Third Order Differential Equations (Mass-Spring-Damper-etc)

In summary, the conversation discussed the concept of damping constant in a third-order equation for mass-damper-spring systems and its interpretation in the real world. The speaker also mentioned the characteristic equations for second, third, and fourth order systems and their respective damping ratios and frequencies.
  • #1
Dnandrea
1
0
So I'm not sure whether this should be posted in the Physics forum or not, but here goes:

I'm a Junior ME student at NC State taking a Vibrations course. We've gone over the general differential equation for mass-damper-spring systems where

Meq⋅x'' + Ceq⋅x' + keq⋅x = Fextermal

What I was wondering is, if you had a third order equation, where x''' (jerks, I believe) was added in, what would be the "damping constant" equivalent that would be multiplied by x''' and how could this be interpreted in the real world?

If you follow units, I think it should be mass ⋅ time as the coefficient. For keq, if you assume N/m, can be canceled out into kg/s2, Ceq into kg/s, and Meq into kg. Following that trend, the coefficient would have to be kg⋅s.

Mass ⋅ time I've tried researching and came across something called "action" that was really interesting. I'm not sure whether I'm making this much harder than it needs to be. Any explanation?

Thanks!
 
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  • #2
Dnandrea said:
What I was wondering is, if you had a third order equation, where x''' (jerks, I believe) was added in, what would be the "damping constant" equivalent
As for a 2. order system, written in the Laplace domain, the characteristic equation could be expressed:

s2 + 2ζωns + ωn2 = 0

where ωn is the freqency [rad/s] and ζ is the damping ratio of a sinusoidal oscillation. If ζ<1, the roots of the equation will be complex, and as a complex root will have a conjugated root, the roots will have "used" both roots in the polynomium. There is no further to add. ( See fig. 2. )
As for a 3. order system, you could express the characteristic equation:

( s + a )( s2 + 2ζωns + ωn2 ) = 0

so the same ζ is still there, in principle unchanged. Only a factor ( s + a ) has been added, which is just an overlayed exponential function, that has no damping ratio. The interpretation of the ζ is the same. ( See fig. 3. )

The charateristic equation for a 4. order system could be expressed:

( s2 + 2ζ1ωn1s + ωn12 )( s2 + 2ζ2ωn2s + ωn22 ) = 0

thereby having 2 different damping ratios and 2 frequencies. ( See fig. 4. )
 

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Related to Third Order Differential Equations (Mass-Spring-Damper-etc)

1. What is a third order differential equation?

A third order differential equation is a mathematical equation that involves a third derivative of a variable with respect to another variable. It is typically used to model systems that involve acceleration, such as the motion of a mass attached to a spring with a damper.

2. How are third order differential equations used in real-world applications?

Third order differential equations are commonly used in physics and engineering to model systems that involve acceleration, such as the motion of a mass attached to a spring with a damper. They are also used in control theory to design controllers for mechanical systems.

3. What is the general form of a third order differential equation?

The general form of a third order differential equation is:
y''' = f(x, y, y', y''),
where y is the dependent variable, x is the independent variable, and y', y'', and y''' are the first, second, and third derivatives of y, respectively.

4. How do you solve a third order differential equation?

To solve a third order differential equation, you can use techniques such as separation of variables, substitution, or the method of undetermined coefficients. It is important to first determine whether the equation is linear or nonlinear, as this will dictate the appropriate method to use.

5. What are some common applications of third order differential equations?

Some common applications of third order differential equations include modeling the motion of a mass-spring-damper system, analyzing the stability of control systems, and studying the behavior of oscillating systems. They are also used in fields such as mechanics, electronics, and biology to model various physical phenomena.

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