Viscosity Damping 2: UNIZOR.COM - Physics4Teens - Waves - Mechanical Oscillations
Channel: Zor Shekhtman on UNIZOR Education
Date: 2021-06-20

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

Viscosity Damping 2
Critical and Under-Damping

This lecture continues analyzing the damped oscillation in a case of critical damping. It relies on equations and symbols presented in the previous lecture.

The main differential equation that describes the damped oscillation is
m·x"(t) + c·x'(t) + k·x(t) = 0
where
m is a mass of a object attached to a free end of a spring,
c is viscosity of environment surrounding a spring, as it acts against motion of an object,
k is elasticity of a spring,
x(t) is displacement of an object from its neutral position on a spring.
We also consider initial conditions
x(0) = a (initial stretch),
x'(0) = 0 (no initial speed).

In search of two partial solutions of the equation above we used in a previous lecture a function
x(t) = eγ·t
where γ - some (generally, complex) number.
The reason for this is that, substituting this function with its first and second derivatives into the differential equation, we get
x'(t) = γ·eγ·t
x"(t) = γ²·eγ·t
and the differential equation is
γ²·m·eγ·t + γ·c·eγ·t + k·eγ·t = 0

Canceling eγ·t, which never equal to zero, we get a quadratic equation for γ
γ²·m + γ·c + k = 0
with any of its solutions (generally, complex), used in function eγ·t, delivering the solution of the original differential equation.

Case 2
Δ=c²−4k·m EQUALS to ZERO

So, we assume now that the discriminant of the quadratic equation above is zero:
Δ = c²−4k·m = 0.
Then two roots to the quadratic equation above coincide:
γ1,2 = −c/(2m)

As we see, in this case our approach delivered only one partial solution to our differential equation:
x1(t) = e−c·t/(2m)
But, to get to a general solution, we need two partial ones.

The second partial solution we will search among functions x2(t)=t·eγ·t.
x'2(t) = e^(γ·t) + t·γ·e^(γ·t)
x"2(t) = γ·e^(γ·t) + γ·e^(γ·t) + t·γ²·e^(γ·t)
and the differential equation is
m·[γ·e^(γ·t) + γ·e^(γ·t) + t·γ²·e^(γ·t)] +
+ c·[e^(γ·t) + t·γ·e^(γ·t)] + k·t·e^(γ·t) = 0

Canceling exponent eγ·t that never equals to zero, we obtain
m·[γ + γ + t·γ²] +
+ c·[1 + t·γ] + k·t = 0

Let's group the terms of this equation
t·[m·γ² + c·γ + k] + 2m·γ + c = 0

If we can find such γ that the expression above is always zero for any t, the corresponding function x2(t)=t·eγ·t will be a second partial solution we are looking for.

First of all, we should use the fact that the case we are considering now is
Δ = c²−4k·m = 0.
From this follows that
k = c²/(4m)
Substituting this into the above equation, we get
t·[m·γ² + c·γ + c²/(4m)] +
+ 2m·γ + c = 0

Now let's choose such γ that nullifies the part not involved in multiplication by t:
2m·γ + c = 0
γ = −c/(2m)
What it does with a coefficient at t is as follows:
m·γ² + c·γ + c²/(4m) =
= m·c²/(4m²) − c²/(2m) +
+ c²/(4m) =
= c²/(4m) − c²/(2m) +
+ c²/(4m) = 0
So, the coefficient at t is also equals to zero.

Therefore, with γ=−c/(2m) the quadratic equation above is satisfied for any t and function x2(t)=t·e−c·t/(2m) is the second partial solution to our differential equations.

Now, having two partial solutions
x1(t) = e^[−c·t/(2m)] and
x2(t) = t·e^[−c·t/(2m]),
we can formulate a general solution to critically damped oscillation, when Δ=c²−4k·m=0:
x(t) = A·x1(t) + B·x2(t) =
= A·e^[−c·t/(2m)] + B·t·e^[−c·t/(2m)]

We can determine coefficients A and B from initial conditions
x(0) = a and
x'(0) = 0
From this we get
A = a and
−A·c/(2m) + B = 0

Therefore, our two coefficients are A=a and B=a·c/(2m).
General solution for of a case of critical damping, when c²=4m·k, is
x(t) = a·e^[−c·t/(2m)] +
+ (a·c/(2m))·t·e^[−c·t/(2m)] =
= a·(1+c·t/(2m))·e^[−c·t/(2m)]