infinity

Wish

I studied Theory of Machines in my undergrad course. Coupler curves are part of this course. My wish is to write computer programs using Python to trace these coupler curves. See output, four bar mechanism is tracing out infinity curve

Here's what the code is doing

Definitions of links and angles

• $O_1$ is the fixed pin on left
• $O_2$ is the fixed pin on right
• line $AO_2$ is a helpful construction that assists in calculation of $\delta$
• $\alpha$ is the angle between $AO_2$ and $O_1O_2$
• $\theta$ is the angle between $AO_2$ and $BO_2$
• $\delta$ is the angle made by the link $BO_2$ with x axis
• Point P is tracing the infinity coupler curve

Links and distance

Links	Distance
Crank $O_1A$	a
Coupler AB	b
Crank $BO_2$	c
Fixed frame $O_2 O_1$	d
point P on link AB (AP)	r

Angles and symbol

Angle	Symbol
$\angle A O_1 O_2$	phi ($\phi$)
$\angle A O_2 O_1$	alpha ($\alpha$)
$\angle A O_2 B$	theta ($\theta$)
$O_2B$ with x-axis	delta ($\delta$)
slope of line AB	gamma ($\gamma$)

The key is, once the angles are known following relation can be used to find the co-ordinates: $$x = cos(angle)$$ $$y = sin(angle)$$

Calculation of angles

1. $\phi$ is defined in the code. 2 $\pi$ radians are discretized into 180 points. So, $O_1 A$ is the driving crank.
2. $\theta$ is calculated using the cosine formula.
   1. $cos(\theta) = \frac{O_2A^2 + c^2 - b^2}{2.O_2A.c}$
3. $\delta$:
   1. When $\phi$ < $\pi$ => $\delta$ = $\pi + (\theta - \alpha)$
   2. When $\phi$ > $\pi$ => $\delta$ = $\pi - (\theta - \alpha)$
4. $\gamma = tan^{-1} (\frac{y_B - y_A}{x_B - x_A})$

Calculation of point's co-ordinates

Point	x co-ordinate	y co-ordinate
$O_1$	$x_{O_1} = 0$	$y_{O_1}$ = 0
$O_2$	$x_{O_2} = d$	$y_{O_2}$ = 0
A	$x_A = x_{O_1} + a.cos(\phi)$	$y_A = y_{O_1} + a.sin(\phi)$
B	$x_B = x_{O_2} + d + c.cos(\delta)$	$y_B = x_{O_2} + c.sin(\delta)$
P	$x_P = x_A + r.cos(\gamma)$	$y_P = y_P + r.sin(\gamma)$

Output

animation saved as mp4

infinity_coupler_curve.mp4

final traced coupler curve: infinity

Work products of executing the code

Executing the code

• will generate a crude animation and
• write a video titled 'infinity_coupler_curve.mp4' in current directory and
• save an image titled 'infinity_coupler_curve.png' in current directory

Dependency for executing the code

ffmpeg.exe is required for saving the animation as mp4. ffmpeg executable is downloaded from https://github.com/BtbN/FFmpeg-Builds/releases for win64