THEORY OF MACHINES (ME-403) - Unit 1 Short Notes

1.0 FUNDAMENTALS OF MECHANISMS & KINEMATICS

1.1 Basic Concepts

• Machine: A combination of mechanisms that transmits or modifies energy to perform a specific task (e.g., engine, compressor).

• Mechanism: A combination of kinematic links and pairs that transmits motion and force but does not necessarily modify energy. It is a kinematic chain with one link fixed.

• Structure: A mechanism with no relative motion between its links (DoF = 0). It is designed to bear loads (e.g., bridge truss, building frame).

• Kinematic Link: A rigid body that forms part of a machine and has relative motion with respect to other links.

• Kinematic Pair: Two links in contact that constrain relative motion. Classified by:

  • Lower Pair: Surface contact (revolute, prismatic, screw, cylindrical, spherical, planar).

  • Higher Pair: Line or point contact (cam-follower, gear-teeth, ball-bearing).

• Kinematic Chain: An assembly of links connected by kinematic pairs. If one link is fixed, it becomes a mechanism.

• Kinematic Diagram/Schematic: A simplified line diagram representing the mechanism, showing only the essential dimensions and joints.

• Constrained Motion: The relative motion between links is restricted by the type of pair.

  • Completely Constrained: Motion is fully defined in one direction (e.g., piston in cylinder - prismatic pair).

  • Incompletely Constrained: Motion is possible in more than one direction (e.g., shaft in a journal bearing - cylindrical pair allows rotation and axial slide).

  • Successfully Constrained: Additional links make the motion determinate (e.g., a shaft with two bearings).

[!TIP]

Common Pitfall: Confusing a structure (DoF=0, no motion) with a mechanism (DoF>0, motion possible). Always check if any link can move relative to the fixed link.

1.2 Degree of Freedom (Mobility)

• Definition: The number of independent inputs required to define the position of all links in a mechanism relative to a fixed link. It determines if the mechanism is workable (DoF = 1 for a simple input).

• Kutzbach's Criterion (for plane mechanisms):

$$m = 3(n - 1) - 2j - h$$

Where:

*   $m$ = Degree of freedom (mobility)

*   $n$ = Number of links (including fixed link)

*   $j$ = Number of **lower pairs** (each contributes 1 constraint)

*   $h$ = Number of **higher pairs** (each contributes 2 constraints)

\boxed{m = 3(n - 1) - 2j - h}

• Gruebler's Criterion (Modified for redundant constraints):

$$m = 3(n - 1) - 2j_1 - j_2$$

Where:

*   $$\displaystyle j_1 $$ = Number of 1-DOF (lower) pairs

*   $$\displaystyle j_2 $$ = Number of 2-DOF (higher) pairs

This form accounts for special cases like parallel joints or links with multiple connections. For most standard mechanisms, Kutzbach and Gruebler yield the same result.

• Calculation Steps:

  1. Count all links ($n$), including the fixed frame.

  2. Identify and count all lower pairs ($j$) and higher pairs ($h$).

  3. Apply the formula.

  4. If $$\displaystyle m < 0 $$, the structure is over-constrained (a structure or statically indeterminate).

  5. If $$\displaystyle m = 0 $$, it's a structure.

  6. If $$\displaystyle m = 1 $$, it's a simple mechanism (one driver).

  7. If $$\displaystyle m > 1 $$, it's a compound mechanism (needs multiple drivers or is redundant).

[!TIP]

Exam Focus: Always include the fixed link in $n$. For a four-bar linkage, $$\displaystyle n=4 $$, $$\displaystyle j=4 $$ (all revolute pairs), $$\displaystyle h=0 $$ → $$\displaystyle m = 3(4-1) - 2(4) = 9 - 8 = 1 $$.

1.3 Grashof's Law & Four-Bar Linkage Classification

• Grashof's Law (for a four-bar chain): For continuous relative rotation of at least one link, the sum of the shortest ($S$) and longest ($L$) link lengths must be less than or equal to the sum of the remaining two links ($P$ and $Q$).

$$S + L \leq P + Q$$

• Link Length Ordering: First, arrange the four links in ascending order: $S \leq P \leq Q \leq L$.

