4.5 Mechanisms

4.5 Mechanisms

1. Gyroscopic Effects

1. Gyroscopic Couple:

   • When axis of spinning rotor is rotated, couple acts perpendicular to both.

   • $T = I \omega \omega_p$

   • $I$ = moment of inertia, $\omega$ = spin speed, $\omega_p$ = precession speed.

2. Applications:

   • Stabilization: Ships, aircraft, satellites.

   • Steering Effects:

     • Motorcycle/bicycle turning.

     • Ship rolling causes turning (gyroscopic action).

3. Reactive Forces:

   • Act on bearings supporting rotating shaft.

   • Direction depends on spin and precession directions.

2. Governors and Flywheels

1. Governors:

   • Speed-regulating devices for engines.

   • Types:

     • Centrifugal: Watt, Porter, Proell, Hartnell.

     • Inertia: More sensitive but less stable.

   • Terminology:

     • Sensitiveness: $\frac{N_1 - N_2}{N}$.

     • Isochronous: Constant speed at all radii.

     • Hunting: Continuous speed fluctuations.

2. Flywheels:

   • Energy storage devices to minimize speed fluctuations.

   • Function: Store energy during excess, release during deficit.

   • Coefficient of Fluctuation of Speed:

     • $C_s = \frac{\omega_{max} - \omega_{min}}{\omega_{avg}}$

   • Energy Storage:

     • $\Delta E = \frac{1}{2} I (\omega_{max}^2 - \omega_{min}^2) = I \omega^2 C_s$

   • Mass Calculation: $I = mk^2$, $k$ = radius of gyration.

3. Balancing

1. Static Balancing:

   • For objects in single plane of rotation.

   • Center of mass on axis of rotation.

   • $\sum m r = 0$.

2. Dynamic Balancing:

   • For objects with width/length.

   • Requires multiple correction planes.

   • Conditions:

     • $\sum m r = 0$ (static balance).

     • $\sum m r l = 0$ (couple balance).

3. Balancing of Rotating Masses:

   • Single Plane: Vector polygon method.

   • Multiple Planes: Two-plane balancing required.

4. Balancing of Reciprocating Masses:

   • Primary Force: $F_p = m r \omega^2 \cos \theta$.

   • Secondary Force: $F_s = m r \omega^2 \frac{\cos 2\theta}{n}$.

   • Complete Balance impossible with single cylinder.

   • Multi-cylinder arrangements (inline, V, radial) provide partial balance.

4. Cam and Follower

1. Components:

   • Cam: Rotating/oscillating member with shaped profile.

   • Follower: Reciprocating/oscillating output member.

2. Follower Types:

   • Knife-edge, Roller, Flat-face, Spherical.

   • Motion Path: Translating or oscillating.

3. Cam Terminology:

   • Base Circle: Smallest circle to cam profile.

   • Trace Point: Reference point on follower.

   • Pressure Angle: Angle between follower motion and normal to cam profile.

4. Follower Motions:

   • SHM: Smooth acceleration but infinite jerk at ends.

   • Uniform Velocity: Discontinuous acceleration.

   • Uniform Acceleration: Constant acceleration, finite jerk.

   • Cycloidal: No abrupt changes in acceleration (preferred for high speed).

5. Design Considerations:

   • Pressure angle should be small (<30°) to prevent jamming.

   • Smaller base circle increases pressure angle.

5. SHM and Cycloidal Motion

1. Simple Harmonic Motion (SHM):

   • Displacement: $s = \frac{h}{2}(1 - \cos \theta)$.

   • Velocity: $v = \frac{h\omega}{2} \sin \theta$.

   • Acceleration: $a = \frac{h\omega^2}{2} \cos \theta$.

   • Characteristics: Smooth acceleration, infinite jerk at ends.

2. Cycloidal Motion:

   • Displacement: $s = h\left(\frac{\theta}{\beta} - \frac{1}{2\pi} \sin\frac{2\pi\theta}{\beta}\right)$.

   • Velocity: $v = \frac{h\omega}{\beta}\left(1 - \cos\frac{2\pi\theta}{\beta}\right)$.

   • Acceleration: $a = \frac{2\pi h\omega^2}{\beta^2} \sin\frac{2\pi\theta}{\beta}$.

   • Advantage: Zero acceleration and jerk at ends (smooth).

6. Belt, Rope, Chain Drives

1. Belt Drives:

   • Types: Flat, V-belt, Timing (synchronous).

   • Velocity Ratio: $\frac{N_1}{N_2} = \frac{D_2}{D_1}$ (neglecting slip).

   • Slip: Difference between theoretical and actual speed.

   • Creep: Due to elasticity of belt.

   • Power Transmission: $P = (T_1 - T_2)v$.

   • Centrifugal Tension: $T_c = m v^2$.

2. Rope Drives:

   • For larger power and center distances.

   • Grooved pulleys for better grip.

3. Chain Drives:

   • Positive drive (no slip).

   • Velocity Ratio: $\frac{N_1}{N_2} = \frac{T_2}{T_1}$.

   • Length Calculation: $L = 2C + \frac{T_1+T_2}{2} + \frac{(T_1-T_2)^2}{4\pi^2 C}$.

   • Polygon Effect: Causes speed variations.

7. Gears and Gear Trains

1. Gear Terminology:

   • Module (m): $m = \frac{D}{T}$.

   • Circular Pitch (p): $p = \pi m$.

   • Pressure Angle ($\phi$): Standard 14.5°, 20°, 25°.

   • Addendum: Radial height above pitch circle.

   • Dedendum: Radial depth below pitch circle.

2. Gear Types:

   • Spur: Parallel axes, teeth parallel to axis.

   • Helical: Smoother, quieter, axial thrust.

   • Bevel: Intersecting axes (usually 90°).

   • Worm: High reduction ratio, self-locking.

   • Rack and Pinion: Rotary to linear motion.

3. Law of Gearing:

   • Common normal at point of contact must pass through pitch point.

   • Ensures constant velocity ratio.

4. Gear Trains:

   • Simple: $\frac{N_{last}}{N_{first}} = \pm \frac{\text{Product of driving teeth}}{\text{Product of driven teeth}}$.

   • Compound: Multiple gears on same shaft.

   • Epicyclic (Planetary):

     • $VR = 1 + \frac{T_{sun}}{T_{ring}}$ (for simple planetary).

     • Solved using tabular method or relative velocity.

   • Reverted: Input and output shafts co-axial.