Where does the equation C=2*sqrt(km) for Critical Damping come from?

[physics51]

Homework Statement

what's the background on this equation, I can't find anything about I, what can you guys tell me about it related to damping, when to use it, how to derive it, what it means, etc.? Anything is useful!! Thank you!!

Relevant Equations

C= 2 sqrt km

Im using this equation to find the damping from a ruler cantilever experiment. Any information about what critical damping really means and how it reflects in a ruler cantilever is also really helpful. Thank you again.

 

[berkeman]

Welcome to PF.

Does this help? It's the frirst hit on the Google search list for critically damped spring mass system

https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=&cad=rja&uact=8&ved=2ahUKEwjxuqq-2Y72AhUSIEQIHT_7Dr4QFnoECAMQAQ&url=https://ocw.mit.edu/courses/mathematics/18-03sc-differential-equations-fall-2011/unit-ii-second-order-constant-coefficient-linear-equations/damped-harmonic-oscillators/MIT18_03SCF11_s13_2text.pdf&usg=AOvVaw2kSZcTAr8A9gOV6ZAwZHIf

 

[vanhees71]

From the context I guess that you are discussing a damped spring pendulum with the equation of motion
$$m \ddot{x}=-C \dot{x} - k x.$$
To solve this equation a nice trick is to make the substitution
$$x(t)=\exp(\lambda t) y(t).$$
Plug this into the equation and check that then $y$ fulfills
$$m \ddot{y}+(C+2 \lambda m) \dot{y} + [k+\lambda(C+\lambda m)]y=0.$$
Now make $\lambda=-C/(2m)$ to get rid of the term with $\dot{y}$. Then the equation of motion for $y$ simplifies to
$$m\ddot{y} + \left (k-\frac{C^2}{4m} \right) y=0.$$
Now discuss what happens for the different values of the expression in the bracket,
$$k-\frac{C^2}{4m}<0, \quad k-\frac{C^2}{4m}=0, \quad k-\frac{C^2}{4m}>0.$$

 

[Orodruin]

For an ODE with constant coefficients I’d just make the ansatz $x(t) = A \exp(\lambda t)$ and solve the characteristic equation, but any way that works works.

 

[vanhees71]

Well, then you've the trouble with exactly the case of critical damping ;-).

 

[Mister T]

what's the background on this equation, I can't find anything about I,

Where did you look? Most college-level introductory physics textbooks will have a discussion of critical damping and will derive this formula.