Mechatronics (ME-503 (A)) - Unit 1 Short Notes

1. Kinematics and Dynamics of Mechanisms

Four-Bar Mechanisms

• Velocity Analysis (Relative Velocity Method)

  • For link BC: $$\displaystyle v_C = v_B + v_{C/B} $$, where $$\displaystyle v_{C/B} = \omega_{BC} \times r_{C/B} $$.

  • Instantaneous Center (IC) Method: $$\displaystyle \omega = \frac{v}{r} $$ from IC.

• Acceleration Analysis

  • $$\displaystyle a_C = a_B + \alpha_{BC} \times r_{C/B} - \omega_{BC}^2 \cdot r_{C/B} $$.

  • Coriolis Component: $$\displaystyle a_c = 2 \omega \cdot v_{rel} $$ when a point moves on a rotating link.

• Instantaneous Center Method

  • Number of ICs for n-link mechanism: $$\displaystyle N = \frac{n(n-1)}{2} $$.

  • Kennedy’s Theorem: Three ICs lie on a straight line.

[!TIP] Common Pitfall: In acceleration, always include both tangential ($\alpha \times r$) and centripetal ($$\displaystyle \omega^2 r $$) components. Coriolis term appears only if the point has relative velocity on a rotating link.

Cams and Followers

• Offset Circular Cam Acceleration

  • For a follower with line of action offset by $e$: $$\displaystyle x = r(1 - \cos\theta) + e\sin\theta $$.

  • Acceleration: $$\displaystyle a = r\omega^2\cos\theta + e\omega^2\cos\theta = \omega^2(r+e)\cos\theta $$.

• Condition for Follower Lift

  • Lift occurs when normal reaction $$\displaystyle R_n > 0 $$: $$\displaystyle R_n = F_{spring} - m a_n > 0 $$.

  • Critical Speed: $$\displaystyle \omega_{crit} = \sqrt{\frac{F_{spring}}{m(r+e)\cos\theta}} $$.

2. Governors

Function and Classification

• Purpose: Maintain constant engine speed under load variation (unlike flywheels, which reduce speed fluctuation).

• Types: Centrifugal (Watt, Porter, Proell), Inertia (governor balls on shaft).

Key Terminology

• Sensitiveness: $$\displaystyle \frac{\text{Change in speed}}{\text{Change in radius}} = \frac{N_1 - N_2}{N_{mean}} $$. High sensitiveness → large speed variation for small load change.

• Isochronism: Infinite sensitiveness ($$\displaystyle N_1 = N_2 $$); speed constant for all radii.

• Hunting: Oscillations about equilibrium position due to over-sensitivity.

• Stability: Governor returns to equilibrium after disturbance.

• Coefficient of Insensitiveness: $$\displaystyle K = \frac{1}{S} - 1 $$, where $S$ = sensitiveness.

Watt Governor

• Derivation of Height:

$$h = \frac{g}{\omega^2} = \frac{895.3}{N^2} \text{ (if } h \text{ in m, } N \text{ in rpm)}$$

\boxed{h \propto \frac{1}{N^2}}

• Proof: From force balance, $$\displaystyle mg = m\omega^2 r \cdot \frac{h}{r} \Rightarrow h = \frac{g}{\omega^2} $$.

Porter Governor

• Construction: Upper arms pivot on axis; lower arms pivot on sleeve.

• Speed Range with Friction:

$$N_{max} = \sqrt{\frac{(m+M)g}{m r_1} \cdot \frac{a+b}{a}} \quad ; \quad N_{min} = \sqrt{\frac{(m+M)g}{m r_2} \cdot \frac{a+b}{a}}$$

where $a$ = distance from pivot to sleeve, $b$ = length of lower arm.

• Comparison with Proell: Proell has lower arms pivoted on a separate frame, giving higher sensitiveness.

Proell Governor

• Construction: Lower arms pivoted at offset $h$ from axis; extensions parallel to axis at min radius.

• Minimum Speed (when extensions parallel):

$$N_{min} = \frac{1}{2\pi} \sqrt{\frac{g(m+M)}{r \left(1 + \frac{M}{2m}\right)}}$$

• Sensitiveness: Greater than Porter because $r$ increases more for same sleeve lift.

Stability Analysis

• Controlling Force vs. Radius Diagram:

  • Stable: $$\displaystyle F_c $$ increases with $r$ (slope $$\displaystyle > 0 $$).

  • Unstable: $$\displaystyle F_c $$ decreases with $r$ (slope $$\displaystyle < 0 $$).

  • Isochronous: $$\displaystyle F_c $$ constant (horizontal line).

• Condition for Stability: $$\displaystyle \frac{dF_c}{dr} > 0 $$.

[!TIP] Exam Focus: Derive Porter/Proell speed formulas from force triangles. Compare sensitiveness analytically: Proell’s $r$ change is larger for same sleeve lift → higher $S$.

