UNIT 2: MECHANICAL VIBRATIONS

I. INTRODUCTION TO VIBRATIONS

A. Fundamental Concepts

• Vibration: Oscillatory motion of a body about its equilibrium position.

• Classification:

  | Type | Description | Example | |------|-------------|---------| | Free vs Forced | No external force vs持续外力 | Plucked string vs machine on foundation | | Undamped vs Damped | No energy loss vs energy dissipation | Ideal spring-mass vs real system with damper | | Linear vs Nonlinear | Restoring force ∝ displacement vs not | Simple spring vs large deflection | | Deterministic vs Random | Predictable vs unpredictable | Harmonic force vs road roughness |

• Main Causes: Unbalance, misalignment, looseness, backlash, manufacturing errors, external forces.

• Simple Harmonic Motion (SHM):

  • Mathematical definition: $$\displaystyle \ddot{x} + \omega_n^2 x = 0 $$

  • Vector Representation:

    DiagramCANVAS: Show rotating vector (phasor) with projections on x-axis for displacement, derivative for velocity, second derivative for acceleration. Label amplitude A, angular frequency ω, phase φ.

    • Displacement: $$\displaystyle x = A \sin(\omega_n t + \phi) $$

    • Velocity: $$\displaystyle \dot{x} = A \omega_n \cos(\omega_n t + \phi) $$

    • Acceleration: $$\displaystyle \ddot{x} = -A \omega_n^2 \sin(\omega_n t + \phi) $$

B. Degrees of Freedom (DOF)

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

• Significance: Determines complexity; SDOF → 1 eqn, MDOF → coupled eqns, continuous → PDE.

• Examples:

  • SDOF: Mass on spring, pendulum (small angle).

  • MDOF: Two-mass-three-spring system.

  • Continuous: Beam, shaft, plate.

C. Modeling Approach

• Free Body Diagram (FBD): Isolate mass, apply Newton's 2nd law → equation of motion.

• Energy Methods:

  • Rayleigh's Principle: For conservative systems, $$\displaystyle \text{KE}_{\text{max}} = \text{PE}_{\text{max}} $$.

  • Natural frequency approximation: $$\displaystyle \omega_n^2 = \frac{\text{Stiffness at equilibrium}}{\text{Mass}} $$ for assumed mode shape.

  [!TIP] Rayleigh's method gives upper bound for $$\displaystyle \omega_n $$; use exact mode shape for accuracy.

II. SINGLE DEGREE OF FREEDOM (SDOF) SYSTEMS

A. Free Vibration

1. Undamped Systems

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

• Natural frequency: $$\displaystyle \omega_n = \sqrt{\frac{k}{m}} $$ \quad (rad/s)

• General solution: $$\displaystyle x(t) = A \sin(\omega_n t + \phi) $$

  where $A$, $\phi$ from initial conditions $$\displaystyle x(0)=x_0 $$, $$\displaystyle \dot{x}(0)=\dot{x}_0 $$.

2. Viscously Damped Systems

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

• Standard form: $$\displaystyle \ddot{x} + 2\zeta\omega_n\dot{x} + \omega_n^2 x = 0 $$

  where $$\displaystyle \zeta = \frac{c}{c_c} $$, \quad $$\displaystyle c_c = 2\sqrt{km} $$ (critical damping)

• Damped natural frequency: $$\displaystyle \omega_d = \omega_n \sqrt{1-\zeta^2} $$

• Response types:

  • Underdamped ($$\displaystyle \zeta < 1 $$): Oscillatory with decaying amplitude.

    $$\displaystyle x(t) = A e^{-\zeta\omega_n t} \sin(\omega_d t + \phi) $$

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

    $$\displaystyle x(t) = (A + B t) e^{-\omega_n t} $$

  • Overdamped ($$\displaystyle \zeta > 1 $$): Slow return, no oscillation.

    $$\displaystyle x(t) = A e^{s_1 t} + B e^{s_2 t} $$, $$\displaystyle s_{1,2} = -\zeta\omega_n \pm \omega_n\sqrt{\zeta^2-1} $$

• Logarithmic decrement: $$\displaystyle \delta = \ln\left(\frac{x(t)}{x(t+T_d)}\right) = \frac{2\pi\zeta}{\sqrt{1-\zeta^2}} $$

  where $$\displaystyle T_d = \frac{2\pi}{\omega_d} $$ (damped period).

