Problem Definition

Single degree of freedom systems contain only one type of repetitive motion and therefore equation of motion is a single second order differential equation.

m\ddot{x} + c\dot{x} + kx = 0

Solving this equation often uses assumption that solution is in the form of $x = X_0e^{st}$ and substituting this will give following quadratic equation.

ms^2 + cs + k = 0

Solutions to this can be calculated from usual quadratic solution equation:

s_{1,2} = \frac{-c \pm \sqrt{c^2-2mk}}{2m}

Depending on the nature of the solutions of this quadratic equation, there are three ways the system can behave.

Damping Situations

1) No Damping

When damping coefficient is equal to zero ( $c = 0$ ), there is no damping happening and harmonic motion continues forever.

2) Under Damped

When $\beta < 1$ , $s$ has two complex solutions and it leads to following form of solution known as underdamped system.

x(t) = X_0e^{-\beta\omega_n t} \sin(\omega_d t + \phi_0)

Where $X_0, \phi_0$ , and $\omega_d$ are amplitude, phase angle and frequency of oscillation and:

\omega_d = \omega_n\sqrt{1-\beta^2}

3) Critically Damped

When $\beta = 1$ , $s_1=s_2$ . Because of that system is critically damped and it reaches the stability within shortest possible time. The solution to DE is in the following form:

x(t) = (c_1 + c_2t)e^{\omega_n t}

4) Overdamped

When $\beta > 1$ , system has two real roots and solution has the following form:

x(t) = c_1e^{s_1t} + c_2e^{s_2t}

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