Question

Consider the initial value problem $$ m u^{\prime \prime}+\gamma u^{\prime}+k u=0, \quad u(0)=u_{0}, \quad u^{\prime}(0)=v_{0} $$ Assume that \(\gamma^{2}<4 k m .\) (a) Solve the initial value problem, (b) Write the solution in the form \(u(t)=R \exp (-\gamma t / 2 m) \cos (\mu t-\delta) .\) Determine \(R\) in terms of \(m, \gamma, k, u_{0},\) and \(v_{0}\). (c) Investigate the dependence of \(R\) on the damping coefficient \(\gamma\) for fixed values of the other parameters.

Short answer

Question: Investigate how the amplitude of the solution of the given differential equation with initial conditions \(u(0) = u_0\) and \(u'(0) = v_0\) depends on the damping coefficient \(\gamma\). Answer: The amplitude of the solution depends on the damping coefficient \(\gamma\) through the expression for \(R\). For small values of \(\gamma\), the solutions will oscillate more and the amplitude of oscillations, represented by the value of R, will be larger. As \(\gamma\) increases, the oscillations will dampen more quickly, and the amplitude of the oscillations (R) will decrease. In the limiting case as \(\gamma \rightarrow \infty\), the roots will be extremely large negative numbers and the oscillations will die out very quickly, resulting in \(R \rightarrow 0\).

Step-by-step solution

Find the characteristic equation and roots

Start by finding the characteristic equation for the differential equation: $$ m\lambda^2 + \gamma\lambda + k = 0 $$ We are given that \(\gamma^2<4km\), so the discriminant of the quadratic equation is positive: $$ D = \gamma^2-4mk > 0 $$ Thus, the characteristic equation has two distinct, real roots: $$ \lambda_{1,2} = \frac{-\gamma \pm \sqrt{\gamma^2 - 4mk}}{2m} $$

Form the general solution

Now that we have the characteristic roots, we can form the general solution: $$ u(t) = C_1 e^{\lambda_1 t} + C_2 e^{\lambda_2 t} $$ where \(C_1\) and \(C_2\) are constants to be determined using the initial conditions.

Apply the initial conditions

Using the initial conditions \(u(0)=u_0\) and \(u'(0)=v_0\), we can solve for the constants \(C_1\) and \(C_2\): $$ u_0 = C_1 e^{\lambda_1 \cdot 0} + C_2 e^{\lambda_2 \cdot 0} \implies u_0 = C_1 + C_2 $$ and $$ u'(t) = C_1\lambda_1 e^{\lambda_1 t} + C_2\lambda_2 e^{\lambda_2 t} \implies v_0 = C_1\lambda_1 + C_2\lambda_2 $$ Solve this system of linear equations to determine the values of \(C_1\) and \(C_2\).

Rewrite the solution in the given form

The given form is: $$ u(t) = R e^{-\gamma t / (2m)} \cos(\mu t - \delta) $$ To rewrite the general solution in this form, first group together the exponential terms in the solution: $$ u(t) = e^{(-\gamma t / 2m)}(C_1 e^{(\sqrt{\gamma^2 - 4mk} t)/(2m)} + C_2 e^{(-\sqrt{\gamma^2 - 4mk} t)/(2m)}) $$ We know that \(\mu = \frac{\sqrt{\gamma^2 - 4mk}}{2m}\). Define \(\omega = \sqrt{k/m}\), and rewrite the solution as: $$ u(t) = e^{-\gamma t / (2m)}( C_1 e^{(\mu t)} + C_2 e^{(-\mu t)}) $$ Using the sum-to-product formulas, we can rewrite this as: $$ u(t) = R e^{-\gamma t / (2m)} \cos(\mu t - \delta) $$ where \(R = 2\sqrt{C_1C_2}\) and \(\delta = \arctan\left(\frac{C_1-C_2}{C_1+C_2}\right)\).