  [!TIP] $\delta$ only valid for $$\displaystyle \zeta < 1 $$ (oscillatory response).

3. Coulomb Damping (Dry Friction)

• Equation: $$\displaystyle m\ddot{x} + kx = \pm F_f $$, where $$\displaystyle F_f = \mu N $$ (constant magnitude).

• Solution: Amplitude decreases linearly per half-cycle: $$\displaystyle \Delta A = \frac{2F_f}{k} $$

• Frequency: Independent of amplitude, $$\displaystyle \omega_n = \sqrt{k/m} $$ (same as undamped).

• Comparison with viscous damping:

  | Feature | Viscous Damping | Coulomb Damping | |---------|----------------|----------------| | Force | $c\dot{x}$ | $$\displaystyle \pm F_f $$ (constant) | | Amplitude decay | Exponential | Linear | | Frequency | Slightly less than $$\displaystyle \omega_n $$ | Equal to $$\displaystyle \omega_n $$ | | Energy dissipation per cycle | $$\displaystyle \Delta E = 2\pi c \omega A^2 $$ | $$\displaystyle \Delta E = 4F_f A $$ |

4. Hysteresis and Structural Damping

• Hysteresis loop: Force-displacement loop for cyclic loading; area = energy dissipated per cycle.

• Equivalent viscous damping: $$\displaystyle c_{eq} = \frac{\text{Area}}{2\pi \omega A^2} $$

• Structural damping ratio: Often expressed as loss factor $$\displaystyle \eta = \frac{\Delta E}{2\pi E} $$.

5. Rotational Systems (Torsional)

• Equation: $$\displaystyle J\ddot{\theta} + c_\theta \dot{\theta} + k_\theta \theta = 0 $$

• Natural frequency: $$\displaystyle \omega_n = \sqrt{\frac{k_\theta}{J}} $$

• Combined systems (e.g., rigid bar with springs):

  • Use Newton's 2nd law for rotation: $$\displaystyle \sum M = J\ddot{\theta} $$

  • Equivalent stiffness and inertia about pivot.

B. Forced Vibration

1. Harmonic Forcing

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

• Steady-state solution: $$\displaystyle x_p(t) = X \sin(\omega t - \phi) $$

• Magnification factor (Dynamic Amplification):

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

where $$\displaystyle r = \frac{\omega}{\omega_n} $$, $$\displaystyle X_s = \frac{F_0}{k} $$ (static deflection).

• Phase angle: $$\displaystyle \phi = \tan^{-1}\left(\frac{2\zeta r}{1-r^2}\right) $$

  • $$\displaystyle \phi = 0 $$ for $r \ll 1$, $$\displaystyle \phi = \pi $$ for $r \gg 1$, $$\displaystyle \phi = \pi/2 $$ at $$\displaystyle r=1 $$.

• Resonance:

  • For $$\displaystyle \zeta < \frac{1}{\sqrt{2}} $$, peak $M$ occurs at $$\displaystyle r_r = \sqrt{1-2\zeta^2} $$

  • At resonance ($$\displaystyle r=1 $$): $$\displaystyle M = \frac{1}{2\zeta} $$

  [!TIP] Resonance amplitude inversely proportional to $\zeta$; low damping → high amplification.

2. Base Excitation

• Equation: $$\displaystyle m\ddot{x} + c(\dot{x}-\dot{y}) + k(x-y) = 0 $$, where $$\displaystyle y = Y \sin \omega t $$.

• Transmissibility (ratio of transmitted force to applied force):

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

• Relative motion amplitude: $$\displaystyle |x-y| = Y \cdot \frac{r^2}{\sqrt{(1-r^2)^2 + (2\zeta r)^2}} $$

• Isolation: For $$\displaystyle r > \sqrt{2} $$, $$\displaystyle M < 1 $$ (vibration isolation effective).

