9.1 Simple Mechanisms, Kinematic Inversions & Velocity/Acceleration Diagrams

Simple Mechanisms, Kinematic Inversions & Velocity/Acceleration Diagrams

Quick Reference: A kinematic link is a resistant body possessing relative motion. Planar mechanism mobility is given by Kutzbach's criterion $F = 3(n - 1) - 2j - h$. Continuous rotation in a four-bar chain requires Grashof's condition $s + l \le p + q$. The Coriolis acceleration component $a_c = 2\omega v$ arises when a slider moves along a rotating link and acts perpendicular to the link.

In Coal India Limited (CIL) open-cast and underground mechanized mining operations, mechanisms form the core of heavy machinery—from dragline bucket linkages, shovel crowd motions, and continuous miners to conveyor tensioning drives and reciprocating screening decks. A rigorous understanding of kinematics is essential for mechanical engineers to design, troubleshoot, and evaluate dynamic loads in these heavy-duty systems.

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1. Kinematic Links and Pairs

Kinematic Links (Elements)

A kinematic link is defined as a resistant body (or an assembly of resistant bodies) that provides constrained relative motion to other parts of a machine while transmitting forces. A body is classified as "resistant" if it does not undergo permanent deformation that impairs its functional operation under operating loads.

• Rigid Link: Experiences negligible deformation (e.g., connecting rods, engine cranks, machine tool frames).
• Flexible Link: Transmits tensile forces while deforming elastically in bending (e.g., belts, ropes, chains).
• Fluid Link: Transmits power through fluid hydrostatic or hydrodynamic pressure (e.g., hydraulic jacks, fluid couplings, hydraulic cylinders in mining excavators).

Classification of Kinematic Pairs

A kinematic pair consists of two interconnected links maintaining continuous physical contact with specified relative motion.

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2. Mobility and Degree of Freedom (DOF)

Kinematic Chains and Mechanisms

When kinematic pairs are assembled, they form a kinematic chain.

• Locked Chain (Structure / Frame): $F \le 0$ (No relative motion possible).
• Constrained Mechanism: $F = 1$ (A single input produces unique, predictable output motions).
• Unconstrained Mechanism: $F > 1$ (Requires multiple independent inputs to define the configuration).

+-------------------------------------------------------------------------+
|                        KINEMATIC MOBILITY SPECTRUM                      |
|                                                                         |
|    F < 0               F = 0               F = 1               F > 1    |
|  Superstructure     Determinate        Constrained         Unconstrained|
|  (Pre-stressed)      Structure          Mechanism           Mechanism   |
|  (Over-locked)     (Zero Mobility)    (1 Input -> 1 Out)   (Multi-Input)|
+-------------------------------------------------------------------------+

Kutzbach Criterion for Planar Mechanisms

For any mechanism operating in a two-dimensional plane:

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

Where:

• $n = \text{Total number of links (including the fixed frame)}$
• $j = \text{Number of binary joints (lower pairs with } 1 \text{ DOF, such as turning and sliding pairs)}$
• $h = \text{Number of higher pairs with } 2 \text{ DOF (such as cam-roller or gear tooth contacts)}$

Special Joint Handling:

1. A ternary joint (connecting 3 links at a single pivot) is equivalent to $2$ binary joints ($j = 2$).
2. A quaternary joint (connecting 4 links) is equivalent to $3$ binary joints ($j = 3$).
3. Generally, an $m$-order joint connecting $m$ links equals $(m - 1)$ binary joints.

Grubler's Criterion for Plane Mechanisms

Grubler's criterion is a specialized formulation of Kutzbach's equation applied to mechanisms having only lower pairs ($h = 0$) and requiring a single degree of freedom ($F = 1$):

$$3(n - 1) - 2j = 1 \implies 3n - 2j - 4 = 0$$

For Grubler's criterion to yield a physically valid mechanism with lower pairs:

1. The minimum number of links required is $n = 4$.
2. The number of links $n$ must always be an even number ($n = 4, 6, 8, \dots$). An odd number of links with all lower pairs cannot form a single-degree-of-freedom planar mechanism.

