Question

Problems 17 through 22 deal with the effect of a sequence of impulses on an undamped oscillator. Suppose that $$ y^{\prime \prime}+y=f(t), \quad y(0)=0, \quad y^{\prime}(0)=0 $$ For each of the following choices for \(f(t)\) : (a) Try to predict the nature of the solution without solving the problem. (b) Test your prediction by finding the solution and drawing its graph. (c) Determine what happens after the sequence of impulses ends. $$ f(t)=\sum_{k=1}^{40}(-1)^{k+1} \delta(t-11 k / 4) $$

Short answer

Based on the step by step solution provided above, create a short answer. The given second-order homogeneous linear differential equation has an oscillatory solution due to the sine and cosine terms. The forcing function of impulses introduces different frequency components in the solution, which can result in resonance effects. After calculating the general solution and applying the initial conditions, we find the particular solution as: $$ y_p(t) = \sum_{k=1}^{40}(-1)^{k+1} \sin\left(t-11 \frac{k}{4}\right) $$ After plotting this solution, we observe oscillatory behavior with varying frequencies and resonance effects. After the impulses end, the oscillator will keep oscillating with the frequencies and amplitudes acquired from the last impulse, given that the system is undamped.

Step-by-step solution

Analyze the equation and given values

We're given a homogeneous second-order linear differential equation: $$ y^{\prime \prime} + y = f(t) $$ with the initial conditions $$y(0) = 0$$ and $$y^{\prime}(0) = 0$$. The forcing function is given by: $$ f(t) = \sum_{k=1}^{40}(-1)^{k+1} \delta(t-11k/4) $$ where $$\delta(t)$$ is the Dirac delta function, representing impulse forces at different time points.

Predict the nature of the solution

Given the equation is a undamped (without any damping coefficient), the solution can be expected to oscillate. The impulses will introduce different frequency components to the solution. Since the impulse sequence is periodical with period $$11/4$$, it is possible that resonance might occur.

Find the general solution of the equation

The general solution of the homogeneous equation $$y^{\prime \prime} + y = 0$$ can be written in the form: $$ y_h(t) = A \cos(t) + B \sin(t) $$ To find a particular solution, we need to use the convolution integral: $$ y_p(t) = \int_{0}^{t} y_h(t-\tau) f(\tau) d\tau $$ Insert the given forcing function: $$ y_p(t) = \int_{0}^{t} (A \cos(t - \tau) + B \sin(t - \tau)) \sum_{k=1}^{40}(-1)^{k+1} \delta(\tau - 11k/4) d\tau $$

Apply the initial conditions to find the particular solution

Since $$y(0) = 0$$ and $$y^{\prime}(0) = 0$$, we have: $$ 0 = A \cos(0) + B \sin(0) \Rightarrow A=0 $$ $$ 0 = -A \sin(0) + B\cos(0) \Rightarrow B=0 $$ Thus, the homogeneous solution is the zero function, and the particular solution is the only relevant part of the solution. By evaluating the convolution integral, considering the Dirac delta function properties, we get: $$ y_p(t) = \sum_{k=1}^{40}(-1)^{k+1} \sin\left(t-11 \frac{k}{4}\right) $$

Plot and analyze the solution

Plotting the particular solution $$y_p(t)$$ shows oscillatory behavior with varying frequencies due to the impulses. Resonance effects are visible as increases in oscillation amplitude at multiples of the impulse period.

Determine the behavior after impulses end

When the sequence of impulses ends, there will be no more external forces acting on the oscillator. As the equation is undamped, the oscillator will keep oscillating freely with the frequencies and amplitudes acquired from the last impulse.

Key concepts

Differential Equations

Differential equations are mathematical equations that involve functions and their derivatives. They are a powerful tool to model physical phenomena, describing how a certain quantity changes over time. In the context of oscillators, differential equations help us understand how oscillations evolve under different forces.

For example, in the given problem, the differential equation is \(y'' + y = f(t)\), where \(y(t)\) represents the position or displacement of the oscillator. The term \(f(t)\) is a forcing function, influencing the system's dynamics. The double prime \(y''\) denotes the second derivative, implying acceleration. Solving such equations often involves finding both the homogeneous and particular solutions, which together provide a complete picture of the system's motion.

Differential equations can vary in complexity, but they often hold the key to understanding the behavior of systems across physics, engineering, and other scientific fields. Their solutions unveil much about the response, stability, and long-lasting behavior of oscillatory systems.

Dirac Delta Function

The Dirac delta function, denoted as \(\delta(t)\), is a mathematical function that plays a crucial role when modeling instant impacts or impulses in systems. It's not a conventional function but more of a symbolic representation used in theoretical physics and engineering.

Its defining characteristic is that it is zero everywhere except at one point, often at \(t = 0\), where it is infinitely tall but has an integral of one. This makes it ideal for representing a sudden, concentrated impulse. In our problem, the function \(f(t) = \sum_{k=1}^{40}(-1)^{k+1} \delta(t-11k/4)\) involves multiple Dirac delta functions, simulating a sequence of impulses acting at various time intervals.

• They appear at times \(t = \frac{11k}{4}\) for \(k = 1, 2, \ldots, 40\).
• The alternating sign in front of \(\delta(t-11k/4)\) indicates a pattern of push and pull effects on the oscillator.

The use of the Dirac delta function in differential equations like this enables precise modeling of systems subject to sudden changes.

Oscillatory Behavior

Oscillatory behavior refers to the repetitive back and forth motion often seen in physical systems. In the context of our problem, the oscillator moves about equilibrium in response to an initial disturbance. This type of motion is prominently characterized by sine and cosine functions resulting from differential equation solutions.

For undamped oscillators, like the one we are dealing with, the energy does not dissipate, meaning the oscillation continues indefinitely. The solution to our differential equation contains a sum of sine terms due to the impulses, leading to complex oscillatory patterns. The impulses cause the system to oscillate at various frequencies, which can be observed in the solution \(y_p(t) = \sum_{k=1}^{40}(-1)^{k+1} \sin(t-11\frac{k}{4})\).

Complex oscillatory behavior often results in the phenomenon known as beats, where waves of different frequencies interact. In applications, understanding these oscillations helps in designing systems like suspension bridges or tuning musical instruments to avoid undesirable oscillation frequencies or modes.

Resonance

Resonance is a powerful phenomenon occurring when a system is subjected to oscillations at its natural frequency. When this happens, even small periodic forces can result in large amplitude oscillations due to energy transfer efficiency. In our exercise, resonance can occur because the impulses are periodic and synchronized in such a way that they could potentially match the oscillator's natural frequency.

For undamped oscillators, the risk of resonance is higher because there is no damping mechanism to dissipate energy, and the response can grow indefinitely. In practice, resonance phenomena must be carefully managed to avoid catastrophic failures in structures like bridges or buildings.

• The impulses in this exercise have a period of \(11/4\), and resonance would amplify oscillations if the natural frequency aligns with this periodicity.
• Visualizing the solution \(y_p(t)\) through plotting helps in identifying resonance as seen by peaks in amplitude.

Understanding resonance helps engineers and scientists design safer and more efficient systems, utilizing or avoiding resonance where appropriate.