• Classification Based on Fixed Link:

  | Fixed Link (Ground) | Mechanism Type | Crank Possibility | | :--- | :--- | :--- | | Shortest link ($S$) | Double Crank (Drag Link) | Both adjacent links (crank and coupler) can rotate fully. | | Longest link ($L$) | Crank-Rocker | Shortest link (crank) rotates fully; opposite link (rocker) oscillates. | | One of the other links ($P$ or $Q$) | Double Rocker | Both cranks oscillate; no link can rotate 360°. | | $$\displaystyle S + L = P + Q $$ | Change Point Mechanism | All links can theoretically rotate 360°, but mechanism changes type at dead-center positions. |

• Key Implications:

  • If $$\displaystyle S + L < P + Q $$: At least one link can make a full revolution.

  • If $$\displaystyle S + L > P + Q $$: No link can rotate fully; it is a double rocker for any fixed link.

  • The crank is the link that can rotate 360°. It must be adjacent to the fixed link.

[!TIP]

Common Pitfall: Misidentifying $S$ and $L$. Always sort all four link lengths first. Also, the link opposite the fixed link is the output rockpper/crank.

1.4 Inversions of Mechanisms

• Inversion: Obtaining a different mechanism by fixing a different link in a kinematic chain. The relative motion between links remains unchanged.

• Four-Bar Chain Inversions (4 links, 4 revolute pairs):

  1. Fixed $S$ (Shortest): Double Crank (Drag Link) - used in pantographs.

  2. Fixed $L$ (Longest): Crank-Rocker - common in pump handles, windshield wipers.

  3. Fixed $P$ or $Q$: Double Rocker - used in steering linkages.

• Slider-Crank Chain Inversions (4 links: 3 turning, 1 sliding pair):

  1. Fixed crank (1): Reciprocating engine/compressor (slider is output, crank is input).

  2. Fixed connecting rod (3): Oscillating engine (rocker is output, crank is input).

  3. Fixed slider (4): Hand pump, Whitworth quick return (crank is output, slider is input).

  4. Fixed ground (2): Slider-crank mechanism (standard form).

• Double Slider Crank Chain Inversions (4 links: 2 turning, 2 sliding pairs):

  1. Fixed link 1 (adjacent to both sliders): Elliptical Trammel - used to draw ellipses.

  2. Fixed link 2 (opposite to fixed link 1): Scotch Yoke - converts rotary to reciprocating motion with sinusoidal displacement.

  3. Fixed link 3 or 4 (a slider): Oldham's Coupling - connects two parallel shafts with constant velocity ratio, accommodating slight misalignment.

[!TIP]

Exam Focus: Be able to sketch the inversion and state its application. For example, "Inversion of slider-crank with fixed connecting rod gives an oscillating engine mechanism."

1.5 Instantaneous Center (IC) Method

• Instantaneous Center (IC): The point on a body (or in a plane) that has zero instantaneous velocity relative to another body. It is the instantaneous center of rotation for that body.

• Types of ICs:

  • Fixed IC: Permanently fixed in the fixed link (e.g., revolute joint between a link and ground).

  • Permanent IC: Remains in the same relative position on both links (e.g., IC between two meshing gears at the pitch point).

  • Instantaneous IC: Exists only for a particular instant; its location changes.

  • Virtual IC: The point on one link that coincides with the IC of the other link at that instant.

• Kennedy's Theorem (Three IC Theorem): For any three rigid bodies in relative motion, the three instantaneous centers (IC12, IC23, IC31) are always collinear.

• Locating ICs in a Slider-Crank Mechanism:

  Consider standard slider-crank with links: 1 (fixed), 2 (crank), 3 (connecting rod), 4 (slider).

  • IC12: At the fixed revolute joint O₂ (permanent).

  • IC23: At the revolute joint B (permanent).

  • IC34: At the slider-guide contact. Since slider 4 moves linearly, IC34 is at infinity along a line perpendicular to the slider's path.

  • IC14: At infinity along the slider's path (since link 1 is fixed and slider moves linearly relative to it).

  • IC13: Found using Kennedy's theorem on bodies 1, 2, 3. IC13 lies on the line joining IC12 and IC23 (i.e., line O₂B). It is the point where velocity of link 3 relative to link 1 is zero.