Redundant Degrees of Freedom and Redundant Links

• Redundant Degree of Freedom ($F_r$): Motion that does not affect the input-output kinematic relationship of the mechanism (e.g., rotation of a roller follower about its own pin in a cam system). The corrected formula is: $$F = 3(n - 1) - 2j - h - F_r$$
• Redundant Link ($n_r$): A link that introduces no additional kinematic constraints (e.g., a parallel tie rod in a five-bar double-crank mechanism). Redundant links must be subtracted prior to applying mobility criteria.

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3. Grashof's Law for Four-Bar Chains

Consider a planar four-bar kinematic chain comprising four revolute pairs with link lengths:

• $s = \text{Length of shortest link}$
• $l = \text{Length of longest link}$
• $p, q = \text{Lengths of the two intermediate links}$

                   Coupler Link (p)
             B o-----------------------o C
              /                         \
   Input     /                           \  Output
   Crank    /                             \ Rocker
    (s)    /                               \  (q)
          /                                 \
         o-----------------------------------o
         A          Fixed Link (l)           D

Grashof's Condition: $s + l \le p + q$

1. Class-I Mechanisms ($s + l < p + q$): At least one link can make a complete $360^{\circ}$ revolution relative to the other three links.
   • Crank-Rocker Mechanism: Link adjacent to the shortest link is fixed. The shortest link acts as a continuously rotating input crank, while the opposite link oscillates as a rocker.
   • Double-Crank (Drag-Link) Mechanism: The shortest link ($s$) itself is fixed. Both adjacent links make complete $360^{\circ}$ rotations at varying relative angular velocities.
   • Double-Rocker Mechanism: Link opposite to the shortest link is fixed. The shortest link acts as the coupler (floating link), and both grounded links oscillate as rockers.
2. Class-II Mechanisms ($s + l > p + q$): No link can rotate through a complete $360^{\circ}$ cycle. All inversions yield Double-Rocker (Rocker-Rocker) mechanisms.
3. Change-Point Mechanisms ($s + l = p + q$): Links become collinear during motion (e.g., parallelogram linkages, deltoid/kite mechanisms). The output can either maintain parallel motion or toggle into an anti-parallel configuration unless guided by momentum or an auxiliary link.

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4. Kinematic Inversions of Fundamental Chains

A kinematic inversion is obtained by fixing a different link in a given kinematic chain as the stationary reference frame. While the relative motions between adjacent links remain identical, the absolute motions of the links relative to the ground change completely.

+-------------------------------------------------------------------------+
|                   KINEMATIC INVERSIONS CLASSIFICATION                   |
+-------------------------------------------------------------------------+
                                     |
     +-------------------------------+-------------------------------+
     |                               |                               |
Four-Bar Chain              Single Slider-Crank             Double Slider-Crank
- Beam Engine               - Reciprocating Engine (Link 1) - Elliptical Trammel (Link 1)
- Locomotive Coupler        - Whitworth / Rotary (Link 2)   - Scotch Yoke (Link 2)
- Watt Indicator Mechanism  - Slotted-Lever / Oscillating  - Oldham Coupling (Link 3)
- Ackermann Steering          Cylinder (Link 3)
                            - Hand / Pendulum Pump (Link 4)

Inversions of Single Slider-Crank Chain

A single slider-crank chain has 3 turning pairs and 1 sliding pair.

Quick-Return Ratio ($QRR$)

In quick-return mechanisms (Whitworth and Slotted-Lever), the cutting stroke occurs during a larger crank rotation angle $\alpha$, while the idle return stroke occurs during a smaller angle $\beta = 360^{\circ} - \alpha$:

$$QRR = \frac{\text{Time of Cutting Stroke}}{\text{Time of Return Stroke}} = \frac{\alpha}{\beta} = \frac{360^{\circ} - \beta}{\beta} > 1$$

For a crank of radius $r$ and fixed link distance $d$ (where $d > r$ for slotted lever):

$$\cos\left(\frac{\beta}{2}\right) = \frac{r}{d}$$

Inversions of Double Slider-Crank Chain

A double slider-crank chain has 2 turning pairs and 2 sliding pairs.

