11 Ordinary Differential Equations III

In this class,

• Questions must be answered in order asked.
• Solutions must be neat.
• Solutions must be exact and fully simplified.
• You must use the given symbols.
• You must show all reasoning.
• Copying is never allowed, even when working together.

1. Resonant forcing of an undamped spring-mass system over some time period $T$ that spans a large number of periods can introduce large-amplitude vibrations. To study the problem, consider the example
   
   $$m \ddot x + k x = F(t) \qquad x(0) = \dot x(0) = 0$$
   
   
   where the mass, spring constant, and applied force are given by
   
   $$m=1 \qquad k=4 \qquad F(t)=\cos(2t)\mbox{ if } t<T \qquad F(t)= 0 \mbox{ if } t>T$$
   
   
   Solve using the Laplace transform method. (Note: from S8 and S11 you can see that
   
   $$\sin(\omega t) - \omega t \cos(\omega t) \Longleftrightarrow \frac{2\omega^3}{(s^2+\omega^2)^2}$$
   
   
   Call it result S15.) Clean up your answer. I find that beyond time $t=T$, the amplitude stays constant at
   
   $${\textstyle\frac{1}{4}} T \sqrt{1 + 2\cos(2T)\frac{\sin(2T)}{2T} + \left(\frac{\sin(2T)}{2T}\right)^2}$$
   
   
   which is approximately proportional to $T$ for large $T$. Do your results agree?

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 a previous question, the natural frequency of the free undamped system is $\omega=\sqrt{k/m}$. 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 $\zeta=4/5$, find $x$ using Laplace transformation. (You must complete the square as explained, for example, in the revised notes on ordinary differential equations. Complex roots in partial fractions are not allowed.) Note that after the kick, there is no further force and the system vibrates as a free system. Plot your solution accurately versus time and so show graphically that the mass keeps vibrating between negative and positive values although 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.

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 in an earlier question, if $\widetilde\omega$ is close to the natural frequency of the system and damping is small, mass $m_1$ may experience severe vibration. If mass $m_1$ is, say, really a building and the force is really an earthquake, that may be very bad news. 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
   
   $$m_2 \ddot x_2 = - k_2 (x_2-x_1)$$
   
   
   Find the Laplace transforms $\widehat x_1$ and $\widehat x_2$. To keep it simple, assume that before the quake,
   
   $$x_1(0) = 0 \quad \dot x_1(0) = v_{10} \quad x_2(0) = 0 \quad \dot x_2(0) = v_{20} \quad$$
   
   
   You do not have to find $x_1$ and $x_2$; you can answer the next questions from what you know about partial fractions. To do so, show first that
   
   $$\widehat x_1 = F_1 \frac{s (m_2s^2+k_2)}{Q (s^2+\widetilde\omega^2)} + \frac{m_1v_{10}(m_2 s^2+k_2)+m_2v_{20}k_2}{Q}$$
   
   
   where $Q$ is the quadratic
   
   $$Q = (m_1s^2+c_1s+k_1+k_2)(m_2s^2+k_2)-k_2^2$$
   
   
   Cramers rule works nicely in this case.

   Now you need to find the qualitative form of the partial fraction expansion of $\widehat x_1$. Now the 4 roots of the quadratic $Q$ in the bottom 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. Knowing about the nature of these four roots, you can now ignore the second term in $\widehat x_1$ and look at the first term with $F_1$ nonzero. The terms will correspond to two decaying modes of vibration and one term where $m_1$ vibrates with frequency $\widetilde\omega$. This term will have a large amplitude for small damping. 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. So the building will still experience large transient vibration.)

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 $\vec{x}(0)=(1,7,3)^{\rm T}$ and write $\vec{x}(t)$ for that case.