Laplace transforms: Under-damped Mass-Spring System on an Incline

Brandon.plubell 05:44, 26 October 2009 (UTC)

Under-Damped Mass-Spring System on an Incline

Part 1 - Use Laplace Transformations

Problem Statement

Find the equation of motion for the mass in the system subjected to the forces shown in the Free Body Diagram (FBD). The inclined surface is coated in SAE 30 oil.

Initial Conditions and Values

• A is the area of the box in contact with the surface
• g is the gravitational acceleration field constant
• b_t is the thickness of the fluid covering the inclined surface
• μ is the viscosity constant of the fluid;
• m is the mass of the box
• k is the spring constant

$A={\frac {1}{4}}m^{2}$

$g=9.81{\frac {m}{s^{2}}}$

$b_{t}=1mm{\frac {}{}}$

$\mu =0.06{\frac {N\cdot s}{m^{2}}}$

$m=45kg{\frac {}{}}$

$k=200{\frac {N}{m}}$

$\theta =30^{\circ }{\frac {}{}}$

Let the initial conditions be:

$x(0)=-0.5m{\frac {}{}}$

${\dot {x}}(0)=0{\frac {m}{s}}$

Force Equations

The sum of the moments in the x direction yields the equatiom

$+\swarrow \sum F_{x}=m{\ddot {x}}$

$\Rightarrow \ m{\ddot {x}}=F_{s}+F_{f}+mg\sin \theta$

Where

$F_{s}=-k\,x$

$F_{f}=-{\frac {\mu \,A}{b_{t}}}\,{\dot {x}}$

To make the algebra easier, let

$\lambda ={\frac {\mu \,A}{b_{t}}}$

Then, from the sum of forces equation:

$m\,{\ddot {x}}+\lambda \,{\dot {x}}+k\,x=mg\sin \theta$

$\Rightarrow \ {\ddot {x}}+{\frac {\lambda }{m}}\,{\dot {x}}+{\frac {k}{m}}\,x=g\sin \theta$

Laplace Transform

${\mathcal {L}}\left\{{\ddot {x}}+{\frac {\lambda }{m}}\,{\dot {x}}+{\frac {k}{m}}\,x\right\}={\mathcal {L}}\left\{g\sin \theta \right\}$

$\Rightarrow \ s^{2}\,X(s)-s\,x(0)-{\dot {x}}(0)+{\frac {\lambda }{m}}\,s\,X(s)-{\frac {\lambda }{m}}\,x(0)+{\frac {k}{m}}\,X(s)=g\sin \theta \,\left({\frac {1}{s}}\right)$

$\Rightarrow \ X(s)\left(s^{2}+{\frac {\lambda }{m}}\,s+{\frac {k}{m}}\right)=g\sin \theta \,\left({\frac {1}{s}}\right)+s\,x(0)+{\dot {x}}(0)+{\frac {\lambda }{m}}\,x(0)$

If we let $x(0){\text{ and }}{\dot {x}}(0)$ be 0 and rearrange the equation,

${\frac {X(s)}{X_{in}(s)}}={\frac {X(s)}{g\sin \theta \left({\frac {1}{s}}\right)}}=H(s)$

$\Rightarrow H(s)={\frac {1}{s^{2}+{\frac {\lambda }{m}}\,s+{\frac {k}{m}}}}$

The above is the transfer function that will be used in the Bode plot and can provide valuable information about the system.

$X(s)=g\sin \theta \left({\frac {1}{s}}\right)\left({\frac {1}{s^{2}+{\frac {\lambda }{m}}\,s+{\frac {k}{m}}}}\right)+x(0)\,\left({\frac {s}{s^{2}+{\frac {\lambda }{m}}\,s+{\frac {k}{m}}}}\right)+\left({\dot {x}}(0)+{\frac {\lambda }{m}}\,x(0)\right)\,\left({\frac {1}{s^{2}+{\frac {\lambda }{m}}\,s+{\frac {k}{m}}}}\right)$

Inverse Laplace Transform

Since the Laplace Transform is a linear transform, we need only find three inverse transforms. All of the these have complex roots, since ${\left({\frac {\lambda }{m}}\right)}^{2}<4\,{\frac {\lambda }{m}}$. Because I am not yet comfortable finding the inverse with complex roots by hand, I used a laplace transform program for the TI-89.

${\mathcal {L}}^{-1}\left\{{\frac {1}{s\left(s^{2}+{\frac {\lambda }{m}}\,s+{\frac {k}{m}}\right)}}\right\}=e^{{\frac {-1}{6}}\,t}\,\left[{\frac {-9}{40}}\cos {\left({\frac {{\sqrt {159}}\,t}{6}}\right)}-{\frac {3\,{\sqrt {159}}}{2120}}\,\sin {\left({\frac {{\sqrt {159}}\,t}{6}}\right)}\right]+{\frac {9}{40}}$

${\mathcal {L}}^{-1}\left\{{\frac {s}{s^{2}+{\frac {\lambda }{m}}\,s+{\frac {k}{m}}}}\right\}=e^{{\frac {-1}{6}}\,t}\,\left[\cos {\left({\frac {{\sqrt {159}}\,t}{6}}\right)}-{\frac {\sqrt {159}}{159}}\,\sin {\left({\frac {{\sqrt {159}}\,t}{6}}\right)}\right]$

${\mathcal {L}}^{-1}\left\{{\frac {1}{s^{2}+{\frac {\lambda }{m}}\,s+{\frac {k}{m}}}}\right\}=e^{{\frac {-1}{6}}\,t}\,\left[{\frac {2\,{\sqrt {159}}}{53}}\,\sin {\left({\frac {{\sqrt {159}}\,t}{6}}\right)}\right]$

Equation of Motion

Putting it all back together again gives,

$x(0)=g\,\sin {\theta }\,\left(e^{{\frac {-1}{6}}\,t}\,\left[{\frac {-9}{40}}\cos {\left({\frac {{\sqrt {159}}\,t}{6}}\right)}-{\frac {3\,{\sqrt {159}}}{2120}}\,\sin {\left({\frac {{\sqrt {159}}\,t}{6}}\right)}\right]+{\frac {9}{40}}\right)$

$+\,x(0)\,\left(e^{{\frac {-1}{6}}\,t}\,\left[\cos {\left({\frac {{\sqrt {159}}\,t}{6}}\right)}-{\frac {\sqrt {159}}{159}}\,\sin {\left({\frac {{\sqrt {159}}\,t}{6}}\right)}\right]\right)$

$+\left({\dot {x}}(0)+{\frac {\lambda }{m}}\,x(0)\right)\,\left(e^{{\frac {-1}{6}}\,t}\,\left[{\frac {2\,{\sqrt {159}}}{53}}\,\sin {\left({\frac {{\sqrt {159}}\,t}{6}}\right)}\right]\right)$

It is useful to have the equation in the form given above because $x(0){\text{ , }}{\dot {x}}(0){\text{ , }}\theta {\text{ , and even }}g$ can be varied and still give accurate results. The Matlab (or Octave) script below can be edited as described. Take note! $\lambda {\text{ (and all that depend on it), }}m{\text{ , and }}k$ cannot be altered (else the inverse Laplace is false)!

Matlab Script

Part 3 - Bode Plot

The bode plot shows useful information about the system we are analyzing. It has only to do with the transfer function, which means that it does not change based upon the input. However, it can show what a given frequency of a harmonic input will do to the output. For my example, it can be seen that at about $2rad/s$ there is a rise in the magnitude of the transfer function. If it were hit with a corresponding frequency by an input function, it could have very larg oscillations.