# Harmonic systems, system linearity

I have a bucket full of water with a hole in bottom, lying on a spring like in the picture I have attached. Is this system considered to be linear or not? • Welcome to Engineering SE. Is this a HW question? Jun 27 '16 at 10:15
• What is the equation of motion? Have you derived it? You should be able to tell from the equation if it is linear or not. Jun 27 '16 at 18:52
• No real-world system is ever linear, if you model it in enough detail. If you don't tell us what you want to know about the system, it is impossible to say whether it is approximately linear or not. For example, do you want to model the dynamic response to each individual drip of water, including the fact that the drips will not necessarily occur at equal time intervals ..... ??? Jun 27 '16 at 22:18
• A bucket with a hole might be linear, but it is not time-invariant. Most of the simpler techniques for analyzing such systems require both conditions to be true. Jun 28 '16 at 12:13
• I smell a nice truncated infinite series here :-) Jun 28 '16 at 13:18

Look at the size of the nonlinear terms relative to the linear ones; then decide for yourself whether you can throw them away. This is greatly simplified using nondimensional analysis contrived so that all coefficients have a magnitude on the order one.

So set up reference quantities for the terms that appear in the governing equations. The reference mass might be the starting mass or the mass of a half filled bucket. So the mass in the governing equation is the actual mass divided by the reference mass. You can constrain the frequency using nondimensional time. For instance, you might divide the actual period by the natural period of the reference mass. Instantaneous flow rate out the bucket is now expressed in reference masses per reference cycle. Now you can decide under what conditions you are willing to disregard higher order terms (and justify the decisions) because you don't have to worry about huge coefficients counteracting some product of otherwise small terms.

This is a big part of exploiting series expansions such as Taylor series. To do so, you need to know the higher order terms get smaller at a decent rate.

(This problem would be a lot more fun if the bucket was supplying fuel to a rocket engine and the spring represented a vibration mode. Because now you really do want to get the terms on the order of one.)

this system is constituitively linear but contains one time-varying coefficient, so it is technically nonlinear. even if your modeling software does not permit time-varying coefficients, you can do a piecewise model in which you start the run, halt it, change the system mass, and restart it. this will get you close, and by shortening the run time at a given value of mass you will approach the behavior of the "real" system.

• A time-varying coefficient would imply a linear-time-varying system not non-linear. However it is not true that there is a time-varying coefficient, namely the outflow of the water also depends on the acceleration of the bucket, so would this coefficient would also be a function of the states of the system, which does make the system non-linear. Oct 24 '17 at 16:51

It's a nonstationary non-linear system. It can be easily linearized with a large adequate answer around it's working point.