1. First Inversion (Slotted Frame Fixed): Elliptical Trammel.
   • Any point $P$ on the link connecting the two sliders traces an exact ellipse: $$\frac{x^2}{(a + b)^2} + \frac{y^2}{b^2} = 1$$
   • The midpoint of the connecting link traces a true circle of radius $r = a = b$.
2. Second Inversion (One Slider Fixed): Scotch Yoke Mechanism.
   • Converts continuous rotary motion into pure Simple Harmonic Motion (SHM) of the reciprocating slotted yoke without angularity errors.
3. Third Inversion (Slotted Link / Cross-Piece Fixed): Oldham's Coupling.
   • Connects two parallel shafts whose axes are offset by a lateral distance $d$.
   • Velocity ratio is always strictly $1:1$ (constant velocity transmission).
   • Maximum sliding velocity of the central tongue in the slot is $v_s = d \cdot \omega$.

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5. Velocity Analysis & Kennedy's Theorem

Instantaneous Centre of Rotation (I-Centre)

The instantaneous centre of relative rotation ($I$) is a point common to two bodies in planar motion that has the exact same instantaneous linear velocity in both bodies.

• The total number of instantaneous centres $N$ in a mechanism containing $n$ links is: $$N = \frac{n(n - 1)}{2}$$

Aronhold-Kennedy Theorem of Three Centres

Theorem: If three planar bodies ($1, 2, 3$) undergo relative motion with respect to one another, their three relative instantaneous centres of rotation—$I_{12}, I_{23},$ and $I_{13}$—must lie on a single straight line.

                 Body 2
              o----------o
             /  I_23      \
            /              \
     I_12  o                o  I_13
          /                  \
         o--------------------o
     Body 1                  Body 3
     
     Kennedy's Collinear Line: [ I_12 ] ===== [ I_23 ] ===== [ I_13 ]

Using Kennedy's theorem, the linear velocity of any common point $P$ (such as $I_{23}$) is given by:

$$v_P = \omega_2 \cdot (I_{12} I_{23}) = \omega_3 \cdot (I_{13} I_{23})$$

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6. Acceleration Analysis & Coriolis Acceleration

Acceleration Components of a Rigid Link

When a point $B$ rotates about a fixed point $A$ with angular velocity $\omega$ and angular acceleration $\alpha$:

1. Centripetal (Radial) Acceleration ($a_r$): Acts along the link directed toward the center of rotation: $$a_r = \omega^2 r = \frac{v^2}{r}$$
2. Tangential Acceleration ($a_t$): Acts perpendicular to the link in the sense of $\alpha$: $$a_t = \alpha \cdot r$$
3. Total Acceleration ($a$): $$a = \sqrt{a_r^2 + a_t^2}$$

Coriolis Component of Acceleration ($a_c$)

When a slider moves along a guide path that is simultaneously rotating in a plane, the slider experiences an additional acceleration component called the Coriolis acceleration.

$$a_c = 2\omega v$$

Where:

• $\omega = \text{Angular velocity of the rotating guide link (rad/s)}$
• $v = \text{Linear velocity of the slider relative to the rotating guide (m/s)}$

                     ^ v (Sliding outward)
                     |
                     |     Direction of Coriolis Acceleration (a_c):
       Rotating      |     Rotate sliding vector v by 90 deg
       Guide (w)     |     in the direction of w (Counter-Clockwise)
          \          |
           \         |       <=========== a_c (Acts to the left)
            \     [Slider]
             \       |
              \      |
               \     |
                \    |
                 o   O (Pivot)

Rule for Direction of Coriolis Acceleration: Take the vector representing the relative sliding velocity $\vec{v}$ of the slider along the link, and rotate it by $90^{\circ}$ in the direction of the angular velocity $\vec{\omega}$ of the guide link. The resulting vector direction gives the exact sense of the Coriolis acceleration.

