Laplace Transforms: Vertical Motion of a Coupled Spring System

Figure 1. Coupled Spring System.

Problem Statement

Derive the system of differential equations describing the straight-line vertical motion of the coupled spring shown in Figure 1. Use Laplace transform to solve the system when Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_1=k_2=k_3=1} , Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m_1=m_2=1} , and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_1(0)=0} , Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x'1(0)=-1} , Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_2(0)=0} , and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x'_2(0)=1} .

Solution

At positions Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_1} and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_2} , the masses Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m_1} and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m_2} are in equilibrium. Thus, the motion equations for Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m_1} and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m_2} are,

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m_1\ddot{x}_1=-k_1x_1+k_2(x_2-x_1)}

∴Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m_1\ddot{x}_1+k_1x_1-k_2(x_2-x_1)=0}

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m_2\ddot{x}_2=-k_2(x_2-x_1)-k_3x_2}

∴ Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m_2\ddot{x}_2+k_2(x_2-x_1)-k_3x_2=0}

where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m_1x''_1} and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m_2x''_2} represent the Newton's Second Law of Motion and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -k_1x_1+k_2(x_2-x_1)} and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -k_2(x_2-x_1)-k_3x_2} represent the net forces acting in the masses.

Laplace Transform

Applying the Laplace Transform to the motion equations and plugging the values of Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_1} , Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_2} , Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_3} , Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m_1} , Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m_2} , Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_1(0)} , Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x'1(0)} , Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_2(0)} , and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x'_2(0)} for this systems, we obtain,

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{L}[m_1\ddot{x}_1+k_1x_1-k_2(x_2-x_1)]=m_1[s^2X_1(s)-sx_1(0)-\dot{x}_1(0)]+k_1X_1(s)-k_2(X_2(s)-X_1(s))=0}

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X_1(s)(m_1s^2+k_1+k_2)=m_1(sx_1(0)-\dot{x}_1(0))+k_2X_2(s)}

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X_1(s)=\dfrac{m_1(sx_1(0)+\dot{x}_1(0))+k_2X_2(s)}{(m_1s^2+k_1+k_2)}}

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X_1(s)=\dfrac{1(s0+(-1)]+1X_2(s)}{(1s^2+1+1)}}

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X_1(s)=\dfrac{X_2(s)-1}{(s^2+2)}}

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{L}[m_2\ddot{x}_2+k_2(x_2-x_1)+k_3x_2]=m_2[s^2X_2(s)-sx_2(0)-\dot{x}_2(0)]+k_2(X_2(s)-X_1(s))+k_3X_2(s)=0}

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X_2(s)(m_2s^2+k_2+k_3)=m_2(sx_2(0)-\dot{x}_2(0))+k_2X_1(s)}

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X_2(s)=\dfrac{m_2(sx_2(0)+\dot{x}_2(0))+k_2X_1(s)}{(m_2s^2+k_2+k_3)}}

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X_2(s)=\dfrac{1(s0+1)+1X_1(s)}{(1s^2+1+1)}}

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X_2(s)=\dfrac{X_1(s)+1}{(s^2+2)}}

Finally, solving for Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X_1(s)} and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X_2(s)} yields,

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X_1(s)=\dfrac{-1}{s^2+3}}

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X_2(s)=\dfrac{1}{s^2+3}}

Inverse Laplace Transform

Figure 2. Coupled Spring System Motion.

First, we recognize that

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle sin(kt) = \mathcal{L}^{-1} \left \{ \dfrac{k}{s^2+k^2} \right \}}

On the other hand, we identify that Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k^2=3} , and so Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k=\sqrt{3}} . Hence, we fix the expression by multiplying and dividing by Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sqrt{3}} ,

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_1(t)=\mathcal{L}^{-1}[X_1(s)] = \mathcal{L}^{-1} \left [ \dfrac{-1}{s^2+3} \right ] = -\dfrac{1}{\sqrt{3}} \mathcal{L}^{-1} \left [ \dfrac{\sqrt{3}}{s^2+3} \right ] = -\dfrac{1}{\sqrt{3}} sin(\sqrt{3}t)}

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_2=(t)\mathcal{L}^{-1}[X_2(s)] = \mathcal{L}^{-1} \left [ \dfrac{1}{s^2+3} \right ] = \dfrac{1}{\sqrt{3}} \mathcal{L}^{-1} \left [ \dfrac{\sqrt{3}}{s^2+3} \right ] = \dfrac{1}{\sqrt{3}} sin(\sqrt{3}t)}

A plot of the oscillatory motion is shown on Figure 2.

Initial-Value & Final-Value Theorem

The initial-value and final-value theorem can be useful the finding the behavior of a functionat small and large times respectively. By definition, the Initial-Value Theorem is,

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{s\rightarrow \infty} sX(s)=x(0)\,}

and the Final-Value Theorem is,

$\lim _{s\rightarrow 0}sX(s)=x(\infty )\,$

Thus, applying both theorems to our the Laplace Transforms,

$\lim _{s\rightarrow \infty }sX_{1}(s)=\lim _{s\rightarrow \infty }s{\dfrac {-1}{s^{2}+3}}=\lim _{s\rightarrow \infty }{\dfrac {-s}{s^{2}+3}}={\dfrac {-\infty }{\infty ^{2}+3}}=0=x_{1}(0)\,$

$\lim _{s\rightarrow 0}sX_{1}(s)==\lim _{s\rightarrow 0}s{\dfrac {1}{s^{2}+3}}=\lim _{s\rightarrow 0}{\dfrac {s}{s^{2}+3}}={\dfrac {0}{0^{2}+3}}=0=x_{1}(\infty )\,$

$\lim _{s\rightarrow \infty }sX_{2}(s)=\lim _{s\rightarrow \infty }s{\dfrac {1}{s^{2}+3}}=\lim _{s\rightarrow \infty }{\dfrac {s}{s^{2}+3}}={\dfrac {\infty }{\infty ^{2}+3}}=0=x_{2}(0)\,$

$\lim _{s\rightarrow 0}sX_{2}(s)==\lim _{s\rightarrow 0}s{\dfrac {1}{s^{2}+3}}=\lim _{s\rightarrow 0}{\dfrac {s}{s^{2}+3}}={\dfrac {0}{0^{2}+3}}=0=x_{2}(\infty )\,$

Bode Plot

Figure 3. Bode Diagram.

