To solve this problem, we’ll follow these steps:

1. Set up the differential equation for the damped, driven oscillator.
2. Solve for the unique solution using the initial conditions.
3. Identify the steady-state and transient terms.

The given data includes:

• Forcing function: $F_{\text{drive}}(t) = -40 \sin(t) \, \text{N}$
• Mass ($m$): $1 \, \text{kg}$
• Spring constant ($k$): $5 \, \text{N/m}$
• Damping coefficient ($c$): $4 \, \text{N·s/m}$
• Initial position: $x(0) = -5 \, \text{m}$
• Initial velocity: $\dot{x}(0) = 2 \, \text{m/s}$

Step 1: Set Up the Differential Equation

The equation of motion for a damped, driven harmonic oscillator is given by: $m \ddot{x} + c \dot{x} + k x = F_{\text{drive}}(t)$ Substituting the values:

$1 \ddot{x} + 4 \dot{x} + 5 x = -40 \sin(t)$ or simply: $\ddot{x} + 4 \dot{x} + 5 x = -40 \sin(t)$

Step 2: Solve for the Complementary (Homogeneous) Solution

To find the complementary solution $x_c(t)$, we solve the homogeneous equation: $\ddot{x} + 4 \dot{x} + 5 x = 0$

1. Assume a solution of the form $x = e^{\lambda t}$.
2. Substitute into the equation to get the characteristic polynomial: $\lambda^2 + 4\lambda + 5 = 0$
3. Solve for $\lambda$: $\lambda = -2 \pm i$

The complementary solution is: $x_c(t) = e^{-2t} (A \cos t + B \sin t)$ where $A$ and $B$ are constants determined by initial conditions.

Step 3: Solve for the Particular Solution

To find the particular solution $x_p(t)$ for the non-homogeneous equation $\ddot{x} + 4 \dot{x} + 5 x = -40 \sin(t)$, assume a solution of the form: $x_p(t) = C \cos t + D \sin t$

Substitute $x_p(t)$ into the original differential equation and equate terms of $\cos t$ and $\sin t$ to solve for $C$ and $D$. After calculation, we find that: $C = 0 \quad \text{and} \quad D = -10$ Thus, the particular solution is: $x_p(t) = -10 \sin(t)$

Step 4: Form the General Solution

The general solution is the sum of the complementary and particular solutions: $x(t) = x_c(t) + x_p(t) = e^{-2t} (A \cos t + B \sin t) - 10 \sin(t)$

Step 5: Apply Initial Conditions

1. Position at $t = 0$: $x(0) = -5$ $x(0) = e^{-2 \cdot 0} (A \cdot 1 + B \cdot 0) - 10 \cdot 0 = A = -5$

2. Velocity at $t = 0$: $\dot{x}(0) = 2$ $\dot{x}(t) = -2 e^{-2t} (A \cos t + B \sin t) + e^{-2t} (-A \sin t + B \cos t) - 10 \cos t$ At $t = 0$: $\dot{x}(0) = -2 \cdot (-5) + B - 10 = 2$ Solving, $B = -8$.

So, the unique solution is: $x(t) = e^{-2t} (-5 \cos t - 8 \sin t) - 10 \sin t$

Step 6: Identify Steady-State and Transient Terms

• Transient Term: $e^{-2t} (-5 \cos t - 8 \sin t)$ (decays over time due to the exponential factor)
• Steady-State Term: $-10 \sin t$ (persists as $t \to \infty$)

Would you like more detailed steps on solving the initial conditions or particular solution? Here are some related questions:

1. How does the damping coefficient affect the transient term in this system?
2. What would happen to the steady-state solution if the forcing function were changed?
3. How would the solution differ if the mass were different?
4. Why does the transient term decay over time?
5. How would this solution change with different initial conditions?

Tip: In forced oscillation problems, identifying whether the solution is underdamped, critically damped, or overdamped helps predict the behavior of the transient response.