Let us look at some applications of linear second order constant coefficient equations.
Subsection2.4.1Some examples
Our first example is a mass on a spring. Suppose we have a mass \(m > 0\) (in kilograms) connected by a spring with spring constant \(k > 0\) (in newtons per meter) to a fixed wall. There may be some external force \(F(t)\) (in newtons) acting on the mass. Finally, there is some friction measured by \(c \geq 0\) (in newton-seconds per meter) as the mass slides along the floor (or perhaps a damper is connected).
Let \(x\) be the displacement of the mass (\(x=0\) is the rest position), with \(x\) growing to the right (away from the wall). The force exerted by the spring is proportional to the compression of the spring by Hooke’s law. Therefore, it is \(kx\) in the negative direction. Similarly the amount of force exerted by friction is proportional to the velocity of the mass. By Newton’s second law we know that force equals mass times acceleration and hence \(mx'' = F(t)-cx'-kx\) or
This is a linear second order constant coefficient ODE. We say the motion is
forced, if \(F \not\equiv 0\) (if \(F\) is not identically zero),
unforced or free, if \(F \equiv 0\) (if \(F\) is identically zero),
damped, if \(c > 0\text{,}\) and
undamped, if \(c = 0\text{.}\)
This system appears in lots of applications even if it does not at first seem like it. Many real-world scenarios can be simplified to a mass on a spring. For example, a bungee jump setup is essentially a mass and spring system (you are the mass). It would be good if someone did the math before you jump off the bridge, right? Let us give two other examples.
Here is an example for electrical engineers. Consider the pictured RLC circuit. There is a resistor with a resistance of \(R\) ohms, an inductor with an inductance of \(L\) henries, and a capacitor with a capacitance of \(C\) farads. There is also an electric source (such as a battery) giving a voltage of \(E(t)\) volts at time \(t\) (measured in seconds). Let \(Q(t)\) be the charge in coulombs on the capacitor and \(I(t)\) be the current in the circuit. The relation between the two is \(Q' = I\text{.}\) By elementary principles we find \(L I' + RI + \nicefrac{Q}{C} = E\text{.}\) We differentiate to get
\begin{equation*}
L I''(t) + R I'(t) + \frac{1}{C} I(t) = E'(t) .
\end{equation*}
This is a nonhomogeneous second order constant coefficient linear equation. As \(L, R\text{,}\) and \(C\) are all positive, this system behaves just like the mass and spring system. Position of the mass is replaced by current. Mass is replaced by inductance, damping is replaced by resistance, and the spring constant is replaced by one over the capacitance. The change in voltage becomes the forcing function—for constant voltage this is an unforced motion.
Our next example behaves like a mass and spring system only approximately. Suppose a mass \(m\) hangs on a pendulum of length \(L\text{.}\) We seek an equation for the angle \(\theta(t)\) (in radians). Let \(g\) be the force of gravity. Elementary physics mandates that the equation is
Let us derive this equation using Newton’s second law: force equals mass times acceleration. The acceleration is \(L \theta''\) and mass is \(m\text{.}\) So \(mL\theta''\) has to be equal to the tangential component of the force given by the gravity, which is \(m g \sin \theta\) in the opposite direction. So \(mL\theta'' = -mg \sin \theta\text{.}\) The \(m\) curiously cancels from the equation.
Now we make our approximation. For small \(\theta\) we have that approximately \(\sin \theta \approx \theta\text{.}\) This can be seen by looking at the graph. In Figure 2.1 we can see that for approximately \(-0.5 < \theta < 0.5\) (in radians) the graphs of \(\sin \theta\) and \(\theta\) are almost the same.
Figure2.1.The graphs of \(\sin \theta\) and \(\theta\) (in radians).
Therefore, when the swings are small, \(\theta\) is small and we can model the behavior by the simpler linear equation
The errors from this approximation build up. So after a long time, the state of the real-world system might be substantially different from our solution. Also we will see that in a mass-spring system, the amplitude is independent of the period. This is not true for a pendulum. Nevertheless, for reasonably short periods of time and small swings (that is, only small angles \(\theta\)), the approximation is reasonably good.
In real-world problems it is often necessary to make these types of simplifications. We must understand both the mathematics and the physics of the situation to see if the simplification is valid in the context of the questions we are trying to answer.
Subsection2.4.2Free undamped motion
In this section we only consider free or unforced motion, as we do not know yet how to solve nonhomogeneous equations. Let us start with undamped motion where \(c=0\text{.}\) The equation is
\begin{equation*}
mx'' + kx = 0 .
\end{equation*}
We divide by \(m\) and let \(\omega_0 = \sqrt{\nicefrac{k}{m}}\) to rewrite the equation as
\begin{equation*}
x'' + \omega_0^2 x = 0 .