3. Flywheels

Purpose vs. Governor

• Flywheel: Reduces speed fluctuation by storing/releasing kinetic energy.

• Governor: Controls mean speed by regulating energy supply.

Energy and Speed Fluctuations

• Fluctuation of Energy ($\Delta E$): Maximum energy deviation from mean.

• Fluctuation of Speed ($\Delta N$): $$\displaystyle N_{max} - N_{min} $$.

• Coefficients:

$$C_E = \frac{\Delta E}{E_{mean}} \quad ; \quad C_S = \frac{\Delta N}{N_{mean}}$$

Relation: $$\displaystyle C_S = \sqrt{C_E} $$ (for simple flywheel).

Turning Moment Diagrams

• Interpretation: Torque vs. crank angle. Area above mean = excess energy; below = deficit.

• Area Method: $$\displaystyle \Delta E = \text{Maximum accumulated energy} - \text{Minimum accumulated energy} $$.

• Scale Conversion:

$$\text{Energy scale} = \frac{\text{Torque scale} \times \text{Angle scale}}{2\pi}$$

Design Parameters

• Mass Moment of Inertia:

$$I = \frac{\Delta E}{\omega_{mean}^2 \cdot C_S}$$

• Radius of Gyration: $$\displaystyle k = \sqrt{I/m} $$.

• Speed Constraints: $$\displaystyle N_{max} = N_{mean}(1 + C_S/2) $$; $$\displaystyle N_{min} = N_{mean}(1 - C_S/2) $$.

[!TIP] Always convert turning moment diagram areas to energy using $$\displaystyle \Delta E = \text{Area} \times (\text{scale factors}) $$. Use $$\displaystyle \Delta E = I \omega_{mean}^2 \cdot C_S $$ for $I$ or $k$.

4. Balancing of Engines

Fundamentals

• Primary Balancing: Balance forces at crank radius (1st order).

• Secondary Balancing: Balance forces due to connecting rod obliquity (2nd order).

• Partial Balancing: Only a fraction of reciprocating masses balanced (e.g., 2/3 in locomotives).

Forces in Reciprocating Engines

• Inertia Force: $$\displaystyle F_I = m \omega^2 r \left( \cos\theta + \frac{r}{l}\cos2\theta \right) $$.

  • Primary: $$\displaystyle m \omega^2 r \cos\theta $$.

  • Secondary: $$\displaystyle m \omega^2 \frac{r}{l} \cos2\theta $$.

• Piston Effort ($$\displaystyle F_P $$): Net force on piston = gas force $$\displaystyle - F_I $$.

• Crank Effort ($$\displaystyle F_T $$): Tangential component on crank: $$\displaystyle F_T = F_P \frac{\sin(\theta + \phi)}{\cos\phi} $$.

• Side Thrust ($$\displaystyle F_S $$): $$\displaystyle F_S = F_P \frac{\cos(\theta + \phi)}{\cos\phi} $$.

• $$\displaystyle \phi = \tan^{-1}\left(\frac{\sin\theta}{\cos\theta + r/l}\right) $$.

[!TIP] Neglecting obliquity ($l \gg r$) simplifies: $$\displaystyle F_I \approx m\omega^2 r \cos\theta $$, $$\displaystyle F_T \approx F_P \sin\theta $$.

Unbalanced Effects in Locomotives

• Hammer Blow (Unbalanced Vertical Force):

$$F_V = (1 - c) m \omega^2 r \cos\theta$$

where $c$ = fraction balanced. Max at $$\displaystyle \theta=0,\pi $$.

• Swaying Couple (Unbalanced Couple about vertical axis):

$$F_{SC} = (1 - c) m \omega^2 r \cdot d \cdot \sin\theta$$

where $d$ = distance between cylinder centerlines.

• Tractive Effort Variation:

$$\Delta F = \pm (1 - c) m \omega^2 r \cdot \frac{D}{2}$$

where $D$ = wheel diameter.

Multi-Cylinder Balancing

• In-line Engines: Primary forces balanced if cranks equally spaced; secondary always unbalanced unless special arrangements.

• Radial Engines (e.g., 3-cylinder, 120° apart): Primary balanced; secondary partially balanced.

• Locomotive Balancing (2-cylinder, crank angles $$\displaystyle 90^\circ $$ or $$\displaystyle 180^\circ $$):

  • Balance fraction $c$ chosen to limit hammer blow at max speed.

  • Balancing mass: $$\displaystyle m_b = c \cdot m \cdot \frac{r_b}{r} $$.

• Balancing Mass Determination:

  • Magnitude: $$\displaystyle m_b r_b = c \cdot m r $$.

  • Position: Opposite to crank angle at which unbalanced force is max.

