11 Ordinary Differential Equations III

In this class,

• Questions must be answered in order asked.
• Solutions must be neat.
• 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 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? Also verify that for nonresonant forcing, $\widetilde\omega\ne\omega$, both integrands will periodically change sign, so that both $A$ and $B$ stay finite.

3. 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 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.

   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
   
   $$\frac{A s + B}{(s-s_r)^2+s_i^2} = \frac{A' (s-s_r) + B'}{(s-s_r)^2+s_i^2}$$
   
   
   and that that must give rise to an exponential multiplied by a sine and/or cosine. Those things are true for any homogeneous 2nd order constant coefficient ODE, not just spring-mass systems. However, give a clear and solid physical reason why 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 nonoscillatory exponentials, whether 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 a 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 in controls, where you look where the real part of roots are to determine the stability of your system.

4. 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. 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.)

5. 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.