\end{equation*}
The general solution to this equation is
\begin{equation*}
x(t) = A \cos (\omega_0 t) + B \sin (\omega_0 t) .
\end{equation*}
By a trigonometric identity
\begin{equation*}
A \cos (\omega_0 t) + B \sin (\omega_0 t) =
C \cos ( \omega_0 t - \gamma ) ,
\end{equation*}
for two different constants \(C\) and \(\gamma\text{.}\) It is not hard to compute that \(C= \sqrt{A^2 + B^2}\) and \(\tan \gamma =
\nicefrac{B}{A}\text{.}\) Therefore, we let \(C\) and \(\gamma\) be our arbitrary constants and write \(x(t) = C \cos ( \omega_0 t - \gamma )\text{.}\)
Exercise2.4.1.
Justify the identity \(A \cos (\omega_0 t) + B \sin (\omega_0 t) =
C \cos ( \omega_0 t - \gamma )\) and verify the equations for \(C\) and \(\gamma\text{.}\) Hint: Start with \(\cos (\alpha-\beta) = \cos (\alpha) \cos
(\beta) + \sin (\alpha)\sin (\beta)\) and multiply by \(C\text{.}\) Then what should \(\alpha\) and \(\beta\) be?
While it is generally easier to use the first form with \(A\) and \(B\) to solve for the initial conditions, the second form is much more natural. The constants \(C\) and \(\gamma\) have nice physical interpretation. Write the solution as
\begin{equation*}
x(t) = C \cos ( \omega_0 t - \gamma ) .
\end{equation*}
This is a pure-frequency oscillation (a sine wave). The amplitude is \(C\text{,}\)\(\omega_0\) is the (angular) frequency, and \(\gamma\) is the so-called phase shift. The phase shift just shifts the graph left or right. We call \(\omega_0\) the natural (angular) frequency. This entire setup is called simple harmonic motion.
Let us pause to explain the word angular before the word frequency. The units of \(\omega_0\) are radians per unit time, not cycles per unit time as is the usual measure of frequency. Because one cycle is \(2
\pi\) radians, the usual frequency is given by \(\frac{\omega_0}{2\pi}\text{.}\) It is simply a matter of where we put the constant \(2\pi\text{,}\) and that is a matter of taste.
The period of the motion is one over the frequency (in cycles per unit time) and hence it is \(\frac{2\pi}{\omega_0}\text{.}\) That is the amount of time it takes to complete one full cycle.
Example2.4.1.
Suppose that \(m=\unit[2]{kg}\) and \(k=\unitfrac[8]{N}{m}\text{.}\) The whole mass and spring setup is sitting on a truck that was traveling at 1 \(\nicefrac{\text{m}}{\text{s}}\text{.}\) The truck crashes and hence stops. The mass was held in place 0.5 meters forward from the rest position. During the crash the mass gets loose. That is, the mass is now moving forward at 1 \(\nicefrac{\text{m}}{\text{s}}\text{,}\) while the other end of the spring is held in place. The mass therefore starts oscillating. What is the frequency of the resulting oscillation? What is the amplitude? The units are the mks units (meters-kilograms-seconds).
The setup means that the mass was at half a meter in the positive direction during the crash and relative to the wall the spring is mounted to, the mass was moving forward (in the positive direction) at 1 \(\nicefrac{\text{m}}{\text{s}}\text{.}\) This gives us the initial conditions.
We directly compute \(\omega_0 = \sqrt{\nicefrac{k}{m}} = \sqrt{4} = 2\text{.}\) Hence the angular frequency is 2. The usual frequency in Hertz (cycles per second) is \(\nicefrac{2}{2\pi} = \nicefrac{1}{\pi} \approx 0.318\text{.}\)
The general solution is
\begin{equation*}
x(t) = A \cos (2t) + B \sin (2t) .