  • IC24: Found using Kennedy's theorem on bodies 2, 3, 4. IC24 lies on the line joining IC23 and IC34. Since IC34 is at ∞ perpendicular to slider path, IC24 is at the intersection of line O₂B and a line from B perpendicular to the slider path.

• Velocity Analysis Using IC:

  For any two links $i$ and $j$:

$$v_{i/j} = \omega_j \times r_{i/IC_{ij}} = \omega_i \times r_{j/IC_{ij}}$$

Where $\omega$ is angular velocity and $r$ is the distance from the IC to the point of interest.

*   **Velocity of Slider (Link 4):** $$\displaystyle v_4 = \omega_2 \times r_{4/IC_{24}} $$ (since IC24 is on link 2 and 4).

*   **Angular Velocity of Connecting Rod (Link 3):** $$\displaystyle \omega_3 = v_2 / r_{3/IC_{13}} = v_4 / r_{3/IC_{34}} $$ (but IC34 at ∞, so use IC13).

• Rubbing Velocity at a Pin Joint:

  If two links $i$ and $j$ rotate with angular velocities $$\displaystyle \omega_i $$ and $$\displaystyle \omega_j $$ about a common axis, and the pin radius is $$\displaystyle r_{pin} $$:

$$v_{rub} = |\omega_i \pm \omega_j| \times r_{pin}$$

The sign depends on the direction of rotation (use '+' if same direction, '-' if opposite).

[!TIP]

Common Pitfall: For a slider, its IC with ground (IC14) is at infinity along the direction of slider motion. This is crucial for applying Kennedy's theorem correctly. Always draw the IC diagram for the mechanism first.

2.0 KINEMATIC SYNTHESIS & ANALYSIS (Brief Overview for Context)

• Motion vs. Number Synthesis:

  • Number Synthesis: Determines the type and number of links and pairs needed to achieve a specified DoF.

  • Motion (Type/Dimensional) Synthesis: Determines the geometric dimensions of links to generate a prescribed motion (output vs. input relationship).

• Freudenstein's Equation: Used for dimensional synthesis of a four-bar linkage to satisfy one position with prescribed $$\displaystyle \theta_2 $$, $$\displaystyle \omega_4 $$, $$\displaystyle \alpha_4 $$. Derived from loop closure equation:

$$K_1 \cos \theta_4 - K_2 \cos \theta_2 + K_3 \cos(\theta_2 - \theta_4) = K_4$$

Where $$\displaystyle K_1, K_2, K_3, K_4 $$ are constants based on link lengths $a, b, c, d$ (ground $d$, crank $a$, coupler $b$, rocker $c$). For one position, it relates $$\displaystyle \theta_2 $$ and $$\displaystyle \theta_4 $$. For multiple positions, solve simultaneous equations.

• D'Alembert's Principle: A dynamic equivalent of Newton's second law. It states that the inertia forces (and moments) acting on a body, combined with the external forces, form a system in dynamic equilibrium.

  • Significance: Allows the use of static force analysis equations for dynamic problems by adding fictitious inertia forces ($-m\vec{a}$) and inertia torques ($-I\vec{\alpha}$) to the free-body diagram.

  • Application: E.g., in a slider-crank with piston load, apply D'Alembert's principle to find the input torque on the crank, considering inertia of connecting rod and piston.

• Coriolis Acceleration Component: Arises when a point on a rigid body has relative velocity along a line that is rotating. For a point $P$ on link $B$ moving with relative velocity $$\displaystyle \vec{v}_{rel} $$ along a path on link $B$, and link $B$ rotating with $$\displaystyle \vec{\omega}_B $$ relative to another link $A$:

$$\vec{a}_c = 2 \vec{\omega}_B \times \vec{v}_{rel}$$

*   **Magnitude:** $$\displaystyle a_c = 2 \omega_B v_{rel} $$

*   **Direction:** Perpendicular to $$\displaystyle \vec{v}_{rel} $$, rotated by $$\displaystyle 90^\circ $$ in the direction of $$\displaystyle \vec{\omega}_B $$.

*   **Application:** Slider-crank with an **oscillating follower** or any mechanism with a sliding link on a rotating body.

[!TIP]

Exam Focus: D'Alembert's principle converts a dynamic problem into a static equilibrium problem with added inertia forces. Coriolis term appears only when there is relative sliding motion on a rotating link.

