Question

Consider the forced but undamped system described by the initial value problem $$ u^{\prime \prime}+u=3 \cos \omega t, \quad u(0)=0, \quad u^{\prime}(0)=0 $$ (a) Find the solution \(u(t)\) for \(\omega \neq 1\). (b) Plot the solution \(u(t)\) versus \(t\) for \(\omega=0.7, \omega=0.8,\) and \(\omega=0.9\). Describe how the response \(u(t)\) changes as \(\omega\) varies in this interval. What happens as \(\omega\) takes on values closer and closer to \(1 ?\) Note that the natural frequency of the unforced system is \(\omega_{0}=1\)

Short answer

In conclusion, the general solution for the given forced but undamped system is: $$ u(t) = \frac{3 \cos(\omega t)}{1 - \omega^2} - \frac{3 \cos(t)}{1 - \omega^2} $$ By plotting the solutions for different values of \(\omega\), we observe that as the forcing frequency \(\omega\) approaches the natural frequency of the unforced system (\(\omega_0 = 1\)), the response becomes increasingly amplified, leading to resonance.

Step-by-step solution

Find the particular solution

To find the particular solution, we will try a solution of the form \(u_p = Acos(\omega t) + Bsin(\omega t)\). Taking the second derivative, we get: $$ u_p^{\prime \prime} = -A\omega^2\cos(\omega t) - B\omega^2\sin(\omega t) $$ Substituting the trial solution and its second derivative into the given equation: $$ -(A\omega^2 \cos(\omega t) + B\omega^2 \sin(\omega t)) + (A \cos(\omega t) + B \sin(\omega t)) = 3 \cos(\omega t) $$ For the left side to equal the right side of the equation, the coefficients of the cosine and sine terms must match: $$ A(1 - \omega^2) = 3 \\ B(1 - \omega^2) = 0 $$ Since \(\omega \neq 1\), we can divide by \((1-\omega^2)\): $$ A = \frac{3}{1 - \omega^2} \\ B = 0 $$ Thus, the particular solution is: $$ u_p(t) = \frac{3 \cos(\omega t)}{1 - \omega^2} $$

Find the complementary solution

The complementary equation is the homogeneous equation without the forcing term: $$ u_c^{\prime \prime} + u_c = 0 $$ The general solution for this linear homogeneous equation is in the form of \(u_c = Acos(t) + Bsin(t)\). We will find the values of \(A\) and \(B\) using the initial conditions.

Apply the initial conditions

Applying the initial condition \(u(0) = 0\), we have: $$ u(0) = u_p(0) + u_c(0) = 0 \\ \frac{3}{1 - \omega^2} + A = 0 \\ A = -\frac{3}{1 - \omega^2} $$ Now, apply the initial condition \(u^{\prime}(0) = 0\), and first, find the derivative of \(u_p\) and \(u_c\): $$ u_p^{\prime}(t) = \frac{3\omega \sin(\omega t)}{1 - \omega^2} \\ u_c^{\prime}(t) = -A \sin(t) + B \cos(t) $$ Applying the initial condition: $$ u^{\prime}(0) = u_p^{\prime}(0) + u_c^{\prime}(0) = 0 \\ 0 + B = 0 \\ B = 0 $$ Therefore, the complementary solution is: $$ u_c(t) = -\frac{3 \cos(t)}{1 - \omega^2} $$

Find the general solution

To find the general solution, combine the particular and complementary solutions: $$ u(t) = u_p(t) + u_c(t) = \frac{3 \cos(\omega t)}{1 - \omega^2} - \frac{3 \cos(t)}{1 - \omega^2} $$

Plot the solutions and analyze the changes in the response

You can use a graphing tool like Desmos, GeoGebra, or a programming language like Python with the Matplotlib package. Plot the solution function \(u(t)\) for \(\omega = 0.7, 0.8\), and \(0.9\). Observe the change in the response as \(\omega\) varies in this interval. As the forcing frequency \(\omega\) approaches the natural frequency of the unforced system (\(\omega_0 = 1\)), the response will become increasingly amplified, leading to resonance.

Key concepts

Initial Value Problem

In the realm of differential equations, an initial value problem (IVP) consists of a differential equation coupled with a specified value, called the initial condition, at a given point in the domain of the solution. The dilemma presented here involves a forced but undamped system, where the goal is to determine the function \(u(t)\) that satisfies not only the equation \(u''+u=3\cos\omega t\) but also the initial conditions \(u(0)=0\) and \(u'(0)=0\).

Addressing an IVP begins with understanding the behavior of the solution at the outset and subsequently calculating how it evolves over time. This approach is particularly crucial when dealing with real-world scenarios such as physical systems where initial states often determine subsequent dynamics.

Particular Solution

The particular solution to a differential equation is a specific solution that satisfies not only the differential equation itself but also fits given external or nonhomogeneous conditions, such as a forcing function. In this scenario, we're searching for a particular solution to the equation with the forcing term \(3\cos\omega t\), where \(\omega\) represents the frequency of the external force.

To uncover this particular solution, we assume a form that is similar to the forcing term, leading us to consider \(u_p(t) = \frac{3\cos(\omega t)}{1 - \omega^2}\). This solution mirrors the oscillatory nature of the forcing term and holds true for any value of \(\omega\) that is not equal to 1, the system's natural frequency. This ensures that the solution is well-defined and avoids the singularities that occur at resonance.

Complementary Solution

The complementary solution, also known as the homogeneous solution, is determined by the associated homogeneous differential equation, which lacks the forcing term. It encapsulates the system's natural response without external influences.

In our example, the complementary equation is \(u_c'' + u_c = 0\), which, solved properly, gives \(u_c(t) = -\frac{3\cos(t)}{1 - \omega^2}\). While the particular solution accounts for the external force's effect, the complementary solution conveys the innate motion of the system, given its initial state and free of forced disturbances. The complete dynamics of the system are then described by combining both the particular and the complementary solutions.

Natural Frequency

The natural frequency of a system is a fundamental concept in physics and engineering, denoting the frequency at which a system oscillates when not subjected to continuous or external forces. It's the frequency intrinsic to the system's structure and mass distribution.

For the given undamped system, the natural frequency \(\omega_0\) is the value at which the homogeneous equation \(u_c'' + u_c = 0\) has non-trivial solutions. In this case, \(\omega_0 = 1\) signifies the natural frequency of the unforced system. This frequency is crucial for predicting the behavior of a system, as it is the frequency at which resonance occurs if the system is driven by an external force matching this natural frequency. Recognizing the natural frequency helps prevent destructive resonance in engineering designs and can be exploited in applications like musical instruments to enhance desired tones.

Resonance

Resonance is a phenomenon where a system experiences a dramatic increase in amplitude when the frequency of a periodic external force aligns with the system's natural frequency. This concurrence can lead to large oscillations and potentially catastrophic failures in physical structures, like bridges or buildings, or to desired effects in musical instruments.

Looking at the solved differential equation, as the forcing frequency \(\omega\) approaches the system's natural frequency of \(\omega_0 = 1\), we anticipate a resonance behavior. This case demonstrates a classic example of resonance, where the forced system response amplifies, a scenario well-understood by engineers to avoid structural fatigue and failure. In the plotted response for different values of \(\omega\), we would see the amplitude of the oscillations increase as \(\omega\) comes closer to \(\omega_0\), illustrating this critical concept in practice.