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Why is it that acceleration has multiple components in SHM?

  1. Aug 8, 2013 #1
    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.

    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?
     
  2. jcsd
  3. Aug 8, 2013 #2
    All models are wrong, some are useful.
     
  4. Aug 8, 2013 #3
    It is a bit hard to visualize inertial forces(unbalanced because the gas force exists irregularly- only once in 4 cycles for a 4 stroke) morphing into this whole bunch of primary,secondary,tertiary components.
     
  5. Aug 9, 2013 #4
    How are you using the binomial theorem to expand that into your result? Can you post your math?
     
  6. Aug 10, 2013 #5
    http://www.mech.uq.edu.au/courses/engg1010/dynamics_pracs/prac1/sketch.jpg

    First,I can expand the expression derived from the pythagoras theorem([itex]cos\phi=(1-(\frac{r}{l}sin\theta)^2)^\frac{1}{2}[/itex]) using the binomial theorem as follows:
    $$cos\phi=1-\frac{1}{2}(\frac{r}{l}sin\theta)^2+(\frac{\frac{1}{2}*(\frac{1}{2}-1)}{2!})(\frac{r}{l}sin\theta)^4-....+((\frac{r}{l}sin\theta)^2)^\frac{1}{2}$$

    The powers of sin(theta) can be further expanded using the multiples of angles, like so:
    $$sin^2\theta=\frac{1}{2}-\frac{1}{2}cos2\theta$$
    $$sin^4\theta=\frac{3}{8}-\frac{1}{2}cos2\theta+\frac{1}{8}cos4\theta$$

    How I expand the above expressions can be explained by using the DeMoivre's theorem.For instance:
    $$\begin{align}\sin^4 \theta &= (\sin^2\theta)^2\\ &= \left(\frac12-\frac12\cos(2\theta)\right)^2\\ &= \frac14 \left(1 - \cos(2\theta)\right)^2\\ &= \frac14\left(1 - 2 \cos(2\theta) + \cos^2(2 \theta)\right)\\ &= \frac14\left(1 - 2 \cos(2 \theta) + \frac12(\cos (4\theta) + 1)\right)\\ &= \frac14\left(\frac32 - 2\cos(2\theta) + \frac12\cos(4 \theta)\right)\\ &= \frac38 - \frac12\cos(2\theta) + \frac18\cos(4\theta).\end{align}$$

    So,now you can see the math clearly and as as result of the above expansions,one substitutes [itex]cos\phi[/itex] in the expression for piston stroke(displacement) [itex]s=r(cos\theta +\frac{l}{r}cos\phi)[/itex]from which one can arrive at the expression for acceleration and hence=>intertial forces in a IC engine, which are now split into primary,secondary and more components BECAUSE of the choice of expansion we chose i.e the binomial theorem[itex]F=mr\omega^2(cos\theta+\frac{r}{l}cos2\theta+...)[/itex].
    Note:the coefficient of theta is referred to as 'order of the harmonic'.

    As you might know,engineers choose to balance these forces(+the unbalance due to gas forces also) based on the magnitudes derived from the expressions above. Please tell me what will happen if I choose to expand in another way.

    It looks like mechanical engineers are assuming that the inertial forces in an engine will have different components(how and why these components arise in reality when there are only 4 links in a slider-crank mechanism still intrigues me),all based on the choice of mathematical function expansions-the binomial theorem.
    Was my question clear?
     
    Last edited: Aug 10, 2013
  7. Aug 15, 2013 #6
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