\end{equation*}
Letting \(x(0) = 0.5\) means \(A = 0.5\text{.}\) Then \(x'(t) = - 2(0.5) \sin (2t)
+ 2B \cos (2t)\text{.}\) Letting \(x'(0) = 1\) we get \(B = 0.5\text{.}\) Therefore, the amplitude is \(C = \sqrt{A^2+B^2} = \sqrt{0.25+0.25} = \sqrt{0.5} \approx 0.707\text{.}\) The solution is
In general, for free undamped motion, a solution of the form
\begin{equation*}
x(t) = A \cos (\omega_0 t) + B \sin (\omega_0 t) ,
\end{equation*}
corresponds to the initial conditions \(x(0) = A\) and \(x'(0) = \omega_0 B\text{.}\) Therefore, it is easy to figure out \(A\) and \(B\) from the initial conditions. The amplitude and the phase shift can then be computed from \(A\) and \(B\text{.}\) In the example, we have already found the amplitude \(C\text{.}\) Let us compute the phase shift. We know that \(\tan \gamma = \nicefrac{B}{A} = 1\text{.}\) We take the arctangent of 1 and get \(\nicefrac{\pi}{4}\) or approximately 0.785. We still need to check if this \(\gamma\) is in the correct quadrant (and add \(\pi\) to \(\gamma\) if it is not). Since both \(A\) and \(B\) are positive, then \(\gamma\) should be in the first quadrant, \(\nicefrac{\pi}{4}\) radians is in the first quadrant, so \(\gamma =
\nicefrac{\pi}{4}\text{.}\)
Note: Many calculators and computer software have not only the atan function for arctangent, but also what is sometimes called atan2. This function takes two arguments, \(B\) and \(A\text{,}\) and returns a \(\gamma\) in the correct quadrant for you.
Subsection2.4.3Free damped motion
Let us now focus on damped motion. Let us rewrite the equation
\begin{equation*}
m x'' + c x' + kx = 0,
\end{equation*}
The sign of \(p^2-\omega_0^2\) is the same as the sign of \(c^2 - 4km\text{.}\) Thus we get real roots if and only if \(c^2-4km\) is nonnegative, or in other words if \(c^2 \geq 4km\text{.}\)
Subsubsection2.4.3.1Overdamping
When \(c^2 - 4km > 0\text{,}\) the system is overdamped. In this case, there are two distinct real roots \(r_1\) and \(r_2\text{.}\) Both roots are negative: As \(\sqrt{p^2 - \omega_0^2}\) is always less than \(p\text{,}\) then \(-p \pm \sqrt{p^2 - \omega_0^2}\) is negative in either case.
Since \(r_1, r_2\) are negative, \(x(t) \to 0\) as \(t \to \infty\text{.}\) Thus the mass will tend towards the rest position as time goes to infinity. For a few sample plots for different initial conditions, see Figure 2.3.
Figure2.3.Overdamped motion for several different initial conditions.
No oscillation happens. In fact, the graph crosses the \(x\)-axis at most once. To see why, we try to solve \(0 = C_1 e^{r_1 t} + C_2 e^{r_2 t}\text{.}\) Therefore, \(C_1 e^{r_1 t} = - C_2 e^{r_2 t}\) and using laws of exponents we obtain
This equation has at most one solution \(t \geq 0\text{.}\) For some initial conditions the graph never crosses the \(x\)-axis, as is evident from the sample graphs.
Example2.4.2.
Suppose the mass is released from rest. That is, \(x(0) = x_0\) and \(x'(0) = 0\text{.}\) Then
It is not hard to see that this satisfies the initial conditions.
Subsubsection2.4.3.2Critical damping
When \(c^2 - 4km = 0\text{,}\) the system is critically damped. In this case, there is one root of multiplicity 2 and this root is \(-p\text{.}\) Our solution is
The behavior of a critically damped system is very similar to an overdamped system. After all a critically damped system is in some sense a limit of overdamped systems. Since these equations are really only an approximation to the real world, in reality we are never critically damped, it is a place we can only reach in theory. We are always a little bit underdamped or a little bit overdamped. It is better not to dwell on critical damping.
Subsubsection2.4.3.3Underdamping
When \(c^2 - 4km < 0\text{,}\) the system is underdamped. In this case, the roots are complex.
where \(\omega_1 =\sqrt{\omega_0^2 - p^2}\text{.}\) Our solution is
\begin{equation*}
x(t) = e^{-pt} \bigl( A \cos (\omega_1 t) + B \sin (\omega_1 t) \bigr) ,
\end{equation*}
or
\begin{equation*}
x(t) = C e^{-pt} \cos ( \omega_1 t - \gamma ) .
\end{equation*}
An example plot is given in Figure 2.4. Note that we still have that \(x(t) \to 0\) as \(t \to \infty\text{.}\)
Figure2.4.Underdamped motion with the envelope curves shown.
The figure also shows the envelope curves\(C e^{-pt}\) and \(-C e^{-pt}\text{.}\) The solution is the oscillating line between the two envelope curves. The envelope curves give the maximum amplitude of the oscillation at any given point in time. For example, if you are bungee jumping, you are really interested in computing the envelope curve as not to hit the concrete with your head.
The phase shift \(\gamma\) shifts the oscillation left or right, but within the envelope curves (the envelope curves do not change if \(\gamma\) changes).
