Crank-and-Slotted-Rod Mechanism — Relative-Motion Kinematics: 17-Phys-A1 Classical Mechanics, May 2016, Q3

Question 3: Crank-and-Slotted-Rod Mechanism — Relative-Motion Kinematics (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. Crank $AB$ rotates about the fixed pin $A$; pin $B$ (fixed to the crank) engages the straight slot of rod $CD$, which itself rotates about the fixed pin $C$.

Given data
Quantity	Symbol	Value
Crank length	$AB$	100 mm
Slider position along rod	$CB$	300 mm
Crank orientation (from $+X$)	—	30°
Rod $CD$ orientation (from $+X$)	—	60° (30° from $+Y$)
Crank angular velocity	$\omega_{AB}$	3 rad/s ($+\hat k$)
Crank angular acceleration	$\alpha_{AB}$	$-1$ rad/s² ($+\hat k$)

Crank $AB$ (fixed pivot $A$) drives slider $B$, which rides in the straight slot of rod $CD$ (fixed pivot $C$). $CB=300$ mm.

Find. Velocity and acceleration of slider $B$ relative to the slotted rod $CD$.

Approach. Since $B$ is a point fixed to the rigid crank $AB$ (which rotates about fixed $A$), its absolute velocity/acceleration follow directly from rigid-body rotation. Since $B$ is simultaneously a point sliding within the slot of rigid body $CD$ (rotating about fixed $C$), the same absolute velocity/acceleration are also given by the rotating-reference-frame equations attached to $CD$, with the sliding (relative) term along the slot as the unknown. Equating the two expressions for $\vec v_B$, and then for $\vec a_B$, gives two vector (four scalar) equations for the two unknown pairs $(\omega_{CD},\,v_{\text{rel}})$ and $(\alpha_{CD},\,a_{\text{rel}})$.

Absolute velocity and acceleration of $B$ (via the crank). With $\vec r_{B/A}=0.1(\cos30^\circ,\sin30^\circ)=(0.0866,0.0500)$ m, $$\vec v_B=\vec\omega_{AB}\times\vec r_{B/A} = (-0.150,\ 0.2598)\ \text{m/s},\qquad |\vec v_B| = \omega_{AB}\cdot AB = 0.300\ \text{m/s}.$$ $$\vec a_B=\vec\alpha_{AB}\times\vec r_{B/A}-\omega_{AB}^2\vec r_{B/A} = (-0.7294,\ -0.5366)\ \text{m/s}^2.$$
Velocity equation on the rotating rod $CD$. With $\vec r_{B/C}=0.3(\cos60^\circ,\sin60^\circ)$ m, unit vectors along ($\hat t$) and normal to ($\hat n$) the slot, $$\vec v_B = \vec\omega_{CD}\times\vec r_{B/C} + v_{\text{rel}}\hat t.$$ Resolving into $\hat n,\hat t$ components and solving the $2\times2$ system: $$\boxed{\omega_{CD} = \tfrac{\sqrt3}{2}=0.866\ \text{rad/s (CCW)}}, \qquad \boxed{v_{\text{rel}} = 0.150\ \text{m/s (away from }C\text{, i.e. toward }D\text{)}}.$$
Acceleration equation on the rotating rod $CD$. With the Coriolis term $2\vec\omega_{CD}\times v_{\text{rel}}\hat t$ and centripetal term $-\omega_{CD}^2\vec r_{B/C}$, $$\vec a_B = \vec\alpha_{CD}\times\vec r_{B/C}-\omega_{CD}^2\vec r_{B/C}+2\vec\omega_{CD}\times(v_{\text{rel}}\hat t)+a_{\text{rel}}\hat t,$$ solving the resulting $2\times2$ system for $(\alpha_{CD},a_{\text{rel}})$ gives $$\boxed{\alpha_{CD}= \tfrac32-\tfrac{2\sqrt3}{3}=0.345\ \text{rad/s}^2\ \text{(CCW)}},\qquad \boxed{a_{\text{rel}} = \tfrac{7}{40}-\tfrac{9\sqrt3}{20}=-0.604\ \text{m/s}^2\ \text{(i.e.\ toward }C\text{)}}.$$

Final results
Quantity	Value
$\omega_{CD}$	0.866 rad/s (CCW)
$\alpha_{CD}$	0.345 rad/s² (CCW)
Velocity of $B$ relative to slot, $v_{\text{rel}}$	0.150 m/s, directed from $C$ toward $D$
Acceleration of $B$ relative to slot, $a_{\text{rel}}$	0.604 m/s², directed from $B$ toward $C$

Topic: Rigid-body kinematics using a rotating reference frame (sliding-contact / slotted-link mechanisms).

Key relations: $\vec v_B=\vec v_{\text{coincident pt.\ on }CD}+\vec v_{\text{rel}}$; $\vec a_B=\vec a_{\text{coincident pt.}}+2\vec\Omega\times\vec v_{\text{rel}}+\vec a_{\text{rel}}$ (Coriolis term appears only in acceleration, never velocity).

Why this works: $B$ is a single physical point that can be described two ways — as fixed to crank $AB$, or as sliding along rod $CD$ — and both descriptions must give the identical absolute velocity/acceleration; equating them is exactly two vector equations for the two genuinely unknown scalars at each order (angular rate/acceleration of $CD$, and the slip rate/acceleration along the slot).

Common pitfall: Forgetting the Coriolis term $2\vec\Omega\times\vec v_{\text{rel}}$ in the acceleration equation (it is invisible in the velocity equation and easy to drop by analogy), and mixing up which vector ($\hat t$, along the slot) carries the relative motion versus which ($\hat n$) is used purely to solve for $\omega_{CD},\alpha_{CD}$.

Source: Hibbeler, Engineering Mechanics: Dynamics (14th ed.), Ch. 16 (relative-motion analysis: rotating axes).

Question 3 of 6: Crank-and-Slotted-Rod Mechanism — Relative-Motion Kinematics

Question 3: Crank-and-Slotted-Rod Mechanism — Relative-Motion Kinematics (20 marks)