Comprehensive Theory of Machines Study Notes

Fundamentals of Machine Theory and Kinematic Principles

• Theory of Machines (TOM) Definition: TOM is the branch of engineering science dealing with the study of relative motion between various machine parts and the forces acting upon them.

• Sub-divisions of TOM:     * Kinematics: Deals with the relative motion between various machine parts.     * Dynamics: Deals with forces and their effects while acting on machine parts in motion.

• Resistant Bodies: Bodies that do not suffer appreciable distortion or physical form changes by acting forces, such as springs and belts.

• Kinematic Link (Element): A resistant body that is part of a machine and has relative motion to other connected parts.     * Characteristics: 1. It must have relative motion. 2. It must be a resistant body.

• Functions of Linkages: Link mechanisms convert rotating, oscillating, or reciprocating motion. Specific conversions include:     * Continuous rotation to continuous rotation (constant or variable angular velocity ratio).     * Continuous rotation into oscillation or reciprocation (and reverse).     * Oscillation into oscillation or reciprocation into reciprocation.

• Primary Goals of Mechanisms:     * Function Generation: Managing relative motion between links connected to the frame.     * Path Generation: Managing the path of a tracer point.     * Motion Generation: Managing the motion of the coupler link.

• Types of Links:     * Rigid Link: No deformation while transmitting motion (e.g., Connecting Rod).     * Flexible Link: Partly deformed (e.g., Spring, Belts).     * Fluid Link: Motion transmitted through fluid pressure (e.g., Hydraulic press, brakes).

Kinematic Pairs and Constraints

• Kinematic Pair: Two elements connected such that their relative motion is completely or successfully constrained.

• Lower Pair: Surface contact during motion (e.g., Revolute, Prismatic, Screw, Cylindrical, Spherical, Planar).     * Revolute Pair: Allows relative rotation; expressed by one coordinate angle $\theta$; Degree of Freedom (DOF) = 1.     * Prismatic Pair: Allows relative translation; expressed by coordinate $x$; DOF = 1.     * Screw Pair: Relative movement expressed by $\theta$ or $x$; DOF = 1 (e.g., Lead screw of a lathe).     * Cylindrical Pair: Allows rotation and translation; independent coordinates $\theta$ and $x$; DOF = 2.     * Spherical Pair: Allows three DOF; needs three independent coordinates (e.g., Motorcycle mirror attachment).     * Planar Pair: Allows three DOF ($x$, $y$, and $\theta$ about the z-axis).

• Higher Pair: Point or line contact (e.g., Ball bearings, gear teeth, cam-follower).

• Wrapping Pairs: Devices like belts and chains.

• Kinematic Constraints (Types of Motion):     * Completely Constrained: Limited to a definite direction regardless of force (e.g., Square bar in a square hole).     * Incompletely Constrained: Motion possible in more than one direction (e.g., Circular bar in a round hole).     * Successfully Constrained: Motion not completed by itself but by other means (e.g., IC engine valve kept on seat by a spring).

Kinematic Chains, Mobility, and Grashof's Law

• Kinematic Chain: Series of links connected by kinematic pairs.     * Singular Link: Connected to only one link.     * Binary Link: Connected to two links.     * Ternary Link: Connected to three links.     * Quaternary Link: Connected to four links.

• Chain Link-Pair Relationship: For a chain with only binary links:     * $l = 2p - 4$     * $j = \frac{3}{2}l - 2$     * If LHS > RHS: Structure; LHS = RHS: Constrained chain; LHS < RHS: Unconstrained chain.

• Mechanism vs. Machine: A mechanism is formed when one link of a kinematic chain is fixed. A machine is a mechanism that transmits power or does work.

• Degrees of Freedom (Mobility): The number of independent variables needed to define the system's condition.     * Unconstrained body in a plane: 3 DOF.     * Unconstrained body in space: 6 DOF.

• Kutzbach Criterion: For plane motion mobility $n$:     * $n = 3(l - 1) - 2j - h$     * Where $l$ = links, $j$ = binary joints/lower pairs, $h$ = higher pairs.

• Grubler Criterion: Applies when overall movability is unity ($n = 1$) and $h = 0$:     * $3l - 2j - 4 = 0$

• Grashof's Law: For a four-bar linkage (shortest link $s$, longest $l$, intermediate $p$, $q$), at least one link revolves if:     * $s + l \le p + q$     * Case 1: Shortest bar = Frame $\rightarrow$ Double-crank.     * Case 2: Shortest bar = Side $\rightarrow$ Crank-rocker.     * Case 3: Shortest bar = Coupler $\rightarrow$ Double-rocker.     * Case 4: If s + l > p + q, all mobile links will rock.

Inversions and Specific Mechanisms

• Single Slider-Crank Inversions:     * 1st Inversion: Cylinder fixed (e.g., Reciprocating engine).     * 2nd Inversion: Crank fixed (e.g., Whitworth quick return, Gnome engine).     * 3rd Inversion: Connecting rod fixed (e.g., Slotted lever quick return, oscillating cylinder engine).     * 4th Inversion: Slider fixed (e.g., Hand pump).