3.0 CAMS & FOLLOWER SYSTEMS (Only definitions/classifications as per Unit 1 scope)

• Cam Classification by Shape:

  • Radial (Disc) Cam: Follower moves perpendicular to cam axis.

  • Translating Cam: Cam translates (e.g., face cam).

  • Cylindrical Cam: Follower moves parallel to cam axis.

  • Conical Cam: Cam is conical.

  • Universal Cam: Can produce motion in any direction.

• Cam Classification by Follower:

  • Knife-edge: Simple, high wear, high contact stress.

  • Roller: Low wear, larger size.

  • Flat-faced ( mushroom): Used for heavy loads, limited to low-speed cams.

  • Spherical: For angular motion.

• Cam Classification by Motion: Rise-Dwell-Fall-Dwell (RDRD) cycles. Dwell periods require the cam to maintain contact without moving the follower.

• Key Terminology (with sketches):

  • Base Circle: Smallest circle that can be drawn tangent to the cam profile.

  • Pitch Circle: Circle on which the follower's motion is designed (for roller follower, it's the circle through the roller centers).

  • Prime Circle: For cams with offset or translating followers, the circle from which follower displacement is measured.

  • Lift/Stroke (L): Maximum displacement of the follower.

  • Pressure Angle ($\phi$): Angle between the direction of follower motion and the normal to the cam profile at the point of contact. Should be minimized (<30° for translating, <35° for oscillating) to avoid excessive side thrust.

  • Angle of Action: Cam rotation angle during which the follower is in contact with the cam profile.

  • Undercutting: Occurs when the cam profile is too sharp, causing the follower to lose contact. Prevented by increasing base circle radius or using a roller follower.

• Follower Motions & Displacement Diagrams:

  • Uniform Velocity (UV): Constant velocity → infinite acceleration at start/end → not practical.

  • Simple Harmonic Motion (SHM): $$\displaystyle s = \frac{L}{2} \left(1 - \cos \frac{\pi \theta}{\beta}\right) $$

    • Max velocity: $$\displaystyle v_{max} = \frac{\pi L \omega}{2\beta} $$

    • Max acceleration: $$\displaystyle a_{max} = \frac{\pi^2 L \omega^2}{4\beta^2} $$

  • Uniform Acceleration/Deceleration (UAD): Parabolic segments.

    • Max velocity: $$\displaystyle v_{max} = \frac{2L \omega}{\beta} $$

    • Max acceleration: $$\displaystyle a_{max} = \frac{4L \omega^2}{\beta^2} $$

  Where $\beta$ = cam angle for rise/fall, $\omega$ = cam angular speed (rad/s).

• Cam Profile Construction: For knife-edge and roller followers, use the inverse method: imagine the follower as the cam and the cam as the follower (or use the base circle offset method for roller followers).

[!TIP]

Exam Focus: Be able to draw displacement, velocity, and acceleration diagrams for SHM and UAD motions. Know formulas for max velocity/acceleration. Undercutting is caused by small base circle radius or high lift.

4.0 GEARS & GEAR TRAINS (Only fundamentals as per Unit 1 scope)

• Fundamental Law of Gearing: For two gears to transmit constant angular velocity ratio, the common normal at the point of contact must always pass through a fixed point on the line of centers, called the Pitch Point.

  • Significance: This condition ensures no slip and constant velocity ratio.

  • Conjugate Profiles: Tooth profiles that satisfy the fundamental law. Involute profile is the most common because it satisfies this law and is easy to generate.

• Involute Gear Terminology:

  • Pitch Circle: Imaginary circle that rolls without slip with the pitch circle of the mating gear.

  • Base Circle: Circle from which the involute profile is generated.

  • Addendum ($a$): Radial distance from pitch circle to top of tooth.

  • Dedendum ($b$): Radial distance from pitch circle to bottom of tooth (root).

  • Circular Pitch ($$\displaystyle p_c $$): Distance along pitch circle between corresponding points on adjacent teeth. $$\displaystyle p_c = \frac{\pi m}{} $$ where $m$ = module.

  • Module ($m$): $$\displaystyle m = \frac{\text{Pitch Diameter}}{\text{Number of Teeth}} $$. Standard unit.

  • Pressure Angle ($\phi$): Angle between the line of action (common normal) and the tangent to the pitch circle. Standard values: 14.5°, 20°, 25°.

