	A mass-spring-dashpot system has the following parameters, with an external forcing function F(t)=10 sin⁡3t. Find the steady periodic solution of the position function of the system. (20 pts)
m=1 kg, 	k=4 N/m, 	c=4 Ns/m


To solve for the steady periodic solution of the position function in a mass-spring-dashpot system with the given parameters, we start with the differential equation governing the system. The equation of motion for a damped harmonic oscillator with an external forcing function is:

\[
m\frac{d^2x}{dt^2} + c\frac{dx}{dt} + kx = F(t)
\]

Substituting the given values:

- Mass \( m = 1 \, \text{kg} \)
- Damping coefficient \( c = 4 \, \text{Ns/m} \)
- Spring constant \( k = 4 \, \text{N/m} \)
- External force \( F(t) = 10 \sin(3t) \)

The equation becomes:

\[
\frac{d^2x}{dt^2} + 4\frac{dx}{dt} + 4x = 10\sin(3t)
\]

### Step 1: Find the Homogeneous Solution

The characteristic equation for the homogeneous part (\( F(t) = 0 \)) is:

\[
m r^2 + c r + k = 0 \implies r^2 + 4r + 4 = 0
\]

This factors to:

\[
(r + 2)^2 = 0
\]

Thus, there is a repeated root \( r = -2 \). The homogeneous solution is:

\[
x_h(t) = (C_1 + C_2 t)e^{-2t}
\]

### Step 2: Find the Particular Solution

To find the particular solution \( x_p(t) \) corresponding to the forcing function \( F(t) = 10 \sin(3t) \), we use the method of undetermined coefficients. We assume a solution of the form:

\[
x_p(t) = A \sin(3t) + B \cos(3t)
\]

### Step 3: Compute Derivatives

Calculating the derivatives:

\[
\frac{dx_p}{dt} = 3A \cos(3t) - 3B \sin(3t)
\]

\[
\frac{d^2x_p}{dt^2} = -9A \sin(3t) - 9B \cos(3t)
\]

### Step 4: Substitute into the Differential Equation

Substituting \( x_p(t) \), \( \frac{dx_p}{dt} \), and \( \frac{d^2x_p}{dt^2} \) into the equation:

\[
-9A \sin(3t) - 9B \cos(3t) + 4(3A \cos(3t) - 3B \sin(3t)) + 4(A \sin(3t) + B \cos(3t)) = 10 \sin(3t)
\]

Combining like terms:

\[
(-9A + 4A + 12B) \sin(3t) + (-9B + 12A + 4B) \cos(3t) = 10 \sin(3t)
\]

This leads to the system of equations:

1. \(-5A + 12B = 10\)
2. \(3A - 5B = 0\)

### Step 5: Solve the System

From equation 2:

\[
3A = 5B \implies A = \frac{5}{3}B
\]

Substituting into equation 1:

\[
-5\left(\frac{5}{3}B\right) + 12B = 10
\]

\[
-\frac{25}{3}B + 12B = 10 \implies -\frac{25}{3}B + \frac{36}{3}B = 10
\]

\[
\frac{11}{3}B = 10 \implies B = \frac{30}{11}
\]

Then,

\[
A = \frac{5}{3} \cdot \frac{30}{11} = \frac{50}{11}
\]

### Step 6: Write the Particular Solution

The particular solution is:

\[
x_p(t) = \frac{50}{11} \sin(3t) + \frac{30}{11} \cos(3t)
\]

### Final Solution

The general solution is given by:

\[
x(t) = x_h(t) + x_p(t) = (C_1 + C_2 t)e^{-2t} + \frac{50}{11} \sin(3t) + \frac{30}{11} \cos(3t)
\]

The steady-state periodic solution is thus:

\[
x_{ss}(t) = \frac{50}{11} \sin(3t) + \frac{30}{11} \cos(3t)
\]

### Summary

The steady periodic solution of the position function of the system is:

\[
x_{ss}(t) = \frac{50}{11} \sin(3t) + \frac{30}{11} \cos(3t)
\]

If you have any questions or need further clarification, feel free to ask! 

Here are some related questions to explore:

1. How does changing the damping coefficient affect the system's response?
2. What would the solution look like for a different forcing function, like \( F(t) = 10 \cos(3t) \)?
3. How can we determine the natural frequency of the system?
4. What are the implications of resonance in this type of system?
5. How does the mass value influence the amplitude of the steady-state response? 

**Tip:** Always consider the damping effects in a dynamic system; it can significantly alter the system's behavior.

A mass-spring-dashpot system has the following parameters, with an external forcing function F(t)=10 sin⁡3t. Find the steady periodic solution of the position function of the system. (20 pts) m=1 kg, k=4 N/m, c=4 Ns/m

Explore the steady periodic solution of a mass-spring-dashpot system with parameters m=1 kg, k=4 N/m, c=4 Ns/m, and an external force F(t)=10 sin(3t). This detailed analysis covers differential equations and methods of solving for position functions in harmonic systems. Suitable for advanced high school students.

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