Скачать или смотреть Resonance in a damped oscillator: the derivation your physics book skipped over.

00:00 Introductions: in this video we find the general solution of the underdamped driven oscillator, then we compute the amplitude of the steady-state oscillations as a function of driving frequency.

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We then calculate the resonant frequency of the oscillator and investigate the impact of damping coefficient on the location of the resonant frequency. Your physics book probably skipped this derivation, but knowing the details of this result can resolve some very common misconceptions about resonance in a damped oscillator!

00:46 Apply Newton's second law to set up the differential equation for the damped driven oscillator: mx''+cx'+kx=f(t), where we're going to use a sinusoidal driving force f(t)=Fcos(omega*t).

03:48 Find the homogeneous (transient) solution: by making the usual proposal x(t)=e^(rt), we find the characteristic equation for the oscillator, solve for r and write down the homogenous solution of the equation. Because we're assuming the oscillator is underdamped, the homogeneous, or transient solution has decaying exponentials multiplying sines and cosines. The decaying exponentials die off rapidly, which is why we call this the transient solution.

07:54 Find the particular (steady-state) solution: using the method of undetermined coefficients, we propose a particular solution of the form acos(omega*t)+bsin(omega*t) and run it through the original differential equation. This allows us to generate two equations by comparing coefficients of the trig functions on the left and right sides, and we solve for the undetermined coefficients a and b.

13:13: General solution of the damped driven oscillator: the sum of the homogeneous (transient) and particular (steady-state) solutions.

14:05 Amplitude as a function of driving frequency: we combine the sine and cosine from the steady-state solution into a single cosine function with a phase shift, which makes it easy to see the amplitude of the long-run position function for the damped driven oscillator.

16:58 Find the resonant frequency: we use the derivative to optimize the amplitude with respect to frequency and obtain our formula for the resonant frequency of a damped driven oscillator.

20:57 How resonance changes with damping coefficient: using the parameter c^2/4mk, we see that resonance only occurs when this parameter is between 0 and 1/2. We view a parameter sweep animation showing how the resonance curve changes as the relative size of the damping coefficient increases: the resonant frequency decreases and the maximum amplitude decreases.