  • Path of Contact: Line along which the point of contact moves from start to end of engagement.

  • Arc of Contact: Arc on the pitch circle through which a point on a tooth remains in contact.

  • Contact Ratio ($\epsilon$): Average number of teeth in contact. $$\displaystyle \epsilon = \frac{\text{Path of Contact}}{\text{Circular Pitch}} $$. Must be >1 for smooth transmission.

• Interference:

  • Definition: The phenomenon where the tip of a tooth of one gear contacts the root of the mating gear tooth, causing jamming and wear.

  • Cause: Occurs when the addendum angle of the pinion is greater than the angle of approach.

  • Avoidance: Use a minimum number of teeth on the pinion ($$\displaystyle T_{min} $$). For standard 20° full-depth involute gears: $$\displaystyle T_{min} = \frac{2a}{\sin^2 \phi} $$ (where $a$ is addendum). Typically $$\displaystyle T_{min} = 18 $$ for 20° pressure angle.

• Sliding Velocity: At the point of contact, the sliding velocity is the difference between the pitch line velocities. Maximum sliding occurs at the end of engagement.

[!TIP]

Common Pitfall: Interference occurs on the pinion (smaller gear) because its tooth is longer. Increasing the addendum increases the chance of interference.

5.0 POWER TRANSMISSION ELEMENTS (BELTS, BRAKES, CLUTCHES) (Only belt drives and basic friction devices as per Unit 1 scope)

• Belt Drives:

  • Types: Flat, V-belt (wedge action increases friction), Ribbed.

  • Velocity Ratio: $$\displaystyle \frac{N_1}{N_2} = \frac{D_2}{D_1} \left(1 - \frac{s}{100}\right) $$ where $s$ = slip %. Considering belt thickness: $$\displaystyle \frac{N_1}{N_2} = \frac{D_2 + t}{D_1 + t} $$.

  • Centrifugal Tension ($$\displaystyle T_c $$): Due to belt speed $v$, $$\displaystyle T_c = m v^2 $$ (where $m$ = mass per unit length). Reduces effective tension for power transmission.

  • Maximum Power Transmission: Occurs when $$\displaystyle T_1/T_2 = e^{\mu \theta} $$ (Euler's formula) and $$\displaystyle T_c = T_1/3 $$ for flat belts. Power $$\displaystyle P = (T_1 - T_2) v $$.

  • Tension Ratio (Euler's Formula): $$\displaystyle T_1 = T_2 e^{\mu \theta} $$ (for flat belt, $\theta$ in radians).

  • Open vs. Cross Belt:

    • Open: Same direction of rotation. Angle of lap (θ) for smaller pulley: $$\displaystyle \theta = 180^\circ - 2\alpha $$, $$\displaystyle \sin \alpha = \frac{R_1 - R_2}{C} $$.

    • Cross: Opposite direction. Angle of lap for each pulley: $$\displaystyle \theta = 180^\circ - 2\alpha + 2\beta $$, but effective angle is $$\displaystyle 180^\circ + 2\alpha $$ (due to crossing). Tension ratio same formula.

  • Stress in Belt: $$\displaystyle \sigma = \frac{T_1 + T_c}{A} $$ (for tight side), considering centrifugal tension.

• Friction Devices:

  • Brake: Absorbs kinetic energy of a moving system to stop or slow it down.

  • Clutch: Transmits torque between two rotating shafts, allowing engagement/disengagement.

  • Factors Affecting Braking: Braking torque, heat dissipation, wear, fade, stability.

• Band & Block Brake:

  • Derivation: Consider a band with $n$ blocks, each subtending $2\theta$ at the drum center. Tension increases from $$\displaystyle T_n $$ (slack side) to $$\displaystyle T_0 $$ (tight side) across each block.

    For one block: $$\displaystyle \frac{T_0}{T_1} = \frac{1 + \mu \tan \theta}{1 - \mu \tan \theta} $$

    For $n$ blocks: $$\displaystyle \frac{T_0}{T_n} = \left(\frac{1 + \mu \tan \theta}{1 - \mu \tan \theta}\right)^n $$

    \boxed{\frac{T_0}{T_n} = \left(\frac{1 + \mu \tan \theta}{1 - \mu \tan \theta}\right)^n}

  • Braking Torque: $$\displaystyle T_b = (T_0 - T_n) r $$ (for band on drum of radius $r$).