The Bode plot for the Transfer Functions

$H_{1}(s)={\dfrac {-1}{s^{2}+3}}$

and

$H_{2}(s)={\dfrac {1}{s^{2}+3}}$

can be easily done using a program like Octave or MATLAB. The code is displayed below. From Figure 3 we may notice that the Amplitude vs. Frequency plot for both functions overlaps. The peak amplitude occurs at ${\sqrt {3}}\approx 1.73$ seconds, as well as the phase switching as shown in the Phase vs. Frequency plot.

h1=tf([-1],[1 0 3]); 
h2=tf([1],[1 0 3]); 
bode(h1,'b',h2,'-.r');
legend('H_1(s)','H_2(s)')
grid on;

Magnitude Frequency Response

Considering the Transfer Functions $H_{1}(s)$ and $H_{2}(s)$ described in the Bode Plot section, we notice that there are no values for the s-variable that make $H_{1}(s)$ and $H_{2}(s)$ equal zero.

From the Bode Plot, we notice that the break point occurs at ≈ 1.73 seconds, so the

Convolution

By definition, the convolution of two functions is,

$y(t)=f(t)*h(t)=\int _{-\infty }^{\infty }{f(\tau )h(t-\tau )d\tau }=\int _{\infty }^{-\infty }{f(t-\tau )h(\tau )d\tau }$

where $h_{1}$ refers to the inverse Laplace Transform. Assuming that $\tau$ is a dummy variable of integration, and $f(t)$ is the impulse function ( $f_{1}(t)=f_{2}(t)=\delta (t)$ ), the convolution for our system is,

$y_{1}(t)=\int _{0}^{t}{\delta (t-\tau )h_{1}(\tau )d\tau }=-{\dfrac {1}{\sqrt {3}}}\int _{0}^{t}{\delta (t-\tau )sin({\sqrt {3}}\tau )d\tau }=\left[-{\dfrac {1}{\sqrt {3}}}sin({\sqrt {3}}\tau )\right]_{\tau =t}=-{\dfrac {1}{\sqrt {3}}}sin({\sqrt {3}}t)$

$y_{2}(t)=\int _{0}^{t}{\delta (t-\tau )h_{2}(\tau )d\tau }={\dfrac {1}{\sqrt {3}}}\int _{0}^{t}{\delta (t-\tau )sin({\sqrt {3}}\tau )d\tau }=\left[{\dfrac {1}{\sqrt {3}}}sin({\sqrt {3}}\tau )\right]_{\tau =t}={\dfrac {1}{\sqrt {3}}}sin({\sqrt {3}}t)$

which is the same solution yields by the inverse Laplace Transform

State Equation Model

By definition, the the state equation is stated as

${\underline {\dot {x}}}(t)={\widehat {A}}\,{\underline {x}}(t)+{\widehat {B}}\,{\underline {u}}(t)$

Now, consider the motion equations described in the Solution section,

$m_{1}{\ddot {x}}_{1}+k_{1}x_{1}-k_{2}(x_{2}-x_{1})=m_{1}{\ddot {x}}_{1}+k_{1}x_{1}-k_{2}x_{2}+k_{2}x_{1}=0$

$m_{2}{\ddot {x}}_{2}+k_{2}(x_{2}-x_{1})-k_{3}x_{2}=m_{2}{\ddot {x}}_{2}+k_{2}x_{2}-k_{2}x_{1}-k_{3}x_{2}=0$

Solving for ${\ddot {x}}_{1}$ and ${\ddot {x}}_{2}$ yields,

${\ddot {x}}_{1}=-{\dfrac {k_{1}}{m_{1}}}x_{1}+{\dfrac {k_{2}}{m_{1}}}x_{2}-{\dfrac {k_{2}}{m_{1}}}x_{1}$

${\ddot {x}}_{2}={\dfrac {-k_{2}}{m_{2}}}x_{2}+{\dfrac {k_{2}}{m_{2}}}x_{1}+{\dfrac {k_{3}}{m_{2}}}x_{2}$

Finally, we let $x_{1}{\frac {}{}}$, ${\dot {x}}_{1}{\frac {}{}}$, $x_{2}{\frac {}{}}$, and ${\dot {x_{2}}}{\frac {}{}}$ be the state variables. Thus,

${\begin{bmatrix}{\dot {x}}_{1}\\{\ddot {x}}_{1}\\{\dot {x}}_{2}\\{\ddot {x}}_{2}\end{bmatrix}}={\begin{bmatrix}0&1&0&0\\-{\frac {1}{m_{1}}}(k_{1}+k_{2})&0&{\frac {k_{2}}{m_{1}}}&0\\0&0&0&1\\{\frac {k_{2}}{m_{2}}}&0&{\frac {1}{m_{2}}}(k_{3}-k_{2})&0\end{bmatrix}}{\begin{bmatrix}x_{1}\\{\dot {x}}_{1}\\x_{2}\\{\dot {x}}_{2}\end{bmatrix}}$