# Anomaly in viscous force - vs - velocity relationships

It is well known that the viscous force (or drag force) is directionally proportional to the square of the velocity in many cases. For example the torque on motors driving a fan-type load (pump) is proportional (approximately) to the square of the rotational speed. The air drag force on a car or a plane has a similar relation between linear drag force and straight line velocity.

However in a mass-spring-damper system common in mechanics and control theory, the viscous force F is proportional to the velocity itself, as in the relation is F = -Bv, where v is the velocity and B is the viscous damping constant.

What is the reason for such an anomaly? Are they for different conditions/situations? If so, how? Or am I missing something entirely here?

## 2 Answers

The examples you mentioned both occur due to turbulent flow of a gas. However in a lot of mechanical systems damping often occurs due visco-elastic deformation or due to shearing of the lubricant inside a bearing, which due to different scales can often be described as a laminar flow. In these cases a good approximation for the "drag forces" would be a linear model.

Another reason linear models are often used is that the differential equations always have algebraic solutions and thus easy to calculate.

• So in essence the actual variation is a quadratic one, and the linear variation is a good approximation in some cases? – Transistor Overlord May 20 '17 at 17:15

So first, there things known as dashpots, for which the force really is proportional to velocity. These are the things that intended to be represented by the damping term in the classical mass-spring-damper systems that you find in the textbooks.

But there is a much bigger picture idea here. The mass-spring-damper systems in textbook are just mathematical models. The important thing to realize about models is this: All models are wrong, some models are useful. We use the mass-spring-damper model not because it is exactly correct for real physical systems, but because it is simple to understand and is close enough for many purposes.

Specifically regarding damping, there are some systems for which the damping really is proportional to velocity squared (or maybe some other relation such as dry friction damping, aka coloumb frction, for which the force is proportional to the sign of the velocity). But including a velocity squared term can make the resulting equations much more difficult to solve. e.g. it could go from 30s to solve the equation to 3 hours to solve the equation. If we absolutely must know the exact answer to within 0.1%, we will spend that 3 hours. But if we only want an approximate answer to within, say, 10%, we will take the easier route and just model the damping as if it was a dashpot. We should remember that neither model is exactly correct, both of them are wrong, just one is closer to the truth than the other. We pick the one that is most useful for the given situation.