I was wondering how I can estimate the damping coefficient of my car by doing the hand bouncing the car body and watching the motion of the car?

I just need a rough estimate of the damping coefficient for simulation in matlab of my car with a quarter-car model. The spring constant I found by jacking the car up until the tire barely lifted off the ground and measured the difference between a reference and the new height from the reference. Used 1/4 of the weight of the car with the assumption that the CG is in the middle splitting the weight of the car 50/50 front and rear

Mechanics usually bounce the car by applying hand force and watching how the car moves in harmonic motion. Is there a way to estimate the damping coefficient by using this technique?

I was thinking Mythbusters type stuff like camera, but I have an IPHONE 4 if I know the FPS of the camera I might be able to measure it frame by frame, if I have a measurement stick and reference point. Not sure if theres a practical way of doing this, I know the motion will be simple harmonic damped motion, so if you count the half-cycles maybe you can get the damping, not sure of the forumula to use?

Or is there another technique like dropping the car a few inches or something to estimate it? 24 minutes ago - 4 da


1 Answer 1


Assuming your car can be accurately described by a dambed harmonic oscillator when oscillating undisturbed, the differential equation defining its vertical motion is:

$$ \frac{d^2y}{dt^2}+a\frac{dy}{dt}+by=0 $$

Where $a$ is the damping coefficient and $b$ is the spring constant. It's characteristic equation is:

$$ r^2+ar+b=0 $$

In principal, there are three different cases depending on the nature of the roots of this equation, but assuming your car will exhibitit exponentially decreasing oscillations, we can assume that it has two complex roots, $r_1=\alpha+i\beta$ and $r_2=\alpha-i\beta$ and the general solution to the differential equation will be:

$$ y=e^{\alpha t}\left(C_1\cos\beta t+C_2\sin\beta t\right) $$

If you initialize oscillations with large amplitude in your car manually and then record $y(t)$ with a camera, you should be able to estimate $\alpha$ (basically by registering how fast the amplitude decrease between each oscillation). Note that $\alpha$ should be negative. The easiest way to do this might be to plot the natural logarithm of the highest point reached for each oscillation against time. You should get a straight line with $\alpha$ as slope. Just remember using the car's rest position as baseline when measuring $y$.

Once you have $\alpha$, just use that:

$$ (r-\alpha-i\beta)(r-\alpha+i\beta)=r^2+ar+b\\ \Rightarrow r^2-2\alpha r+\alpha^2+\beta^2=r^2+ar+b\\ \Rightarrow a=-2\alpha $$

to calculate your damping coefficient.

  • $\begingroup$ Wow thanks! very informative and well formatted response :) appreciate your time and effort. I'm confused about this part: "the easiest way to do this might be to plot the natural logarithm of the highest point reached for each oscillation against time" Instead of using a camera would it be possible to use an accelerometer and find the highest points if you graph the output of the acceleration? I remember a concept known as logarithm decrements from school briefly, is this what you are talking about? $\endgroup$ Jan 14, 2013 at 17:29
  • $\begingroup$ Yes, the accelerometer would probably work even better, since it's a more accurate instrument. You can basically deduce $\alpha$ from the acceleration time series exactly as you would from the position time series. I hadn't heard about the concept "logarithm decrements" before, but after looking at the wiki-page on the subject, I think it should work. $\endgroup$
    – jkej
    Jan 14, 2013 at 18:09
  • $\begingroup$ Would I have to integrate the data to get velocity or the acceleration time series would actually be able to be used? I Always thought that car suspensions would be critically damped to return to reference as fast as possible but I keep reading that they are underdamped with damping coefficients of 0.2 to 0.4ish $\endgroup$ Jan 14, 2013 at 18:22
  • $\begingroup$ You should be able to use the acceleration time series. If you differentiate the expression for $y$ above twice, you will get a messier expression, but it will still basically be the same exponential multiplied with a sum of a bunch of sine and cosine terms with different factors. You would still be able to rewrite it as the formula above but with other values for $C_1$ and $C_2$. I don't know if car suspensions are generally underdamped or not. I assumed they were, but you probably know more than me about this. If they are indeed critically damped you need to do this differently. $\endgroup$
    – jkej
    Jan 14, 2013 at 18:39

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