h, etc.).

    • NIOSH: 85 dBA for 8 hours (3 dB exchange rate: more protective).

  • Precautions and Remedies (Nov 2023):

    1. Engineering Controls: Enclose noisy machines, use quieter processes, vibration isolation, mufflers.

    2. Administrative Controls: Rotate workers, limit exposure time, regular maintenance.

    3. Personal Protective Equipment (PPE): Earplugs, earmuffs (check Noise Reduction Rating - NRR).

    4. Hearing Protection Program: Audiometric testing, training, monitoring.

Sound Intensity & Pressure Calculations

• Change in SPL with Distance (Inverse Square Law for point source in free field):

$$\Delta SPL = 20 \log_{10}\left(\frac{r_2}{r_1}\right)$$

*   **Doubling Distance**: $$\displaystyle r_2 = 2r_1 $$ → $$\displaystyle \Delta SPL = 20 \log_{10}(2) \approx 6 \ \text{dB} $$ decrease.

• Conversion between SPL (dB) and Actual Pressure (May 2024):

  Given $$\displaystyle SPL = 11.23 \ \text{dB} $$, $$\displaystyle p_{ref}=20 \ \mu\text{Pa} $$.

$$p = p_{ref} \times 10^{SPL/20} = 20 \times 10^{-6} \times 10^{11.23/20}$$

$$10^{11.23/20} = 10^{0.5615} \approx 3.64$$

$$p \approx 20 \times 10^{-6} \times 3.64 = 72.8 \ \mu\text{Pa}$$

> [!TIP] Remember: 0 dB = $$\displaystyle p_{ref} $$, 20 dB = $$\displaystyle 10 p_{ref} $$, 40 dB = $$\displaystyle 100 p_{ref} $$, etc. (20 dB per factor 10 in pressure).

VII. MISCELLANEOUS & DEFINITIONS (Short Note Topics)

Work Done by Harmonic Force (Nov 2023)

• Force: $$\displaystyle F(t) = F_0 \sin \omega t $$.

• Displacement: $$\displaystyle x(t) = x_0 \sin(\omega t - \phi) $$.

• Work over time $$\displaystyle t_1 $$ to $$\displaystyle t_2 $$:

$$W = \int_{t_1}^{t_2} F \dot{x} \ dt$$

• First Cycle ($$\displaystyle t_1=0, t_2=T=2\pi/\omega $$):

$$W_{cycle} = \frac{\pi F_0 x_0}{2} \cos\phi$$

*   If $$\displaystyle \phi=0 $$ (in phase), $$\displaystyle W_{max} = \frac{\pi F_0 x_0}{2} $$.

*   If $$\displaystyle \phi=\pi/2 $$ (90° lag), $$\displaystyle W=0 $$ (no net work, energy stored/returned).

• First Second: Plug limits $$\displaystyle t_1=0, t_2=1 $$ into integral. May not be integer cycles.

• First Quarter Second: $$\displaystyle t_1=0, t_2=0.25 $$ s. Compute definite integral numerically or analytically if $\omega$ known.

Fourier Series Expansion of Non-Periodic/Impact Forces (Jun 2025)

• Concept: Represent aperiodic force $F(t)$ (e.g., impact) as sum of sinusoidal components over a finite interval $[0, T]$, where $T$ is the "period" of the repeated pulse.

• Fourier Series:

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

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

Coefficients:

$$a_0 = \frac{1}{T} \int_0^T F(t) dt$$

$$a_n = \frac{2}{T} \int_0^T F(t) \cos(n\omega_0 t) dt$$

$$b_n = \frac{2}{T} \int_0^T F(t) \sin(n\omega_0 t) dt$$

• Application: For forging hammer impact (rectangular pulse of force $$\displaystyle F_0 $$ duration $\tau$), compute integrals over $[0,T]$.

Node in Vibration (May 2024)

• Definition: A point or line in a vibrating structure that remains stationary (zero displacement amplitude) during a particular mode of vibration.

• Sketch: For a simply supported beam, first mode has one node at center? Actually, simply supported has no internal nodes for first mode. Fixed-fixed beam first mode has two nodes (points of zero displacement) between ends.

  DiagramCANVAS: Sketch of a fixed-fixed beam in first bending mode, showing endpoints fixed, maximum deflection at center, and two nodes (points of zero deflection) symmetrically located between center and ends.

• Significance: Nodes indicate where structure does not move; useful for placing supports or isolators without affecting mode shape.

Whirling of Shafts (May 2024, Nov 2022)

• Explanation: Whirling is the precession (rotating bending deflection) of a shaft rotating at high speed. The shaft centerline describes a circle.

• Cause: Unbalance, gyroscopic moments, or external forces. At critical whirling speed, the rotational speed matches a natural bending frequency of the shaft-disc system, causing large amplitudes.

• Critical Speed: $$\displaystyle \omega_{crit} = \omega_n $$ (natural transverse frequency). Must operate below ~0.7 $$\displaystyle \omega_{crit} $$ for safety.

Energy Method in Vibration Analysis (Nov 2023)

• Rayleigh's Method:

  • Principle: For conservative system, max potential energy = max kinetic energy.

$$\frac{1}{2} k_{eq} (\text{max deflection})^2 = \frac{1}{2} m (\text{max velocity})^2$$

For assumed mode shape $y(x)$, compute strain energy $$\displaystyle V = \int \frac{1}{2} EI (y'')^2 dx $$ and kinetic energy $$\displaystyle T = \frac{1}{2} \omega^2 \int \rho A y^2 dx $$ (or with concentrated masses). Set $$\displaystyle V = T $$ to get $$\displaystyle \omega^2 $$.

*   **Use**: Approximate natural frequency; exact if assumed shape is exact mode shape.

• Principle of Virtual Work / D'Alembert's Principle: Alternative to Newton's law for deriving EOM. Include inertial forces $-m\ddot{x}$ as "virtual" forces.

Distinction between Viscous and Coulomb Damping (Jun 2025)

• Viscous:

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

  • Energy dissipated per cycle: $$\displaystyle \pi c \omega X^2 $$ (depends on frequency $\omega$ and amplitude $X$).

  • Amplitude decay: Exponential.

  • Frequency of damped oscillation: Slightly less than $$\displaystyle \omega_n $$, depends on $\zeta$.

• Coulomb:

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

  • Energy dissipated per half-cycle: $4\mu N X$ (independent of $\omega$, proportional to $X$).

  • Amplitude decay: Linear (decreases by constant $$\displaystyle \Delta X = 4\mu N/k $$ per half-cycle).

  • Frequency: Approximately $$\displaystyle \omega_n $$, but slightly less and depends on initial amplitude.

  • Motion: Non-sinusoidal (sharp corners).

END OF UNIT 5 SHORT NOTES
Aligned with RGPV past papers (Jun 2025, May 2024, Nov 2023, Nov 2022).