Is my formula for piston height and velocity in an engine correct?

[Townsend]

I wasn't sure of where this question is best suited but since my interest is only mathematical I figured this is a good forum.

Using vectors I came up with a formula for the height of a piston above the center of the crankshaft for any given angle x (in radians). I set an x,y coordinate system with the origin at the center of the crank. With x=0 radians the crankshaft has the lower end of the connecting rod along the positive x-axis.

So if a is the radius of the crankshaft and b is the length of the connecting rod we have

$$f(x)=asin(x)+b \sqrt{ \frac{1-a^2cos^2(x)}{b^2} }$$

So if that formula is correct then if I differentiate it I should have a formula for the change of height with respect to any given angle. This is
$$f'(x)= \frac{a^2*b*sin(x)*cos(x)}{ \sqrt{ b^2-a^2*cos^2(x) }} +a*cos(x)$$

So to get the velocity I need to take $$\frac{f'*2 \pi*t}{minute}$$ where t is any given rpm.

So if I did everything correctly I should be able to find the speed in units of length per minute of the piston for any given rpm t.

So did I make any mistakes so far? Or does everything seem ok?

Thanks for your time...

 

[Fermat]

Apart from two typos, both involving b, your working is OK.

I derived the eqn for the piston speed using a different method and got the same eqn as yourself, so that provides some confrmation.

 

[tacman]

Based on the information provided, it seems like your formula for piston height and velocity in an engine is correct. You have properly set up the coordinate system and have used vectors to come up with a formula for the height of the piston. Your differentiation of the formula also seems correct, which would give you the velocity of the piston at any given angle.

However, it is always a good idea to double check your calculations and equations, especially when they involve complex mathematical concepts like vectors and derivatives. It might be helpful to have someone else review your work and provide feedback to ensure that everything is correct.

Additionally, it is important to keep in mind that in real-world scenarios, there may be other factors that could affect the height and velocity of the piston, such as friction and air resistance. So while your formula may provide an accurate estimation, it may not be entirely accurate in all situations.

Overall, it seems like you have a good understanding of the mathematical concepts involved in calculating piston height and velocity. Keep up the good work and continue to refine your formula as needed. Good luck!