11 Ordinary Differential Equations III

1. The generic undamped spring-mass system with external forcing is
   
   $$m \ddot x + k x = F(t) \qquad x(0)=x_0 \quad \dot x(0)=v_0$$
   
   
   where the mass m and spring constant $k$ are given positive constants, $F(t)$ is the given external force, and the initial displacement $x_0$ and velocity $v_0$ are given constants. Give the solution using Laplace transformation, as always restricting use of convolution to the bare minimum. Write the solution in the form
   
   $$A(t) \sin(\omega t) + B(t) \cos(\omega t)$$
   
   
   Identify the natural frequency $\omega$.

   Next check that for a resonant force $F=F_0\cos(\widetilde\omega t)$ with $F_0$ a constant and $\widetilde\omega=\omega$, the integrand in the $A$ integral is always positive, so $A$ will grow without bound in time. However, the integrand in the $B$ integral will periodically change sign, so that $B$ stays finite. Address the question why if I apply a cosine forcing, it is the magnitude of the sine that keeps increasing instead of the cosine. Does that make physical sense? For nonresonant forcing, $\widetilde\omega\ne\omega$, both integrands will periodically change sign, so that both $A$ and $B$ stay finite.

   To see these things more precisely, find $\widehat x$ for the nonresonant case $\widetilde\omega\ne\omega$, assuming $x_0=v_0=0$. Its partial fraction expansion will be
   

   $$\widehat x = \frac{Cs+D}{s^2+\widetilde\omega^2} + \frac{Es+G}{s^2+\omega^2}$$
   
   
   To quickly find $C$ and $D$, you can use a trick. Multiple $\widehat x$ as found (not the partial fraction expansion) by $s^2+\widetilde\omega^2$, simplify, and evaluate at $s=i\widetilde\omega$ with $i=\sqrt{-1}$. That will give you the value of $i\tilde\omega C+D$. Also evaluate at $s=-i\widetilde\omega$ to get $-i\tilde\omega C+D$. The sum and difference of the equations directly give $D$ and $C$. Show that you get a $\ldots\cos(\widetilde\omega t)$ term (why now a cosine?) with an amplitude that will be large if $\widetilde\omega\approx\omega$. Also find the solution for $\widetilde\omega=\omega$ exactly, and show that it is indeed a growing sine.

2. The generic linearly damped spring-mass system that is initially at rest but receives a kick with momentum $I_0$ at a time $T$ is described by
   
   $$m \ddot x + c\dot x + k x = I_0 \delta(t-T) \qquad x(0) = \dot x(0) = 0$$
   
   
   Here the mass m, damping constant $c$, and spring constant $k$ are given positive constants. As seen in the previous question, the natural frequency of the free undamped system $\omega=\sqrt{k/m}$. It is also useful to define a nondimensional damping constant $\zeta\equiv c/2m\omega$ which is called the damping ratio. Check that in those terms, the ODE can be written as
   
   $$\ddot x + 2\zeta \omega \dot x + \omega^2 x = \frac{I_0}{m} \delta(t-T)$$
   
   
   Assuming that $\omega=5$ rad/s and the damping ratio $4/5$, find $x$ using Laplace transformation. (You will need to complete the square as explained on the handout.)

   Note that after the kick, there is no further force and the system vibrates as a free system. Show graphically from your result that the mass keeps vibrating between negative and positive values though the amplitude of vibration after the kick decreases with time. More generally, it can be seen that if the damping ratio is less than one, the mass keeps vibrating. For damping ratio greater than 1, the amplitude changes sign at most once.

   To understand what is going on in vibrations in more general terms, note that the solution of any homogeneous second order constant coefficient equation is always of the form
   

   $$\frac{A s + B}{(s-s_1)(s-s_2)}$$
   
   
   Ignoring the degenerate cases like $s_1=s_2$, there are two possibilities. The first is that $s_1$ and $s_2$ are distinct real numbers. Show from partial fractions that such a solution will give rise to two exponentials in time. The second main possibility is that $s_1=s_r+is_i$ and $s_2=s_r-is_i$ are a complex conjugate pair. In that case, check that the above expression multiplies out as
   
