Viscosity Damping 3: UNIZOR.COM - Physics4Teens - Waves - Mechanical Oscillations
Channel: Zor Shekhtman on UNIZOR Education
Date: 2021-06-26

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

Viscosity Damping 3
Underdamping

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

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 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 partial solution of the original differential equation.

Case 3

Discriminant of this quadratic equation Δ=c²−4k·m is NEGATIVE

Let
ω² = −Δ /4m² = k/m − (c/(2m))²
where ω - some positive real number.

Then the solutions to our equations are complex numbers
γ1,2 = −c/(2m) ± ω·i
where i²=−1 is an imaginary unit.

Two partial solutions of our differential equation in the domain of complex functions of real argument t are
x1(t) = e(−c/(2m)+ω·i)·t and
x2(t) = e(−c/(2m)−ω·i)·t]

Recall the Euler's formula
ei·x = cos(x) + i·sin(x)
that you can find in the "Math 4 Teens" course on this site UNIZOR.COM - Math 4 Teens - Trigonometry - Complex Numbers and Trigonometry - Euler's Formula.

Using Euler's formula, we can split two partial solutions of our differential equation into real and imaginary parts:
x1(t) = x1Re(t) + i·x1Im(t)
where
x1Re(t) = e−c·t/(2m)·cos(ω·t)
and
x1im(t) = e−c·t/(2m)·sin(ω·t)
Similarly,
x2(t) = x2Re(t) + i·x2Im(t)
where
x2Re(t) = e−c·t/(2m)·cos(−ω·t)
and
x2im(t) = e−c·t/(2m)·sin(−ω·t)

If a complex function x(t) satisfies a linear differential equation, its real and imaginary parts must separately satisfy it as well. This obvious statement follows from the fact that, if a(t)+i·b(t)=0 for all t, then a(t)=0 and b(t)=0 for all t.

Therefore, if
x(t) = xRe(t) + i·xIm(t)
is a solution of our differential equation, then
xRe(t) and xIm(t)
are solutions in their own rights.

Recall that sin(−α)=−sin(α) and cos(−α)=cos(α).
Therefore, we have two linearly independent real functions that are partial solutions to our differential equation:
x1(t) = e−c·t/(2m)·cos(ω·t)
x2(t) = e−c·t/(2m)·sin(ω·t)

Any linear combination of these two functions also is a solution to our differential equation.
This allows to express a general solution as
x(t) = e−c·t/(2m) ·
· [C1·cos(ω·t)+C2·sin(ω·t)]