Basics of SHM (undamped, under-driven)

In summary, the conversation discusses the placement of maximum points on a graph depicting undamped, under-driven harmonic motion. It is initially believed that the maxima occur where the displacement curve and exponential envelope curve intersect, but the textbook states that they actually correspond to points where the velocity is equal to zero. The conversation then questions the constancy of the maxima ratios and the assertion that the exponential curve is an envelope for the graph. The conversation ends with a clarification on the correct terminology for undamped, underdamped motion.
  • #1
cj
85
0
My textbook says the an object undergoing undamped, under-driven
harmonic motion (http://romano.physics.wisc.edu/lab/manual/img279.gif)
does NOT have its maxima at the points where the displacement
curve makes contact with the exponential envelope curve.

How can this be the case?? Doesn't the graph clearly imply that
the maxima are indeed the peaks of the decaying cosine curve (that
do make contact with the exponential wrapper)??

The text goes on to say that the maxima actually correspond not
to the x(t) vs. t plot -- but to the dx(t)/dt (the velocity) plot,
specifically where dx(t)/dt = 0. I can partially understand this since
at the maxima -- velocity does equal 0!


It then states that the displacement ratios between successive
maxima are constant.

I can see the constancy of the maxima ratios, but not the
basis on dx(t)/dt over the visual interpretation -- let alone
the assertion that successive maxima ratios are constant.


Comments? Thanks!
 
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  • #2
That exponential curve is an envelope for that graph if the tangents of the curve and the graph agree at all contact points. No point on the exponential curve has a tangent line with zero slope.
 
  • #3
Is the displacement maxima (which occurs at the time
where v(t) = dx(t)/dt = 0) the same point as the displacement
at t=0, T, 2T, etc., where T = 2*pi/omega (the under-damped
version of omega)?

robphy said:
That exponential curve is an envelope for that graph if the tangents of the curve and the graph agree at all contact points. No point on the exponential curve has a tangent line with zero slope.
 
  • #4
graph

Just try drawing an exponentially damped sine wave and then the smooth exponential -- you will see that's correct.
ymax is dy/dt =0 for the sine , byt dy/dt is never =0 for the exponential.
 
  • #5
My textbook says the an object undergoing undamped, under-driven

Sorry to be nit picky but this is bugging me. Could you possibly have meant to say.

Undriven, under damped?

That is what your graph looks like.
 
  • #6
Apparently, per rigorous research, the correct form is "undriven, underdamped."

Integral said:
Sorry to be nit picky but this is bugging me. Could you possibly have meant to say.

Undriven, under damped?

That is what your graph looks like.
 

1. What is simple harmonic motion (SHM)?

Simple harmonic motion (SHM) is a type of periodic motion in which an object oscillates back and forth around a central equilibrium position with a constant amplitude and frequency. This type of motion can be seen in various systems such as a mass-spring system, a pendulum, or an oscillating fan.

2. What is the equation for SHM?

The equation for SHM is x = A cos(ωt + φ), where x is the displacement of the object from its equilibrium position, A is the amplitude of the motion, ω is the angular frequency, and φ is the phase constant. This equation can also be written in terms of velocity and acceleration as v = -Aωsin(ωt + φ) and a = -Aω^2cos(ωt + φ), respectively.

3. What is the difference between damped and undamped SHM?

In undamped SHM, the system is free from any external forces or friction, and thus the amplitude of the oscillations remains constant. In damped SHM, there is an external force or friction acting on the system, causing the amplitude of the oscillations to decrease over time. This results in a slower rate of oscillations and eventual decay of the motion.

4. What is meant by under-driven SHM?

Under-driven SHM refers to a system where the driving force (a force applied to the system to maintain its motion) is less than the natural frequency of the system. In this case, the system will oscillate with a smaller amplitude and a phase shift compared to the driving force.

5. How does the amplitude affect the period of SHM?

The amplitude does not affect the period of SHM. The period, which is the time it takes for one complete oscillation, is solely determined by the frequency of the system. The amplitude only affects the maximum displacement of the oscillating object from its equilibrium position.

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