My favorite topic in an introductory differential equations course is mechanical and electrical vibrations. I enjoyed learning about it as a student and I enjoyed teaching it later. (Or more accurately, I enjoyed being exposed to it as a student and really learning it later when I had to teach it.)
I find this subject interesting for three reasons.
- The same equations describe a variety of mechanical and electrical systems.
- You can get practical use out of some relatively simple math.
- The solutions display wide variety of behavior as you vary the coefficients.
This is the first of a four-part series of posts on mechanical vibrations. The posts won’t be consecutive: I’ll write about other things in between.
Simple mechanical vibrations satisfy the following differential equation:
We could simply write down the general solution be done with it. But the focus here won’t be finding the solutions but rather understanding how the solutions behave.
We’ll think of our equation as modeling a system with a mass attached to a spring and a dash pot. All coefficients are constant. m is the mass, γ is the damping from the dash pot, and k is the restoring force from the spring. The driving force has amplitude F and frequency ω. The solution u(t) gives the position of the mass at time t.
More complicated vibrations, such as a tall building swaying in the wind, can be approximated by this simple setting.
The same differential equation could model an electrical circuit with an inductor, resistor, and capacitor. In that case replace mass m with the inductance L, damping γ with resistance R, and spring constant k with the reciprocal of capacitance C. Then the equation gives the charge on the capacitor at time t.
We will assume m and k are positive. The four blog posts will correspond to γ zero or positive, and F zero or non-zero. Since γ represents damping, the system is called undamped when γ = 0 and damped when γ is greater than 0. And since F is the amplitude of the forcing function, the system is called free when F = 0 and forced otherwise. So the plan for the four posts is
- γ = 0, F = 0. Free, undamped vibrations (this post)
- γ > 0, F = 0. Free, damped vibrations
- γ = 0, F > 0. Forced, undamped vibrations
- γ > 0, F > 0. Forced, damped vibrations
Free, undamped vibrations
With no damping and no forcing, our equation is simply
'' + k u = 0
and we can write down the solution
u(t) = A sin ω0t + B cos ω0t
ω02 = k/m.
The value ω0 is called the natural frequency of the system because it gives the frequency of vibration when there is no forcing function.
The values of A and B are determined by the initial conditions, i.e. u(0) = B and u
’(0) = A ω0.
Since the sine and cosine components have the same frequency ω0, we can use a trig identity to combine them into a single function
u(t) = R cos(ω0t – φ)
The amplitude R and phase φ are related to the parameters A and B by
A = R cos φ, B = R sin φ.
That’s it for undamped free vibrations: the solutions are just sine waves. The next post in the series will make things more realistic and more interesting by adding damping.