  [!TIP] Isolation requires damping ratio $$\displaystyle \zeta < 0.2 $$ for best performance.

3. Rotating Unbalance

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

• Response amplitude: $$\displaystyle X = \frac{m_e e \omega^2 / m}{\sqrt{(\omega_n^2 - \omega^2)^2 + (2\zeta\omega_n\omega)^2}} $$

• Force transmitted to base: $$\displaystyle F_T = kX \sqrt{1 + (2\zeta r)^2} $$

4. Transient Response (Total Response)

• General solution: $$\displaystyle x(t) = x_h(t) + x_p(t) $$

  • $$\displaystyle x_h(t) $$: Homogeneous (free vibration) part, depends on $\zeta$.

  • $$\displaystyle x_p(t) $$: Particular (steady-state) part.

• Initial conditions: $$\displaystyle x(0)=x_0 $$, $$\displaystyle \dot{x}(0)=\dot{x}_0 $$ determine constants in $$\displaystyle x_h $$.

• Work done by harmonic force:

  • Over one complete cycle: $$\displaystyle W_{\text{cycle}} = \pi F_0 X \sin\phi $$

  • Over first quarter cycle: Integrate $F\dot{x}dt$ from $0$ to $T/4$.

C. Special Applications

1. Gun Recoil Mechanism

• Design criterion: Critically damped ($$\displaystyle \zeta=1 $$) to minimize recoil time.

• Recoil distance $$\displaystyle x_{\text{max}} $$ from energy: $$\displaystyle \frac{1}{2}mv_0^2 = \frac{1}{2}kx_{\text{max}}^2 $$ (if no damping).

• With damping: $$\displaystyle x_{\text{max}} = v_0 / \omega_n $$ for critical damping.

• Spring stiffness: $$\displaystyle k = m\omega_n^2 $$, with $$\displaystyle \omega_n $$ chosen to limit $$\displaystyle x_{\text{max}} $$.

2. Vehicle Suspension System

• Base excitation: Road profile $$\displaystyle y = Y \sin\left(\frac{2\pi}{\lambda} v t\right) $$, where $\lambda$ = wavelength, $v$ = speed.

• Critical speed: Speed at which $$\displaystyle r=1 $$ (resonance), $$\displaystyle v_{\text{crit}} = \frac{\lambda \omega_n}{2\pi} $$.

• Amplitude at speed $v$: Use transmissibility $M$ with $$\displaystyle r = \frac{v}{v_{\text{crit}}} $$.

3. Machine on Resilient Foundation

• Static deflection: $$\displaystyle \delta_s = \frac{W}{k} $$, where $W$ = weight.

• Natural frequency: $$\displaystyle \omega_n = \sqrt{\frac{g}{\delta_s}} $$ (since $$\displaystyle k = W/\delta_s $$, $$\displaystyle m = W/g $$).

• Amplitude at resonance ($$\displaystyle \omega = \omega_n $$): $$\displaystyle X = \frac{Y}{2\zeta} $$, where $Y$ = base amplitude.

• Damping ratio determination: $$\displaystyle \zeta = \frac{Y}{2X} $$ from resonance data.

• Dynamic force on base: $$\displaystyle F_{\text{dyn}} = kX \sqrt{1+(2\zeta)^2} $$ at $$\displaystyle \omega=\omega_n $$.

• Relative amplitude: $$\displaystyle |x-y| = Y \cdot \frac{1}{2\zeta\sqrt{1-\zeta^2}} $$ at $$\displaystyle \omega=\omega_n $$.

4. Dynamic Vibration Absorber

• Principle: Add SDOF (mass $$\displaystyle m_a $$, stiffness $$\displaystyle k_a $$) to primary system; tuned so $$\displaystyle k_a/m_a = \omega^2 $$ at excitation frequency $\omega$.

• Tuning condition: $$\displaystyle k_a = m_a \omega^2 $$.

• Effect: Primary system amplitude becomes zero at $\omega$ if no damping in absorber.

• With damping in primary system, amplitude reduced over a frequency range.

