# How to determine the critical damping factors for a multi-DOF mass-spring system [closed]

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: $h>y>z$ $$k_3 = \frac{m_3*g}{h-y}$$ $$k_2 = \frac{m_3*g+m_2*g}{y-z}$$ $$k_1 = \frac{m_3*g+m_2*g+m_1*g}{z}$$ ## closed as off-topic by Kyle Kanos, peterh, alephzero, Yashas, ZeroTheHeroJun 8 '17 at 12:35

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• – sammy gerbil Jun 8 '17 at 6:33
• How? I can do that with one mass. But I don't know how to do that if there are multiple masses in the system. – Daniel Mårtensson Jun 8 '17 at 6:57
• Exactly. Your related question (answered by Alephzero) says you can do it for one mass, then asks "But the question is: If I have two or more ODE:s with diffrent spring stiffness and dampness ...". You seem to be asking the same question multiple times : eg Get the damping coefficient if I know the natural frequency - MDOF – sammy gerbil Jun 8 '17 at 9:10
• I assume that is not possible to get critical damping for a MDOF system. – Daniel Mårtensson Jun 8 '17 at 16:03

• 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. – Daniel Mårtensson Jun 7 '17 at 20:04