91Ó°ÊÓ

Critical damping is an important concept in the study of differential equations, especially in the context of mechanical and electrical systems. It occurs when a system returns to equilibrium as quickly as possible without oscillating. To achieve critical damping, the damping coefficient, denoted as \( \gamma \), must be equal to the natural frequency of the system. For the given differential equation, \( \frac{d^{2} x}{d t^{2}} + 2 \gamma \frac{d x}{d t} + 169 x = 0 \), critical damping occurs when \( \gamma^2 = 169 \). Solving this gives \( \gamma = \sqrt{169} = 13 \). This value ensures that the system is critically damped, allowing it to return to its equilibrium position in the shortest possible time without passing it. In contrast to underdamping or overdamping, critical damping is unique because it eliminates any oscillations, thus providing a stable and swift response.

The characteristic equation is a crucial part of solving differential equations, such as those involved in describing motion. It is derived from the given differential equation and is used to find the roots, which help form the solution.For the equation \( \frac{d^{2} x}{d t^{2}} + 2 \gamma \frac{d x}{d t} + 169 x = 0 \), with \( \gamma = 13 \) for critical damping, the characteristic equation is formed as \( r^2 + 26r + 169 = 0 \). This simplifies to \( (r + 13)^2 = 0 \), indicating a double root at \( r = -13 \). The roots indicate the behavior of the system:

• Real double roots like \( r = -13 \) result in responses where the solution has the form \( x(t) = (c_1 + c_2 t) e^{-13t} \).
• Different types of roots (real distinct or complex) modify the behavior and form of the solution differently.

Understanding the roots helps determine the nature of the solution and the motion it represents.

Initial conditions in differential equations provide the necessary information to find a unique solution. They specify the state of the system at a specific time, often at \( t = 0 \). For this exercise, we know that \( x(0) = 0 \) and the initial velocity \( v(0) = \frac{dx}{dt}(0) = 8 \text{ ft/sec} \).These conditions help solve for any constants in the general solution derived from solving the characteristic equation. With critical damping, where \( x(t) = (c_1 + c_2 t) e^{-13t} \), applying \( x(0) = 0 \) results in \( c_1 = 0 \). Then, using the derivative \( v(t) = \frac{dx}{dt} = (c_2 - 13c_2 t)e^{-13t} - 13c_1e^{-13t} \) and \( v(0) = 8 \), we find \( c_2 = 8 \).Thus, initial conditions enable us to tailor the general solution to fit the specific scenario described by the problem.

Graphing functions derived from differential equations is essential to visualize the behavior of solutions over time. For this problem, graphing \( x(t) \) over a specified interval helps students understand how the system evolves under different damping scenarios.When \( \gamma = 13 \) (critical damping), the solution is \( x(t) = 8t e^{-13t} \). This plots as an exponential decay without oscillation, stabilizing quickly.For \( \gamma = 12 \), where \( x(t) = \frac{8}{5} e^{-12t} \sin(5t) \), the graph shows a damped oscillation pattern. At \( \gamma = 14 \), with its real distinct roots resulting in a form like \( x(t) = c_1 e^{r_1 t} + c_2 e^{r_2 t} \), the graph depicts a system that returns to equilibrium more slowly than critically damped, with no oscillations.Graphing offers a visual insight into these damping effects, helping students connect mathematical solutions to real-world system behaviors.