Undamped And Damped Vibrations
January 28, 2019
In undamped vibrations, no resistive force acts on the vibrating object. As the object oscillates, the energy in the object is continuously transformed from kinetic energy to potential energy and back again, and the sum of kinetic and potential energy remains a constant value. In practice, it’s extremely difficult to find undamped vibrations. For instance, even an object vibrating in air would lose energy over time due to air resistance.
Let us consider an object undergoing simple harmonic motion. Here, the objet experiences a restoring force towards the equillibrium point, and the size of this force is proportional to displacement. If the displacement of the object is given by x, then for an object with mass m in simple harmonic motion, we can write:
This is a differential equation. A solution to this equation can be written in the form:
If vibration is undamped, the object continues to oscillate sinusoidally.
2.) Damped vibrations:-
In damped vibrations, external resistive forces act on the vibrating object. The object loses energy due to resistance and as a result, the amplitude of vibrations decreases exponentially.
We can model the damping force to be directly proportional to the speed of the object at the time. If the constant of proportionality for the damping force is b, then we can write:
The solution to this differential equation can be given in the form:
We can write this as:
Writing the equation in this form is useful because the quantity can be used to determine the nature of a particular oscillation. Often, this quantity is called the damping coefficient,
If , then we have critical damping. Under this condition, the oscillating object returns to its equilibrium position as soon as possible without completing any more oscillations. When , we have underdamping. In this case, the object continues to oscillate, but with an ever-reducing amplitude. For the resistive forces are very strong. The object would not oscillate again, but the object is slowed down so much, that it goes towards the equilibrium much more slowly compared to an object that is critically damped. Overdamping is the name given to this type of scenario. When , there is no resistive force and the object is undamped. Theoretically, the object continues to carry out simple harmonic motion without any reduction in amplitude.