7. Worked Numerical Examples

Example 1: Quick-Return Mechanism Kinematics

Problem: In a crank and slotted-lever quick-return motion mechanism used in a workshop shaper, the distance between the fixed centers is $d = 450\text{ mm}$ and the driving crank length is $r = 225\text{ mm}$. The length of the slotted lever is $L = 900\text{ mm}$, and the line of stroke passes through the extreme upper position of the slotted lever pivot. Calculate:

1. The ratio of time of cutting stroke to time of return stroke ($QRR$).
2. The total length of the cutting stroke.

Solution:

1. Determine Return Angle $\beta$: At the extreme limiting positions of the slotted lever, the crank is tangent to the lever:

   $$\cos\left(\frac{\beta}{2}\right) = \frac{r}{d} = \frac{225}{450} = 0.5$$ $$\frac{\beta}{2} = \cos^{-1}(0.5) = 60^{\circ} \implies \beta = 120^{\circ}$$

2. Calculate Cutting Angle $\alpha$ and $QRR$:

   $$\alpha = 360^{\circ} - \beta = 360^{\circ} - 120^{\circ} = 240^{\circ}$$ $$QRR = \frac{\text{Time of Cutting}}{\text{Time of Return}} = \frac{\alpha}{\beta} = \frac{240^{\circ}}{120^{\circ}} = 2.0$$

3. Calculate Stroke Length ($S$):

   $$S = 2 \times L \times \sin\left(\frac{\beta}{2}\right) \times \left(\frac{r}{d}\right) = 2 \times L \times \frac{r}{d} = 2 \times 900 \times \frac{225}{450} = 900\text{ mm}$$

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Example 2: Coriolis Acceleration on a Rotating Arm

Problem: A radial excavator discharge chute link $OA$ rotates in a vertical plane at a constant speed of $\omega = 12\text{ rad/s}$ counter-clockwise. A coal feed block slides along the chute radially outward away from pivot $O$ with a relative sliding velocity of $v = 4\text{ m/s}$ and an outward sliding acceleration of $a_s = 6\text{ m/s}^2$. When the block is at a radius of $r = 1.5\text{ m}$ from $O$, determine:

1. The Coriolis component of acceleration.
2. The total resultant acceleration of the block.

Solution:

1. Coriolis Acceleration:

   $$a_c = 2\omega v = 2 \times (12\text{ rad/s}) \times (4\text{ m/s}) = 96\text{ m/s}^2$$

   Direction: Rotate the outward velocity vector by $90^{\circ}$ counter-clockwise (in direction of $\omega$) $\implies$ acts perpendicular to the chute to the left.

2. Centripetal (Radial) Acceleration:

   $$a_r = \omega^2 r = (12)^2 \times 1.5 = 144 \times 1.5 = 216\text{ m/s}^2 \quad (\text{directed toward } O)$$

3. Tangential Acceleration: Since $\omega = \text{constant}$, $\alpha = 0 \implies a_t = \alpha r = 0$.

4. Total Radial Acceleration Component ($a_{\text{rad}}$):

   $$a_{\text{rad}} = a_s - a_r = 6 - 216 = -210\text{ m/s}^2 \quad (\text{directed toward } O)$$

5. Total Tangential Acceleration Component ($a_{\text{tan}}$):

   $$a_{\text{tan}} = a_c + a_t = 96 + 0 = 96\text{ m/s}^2 \quad (\text{perpendicular to chute})$$

6. Resultant Acceleration ($a_{\text{total}}$):

   $$a_{\text{total}} = \sqrt{a_{\text{rad}}^2 + a_{\text{tan}}^2} = \sqrt{(-210)^2 + (96)^2} = \sqrt{44100 + 9216} = \sqrt{53316} \approx 230.9\text{ m/s}^2$$