Notice that the angular pseudo-frequency 1 becomes smaller when the damping \(c\) (and hence \(p\)) becomes larger. This makes sense. When we change the damping just a little bit, we do not expect the behavior of the solution to change dramatically. If we keep making \(c\) larger, then at some point the solution should start looking like the solution for critical damping or overdamping, where no oscillation happens. So if \(c^2\) approaches \(4km\text{,}\) we want \(\omega_1\) to approach 0.
On the other hand, when \(c\) gets smaller, \(\omega_1\) approaches \(\omega_0\) (\(\omega_1\) is always smaller than \(\omega_0\)), and the solution looks more and more like the steady periodic motion of the undamped case. The envelope curves become flatter and flatter as \(c\) (and hence \(p\)) goes to 0.
Subsection2.4.4Exercises
Exercise2.4.2.
Consider a mass and spring system with a mass \(m=2\text{,}\) spring constant \(k=3\text{,}\) and damping constant \(c=1\text{.}\)
Set up and find the general solution of the system.
Is the system underdamped, overdamped or critically damped?
If the system is not critically damped, find a \(c\) that makes the system critically damped.
Exercise2.4.3.
Do Exercise 2.4.2 for \(m=3\text{,}\)\(k=12\text{,}\) and \(c=12\text{.}\)
Exercise2.4.4.
Using the mks units (meters-kilograms-seconds), suppose you have a spring with spring constant 4 \(\nicefrac{\text{N}}{\text{m}}\text{.}\) You want to use it to weigh items. Assume no friction. You place the mass on the spring and put it in motion.
You count and find that the frequency is 0.8 Hz (cycles per second). What is the mass?
Find a formula for the mass \(m\) given the frequency \(\omega\) in Hz.
Exercise2.4.5.
Suppose we add possible friction to Exercise 2.4.4. Further, suppose you do not know the spring constant, but you have two reference weights 1 kg and 2 kg to calibrate your setup. You put each in motion on your spring and measure the frequency. For the 1 kg weight you measured 1.1 Hz, for the 2 kg weight you measured 0.8 Hz.
Find \(k\) (spring constant) and \(c\) (damping constant).
Find a formula for the mass in terms of the frequency in Hz. Note that there may be more than one possible mass for a given frequency.
For an unknown object you measured 0.2 Hz, what is the mass of the object? Suppose that you know that the mass of the unknown object is more than a kilogram.
Exercise2.4.6.
Suppose you wish to measure the friction a mass of 0.1 kg experiences as it slides along a floor (you wish to find \(c\)). You have a spring with spring constant \(k=\unitfrac[5]{N}{m}\text{.}\) You take the spring, you attach it to the mass and fix it to a wall. Then you pull on the spring and let the mass go. You find that the mass oscillates with frequency 1 Hz. What is the friction?
Exercise2.4.101.
A mass of \(2\) kilograms is on a spring with spring constant \(k\) newtons per meter with no damping. Suppose the system is at rest and at time \(t=0\) the mass is kicked and starts traveling at 2 meters per second. How large does \(k\) have to be to so that the mass does not go further than 3 meters from the rest position?
Suppose we have an RLC circuit with a resistor of 100 milliohms (0.1 ohms), inductor of inductance of 50 millihenries (0.05 henries), and a capacitor of 5 farads, with constant voltage.
Set up the ODE equation for the current \(I\text{.}\)
a) \(0.05 I'' + 0.1 I' + (\nicefrac{1}{5}) I = 0\) b) \(I = C e^{-t} \cos(\sqrt{3} \, t - \gamma)\) c) \(I = 10 e^{-t} \cos(\sqrt{3} \, t) + \frac{10}{\sqrt{3}} e^{-t}
\sin(\sqrt{3} \, t)\)
Exercise2.4.103.
A 5000 kg railcar hits a bumper (a spring) at 1 \(\nicefrac{\text{m}}{\text{s}}\text{,}\) and the spring compresses by 0.1 m. Assume no damping.
Find \(k\text{.}\)
How far does the spring compress when a 10000 kg railcar hits the spring at the same speed?
If the spring would break if it compresses further than 0.3 m, what is the maximum mass of a railcar that can hit it at 1 \(\nicefrac{\text{m}}{\text{s}}\text{?}\)
What is the maximum mass of a railcar that can hit the spring without breaking at 2 \(\nicefrac{\text{m}}{\text{s}}\text{?}\)
a) \(k=500000\) b) \(\frac{1}{5\sqrt{2}} \approx 0.141\) c) 45000 kg d) 11250 kg
Exercise2.4.104.
A mass of \(m\) kg is on a spring with \(k=\unitfrac[3]{N}{m}\) and \(c=\unitfrac[2]{Ns}{m}\text{.}\) Find the mass \(m_0\) for which there is critical damping. If \(m < m_0\text{,}\) does the system oscillate or not, that is, is it underdamped or overdamped?