[!TIP] For two-cylinder locomotives, hammer blow max at $$\displaystyle \theta=0 $$; swaying couple max at $$\displaystyle \theta=90^\circ $$. Use $$\displaystyle c = 1 - \frac{F_{V,max}}{m\omega^2 r} $$ to find balance fraction.

5. Friction Devices

Friction Circle

• Definition: Circle representing frictional torque in journal bearings; radius = $r \tan\phi$.

• Radius Derivation:

$$r_f = r \cdot \tan\phi \approx r \cdot \mu \quad (\text{for small } \phi)$$

where $\phi$ = friction angle, $$\displaystyle \mu = \tan\phi $$.

Clutches

• Single Plate Clutch

  • Pressure Distribution:

    • Uniform Pressure: $$\displaystyle p = \frac{F}{\pi(R^2 - r^2)} $$.

    • Uniform Wear: $$\displaystyle p \cdot r = \text{const} \Rightarrow p = \frac{F}{2\pi(R - r)r} $$.

  • Torque Capacity:

    • Uniform Pressure: $$\displaystyle T = \mu F \cdot \frac{R^3 - r^3}{3(R^2 - r^2)} \approx \mu F R_m $$ (mean radius $$\displaystyle R_m $$).

    • Uniform Wear: $$\displaystyle T = \frac{2}{3} \mu F \cdot \frac{R^3 - r^3}{R^2 - r^2} \approx \frac{2}{3} \mu F R_m $$.

  • Design: Given $T$, $F$, $\mu$, find $$\displaystyle R_m $$, face width $$\displaystyle b = (R - r) $$, and $r/R$ ratio (often $$\displaystyle r/R = 0.5 $$ to $0.6$).

• Conical Clutch

  • Working: Normal force $$\displaystyle F_N = \frac{F}{\cos\alpha} $$; friction force $$\displaystyle \mu F_N $$.

  • Torque: $$\displaystyle T = \mu F \tan\alpha \cdot \frac{R^3 - r^3}{3(R^2 - r^2)} $$.

  • Effect of Cone Angle $\alpha$: Smaller $\alpha$ → higher normal force → higher torque, but may not self-energize.

Brakes

• Band Brake

  • Torque: $$\displaystyle T = (T_1 - T_2) \cdot r $$, where $$\displaystyle \frac{T_1}{T_2} = e^{\mu\theta} $$ ($\theta$ in radians).

  • Lever Mechanics: $$\displaystyle F \cdot l = T_1 \cdot a - T_2 \cdot b $$ (depends on attachment).

• Internal Expanding Brake

  • Construction: Two shoes inside drum; hydraulic or mechanical actuation.

  • Self-energizing: Leading shoe gets additional force from drum rotation.

• Double Shoe Brake

  • Spring Force: For symmetric shoes, $$\displaystyle T = 2\mu F_s R \cdot \frac{4\sin^2(\theta/2)}{\theta} $$.

  • Shoe Width: $$\displaystyle b = \frac{F}{p \cdot (2R \cdot \theta)} $$ (pressure limit $p$).

Bearings

• Conical Pivot Bearing

  • Power Loss:

    • Uniform Pressure: $$\displaystyle P = \frac{2}{3} \mu W \omega R \cdot \frac{R^3 - r^3}{R^2 - r^2} $$.

    • Uniform Wear: $$\displaystyle P = \frac{1}{2} \mu W \omega \cdot \frac{R^2 + r^2}{R + r} $$.

  • Design: Given $W$, $$\displaystyle p_{max} $$, find $R$, $r$ from $$\displaystyle W = \frac{\pi p (R^2 - r^2)}{\sin\alpha} $$.

• Collar Bearings

  • Power Absorption: $$\displaystyle P = \mu W \omega \cdot \frac{R^2 + r^2}{2(R - r)} $$ (uniform pressure).

  • Number of Collars: $$\displaystyle n = \frac{W}{p \cdot \pi (R^2 - r^2)} $$.

[!TIP] For clutches/brakes, identify pressure distribution (uniform vs. wear). For bearings, uniform wear gives lower power loss than uniform pressure.

6. Dynamometers

Classification

• Absorption Dynamometers: Absorb power as heat (e.g., Prony brake, rope brake).

• Transmission Dynamometers: Measure power while transmitting (e.g., epicyclic, torsion).

Torsion Dynamometers

• Working Principle: Measure angular twist $\theta$ in a shaft: $$\displaystyle T = \frac{GJ}{L} \theta $$.

  • $G$ = shear modulus, $J$ = polar moment, $L$ = length between strain gauges.

• Power Calculation: $$\displaystyle P = T \cdot \omega $$.

• Example: Epicyclic Train-Type:

  • Torque on gear train gives $T$; speed measured separately.

[!TIP] Absorption dynamometers waste energy; transmission types are more efficient for continuous measurement. Torsion type is precise for high-speed shafts.