Problem 14 - Calculus for the Life Sciences: A Modeling Approach Volume II

Question

Consider now the case that there is resistance \(r>0\) in the spring-mass equation with a harmonic forcing function, $$ m y^{\prime \prime}(t)+r y^{\prime}(t)+k y(t)=\cos \omega t $$ a. Show that in the real root case, \(\mu_{1}\) and \(\mu_{2}\) are negative. b. Show that in the repeated root case, the value of \(\mu\) is negative. c. Show that in the and complex roots case, the value of \(\mu\) is negative. d. Show that in all of the cases, $$ \lim _{t \rightarrow \infty} y_{h}(t)=0 $$ The next few exercises examine the importance of \(y_{p},\) the particular solution of Equation \(18.20 .\) We will need $$ \begin{array}{ll} y_{p}=A \cos 0.3 t+B \sin 0.3 t & \text { and one of } \\ y_{h}=C_{1} e^{\mu_{1} t}+C_{2} e^{\mu_{2} t}, & \text { real roots, } \\ y_{h}=C_{1} e^{\mu t}+C_{2} t e^{\mu t}, & \text { repeated root, and } \\ y_{h}=e^{\mu t}\left(C_{1} \cos (\omega t)+C_{2} \sin (\omega t)\right) & \text { complex roots. } \end{array} $$

Step-by-Step Solution

Answer

All roots \( \mu_1, \mu_2, \mu \) are negative, ensuring \( \lim_{t \to \infty} y_h(t) = 0 \).

Step 1: Understand the Problem

We are given a damped harmonic oscillator with a differential equation involving mass, damping resistance, spring constant, and a forcing function. Our goal is to demonstrate that the roots of the characteristic equation are negative in different cases and to show that the homogeneous solution tends to zero as time approaches infinity.

Step 2: Analyze the Characteristic Equation

The characteristic equation is derived from eliminating the forcing function, which is solved by considering the homogeneous part. Replacing all derivatives with their characteristic equation equivalents, \[ m ho^2 + r ho + k = 0 \]where \( \rho \) represents the roots of the characteristic equation.

Step 3: Solve Real Root Case

For real roots, the discriminant \( D = r^2 - 4mk \) must be positive to ensure real roots. Solving \( m \mu^2 + r \mu + k = 0 \) gives two negative roots \( \mu_1 , \mu_2 \) since both roots of a polynomial with a positive leading coefficient have negative real parts if all coefficients are positive.

Step 4: Solve Repeated Root Case

For a repeated root, the discriminant \( D = r^2 - 4mk = 0 \). The double root \( \mu = -\frac{r}{2m} \) is negative since \( r > 0 \) and \( m > 0 \).

Step 5: Solve Complex Root Case

For complex roots, the discriminant \( D = r^2 - 4mk < 0 \). The roots have the form \( \alpha \pm i\beta \) where \( \alpha = -\frac{r}{2m} \), ensuring negative real parts since \( r > 0 \).

Step 6: Confirm Exponential Decay of Homogeneous Solutions

In all cases, the homogeneous solution includes terms of the form \( e^{\mu t} \) where \( \mu \) or its real component is negative, causing the solution to decay to zero as \( t \to \infty \). Thus, \[\lim_{t \to \infty} y_h(t) = 0\]is confirmed.

Key Concepts

Differential Equations

Differential equations play a crucial role in understanding the behavior of damped harmonic oscillators. They connect the physical parameters of the system, such as mass, damping resistance, spring constant, and the external force applied, through mathematical expressions. In the given problem, the differential equation is derived from the physical principles governing a spring-mass system with damping and forcing function: \[ m y''(t) + r y'(t) + k y(t) = \cos \omega t \] where \(m\) is the mass, \(r\) is the damping coefficient, \(k\) is the spring constant, and \(\omega\) represents the frequency of the external force.
This equation is known as a second-order linear differential equation with constant coefficients.

• The left-hand side involves the system's inherent properties, while
• the right-hand side represents the external input driving the system.

To solve such equations, we typically separate them into a homogeneous part (ignoring external force) and a particular solution (accounting for external force). This separation helps in understanding how the system would behave without any external influence versus how it accommodates external influences.

Characteristic Equation

To analyze how the system behaves without external forces, we focus on the characteristic equation derived from the homogeneous part of the differential equation. Simply put, the characteristic equation encapsulates the internal dynamics of the system:\[ m \rho^2 + r \rho + k = 0 \]Here, \( \rho \) represents the roots of the characteristic equation, and these roots determine the behavior, such as oscillation and damping, of the system over time.
Digging deeper, depending on the value of the discriminant \( D = r^2 - 4mk \), the nature of the roots can be categorized into three cases:

• Real Roots: When \( D > 0 \), it indicates two distinct real roots, each contributing to exponential decay based on their respective values.
• Repeated Root: When \( D = 0 \), it gives a double root, implying critical damping, where the system returns to equilibrium without oscillating.
• Complex Roots: When \( D < 0 \), the roots are complex, introducing oscillatory behavior combined with exponential decay, described by a real part indicating damping and an imaginary part indicating oscillation.

The exponential decay in each case leads to a nullifying effect over time, ensuring that the system eventually stabilizes.

Homogeneous Solution

The homogeneous solution captures the response of the system in the absence of external forces and is significantly dictated by the roots of the characteristic equation. Depending on the nature of these roots, the form of the homogeneous solution varies:

• For real roots \( \mu_1 \) and \( \mu_2 \), the solution generally takes the form:

\[ y_h = C_1 e^{\mu_1 t} + C_2 e^{\mu_2 t} \]

• For a repeated root \( \mu \), it modifies to accommodate the repeated factor:

\[ y_h = C_1 e^{\mu t} + C_2 t e^{\mu t} \]

• For complex roots, often expressed as \( \alpha \pm i\beta \), it takes the form:

\[ y_h = e^{\alpha t}(C_1 \cos(\beta t) + C_2 \sin(\beta t)) \]In each formation above, the coefficients \( C_1 \) and \( C_2 \) are determined by initial conditions, and the exponential terms \( e^{\mu t} \) or \( e^{\alpha t} \) drive the solution towards zero as \( t \to \infty \), confirming: \[ \lim_{t \to \infty} y_h(t) = 0 \] This convergence expresses the system's natural tendency to dampen oscillations and stabilize over time, even when perturbed, making it critical to understanding the long-term behavior of oscillating systems.