  • Differential Band Brake: One end of band attached to fixed point, other to lever. Self-energizing; can hold load with small force.

• Plate Clutch (Uniform Wear Theory):

  • Assumption: Pressure $p \propto 1/r$ to ensure uniform wear.

  • $$\displaystyle p = \frac{c}{r} $$, where $c$ is constant.

  • Axial force $$\displaystyle W = \int_{r_1}^{r_2} p \cdot 2\pi r dr = 2\pi c (r_2 - r_1) $$ → $$\displaystyle c = \frac{W}{2\pi (r_2 - r_1)} $$.

  • Frictional Torque: $$\displaystyle T = \int \mu p \cdot r \cdot 2\pi r dr = 2\pi \mu c \int_{r_1}^{r_2} r^2 dr = \frac{2}{3} \pi \mu W \frac{r_2^3 - r_1^3}{r_2 - r_1} $$.

  • Average Pressure: $$\displaystyle p_{avg} = \frac{W}{\pi (r_2^2 - r_1^2)} $$.

  • Maximum Pressure: $$\displaystyle p_{max} = \frac{c}{r_1} = \frac{W}{2\pi (r_2 - r_1) r_1} $$.

  • Minimum Pressure: $$\displaystyle p_{min} = \frac{c}{r_2} = \frac{W}{2\pi (r_2 - r_1) r_2} $$.

• Cone Clutch: Torque $$\displaystyle T = \frac{2}{3} \mu W \frac{r_2^3 - r_1^3}{r_2 - r_1} \cdot \csc \alpha $$, where $\alpha$ = cone angle.

• Centrifugal Clutch: Torque transmitted when centrifugal force on shoes overcomes spring force. $$\displaystyle T = \mu W R \cdot n $$ (for $n$ shoes), where $W$ is normal force due to centrifugal effect.

[!TIP]

Exam Focus: For plate clutch, know both uniform pressure ($$\displaystyle p = constant $$) and uniform wear ($p \propto 1/r$) theories. Uniform wear is more realistic for old clutches. Derive torque expressions. For belt drives, remember the effect of centrifugal tension on power and stress.

6.0 VIBRATIONS & BALANCING (Only balancing as per Unit 1 scope)

• Balancing of Rotating Masses:

  • Static Balancing: Masses in the same plane (single plane). Condition: $$\displaystyle \sum m r = 0 $$ and $$\displaystyle \sum m r \theta = 0 $$ (vector sum of $m r$ must be zero).

  • Dynamic Balancing: Masses in different planes. Requires both force and couple to be balanced. For two planes $A$ and $B$ separated by distance $l$:

    $$\displaystyle \sum (m r)_A = 0 $$, $$\displaystyle \sum (m r)_B = 0 $$, and $$\displaystyle \sum (m r \cdot d)_A = \sum (m r \cdot d)_B $$ (where $d$ is distance from a reference plane).

  • Calculation: Choose a reference plane, write moment equations about it, solve for balancing mass magnitude and angle.

• Balancing of Reciprocating Masses:

  • Partial vs. Complete Balancing: In engines, complete balancing of reciprocating masses is not possible due to horizontal forces. Usually, primary forces are balanced (by rotating masses), but secondary forces (due to obliquity) are often unbalanced.

  • Multi-cylinder In-line Engines: Unbalanced primary force for $n$ cylinders: $$\displaystyle F_p = m r \omega^2 \left[ \cos \theta + \cos(\theta - \phi) + ... \right] $$, where $\phi$ = angular interval between cranks. For even firing, $$\displaystyle \phi = 720^\circ/n $$.

  • Goal: Choose crank angles so that $$\displaystyle \sum \cos \phi_i = 0 $$ and $$\displaystyle \sum \sin \phi_i = 0 $$ for primary balance.

[!TIP]

Exam Focus: For multi-plane balancing, always take a reference plane and write moment equations. For reciprocating engines, remember that only primary forces are balanced by rotating weights; secondary forces remain.

