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. Solve
   
   $$y'' + 9 y = t^2 \qquad y(0) = y'(0)= 0$$
   
   
   That would of course be quick using undetermined coefficients. Unfortunately, you must use Laplace transforms. You may only use the brief Laplace transform table handed out in class. Everything else must be derived. Do not use convolution. In solving the system of 5 equations in 5 unknowns of the partial fraction expansion, you may mess around; this is no longer linear algebra. However, you must substitute your solution into the original ODE and ICs and go back to fix any problem there may be.

2. 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?

3. 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. However, no actual expression for $F(t)$ is given at this time. 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.

4. 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.