Energy of underdamped oscillator

In the presence of a damping force the oscillating system is no longer closed and its energy dissipates to the environment. The total energy of the oscillator becomes converted into thermal energy. For small values of the damping ratio, the equation given in this law approximately describes the total mechanical energy of the underdamped oscillator.

Conditions:

1. Damping ratio \(\zeta\) is small, i.e. \(\zeta \ll 1\).

Links:

1. Similar equation (15-44) on p. 431 of “Fundamentals of Physics” by David Halladay et al., 10th Ed.

energy

energy of the oscillator.

Symbol:

E

Latex:

\(E\)

Dimension:

energy

mass

mass of the oscillator.

Symbol:

m

Latex:

\(m\)

Dimension:

mass

amplitude

Amplitude, or maximum absolute displacement, of the oscillator.

Symbol:

A

Latex:

\(A\)

Dimension:

length

undamped_angular_frequency

angular_frequency of the undamped oscillator.

Symbol:

w_0

Latex:

\(\omega_{0}\)

Dimension:

angle/time

exponential_decay_constant

exponential_decay_constant of the oscilaltor.

Symbol:

lambda

Latex:

\(\lambda\)

Dimension:

1/time

time

time.

Symbol:

t

Latex:

\(t\)

Dimension:

time

law

E = m * w_0^2 * A^2 * exp(-2 * lambda * t) / 2

Latex:

\[E = \frac{m \omega_{0}^{2} A^{2} \exp{\left(- 2 \lambda t \right)}}{2}\]