1. Exploring limiting Cases.
The damping constant characterizes the magnitude of the the drag force. Confirm that when \(b=0 \) you recover the position function and oscillation frequency for the un-damped mass on a spring.
Section 6.11 Models Including Dissipative Forces
The models you have built for simple harmonic motion of spring, simple pendulums, and rigid body oscillators have not included dissipative forces such as friction. You know from experience that a pendulum left to oscillate will lose energy. The amplitude decreases in time until all energy has been dissipated and the system comes to rest. An oscillation whose amplitude decreases in time is called a damped oscillation. Energy dissipation can be caused by dissipative forces such as air resistance and friction.
You can now analyze a spring system and consider the effects of air drag, assuming a drag force that is directly proportional to the velocity.
\begin{equation*}
\vec{F}_{drag} = -b\vec{v}
\end{equation*}
The damping constant, \(b\text{,}\) depends on the shape of the object and the viscosity of the air (or other medium) in which it moves. The damping constant plays the same role as the coefficient of friction. The negative sign reminds you that the force always opposes the velocity of the object.
Again, apply the general physics problem-solving strategy. First, draw a pictorial representation of the situation and choose a coordinate system. Applying the Law of Motion, you find
\begin{equation*}
ma_x = -kx -bv_x
\end{equation*}
Using the relationship between velocity acceleration and position, rewrite the equation in terms of position only.
\begin{equation}
\frac{d^2}{dt^2}x(t) + \frac{b}{m}\frac{d}{dt}x(t) +\frac{k}{m}x(t) = 0\tag{6.11.1}
\end{equation}
This is the equation of motion of a damped mass on a spring. Since you know that the amplitude decreases in time, you can guess a solution of the form:
\begin{equation}
x(t) = A e^{-bt/2m} \cos{(\omega t +\phi_o)}\tag{6.11.2}
\end{equation}
The oscillation frequency is given by,
\begin{equation}
\omega = \sqrt{\frac{k}{m} - \frac{b^2}{4m}}=\frac{2 \pi}{T}\tag{6.11.3}
\end{equation}
You can see that the oscillation frequency is equivalent to the original oscillation frequency modified by the quantity \(b^2/4m^2 \text{.}\) Below is a graph of the position as a function of time for the damped oscillator. The cosine function modified by the exponential decay is shown. You can see that the envelope of the amplitude decays in time. The maximum displacement away from equilibrium decreases according to the function
\begin{equation*}
x_{max}(t) = A e^{-bt/2m}
\end{equation*}
Exercises Activities
2. Is \(x(t)\) a Solution?
Plug the solution back into the equation of motion and confirm that \(x(t)\) is a solution.
3. Friction at the Surface Boundary.
Repeat the analysis above excluding air drag, but including kinetic friction between the surface on which the mass rests. Determine an equation of motion, the position as a function of time and the oscillation frequency.
4. The Damped Pendulum.
Repeat the analysis above for the pendulum including air drag. Determine an equation of motion, the angular position as a function of time and the oscillation frequency.
References References
[1]
Figure 6.11.1 created by Oleg Alexandrov published on Wikipedia under Public Domain.
[2]
Figure 6.11.2 and Figure 6.11.3created by Rebecka Tumblin.