III. MULTI-DEGREE OF FREEDOM (MDOF) SYSTEMS

A. Two-DOF Systems

1. Modeling and Equations of Motion

• Matrix form (undamped): $$\displaystyle \mathbf{M}\ddot{\mathbf{x}} + \mathbf{K}\mathbf{x} = \mathbf{F}(t) $$

  where $$\displaystyle \mathbf{x} = [x_1, x_2]^T $$, $\mathbf{M}$ = mass matrix, $\mathbf{K}$ = stiffness matrix.

• FBD for coupled systems: Draw each mass, show spring forces from adjacent springs.

• Example: Two masses $$\displaystyle m_1,m_2 $$, three springs $$\displaystyle k_1,k_2,k_3 $$:

$$ \begin{aligned} m_1\ddot{x}_1 + (k_1+k_2)x_1 - k_2 x_2 &= 0 \\ m_2\ddot{x}_2 - k_2 x_1 + (k_2+k_3)x_2 &= 0 \end{aligned} $$

2. Natural Frequencies and Mode Shapes

• Eigenvalue problem: $$\displaystyle (\mathbf{K} - \omega^2 \mathbf{M})\boldsymbol{\phi} = \mathbf{0} $$

• Frequency equation: $$\displaystyle \det(\mathbf{K} - \omega^2 \mathbf{M}) = 0 $$ → quadratic in $$\displaystyle \omega^2 $$ for 2-DOF.

• Mode shapes $\boldsymbol{\phi}$: Non-trivial solutions; normalized arbitrarily.

• Orthogonality:

  • $$\displaystyle \boldsymbol{\phi}_i^T \mathbf{M} \boldsymbol{\phi}_j = 0 $$ for $i \neq j$ (mass orthogonality)

  • $$\displaystyle \boldsymbol{\phi}_i^T \mathbf{K} \boldsymbol{\phi}_j = 0 $$ for $i \neq j$ (stiffness orthogonality)

• Symmetric systems ($$\displaystyle k_1=k_2=k $$, $$\displaystyle m_1=m_2=m $$):

  • $$\displaystyle \omega_1^2 = \frac{k}{m} $$ (in-phase mode: $$\displaystyle \phi_1 = [1, 1]^T $$)

  • $$\displaystyle \omega_2^2 = \frac{3k}{m} $$ (out-of-phase mode: $$\displaystyle \phi_2 = [1, -1]^T $$)

• Node: Point in a mode shape where amplitude is zero.

3. Principal Coordinates

• Transformation: $$\displaystyle \mathbf{x} = \mathbf{\Phi} \mathbf{q} $$, where $$\displaystyle \mathbf{\Phi} = [\boldsymbol{\phi}_1, \boldsymbol{\phi}_2] $$ (modal matrix).

• Decoupling: Substitute into equations → $$\displaystyle \ddot{q}_i + \omega_i^2 q_i = 0 $$ (undamped, no forcing).

• Determination: For given spring-mass system, find $$\displaystyle \boldsymbol{\phi}_1,\boldsymbol{\phi}_2 $$ from eigenvalue problem, then compute $$\displaystyle \mathbf{q} = \mathbf{\Phi}^{-1}\mathbf{x} $$.

B. Torsional Systems

1. Modeling

• Elements: Inertias $$\displaystyle J_i $$ (kg·m²), torsional springs $$\displaystyle k_{\theta i} $$ (N·m/rad).

• Equation: $$\displaystyle \mathbf{J}\ddot{\boldsymbol{\theta}} + \mathbf{K}_\theta \boldsymbol{\theta} = \mathbf{T}(t) $$

• Example: Two inertias $$\displaystyle J_1,J_2 $$ connected by shaft $$\displaystyle k_\theta $$:

$$ \begin{aligned} J_1\ddot{\theta}_1 + k_\theta(\theta_1 - \theta_2) &= 0 \\ J_2\ddot{\theta}_2 + k_\theta(\theta_2 - \theta_1) &= 0 \end{aligned} $$

2. Natural Frequencies and Normal Modes

• Eigenvalue problem: $(\mathbf{K}_\theta - \omega^2 \mathbf{J})\boldsymbol