# Kinematic Calculations a Piston Connected To The Crank Mechanism

A crank mechanism basically consists of crank arm, connecting rod , piston and eccentric shaft. You can see a crank mechanism which is drawn below:

We will see in this writing displacement, velocity and acceleration graphs of a piston depend on angle of crank.

Piston is a element which translates explosives force to lineer motion. This motion occurs with while piston moving between Bottom Dead Center (BDC) and Top Dead Center (TDC).

if we need to create a crank connecting rod model, it would be simply as:

We would explain the model as: The circle is crank, BC line is connecting rod lenght and rectangular is piston also AB is crank arm. Theta is angle of crank and connecting rod.

1. DISPLACEMENT OF PISTON

If we take reference of crank axis, we can calculate displacement of a piston as:

According to this formule, piston has 0° and 360° maximum position, 180° and 540° minumum position.

States encoded in Python:

for 40 mm crank arm and 193 mm connecting rod, displacement graph of a piston between 0° and 360° like this:

2. VELOCITY OF PISTON

Velocity of a piston between BDC and TDC is a function of crank angle and this can be formulize that:

in this formule, w is rotation speed which rpm and must translate to rad/sec. About 1 rpm= 0.104 rad/s

Piston has maximum velocity in 74° and increase as rpm increase velocity.

States encoded in Python:

for 40 mm lenght of crank arm, 193 mm connecting rod and 1000 rpm graph velocity of a piston between 0° and 360° like this:

3. ACCELERATION OF PISTON

we would figure out acceleration of a piston as:

According this formule, maximum acceleration of piston is 0, 360, 720 degrees (BDC), minumum acceleration is 141, 501, 579 degrees (TDC).

States encoded in Python:

for 40 mm lenght of crank, 193 mm connecting rod and 1000 rpm, the acceleration of piston like this: