I am a math lecturer and in my teaching of second-order linear differential equations I present, as an application, the classical mass-spring-dashpot system (and its RLC analogue). According to my understanding, if the overdamping case modelled an automobile suspension system, the ride would be uncomfortable. Furthermore, critical damping represents the minimum damping that can be applied to the physical system without causing oscillation.

One textbook I use (Ordinary Differential Equations and Applications by Weighfer and Lindsay) says: (Critical damping) is often the desired configuration for practical aplications since it represents the weakest damping before oscillatory becomes possible.

Nevertheless, another textbook I use (Differential Equations for Engineers by Xie) states that: Most engineering structures fall in this category (i.e. underdamping case) with (the dimensionless) damping coeffient $\zeta$ usually less than $10\%$. [$\zeta=\frac{\gamma}{2\sqrt{mk}}$ for the mechanical system; $\zeta=\frac{2}{R}\sqrt{\frac{C}{L}}$ for the electrical analogue].

Question 1: Which author is right?

Question 2: In general What are some practical applications where each case (heavy, critical and light damping) is desirable or not desirable?

Thank you very much.

  • $\begingroup$ If a car shock fails then it is underdamped and oscillates uncomfortably and dangerously. Why would overdamped be uncomfortable? Usually they bounce rebound and recover. $\endgroup$
    – Solar Mike
    Nov 20, 2022 at 14:30

1 Answer 1


Question 1 constitutes a false dichotomy, since the two books might be talking about completely different applications. They might both be true or both false. To me, they both seem to be way overgeneralizing.

In many systems, the overall desire is for the system to achieve a new equilibrium state quickly following any change to its inputs. Critical damping is the fastest response you can get without any overshoot at all. But sometimes a different measure is used — minimum settling time, which is defined as the system getting to a point that is within some delta (error window) of its final state in the least amount of time. In second order systems, it turns out that this is achieved by allowing a certain amount of overshoot — in other words, slightly under-damped relative to critical damping.

Vehicle suspensions are one example of such a system. But unfortunately, they must deal with a wide range of input conditions with limited ability to modify their own parameters. It's relatively easy to make a passenger car ride comfortably under most conditions, because the change in mass due to passengers, etc. is a small fraction of the total mass. But have you ever ridden in a lightly-loaded truck? It's very uncomfortable, because the springs and shocks are designed to handle a full load. When the mass is much lower than that, the system is heavily over-damped. You feel every bump in the road.


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