Investigate the dependence of R on \(\gamma\)

To investigate the dependence of \(R\) on the damping coefficient \(\gamma\), we will consider how the expression of \(R\) changes with \(\gamma\) while keeping all other parameters fixed. Since \(R = 2\sqrt{C_1C_2}\), we will consider how \(C_1\) and \(C_2\) change with \(\gamma\). For small values of \(\gamma\), the solutions will oscillate more and the amplitude of oscillations (i.e., the value of R) will be larger. As \(\gamma\) increases, the oscillations will dampen more quickly, and the amplitude of the oscillations (R) will decrease. In the limiting case as \(\gamma \rightarrow \infty\), the roots will be extremely large negative numbers and the oscillations will die out very quickly, resulting in \(R \rightarrow 0\).

Key concepts

Characteristic Equation

In the study of differential equations, the characteristic equation plays a pivotal role in understanding the behavior of solutions. It is derived from a linear homogeneous differential equation by assuming a solution of the form \(e^{\lambda t}\), where \(\lambda\) represents a complex number that characterizes the equation's roots.

In the given exercise, the characteristic equation is obtained by substituting \(u(t) = e^{\lambda t}\) into the differential equation and simplifying, yielding \(m\lambda^2 + \gamma\lambda + k = 0\). This is essentially a quadratic equation in terms of \(\lambda\) with coefficients dependent on the physical parameters of the system—mass (m), damping coefficient (\gamma), and stiffness (k). The nature of the roots determined by this equation dictates whether the system exhibits oscillatory motion, overdamped, or critically damped behavior.

The given exercise presents a scenario where the discriminant \(\gamma^2 - 4mk\) is positive, indicating the presence of two distinct, real roots. This implies that the system is underdamped, leading to a motion that is oscillatory but experiences damping over time. Understanding the characteristic equation is fundamental to predicting the system's response over time and serves as the basis for further analysis.

Damping Coefficient

The damping coefficient, represented by \(\gamma\), quantifies the resistive force in a system that dissipates energy, typically through friction or drag. It is an intrinsic property that affects the rate at which oscillations decrease in amplitude.

In our case, a system described by \(m u'' + \gamma u' + k u = 0\) has a damping coefficient which influences how much the system's oscillations diminish over time. A higher \(\gamma\) value denotes a greater damping effect, resulting in lesser oscillatory behavior and a more rapid return to equilibrium. Conversely, a lower \(\gamma\) implies weaker damping, allowing the system to oscillate more freely.

We have also been tasked to investigate the amplitude \(R\) as a function of the damping coefficient \(\gamma\). It is observed that as \(\gamma\) increases, the term \(e^{-\gamma t / (2m)}\) in the solution causes the motion to cease more quickly. Ultimately, when \(\gamma\) becomes very large, the system no longer oscillates—instead it quickly stabilizes, resulting in \(R\) approaching zero. This inverse relationship between the damping coefficient and the oscillation amplitude is crucial for engineers and physicists when designing systems that require specific damping characteristics.

Oscillation Amplitude

The term oscillation amplitude, denoted as \(R\) in the exercise, is a crucial aspect of an oscillating system’s motion. It measures the maximum extent of oscillation from the system's equilibrium position and is indicative of the energy contained in the system.

In the context of our initial value problem, the oscillation amplitude is not constant, but instead, it changes over time due to the exponential damping factor \(e^{-\gamma t / (2m)}\). This means that although the system initially oscillates with a certain amplitude, this amplitude gradually decreases because of the damping effect characterized by \(\gamma\).

The relationship between the amplitude \(R\) and the parameters such as \(\gamma\), \(m\), \(k\), \(u_{0}\), and \(v_{0}\) is non-trivial. Part b of the problem requires expressing \(R\) in terms of these parameters. This equation serves as a model for predicting how different conditions at the beginning of motion—the initial displacement \(u_{0}\) and initial velocity \(v_{0}\)—along with system properties like mass and stiffness, affect the subsequent oscillation amplitude. Such information is vital for designing and controlling systems to ensure desired performance characteristics, such as the ability to absorb shock or maintain stability under oscillatory conditions.