7.0 SPECIAL MECHANISMS & APPLICATIONS (Only Quick Return as per Unit 1 scope)

• Whitworth Quick Return Mechanism:

  • Construction: A crank (O₁P) rotates at constant speed, connected to a slotted lever (AB) via a sliding block (P). The follower (slider) is attached to the lever at B. The crank center O₁ is offset from the lever pivot A.

  • Working: During the forward (cutting) stroke, the block moves from one end of the slot to the other. During the return stroke, the block moves back quickly because the crank rotates at constant speed but the lever angular displacement is less for the return.

  • Time Ratio (Cutting:Return):

$$\text{Time Ratio} = \frac{\beta}{\alpha} = \frac{180^\circ + \psi}{180^\circ - \psi}$$

    Where $$\displaystyle \psi = \angle O_1AP $$ (angle subtended by offset at pivot).

$$\psi = \sin^{-1}\left(\frac{e}{r}\right)$$

(for small angles, $\psi \approx e/r$ in radians), where $e$ = offset, $r$ = crank length.

*   **Design Synthesis:** Given return stroke length $$\displaystyle L_{ret} $$, time ratio $TR$, and crank length $r$, find offset $e$ and distance $$\displaystyle d = AO_1 $$.

$$L_{ret} = d \sin \psi + r \cos \psi \quad \text{(from geometry)}$$

    Solve for $d$ and $$\displaystyle e = d \sin \psi $$.

• Crank and Slotted Lever Quick Return:

  • Similar principle but with a different geometry. Time ratio derived from crank angle for cutting ($\beta$) and return ($\alpha$).

  • $$\displaystyle \beta = 180^\circ + 2\theta $$, $$\displaystyle \alpha = 180^\circ - 2\theta $$, where $$\displaystyle \theta = \cos^{-1}(e/r) $$.

  • Time Ratio = $$\displaystyle \frac{180^\circ + 2\theta}{180^\circ - 2\theta} $$.

[!TIP]

Exam Focus: Be able to derive the time ratio for Whitworth mechanism and solve design synthesis problems (given return stroke, time ratio, crank length → find offset and pivot distance). Sketch is essential.

8.0 MISCELLANEOUS & DEFINITIONS (Frequent Short Notes)

• Kinematic Chain vs. Mechanism vs. Machine: Chain = links & pairs; Mechanism = chain with one link fixed; Machine = combination of mechanisms for energy/work.

• Velocity & Acceleration:

  • Absolute: Measured relative to fixed link.

  • Relative: Measured relative to another moving link.

  • Convective/Coriolis: $$\displaystyle 2\vec{\omega} \times \vec{v}_{rel} $$, appears when a point has relative velocity on a rotating link.

• Applications:

  • Cams & Followers: Engine valves, automatic machine tools, packaging machinery.

  • Gears: Automobile transmissions, gearboxes, clocks, industrial machinery.

  • Brakes: Vehicle braking systems, hoists, machinery stops.

  • Clutches: Automotive transmissions, machine tool drives.

SUMMARY OF HIGH-YIELD FORMULAS:

1. DoF (Kutzbach): $$\displaystyle m = 3(n-1) - 2j - h $$

2. Grashof's Law: $S + L \leq P + Q$

3. Kennedy's Theorem: IC12, IC23, IC31 are collinear.

4. Rubbing Velocity: $$\displaystyle v_{rub} = |\omega_1 \pm \omega_2| \cdot r_{pin} $$

5. Belt Tension Ratio: $$\displaystyle \frac{T_1}{T_2} = e^{\mu \theta} $$

6. Band Brake Tension: $$\displaystyle \frac{T_0}{T_n} = \left(\frac{1+\mu\tan\theta}{1-\mu\tan\theta}\right)^n $$

7. Plate Clutch Torque (Uniform Wear): $$\displaystyle T = \frac{2}{3} \pi \mu W \frac{r_2^3 - r_1^3}{r_2 - r_1} $$

8. Cam Max Velocity/Accel (SHM): $$\displaystyle v_{max} = \frac{\pi L \omega}{2\beta} $$, $$\displaystyle a_{max} = \frac{\pi^2 L \omega^2}{4\beta^2} $$

9. Whitworth Time Ratio: $$\displaystyle \frac{\beta}{\alpha} = \frac{180^\circ + \psi}{180^\circ - \psi} $$, $$\displaystyle \psi = \sin^{-1}(e/r) $$