I am trying to understand why in a SHM, the acceleration has multiple components(when actually we create such expressions mathematically by approximating the trigonometric expressions using the binomial theorem).I will explain what I mean to ask below with an example.(adsbygoogle = window.adsbygoogle || []).push({});

In a internal combustion engine(slider-crank mechanism), when I calculate the expression for displacement,I arrive at a expression like so:

$$s=r(cos \theta +\frac{l}{r}cos \phi)$$

where,s-distance between crank centre and piston centre

http://www.mech.uq.edu.au/courses/engg1010/dynamics_pracs/prac1/sketch.jpg

Therefore, using the Pythagoras theorem,

$$cos \phi=\sqrt{1-(\frac{r}{l})^2 sin^2 \theta}$$

Expanding the above expression using'say'..... the binomial theorem(its my choice and thats the problem) :

$$cos \phi=1-\frac{1}{2}(\frac{r}{l})^2sin^2\theta-\frac{1}{8}(\frac{r}{l})^4sin^4\theta$$

This is the part that makes it look like the "various harmonic components originate from the choice of approximation by the mathematician i.e binomial theorem" .

And hence,also the different orders of forces originate in the ICE. 1st order forces fluctuate at the speed of the crank and 2nd order forces fluctuate at 2 times crank rpm.

So,what is to say a mathematician used some other method of approximation of the cos(phi) expression on top and we arrived at a different magnitude of the accelerations,this would spoil our whole engine balance act,wouldn't it?

Was my question clear enough?

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# Why is it that acceleration has multiple components in SHM?

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