To solve for the steady periodic solution $x_{sp}(t)$ of the position function for this mass-spring-dashpot system, we’ll start by setting up and solving the differential equation.

Step 1: Formulate the Differential Equation

The general differential equation for a forced damped harmonic oscillator (mass-spring-dashpot system) with external forcing function $F(t)$ is: $m \frac{d^2x}{dt^2} + c \frac{dx}{dt} + kx = F(t)$ Given:

• $m = 1$ kg
• $c = 4$ Ns/m
• $k = 4$ N/m
• $F(t) = 10 \sin(3t)$

Substituting these values in, the equation becomes: $\frac{d^2x}{dt^2} + 4 \frac{dx}{dt} + 4x = 10 \sin(3t)$

Step 2: Rewrite the Forcing Function in Terms of Cosine

The forcing function is $F(t) = 10 \sin(3t)$, which we can rewrite using the identity $\sin(3t) = \cos\left(3t - \frac{\pi}{2}\right)$: $F(t) = 10 \cos\left(3t - \frac{\pi}{2}\right)$ This helps in identifying the frequency of the steady-state solution.

Step 3: Assume a Steady Periodic Solution

The steady periodic solution is assumed to be of the form: $x_{sp}(t) = C \cos(\omega t - \alpha)$ where $\omega = 3$ (since the forcing frequency is 3 rad/s).

Step 4: Substitute into the Equation

To find $C$ and $\alpha$, use the amplitude and phase shift formula for a forced damped harmonic oscillator: $C = \frac{F_0}{\sqrt{(k - m \omega^2)^2 + (c \omega)^2}}$ $\tan(\alpha) = \frac{c \omega}{k - m \omega^2}$

Where:

• $F_0 = 10$ (amplitude of the forcing function)
• $\omega = 3$
• $m = 1$
• $c = 4$
• $k = 4$

Calculating $C$:

1. Calculate $k - m \omega^2$: $k - m \omega^2 = 4 - (1)(3)^2 = 4 - 9 = -5$

2. Calculate $c \omega$: $c \omega = 4 \cdot 3 = 12$

3. Substitute into the formula for $C$: $C = \frac{10}{\sqrt{(-5)^2 + (12)^2}} = \frac{10}{\sqrt{25 + 144}} = \frac{10}{\sqrt{169}} = \frac{10}{13}$

So, $C = \frac{10}{13}$.

Calculating $\alpha$:

$\tan(\alpha) = \frac{c \omega}{k - m \omega^2} = \frac{12}{-5}$ $\alpha = \tan^{-1}\left(-\frac{12}{5}\right)$ Calculating $\alpha$: $\alpha \approx -1.176 \text{ radians}$

Step 5: Write the Steady Periodic Solution

The steady periodic solution is: $x_{sp}(t) = \frac{10}{13} \cos\left(3t + 1.176\right)$

Final Answer:

$x_{sp}(t) = \frac{10}{13} \cos(3t + 1.176)$

Would you like additional details or have any questions?

Related Questions

1. How does the damping coefficient affect the steady-state solution?
2. What would happen if there were no external forcing function?
3. How does changing the mass affect the resonance frequency?
4. Can we find the transient solution as well as the steady-state solution?
5. What would the steady-state solution look like if $c = 0$ (no damping)?

Tip:

Always express the forcing function in terms of cosine if the steady-state solution is requested in cosine form, as it simplifies finding the phase shift.