Mechanics of Machines – Week 2 Notes

All mechanical systems discussed are analysed kinematically, i.e., with the assumption that each member behaves as a rigid body (no deformation during motion).
Key vocabulary forms the foundation for later mobility, synthesis and analysis topics.
Terminology in this lecture is aligned with ISO/IFToMM notation, but informal synonyms are also given to ease recollection in exams.

Frame (Link-1 / Ground / Base)
- Serves as the global reference for position, velocity and acceleration.
- Exhibits no intended motion in the mechanism.
- Example: The bolted base in the adjustable-height platform (Fig. 1).
Link
- A single rigid body that connects to other links through joints in order to transmit motion and force.
- Must maintain geometric integrity (ideal rigid body assumption).
- Elastic members (springs, belts) are excluded from kinematic analysis but appear in dynamic/force analysis.
Linkage
- A mechanism in which links form one closed chain. One of the links is fixed and thereby becomes the frame.
- Garage-door arms, wind-screen wiper cranks and gear-shift levers are common everyday linkages.
Point of Interest (POI)
- A specific point on a link whose displacement/velocity/acceleration is sought, e.g., tip of a cutting blade.

A pair = two links + one mechanical constraint that permits relative motion.
Lower Pairs (surface contact; inexpensive to machine)
- Revolute $R$ (hinge, turning pair) → $1\,\text{DOF}$ rotation.
- Prismatic $P$ (slider) → $1\,\text{DOF}$ translation.
- Cylindrical $C$ → $2\,\text{DOF}$ (1 rot + 1 trans).
- Screw/Helical $H$ → $1\,\text{DOF}$ ; motion obeys pitch relation $x = p\theta$ .
- Planar $PL$ → $3\,\text{DOF}$ (2 trans + 1 rot in plane).
- Spherical $S$ (ball-and-socket) → $3\,\text{DOF}$ rotation.
Higher Pairs (line/point contact)
- Cam-follower, Gear tooth pair, Roller-disc, Ball on cylinder, etc.
- Permit more complex relative motions; usually introduce rolling plus sliding.

Simple/Binary Link
- Contains exactly two joints → line segment in kinematic diagram.
- Crank (360° rotation) & rocker (oscillatory) are classical sub-classes.
Complex/Ternary+ Link
- Houses three or more joints; can branch motion paths.
- Rocker arm with three pivots, bell-crank (bent rocker) are examples.

A link in 3-D free space has $6$ DOF (3 trans + 3 rot).
Joints remove specific DOF; the remaining freedom for relative motion equals the pair’s mobility stated earlier.
Typical mapping (letter, graphic symbol, mobility):
- $R$ → hinge symbol → $1$ .
- $P$ → slider symbol → $1$ .
- $C$ → coaxial circle/rectangle → $2$ .
- $H$ → screw thread symbol → $1$ .
- $S$ → sphere-in-socket → $3$ .
- $PL$ → hatched rectangle → $3$ .

General Definition: an assemblage of links connected by joints.
Closed-Loop Chain
- Each link connects to at least two other links → one or more geometric loops.
- Provide structural rigidity but complicate inverse kinematics.
- Simple-closed (all binary links, single loop) vs. compound-closed (contains ternary links or multiple loops).
Open-Loop Chain
- Sequence of links where only the base and distal end (end-effector) have single connections.
- Widely used in serial robots; easier to analyse but structurally flexible.

Strip away mass, shape and aesthetic details—retain only links (lines) and joints (symbols).
Steps to construct:
1. Choose the frame (usually the ground-attached body).
2. Number remaining links $2,3,\dots$ .
3. Identify and letter joints $A,B,\dots$ .
4. Mark Points of Interest (X, Y …).
Lower & higher pairs, rolling contacts, compound joints are all depicted by agreed icons (Table 1).

Revolute (R) – Turning Pair
- Symbol: filled or open circle with line.
- Key use: crankshaft, door hinge.
Prismatic (P) – Sliding Pair
- Symbol: arrowed slot.
- Key use: piston–cylinder, linear guide.
Helical (H) – Screw Pair
- Implicit constraint $x = p\theta$ couples rot.–trans.
- Lead-screw drives, bottle caps.
Cylindrical (C)
- Combined R+P but axes colinear; e.g., telescopic antenna.
Universal (U)
- Two intersecting $R$ joints; constant-velocity shafts.
Spherical (S)
- Hip joint, ball-joint suspensions.
Planar Joint (PJ)
- Drawer slide with rotation about vertical pin; robotics end-effectors with planar glide.
Rolling Pairs
- Pure rolling $\rightarrow$ $1\,\text{DOF}$ (e.g., wheel on rail).
- Roll-slide $\rightarrow$ $2\,\text{DOF}$ (gear mesh).

Electric Motors (AC): low cost, fixed synchronous speeds tied to mains frequency ~ $50$ / $60\,\text{Hz}$ .
Electric Motors (DC): variable speed $\le 30000\,\text{rpm}$ ; need battery/generator supply.
Engines: internal-combustion sources for continuous rotation.
Hydraulic/Pneumatic Cylinders: high-force linear strokes, require fluid power unit.
Screw Actuators: precise but short stroke; expensive.
Rule of thumb: one independent actuator per degree of freedom.

Architecture: serial stack of links & joints; base is fixed; tip carries tool/end-effector.
Typical industrial arm offers $6$ DOF (3 position + 3 orientation).
Forward Kinematics: given joint angles ${q_i}$ , compute end-effector pose ${x,y,z,\phi,\theta,\psi}$ .
Inverse Kinematics: given desired pose, solve for ${q_i}$ ; often multiple or no solutions.
Component blocks (ref. Cincinnati Milacron schematic):
- Shoulder swivel, elbow extension, wrist pitch–yaw–roll, computer controller, hydraulic power pack.

Example 1: PCB Shear (Pin–Slider Four-bar)
- Frame = bolted base (Link 1).
- Moving links: handle (2), cutting blade (3), tie-bar (4).
- Joints: $A,B,C$ – pins; $D$ – prismatic guide.
- POI $X$ at handle tip.
- Resulting kinematic diagram: four-bar with slider.
Example 2: Vice-Grips (Compound Toggle)
- Frame = top handle (1).
- Moving links: bottom handle (2), bottom jaw (3), toggle bar (4).
- Pins $A,B,C,D$ connect links; POIs $X$ (jaw tip), $Y$ (lower handle end).
- Diagram reveals compound closed chain responsible for locking action.

Pin-Slide-Pin via Cam-Rocker: generates an approximately straight-line extraction path; widely used in pick-and-place machines.
Multi-Station Product Pusher
- Six axes, each described by stroke vs. machine angle functions.
- Design objective: synchronise displacements, minimise cycle time, avoid collisions.
Planar vs. Spatial
- Planar: motion confined to a single geometric plane; all $R$ joint axes parallel.
- Spatial: at least one joint axis is skew/out-of-plane, producing 3-D trajectories.

Links = rigid bodies; frame = immobile link.
Primary (lower) joints $\rightarrow$ surface contact; higher joints $\rightarrow$ line/point contact.
Closed vs Open kinematic chains; latter typical of robot manipulators.
Kinematic diagrams abstract real hardware; essential for DOF counting and synthesis.
Actuator count must equal mechanism mobility.
Forward vs. Inverse kinematics govern robot control algorithms.