Forced Oscillation 2 - Resonance: UNIZOR.COM - Physics4Teens - Waves - Mechanical Oscillations
Channel: Zor Shekhtman on UNIZOR Education
Date: 04/07/2021

UNIZOR.COM - Physics through a Looking Glass of Mathematician
Notes to a video lecture on http://www.unizor.com

Forced Oscillation 2 - Resonance

This lecture continues analysis of forced oscillations for a specific case, when the angular frequency of external periodic force equals to an inherent natural frequency of an object on a spring without any external forces.

As we know, in the absence of external force the differential equation describing the displacement x(t) from a neutral position of an object of mass m on a spring of elasticity k follows from the Hooke's Law and the Newton's Second Law
m·x"(t) = −k·x(t)
with a general solution
x(t) = A·cos(ω0·t) + B·sin(ω0·t)
where
A and B are any constants;
ω0 = √k/m is an inherent natural angular frequency of oscillation of this particular object on this particular spring.

The equation above is homogeneous because, together with any its solution x(t), the function C·x(t) will be a solution as well.

Consider a model of forced oscillation of an object on a spring with a periodic external force F(t)=F0·cos(ω·t).

In this lecture we consider a case of angular frequency ω of the periodic external function F(t)=F0·cos(ω·t) to be equal to the inherent natural angular frequency ω0=√k/m of oscillations without external forces:
ω = ω0 = √k/m

If external force F(t), described above, is present, the differential equation that describes the oscillation of an object looks like
m·x"(t)+ k·x(t) = F0·cos(ω·t)
where ω=√k/m

The function on the right of this equation makes this equation non-homogeneous.

We need to find a single partial solution to this equation and then we will add to it a general solution of the corresponding homogeneous equation.

Let's try the following function:
xp(t) = Q·t·sin(ω·t)
Then
x'p(t) = Q·sin(ω·t) + Q·t·ω·cos(ω·t)
x"p(t) = Q·ω·cos(ω·t)+Q·ω·cos(ω·t)−Q·t·ω²·sin(ω·t)

Our original equation
m·x"(t)+ k·x(t) = F0·cos(ω·t)
can be written as
x"(t) + (k/m)·x(t) = (F0/m)·cos(ω·t)
or
x"(t)+ω²·x(t) = (F0/m)·cos(ω·t)

Substitute the partial solution expression into the left part of the differential equation above:
Q·ω·cos(ω·t)+Q·ω·cos(ω·t)−Q·t·ω²·sin(ω·t)+Q·t·ω²·sin(ω·t)
Final equation is
2Q·ω·cos(ω·t) = (F0/m)·cos(ω·t)
from which
Q = F0/(2m·ω)
The partial solution to our non-homogeneous differential equation is
xp(t) = F0·t·sin(ω·t)/(2m·ω)

Now we can express the general solution to a differential equation that describes the forced oscillation by a periodic external function as
x(t) = F0·t·sin(ω·t)/(2m·ω) + A·cos(ω·t) + B·sin(ω·t)
where
A and B are any constants determined by initial conditions x(0) and x'(0);
ω = √k/m is an inherent natural angular frequency of oscillation of this particular object on this particular spring;

As in the previous lecture,
A·cos(ω·t)+B·sin(ω·t) 
can be transformed into 
A·cos(ω·t)+B·sin(ω·t) =
= D·[cos(φ)·cos(ω·t) + sin(φ)·sin(ω·t)] =
= D·cos(ω·t−φ)

Now the general solution to our differential equation that describes forced oscillation looks like
x(t)=F0·t·sin(ω·t)/(2m·ω) + D·cos(ω·t−φ)
where
F0 is an amplitude of a periodic external force;
m - mass of an object on a spring;
k - elasticity of a spring;
ω=√k/m is an inherent natural angular frequency of oscillation of this particular object on this particular spring and the angular frequency of external periodic force;
t is time;
D and φ parameters are determined by initial conditions for x(0) and x'(0).

The main component of the function describing the movement of an object, when a periodic external force acts synchronously with the inherent natural oscillation of an object on a spring, is F0·t·sin(ω·t)/(2m·ω).
This function represents oscillation with gradually, proportionally to time t, increasing to infinity amplitude.
Added to this function is a regular sinusoidal oscillation with the same angular frequency but, potentially, a phase shift φ.
The periodic external force, acting synchronously (that is, with the same angular frequency) with inherent natural oscillation of an object on a spring, causes proportional to time increasing of an amplitude of oscillations to (theoretically) infinity. This is called the resonance.