Why does damping affect the time period of SHM oscillators?

[jsmith613]

There are two equations that can describe the time period of SHM oscillators (springs / pendulums ONLY)

Spring $T = 2π * \sqrt \frac {m}{k}$

Pendulum $T = 2π * \sqrt \frac {l}{g}$

It would seem from these equations that time period is independant of amplitude
therefore we should be able to conclude that the time period for a damped oscillator is the same as that for an undamped oscillator.

BUT if you oscillate something viscous (water, oil, tar) then then time period is massively increased.

My questions are
(a) would an underdamped system oscillate at the same period as an un-damped system?
(b) how come the equations of SHM and reality don't match up (damped systems are still undergoing SHM...I think)
(c) will the time period change in the damped system during the oscillations? i.e: could it start at 0.5s and then increas to 1s

 

[sambristol]

First point - the equations you state are approximations. The spring one assumes the spring is within its elastic limit and Hookes law strictly applies (not to bad a condition in the real world). The pendulum equation is more of a problem. It is only valid if the pendulum oscillation is through an infinitestimaly small angle.
So real a real pendulum does not execute true SHM and its period is affected by its amplitude.

In answer to your questions

a) No and the period changes with time
b) Damped systems are not undergoing true SHM hence the mismatch
c) Yes

If you are familiar with Fourier transforms you can take the transform of the equation of motion of the damped system and see the change in period directly.

Hope this helps

Regards

Sam

 

[AlephZero]

The way the system behaves depends how it is damped. Different physical processes can produce damping forces that are constant amplitude (but change direction every half cycle of oscillation), proportional to velocity, or proportional to veloicty squared - or even more complicated behaviour.

The simplest to analyse is damping proportioal to velocity, and in that case the damped frequency is constant for the whole motion, but is lower than the frequency without damping. For systems with low levels of damping like a mass oscillating on a spring, the change in frequency is small (typically less than 1%) but as the damping increases the "frequency" eventually drops to zero and there is no oscillation at all, just a motion in one direction that slows down as the system approaches its equilibrium position.

All this is well known and covered in college-level courrses on dynamics, but as in any subject you have to learn to walk before you can run, and a first course in dynamics often only includes undamped SHM.

 

[jsmith613]

thanks all

 

[pmacias]

Damping affects the time period of SHM oscillators because it introduces an external force that acts against the restoring force of the oscillator. This external force is known as the damping force and it is proportional to the velocity of the oscillator. In an undamped system, there is no damping force and the oscillator is able to continue oscillating indefinitely with a constant time period.

In an underdamped system, the damping force is less than the restoring force, so the oscillator is able to complete each oscillation in approximately the same amount of time as an undamped system. However, the presence of the damping force causes the amplitude of the oscillations to decrease over time, resulting in a decrease in the overall energy of the system. This decrease in energy leads to a decrease in the time period of the oscillator over time.

The equations for SHM assume ideal conditions where there is no external force acting on the oscillator. In reality, there are always external forces present, such as air resistance or friction, that can affect the behavior of the oscillator. These external forces can cause the time period of the oscillator to deviate from the ideal equation. In the case of damping, the presence of the damping force results in a longer time period for the oscillator.

The time period in a damped system can change during the oscillations. As the amplitude decreases, the damping force also decreases, resulting in a smaller effect on the time period. This can cause the time period to increase slightly as the oscillations continue. However, the overall trend is for the time period to decrease over time as the energy of the system decreases.