When we determine the natural frequency of free vibration for a system, we start of by defining a reference from where displacements will be measured. I have usually seen in books, taking the reference corresponding to the equilibrium position of the system. Is this compulsory? Or the reference can be arbitrary?

  • $\begingroup$ Many times there is an offset to the data, from the measuring device. Or the equilibrium position may be unknown a-priori. But centering it will on the equilibrium pos. will usually be one of the first things to do when analyzing with the data. $\endgroup$
    – Pete W
    Sep 9, 2021 at 12:21
  • 1
    $\begingroup$ A real-world system will have many modes of vibration. In practice you often measure the motion at many different locations, and then use mathematical procedures to identify the individual mode frequencies and mode shapes. So the "origin" of the initial measurements is probably only important in a textbook type of problem. $\endgroup$
    – alephzero
    Sep 9, 2021 at 16:26
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    $\begingroup$ ... for example if I am measuring the vibration of a jet engine rotor spinning at 3000 RPM, the lowest vibration frequency of the blades is about 70 Hz (not 50 Hz which would correspond to 3000 RPM) and I am making the measurements with a laser system located 10 meters away from the engine, I certainly don't know where the "equilibrium position" of anything is until the engine is actually running! $\endgroup$
    – alephzero
    Sep 9, 2021 at 16:30

3 Answers 3


It can be arbitrary. It just simplifies the mathematics significantly. However the end result should be the same.


So you need a point, "zero point" as you really want the amplitude of the displacement as the conditions vary.

if you use a moving point to measure another moving point then you have to recalculate to get the simple displacement. As long as you get the equations of movement correct the result will be the same.

As a different example, we were plowing at night. We were told to"line up" with an identifiable point on the horizon to keep the first plow line straight (this was before all the gps in tractors). My mate picked a white dot then proceeded to plow all night. In the morning his first line was found to be a curve... Lots of hilarity as he had a lot of short work to do (this is where you spend more time positioning compared to working - costs you money as you are paid for ground covered). We realised that the white dot he had picked was a sheep gently grazing across another field :) Not a fixed point then...


Any point in a wave can be arbitrarily chosen as the reference and it wouldn't change the frequency and amplitude. but it would change the phase of the wave which is essential in complex analysis and tensor calculus.

y = A sin(B(x + C)) + D

  • A is th amplitude

  • Period = 2π/B

  • phase shift is C (positive is to the left)

  • vertical shift is D

it is interesting that if you have two identical waves with a half period phase difference and you add them together the sum will be a straight line. This is the basis of noise cancelling.

Other uses of phase angle difference is in electrical motors and the key to Tesla engine.

If you add two identical waves with same phase angle you get the same frequency twice amplitude.

wave diagram


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