Calculate piston position from crank radius, rod length and crank angle, or find crank angle, piston speed and acceleration from RPM.

Ideal in-line slider crank with rigid parts and rod length greater than crank radius. Angle 0ยฐ is top dead center (TDC); positive displacement, velocity and acceleration point away from TDC. Dynamics assumes constant nonnegative RPM.

Piston Position Formula

The calculator measures displacement s from top dead center (TDC), using an ideal in-line slider crank with rod length b greater than crank radius r. Angle a is zero at TDC. The former expression P = r cos(a) + โˆš(bยฒ โˆ’ rยฒ sinยฒ(a)) measures distance from the crank center; s = r + b โˆ’ P. 

s = r ร— (1 - cos(a)) + b - SQRT(bยฒ - rยฒ ร— sin(a)ยฒ)

Variables:

  • s is piston displacement from TDC (in), ranging from 0 to the stroke 2r
  • r is the crankshaft radius (in) 
  • b is the length of the piston rod (in) 
  • a is the angle of the crankshaft (degrees) 

How to Calculate Piston Position?

The following steps outline how to calculate the Piston Position.

  1. First, determine the crankshaft radius (in).
  2. Next, determine the length of the piston rod (in). 
  3. Next, determine the angle of the crankshaft (degrees). 
  4. Next, use s = r(1 โˆ’ cos(a)) + b โˆ’ โˆš(bยฒ โˆ’ rยฒsinยฒ(a)). Convert degrees to radians before evaluating software trigonometric functions. The inverse mode gives the outward and return crank angles. Dynamics uses exact derivatives at constant RPM; the displayed long-rod peak approximation omits finite rod effects.
  5. Finally, calculate the Piston Position.
  6. After inserting the variables and calculating the result, check your answer with the calculator above. 

Example Problem:

Use the following variables as an example problem to test your knowledge.

crankshaft radius (in) = 3

length of the piston rod (in) = 7

angle of the crankshaft (degrees) = 40

s = 3(1 โˆ’ cos(40ยฐ)) + 7 โˆ’ โˆš(7ยฒ โˆ’ 3ยฒsinยฒ(40ยฐ)) = 0.972720 in (0.024707 m). The stroke is 6 in; the rod ratio b/r is 7/3. The corresponding crank-center distance P is 9.027280 in.