![damped harmonic oscillators damped harmonic oscillators](https://i.stack.imgur.com/p7a7K.png)
For the oscillator with a small damping one obtains for the average energy: (10.28) The average power of losses is: (10.29) Therefore the average power of losses is related to the average energy as (10.29a)įor the oscillating system with damping one introduces the dimentionless factor, called quality factor Q, defined as follows: (10.30) For low damping one obtains from the last equation substituting for power of losses from (10.29a) (10.31) Applying the last result to the electrical oscillator L,R,C, and introducing the analogous quantities one obtains for the quality factor (10.32) Examples of quality factors Q resonance radio circuit several hundreds violin string 103 microwave resonator 104 excited atom 107ģ1 10.7.
![damped harmonic oscillators damped harmonic oscillators](https://i.stack.imgur.com/ZPvgy.png)
for ωo> β, solution (26) can be approximated by (27) The amplitude for the damped oscillator decreases exponentially with time.Įnergy losses for the damped oscillator The amplitude for the damped oscillator decreases exponentially with time. The angular frequency ω of the damped oscillator is less than that of undamped oscillator ωo. Solution (26) can be regarded as a cosine function with a time dependent amplitude Time t = τ, after which the amplitude decreases e1/2 times is called the average lifetime of oscillations or the time of relaxation. The liquid exerts a damping force which in many cases is proportional to the velocity (with opposite sign): b – damping constant (22) In this case the equation of motion can be written as (23) After rearrangement we have (24) Figure from HRW,2 Introducing the substitutions: one gets (25) The solution of (25) for a small damping is: (26) where The damped oscillator shown in the figure consists of a mass m, a spring of constant k and a vane submarged in a liquid.