Is there place for higher order derivatives in mechanics?

In summary, higher order derivatives are important for understanding mechanics and high performance servo systems.
  • #1


The building of theoretical mechanics can be constructed using only the first and the second derivatives (those of coordinates in case of kinematics: velocity and acceleration and those of energy in case of dynamics: force and gradient thereof). It is obviously unavoidable if one wants to deal with linear responses. However, it is a feature of mathematical formalism we use.

My question is whether we have any justification or proof that taking into account higher order derivatives do not lead us to deeper levels of understanding mechanics (or may be some other areas of physics whose mathematical description ist based on the 1st and the 2nd derivatives) in general case, at least in principle (it is possible of course to invent a special case where they do play some rôle, but I am not interested here in inventing such cases)?
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  • #3
I have seen the higher derivatives in camshaft - follower analysis. As mentioned in the link provided by jedishrfu.
  • #4
The third derivative of position with respect to time (in other words the first derivative of acceleration with respect to time) is usually called the jerk. When driving a car, for example, you feel acceleration. It might, for example, push you backwards in your seat. If the acceleration is constant in both magnitude and direction, that push that you feel is constant. When the acceleration changes you are jerked around in your seat. The faster you change that acceleration (in either magnitude or direction) the more you are jerked around in your seat.

Want a smooth comfortable ride? Minimize the jerk.
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  • #6
Engineers working high performance servo systems get real good at using jerk. When the control engineer first started up the machine shown in US Patent Application 20120167527, the machine would not run at design speed and many of the parts clanged and banged. The control engineer called an emergency meeting, claiming a mechanical problem. When asked if he had implemented the S-curve (finite jerk) motion profiles, he had not. When he implemented them, the machine ran design speed with no clanging or banging.

Jerk is also important in high performance cam-operated systems, such as in automobile engines.

Generally, jerk must be included in the design of any mechanism where the period of the lowest natural frequency is greater than 5-10% of the fastest move.
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1. What are higher order derivatives in mechanics?

Higher order derivatives in mechanics refer to the derivatives of a function that are taken more than once. In other words, they are the derivatives of derivatives. For example, the second derivative of a position function with respect to time would be a higher order derivative.

2. Why are higher order derivatives important in mechanics?

Higher order derivatives are important in mechanics because they provide more detailed information about the motion of an object. While the first derivative (velocity) tells us the rate of change of position, the second derivative (acceleration) tells us the rate of change of velocity. This additional information can be useful in understanding the behavior and dynamics of a system.

3. Is there a limit to the number of times a derivative can be taken in mechanics?

Technically, there is no limit to the number of times a derivative can be taken in mechanics. However, the practicality of taking higher order derivatives decreases as the order increases. This is because it becomes increasingly difficult to measure and interpret these derivatives accurately.

4. Are higher order derivatives used in real-world applications?

Yes, higher order derivatives are used in real-world applications, particularly in fields such as engineering and physics. They are commonly used in the analysis of motion, such as in the design of vehicles and structures, as well as in the study of complex systems like weather patterns and fluid dynamics.

5. Can higher order derivatives be negative in mechanics?

Yes, higher order derivatives can be negative in mechanics. This means that the rate of change of a quantity is decreasing over time. For example, if the third derivative of a position function is negative, it means that the acceleration is decreasing, which could indicate that the object is slowing down.

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