   $$\frac{A s + B}{(s-s_r)^2+s_i^2}$$
   
   
   and that that gives rise to an exponential multiplied by a sine or cosine. Those things are true for any homogeneous 2nd order CC ODE. However, give a clear and solid physical reason that for a freely vibrating damped spring-mass system, you can ignore the possibility of growing exponentials. Also give a solid physical reason that for the undamped spring-mass system, the solution must be pure sines and cosines. In other words, show that simple exponentials, decaying, constant, or growing are not an option if undamped. And that sines and cosines times a growing or decaying exponential are not an option either. Then for an slightly damped system, you should get an oscillating solution times a slowly decaying exponential. (I do not see a physical reason why you could not have an oscillating solution for very high damping constant. But the mathematical reason is clear: for high enough damping constant, the discriminant of the quadratic in $s$ cannot be negative. So oscillation is then not possible. There can be only one sign change, and only if the exponentials have coefficients of opposite sign.) Note also that it is the real part of the roots ($s_1$ and $s_2$ if real, otherwise $s_r$) that determines whether there is exponential growth. That is the basis for the root-locus method, where you look where the real part of roots are to determine the stability of your system.

3. The generic linearly damped spring-mass system experiencing an external force with frequency $\omega$ can be written as
   
   $$m_1 \ddot x_1 + c_1\dot x_1 + k_1 x_1 = F_1 \cos(\widetilde\omega t)$$
   
   
   Here $F_1$ is a constant. As seen two questions back, if $\widetilde\omega$ is close to the natural frequency of the system and damping is small, mass $m_1$ may experience severe vibration. But suppose you hang a second mass $m_2$ from the first using a spring with constant $k_2$. Then the equation above becomes
   
   $$m_1 \ddot x_1 + c_1\dot x_1 + k_1 x_1 = F_1 \cos(\widetilde\omega t) + k_2 (x_2-x_1)$$
   
   
   while the second mass satisfies the equation
   
   $$m_2 \ddot x_2 = - k_2 (x_2-x_1)$$
   
   
   To keep it simple, assume initial conditions
   
   $$x_1(0) = 0 \quad \dot x_1(0) = {v_1}_0 \quad x_2(0) = 0 \quad \dot x_2(0) = {v_2}_0 \quad$$
   
   
   Find the Laplace transforms $\widehat x_1$ and $\widehat x_2$. You do not have to find $x_1$ and $x_2$; you can answer the next questions from what you know about partial fractions.

   First, your solution should have a quartic in the bottom for which the roots would be difficult to find. But look for a second at the free solution (i.e. with $F_1=0$). Based on your physical arguments in the previous question, you should be able to describe the qualitative nature of the four roots if the damping is low. Then consider the partial fraction nature of the solution for $F_1$ nonzero (but you can further ignore ${v_1}_0$ and ${v_2}_0$). The terms will correspond to two decaying modes of vibration and one term where $m_1$ vibrates with frequency $\widetilde\omega$. However, if you look a bit closer, you see that if you choose the ratio $k_2/m_2$ to be $\widetilde\omega^2$, the third term disappears. Then the building returns to rest after a transition period, despite the ongoing vibrating force on it! The effect of the force has been eliminated!

   You may be astonished by that, since only the ratio of $k_2$ to $m_2$ is specified. So you could eliminate the vibration in your building $m_1$ by suspending a single grain of sand $m_2$ from it using a very weak spring! (Actually, if you do this, and the natural frequency of the building is close to $\widetilde\omega$, and damping is small, then the coefficients of the decaying modes will be very large. That can easily be seen using the same trick as used two questions back.)

4. Solve the system
   $$\begin{eqnarray*} x_1' & = & 2 x_1 + x_2 - 2 x_3 \\ x_2' & = & 3 x_1 -2 x_2 \\ x_3' & = & 3 x_1 + x_2 - 3 x_3 \end{eqnarray*}$$
   
   
   Find the general solution to this system in vector form and in terms of a fundamental matrix. Then find the vector of integration constants assuming that $x(0)=(1,7,3)^{\rm T}$ and write $\vec x$ for that case.

5. Given the system
   
   $$\dot {\vec x} = A \vec x \qquad A = \left(\begin{array}{cc} 0 & 5 -1 & -2 \end{array} \right)$$
   
   
   Find the general solution to this system in vector form and in terms of a fundamental matrix.