Question

Consider a vibrating system described by the initial value problem $$ u^{\prime \prime}+\frac{1}{4} u^{\prime}+2 u=2 \cos \omega t, \quad u(0)=0, \quad u^{\prime}(0)=2 $$ (a) Determine the steady-state part of the solution of this problem. (b) Find the amplitude \(A\) of the steady-state solution in terms of \(\omega\). (c) Plot \(A\) versus \(\omega\). (d) Find the maximum value of \(A\) and the frequency \(\omega\) for which it occurs.

Short answer

#Answer# The solution to the given problem involves solving the system of equations, determining the amplitude of the steady-state solution, plotting the amplitude as a function of the frequency, and finding the maximum value of the amplitude and the corresponding frequency. The amplitude function is given by: $$ A_s(\omega) = \sqrt{A(\omega)^2 + B(\omega)^2} $$ The maximum amplitude and corresponding frequency can be found by analyzing the plot of this function.

Step-by-step solution

Write down the particular solution

We want to find the particular solution of: $$ u^{\prime \prime}+\frac{1}{4} u^{\prime}+2 u=2 \cos \omega t $$ Assume the particular solution has the form: $$ u_p(t)=A \cos(\omega t) + B \sin(\omega t) $$ To find A and B we can differentiate \(u_p\) twice and substitute it back into the given equation.

Compute the steady-state part of the solution

Differentiating \(u_p\) twice: $$ u_p^\prime(t)=-A\omega \sin(\omega t) + B\omega \cos(\omega t) $$ $$ u_p^{\prime\prime}(t)=-A\omega^2 \cos(\omega t) - B\omega^2 \sin(\omega t) $$ Substituting these expressions into the given equation, we have: $$ -A\omega^2 \cos(\omega t) - B\omega^2 \sin(\omega t) +\frac{1}{4}(-A\omega \sin(\omega t) + B\omega \cos(\omega t)) + 2(A \cos(\omega t) + B \sin(\omega t)) = 2 \cos(\omega t) $$ Comparing coefficients, we get two equations: $$ -\omega^2A + \frac{1}{4}\omega B + 2A = 2 \\ -\omega^2B -\frac{1}{4}\omega A + 2B = 0 $$ Solve this system of equations for A and B.

Determine the amplitude of the steady-state solution

After solving the system of equations, we can rewrite the particular solution as: $$ u_p(t)=A(\omega) \cos(\omega t) + B(\omega) \sin(\omega t) $$ The amplitude \(A_s\) of the steady-state solution can be determined as: $$ A_s(\omega) = \sqrt{A(\omega)^2 + B(\omega)^2} $$

Plot the amplitude as a function of the frequency

Using the expression of \(A_s\) derived in the previous step, plot the amplitude \(A_s\) as a function of the frequency \(\omega\).

Find the maximum value of the amplitude and the frequency at which this maximum value occurs

Analyze the plot and find the maximum value of the amplitude \(A_s\). Then, determine the frequency \(\omega\) at which this maximum value occurs.

Key concepts

Differential Equations

Differential equations are powerful mathematical tools that describe relationships involving rates of change. They are essential in modeling real-world phenomena in physics, engineering, biology, economics, and many other fields. In the exercise, you encountered a second-order linear differential equation.
This type of equation includes terms up to the second derivative of the unknown function, which, in this case, is denoted by u. The equation also includes a forcing term, here given by 2 cos(ωt), which represents an external influence, typical in physical systems like oscillating bodies or electrical circuits.
Finding the solution of differential equations often involves two steps: First, solve the homogeneous equation (where the right-hand side is zero) to find the complementary solution. Second, find a particular solution that satisfies the non-homogeneous equation. The complete solution is then the sum of these two. In applications like vibrating systems, we're frequently interested in the steady-state behavior - the long-term pattern of response that sets in after any initial disturbances have dissipated.

Initial Value Problem

An initial value problem is a type of differential equation along with specified values at a given point in time, which are used to find a unique solution. Such problems are formulated to predict future behavior based on these initial conditions.
In the given exercise, you have the initial conditions u(0) = 0 and u'(0) = 2, specifying the state of the system at t = 0. These conditions are crucial for determining the constants in the complementary solution of the differential equation.
When solving an initial value problem, particularly for systems in motion or electrical circuits, determining the behavior immediately after t = 0 is crucial in understanding the system's dynamics. After a period, the transient effects due to initial conditions often die out, and the steady-state solution becomes dominant, especially in linear systems with damping.

Amplitude of Steady-State Solution

The amplitude of the steady-state solution in a differential equation-driven system reflects the magnitude of oscillations or waves within the system's long-term behavior. It is particularly important in engineering and physics since it describes the maximum extent of vibrations or signal variations over time.
The amplitude in our exercise depends on the frequency ω, which determines how rapidly the input to the system—the forcing term—is oscillating. By solving the system of equations for the coefficients A and B, we can express the particular solution in a form that clearly shows the impact of frequency on the system's response.
The final expression for amplitude involves the square root of the sum of the squares of coefficients A(ω) and B(ω), encapsulating both the cosine and sine components of the response. This resulting amplitude as a function of frequency can reveal resonant behaviors, where the amplitude peaks at certain frequencies, agiving insights into tuning or mitigating such effects in practical applications like mechanical design or signal processing.