Given the dynamic system model of the vertical displacement y(t) of Link on a rocket-powered pogo stick, mẍ + cẋ + ky = bδ(t) - mg with initial conditions y(0) = 0 and ẋ(0) = -4 m/s, and associated parameters: m = 80 kg, b = 1200 N, complete the objectives: rewrite as a linear system using substitution, take the Laplace transform, and solve for y(t).
Solution
The problem involves analyzing a dynamic system and requires completing several tasks:
Step-by-Step Solution Outline:
2.A: Rewrite the model as a linear system
The given equation is:
my¨+cy˙+ky=bδ(t)−mg
Substituting x=[yy˙], we can express the system as:
\dot{x}_1 \\ \dot{x}_2
\end{bmatrix}
=
\begin{bmatrix}
x_2 \\ -\frac{k}{m}x_1 - \frac{c}{m}x_2
\end{bmatrix}
+
\begin{bmatrix}
0 \\ \frac{b}{m}\delta(t)
\end{bmatrix}.$$
This gives:
$$\dot{x} = Ax + b$$
where:
$$A =
\begin{bmatrix}
0 & 1 \\
-\frac{k}{m} & -\frac{c}{m}
\end{bmatrix},
\quad
b = \begin{bmatrix}
0 \\ \frac{b}{m}
\end{bmatrix}.$$
To determine $$c$$ (damping constant) and $$k$$ (spring constant), analyze the eigenvalues of $$A$$. Stability depends on the eigenvalues being negative real numbers or having negative real parts.
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### 1.E: Take the Laplace transform
The Laplace transform of the system will transform the equation into the $$s$$-domain:
$$m(s^2Y(s) - sy(0) - \dot{y}(0)) + c(sY(s) - y(0)) + kY(s) = b\delta(t) - mg.$$
Given the initial conditions $$y(0) = 0$$ and $$\dot{y}(0) = -4$$, simplify:
$$m(s^2Y(s) + 4s) + c(sY(s)) + kY(s) = b - mg/s.$$
Solve for $$Y(s)$$:
$$Y(s) = \frac{b - mg/s - 4ms}{ms^2 + cs + k}.$$
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### 1.L: Inverse Laplace transform and motion $$y(t)$$
Use partial fraction decomposition and lookup tables to find the inverse Laplace transform. This will provide $$y(t)$$ in the time domain.
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Would you like a detailed step-by-step computation of any specific part? Here are follow-up questions that you may find useful:
1. Would you like me to calculate specific values for $$c$$ and $$k$$ for a stable system?
2. Should I solve the Laplace equation in detail and find $$Y(s)$$?
3. Do you want to visualize the eigenvalues of the system's $$A$$-matrix to determine stability?
4. Should I compute and plot the time-domain motion $$y(t)$$?
5. Do you need a general explanation of how the Laplace transform applies to dynamic systems?
**Tip:** Always check the stability of the system by analyzing the eigenvalues of $$A$$ or the characteristic equation roots.
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Math Problem Analysis
Mathematical Concepts
Dynamic Systems Linear Algebra Laplace Transform Second-Order Differential Equations
Formulas
mẍ + cẋ + ky = bδ(t) - mg x = [y ẋ]^T A = [[0, 1], [-k/m, -c/m]], b = [0, b/m] Laplace Transform: L{mẍ + cẋ + ky} = ms^2Y(s) + csY(s) + kY(s)