• Double Slider-Crank Inversions:     * Elliptical Trammel: Used for drawing ellipses. If a point is the midpoint of the bar, it traces a circle.     * Scotch Yoke: Converts rotary to reciprocating motion.     * Oldham’s Coupling: Transmits angular velocity between parallel but eccentric shafts. Maximum sliding speed $v = \omega \times r$, where $r$ is axis distance.

• Hooke’s Joint (Universal Coupling): Connects non-parallel intersecting shafts. Velocity ratio:     * $\frac{\omega_2}{\omega_1} = \frac{\cos(\alpha)}{1 - \cos^2(\theta) \sin^2(\alpha)}$     * Max Ratio: $\frac{1}{\cos(\alpha)}$ at $\theta = 0^{\circ}, 180^{\circ}$.     * Min Ratio: $\cos(\alpha)$ at $\theta = 90^{\circ}, 270^{\circ}$.

• Steering Gears:     * Condition for Correct Steering: $\cot(\phi) - \cot(\theta) = \frac{c}{b}$.     * Davis Gear: Uses sliding pairs.     * Ackerman Gear: Simpler; uses turning pairs; placed at back of front wheels.

• Pantograph: Reproduces paths to an enlarged or reduced scale.

Velocity, Acceleration, and IC Engine Kinematics

• Instantaneous Centre (IC): The point about which a body has pure rotation. For straight-line motion, ICR is at infinity.

• Kennedy’s Theorem: If three links have relative motion, their three relative instantaneous centres must lie on a straight line.

• IC Count: $N = \frac{n(n - 1)}{2}$.

• Coriolis Component of Acceleration: Occurs when a slider moves on a rotating link.     * $a = 2\omega V$     * Direction is the sliding velocity vector rotated by $90^{\circ}$ in the direction of $\omega$.

• Piston Kinematics:     * Displacement: $x_P = r \left[ (1 - \cos(\theta)) + n - \sqrt{n^2 - \sin^2(\theta)} \right]$.     * Velocity: $V_P = \omega r \left( \sin(\theta) + \frac{\sin(2\theta)}{2n} \right)$.     * Acceleration: $f_P = \omega^2 r \left( \cos(\theta) + \frac{\cos(2\theta)}{n} \right)$.

Flywheel and Governor

• Flywheel: Reduces speed fluctuations during a cycle for constant load by storing energy. Does not control mean speed for load changes.

• Governor: Controls mean speed of the engine when load changes by regulating fuel supply.     * Sensitiveness: $\frac{N_1 - N_2}{N_{mean}} = \frac{2(N_1 - N_2)}{N_1 + N_2}$.     * Isochronous Governor: Equilibrium speed is constant for all radii ($N_1 = N_2$); sensitivity is infinity.     * Hunting: When a governor is too sensitive, it fluctuates continuously.

• Flywheel Energy: $\Delta E = I \omega_{mean}^2 C_s$, where $C_s$ is the coefficient of fluctuation of speed.

Vibration Analysis and Balancing

• Balancing:     * Static Balancing: Centre of mass lies on axis of rotation; resultant dynamic forces are zero.     * Dynamic Balancing: Resultant dynamic forces and resultant couples are zero.     * Reciprocating Unbalance: Primary force $m \omega^2 r \cos(\theta)$; Secondary force $m \omega^2 r \frac{\cos(2\theta)}{n}$.

• Vibrations:     * Natural Frequency ($f_n$): $f_n = \frac{1}{2\pi} \sqrt{\frac{k}{m}}$.     * Logarithmic Decrement (\delta): $\delta = \ln \left( \frac{x_n}{x_{n+1}} \right) = \frac{2\pi \zeta}{\sqrt{1 - \zeta^2}}$, where $\zeta$ is the damping ratio.     * Critical Damping Coefficient ($C_c$): $C_c = 2\sqrt{km}$.     * Magnification Factor (M.F.): M.F. = $\frac{1}{\sqrt{[1 - (\frac{\omega}{\omega_n})^2]^2 + [2\zeta \frac{\omega}{\omega_n}]^2}}$.

Gear Systems

• Terminology:     * Module ($m$): $m = \frac{D}{T}$ (mm).     * Circular Pitch ($p$): $p = \pi m$.     * Pressure Angle (\phi): Typically $20^{\circ}$.     * Contact Ratio: Must be > 1; typically $1.25$ to $2.00$.

• Interference Avoidance: Use more teeth, larger $\phi$, or modified addendum. Min pinion teeth ($T_p$) for no interference:     * $T_p = \frac{2A_w}{\sqrt{1 + G(G + 2)\sin^2(\phi)} - 1}$.

• Gears Types: Spur (parallel), Helical (smooth, high speed), Bevel (intersecting axes), Worm (high reduction, non-intersecting), Hypoid (offset axes).

• Lewis Equation (Beam Strength): $W_T = \sigma_w b m y$.