Differential equations are used for various applications. When a mass on a vertical spring is released, the governing equation is:
\( \sum F = ma = m \cfrac{dx^2}{dt^2} = F_g + F_s + F_d + F(t) \)
\( m \cfrac{dx^2}{dt^2} = mg - k(s + x) -\beta \cfrac{dx}{dt} + F(t)\)
Where \(k\) is the spring constant, \(s\) is the distance the spring moves when the mass is applied and \(\beta\) is a damping constant. When the spring is at the equilibrium position \( mg - ks = 0 \).
There are different scenarios which will lead us to use different strategies to solve the DEs.
Free motion occurs when there is no external force \(F(t)\).
Undamped motion occurs when \( \beta = 0 \).
For free undamped motion the DE is:
\( m \cfrac{dx^2}{dt^2} = mg - k(s + x) = mg -ks -kx\)
\( m \cfrac{dx^2}{dt^2} + kx = 0\)
This is a homogeneous linear DE with constant coefficients that can be solved using by solving a characteristic equation.
\( m r^2 + k = 0 \)
\( r = \sqrt{-\frac{k}{m}} = \sqrt{\frac{k}{m}} i\)
The general solution is:
\( x(t) = c_1 \sin {\sqrt{\frac{k}{m}} t} + c_1 \cos {\sqrt{\frac{k}{m}} t} \)
It is common to let \( \omega ^2 = \frac{k}{m} \):
\( x(t) = c_1 \sin {\omega t} + c_1 \cos {\omega t} \)
For free motion the DE is:
\( m \cfrac{dx^2}{dt^2} + \beta \cfrac{dx}{dt} + kx = 0\)
This is also homogeneous linear DE with constant coefficients that can be solved using by solving a characteristic equation. There are 3 different cases:
| Roots | Scenario | Determinant | Solution |
| 2 Real Roots | Overdamping | \(\beta ^2 - 4mk > 0\) | \( x (t) = e^{\frac{\beta}{2m}t} (c_1 e^{\sqrt{(\frac{\beta}{2m})^2 -\frac{k}{m}t}} + c_2 e^{-\sqrt{(\frac{\beta}{2m})^2 -\frac{k}{m}t}}) \) |
| Repeated Real Roots | Critical Damping | \(\beta ^2 - 4mk = 0\) | \( x (t) = e^{\frac{\beta}{2m}t} (c_1 + c_2 t) \) |
| Complex Roots | Underdamping | \(\beta ^2 - 4mk < 0\) | \( x (t) = e^{\frac{\beta}{2m}t} (c_1 \cos {\sqrt{\frac{k}{m}t - (\frac{\beta}{2m})^2}} + c_2 \sin {\sqrt{\frac{k}{m}t - (\frac{\beta}{2m})^2 }})\) |
For driven motion the DE is:
\( m \cfrac{dx^2}{dt^2} + \beta \cfrac{dx}{dt} + kx = F(t)\)
Here, we can solve the homogeneous equation (complementary solution) and then find the particular solution using variation of parameter method.