How do I choose $ b_1 $, $ b_2 $ and $b_3$ to make the masses $m_1$, $m_2$ and $m_3$ have a critical damping behavior? If I have just one mass $m$ and one spring $k$ and one damper $b$, the damper would have the value $$ b = 2*\sqrt{m*k} $$

to give the mass a critically damping behaviour. But now I have a system of ODEs. The stiffness is very easy to determine because as the developer I can choose the static equilibrium point. My picture (and accompanying equations) below will explain.

System of ODE on second order: $$m_3*ddh = m_3g - k_3(h-y)-b_3(dh-dy)$$ $$m_2*ddy = m_2*g +k_3(h-y)+b_3(dh-dy)-k_2(y-z)-b_2(dy-dz)$$ $$m_1*ddz = m_1*g + k_2(y-z)+b_2(dy-dz)-k_1*z-b_1*dz$$

Determine the stiffness if $h$, $y$, and $z$ are known: $h>y>z$ $$k_3 = \frac{m_3*g}{h-y}$$ $$k_2 = \frac{m_3*g+m_2*g}{y-z}$$ $$k_1 = \frac{m_3*g+m_2*g+m_1*g}{z}$$

enter image description here


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This is a 'trick' question since critical damping is strictly defined for a second order (2 state ) linear dynamic system. Your diagram results in a 6th order system. And while you might consider separately solving each platform independently, as they are 2nd order subsystems, the subsystems 'couple' and so do the damping factors.

One may be able to solve a linear transfer function for the 6th order system and use model reduction techniques to acquire an appropriate 2nd order model, but a faster, simpler approach would be to simulate the system, and 'tune' your three factors by trial and error so that the response of the 6th order system approaches what appears closest to critical damping.

If the simulation tool has optimization functions, like Vissim, you can automatically determine a set of parameters such that your system approaches a second order critical damped response of a reference model. Maybe 30 min and you have an answer.

  • $\begingroup$ So you mean, simulate with MATLAB (or other software) and try diffrent values for $b1$, $b2$ and $b3$ so the system will act as the behaviour I want it to do, will solve my problem, even if it's not and an exactly solution for the system? I have heard many diffrent ways to solve a damping matrix, but thoose way seems to be quite hard to apply. $\endgroup$ – Daniel Mårtensson Jun 7 '17 at 20:04
  • $\begingroup$ Critical damping precisely defined, and although you may be able to achieve in the independent 2nd order subsystems of your system, coupling will lead to damping other than critical. Critical damping simply not defined for higher than 2nd order systems $\endgroup$ – docscience Jun 7 '17 at 20:16
  • $\begingroup$ Matlab, simulink, vissim, yes. There is no exact solution since your system, 6th order. Critical damping is not defined. $\endgroup$ – docscience Jun 7 '17 at 20:17
  • $\begingroup$ You can however get close to the response of a second order critically damped system $\endgroup$ – docscience Jun 7 '17 at 20:19
  • $\begingroup$ Thanks! If critical damping is not defined for MDOF-system, I will no longer think about to solve it on exact form. I going to simulate this with ODE45 from Octave. It's a free "version" of MATLAB. By the way! This is the "many diffrent ways" I was talking about donkey2ft.files.wordpress.com/2014/03/damping-matrix.pdf I was trying to do the "Proposed damping matrix"-method, but it did not work beacuse I have to add in and remove dampers that should not exist. Just to make the C damping matrix fit my system. $\endgroup$ – Daniel Mårtensson Jun 7 '17 at 20:56

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