Error tolerant normal mode frequency

[hilbert2]

TL;DR Summary

Making a coupled oscillator system with a stable eigenfrequency for timekeeping purpose.

If a Hookean spring-mass system is made from one mass and a spring, to produce a system with a particular oscillation frequency, it's not a problem to use the propagation of errors concept to find how this frequency responds to small errors in the mass and spring constant. If a chain of oscillators with similar masses and springs (from same supplier) is used instead, any random errors in the masses and $k$:s should cancel out the better the more masses there are in the chain.

I haven't yet calculated how systematic errors in the spring constants (such as from wearing out with time) affect the eigenfrequencies of a coupled oscillator system, but this would be useful when making a mechanical clock with a timekeeping frequency that remains more constant despite springs becoming more or less stiff than before in long-term use. I would expect that some normal modes are more resilient in this sense than others, and some coupled systems could be better than just a simple linear chain with two springs connected to each mass. Have any of you seen this idea anywhere, before? I'm not sure what to call this in a Google search, but writing "stable oscillator normal mode" as search terms didn't return anything obviously related to this. The problem is clearly related to the stability of the eigenvalues of the matrix in the equation of motion, when small perturbations are added.

 

[anorlunda]

It sounds like you could benefit from studies of clockmaking. For centuries, until the advent of digital clocks, countless people struggled with exactly the kind of considerations that you mention. Their practical solutions are almost so numerous to mention. A visit to any clock museum can expose you to several.

Summary: Making a coupled oscillator system with a stable eigenfrequency for timekeeping purpose.

I'm not sure what to call this in a Google search, but writing "stable oscillator normal mode" as search terms didn't return anything obviously related to this.

I think the proper search terms look for clockmaking hits, because long before people had the mathematical tools or the language to describe your approach, they may have tried your ideas intuitively.

 

[hilbert2]

I'll try to find some pdf e-book about clock engineering. One intuitive way to minimize the effect of systematic error could be to have half of the springs made from something that becomes more elastic with time and the other half from something where the opposite happens

 

[anorlunda]

One problem you will have is that most clockmaking technology ceased before the Internet started. No matter what you do, it will be very difficult to prove that it was not tried before because the proof will not be posted on the Internet.

Patent searches may find more useful data than random Internet searches. However, the language used in patent applications is frequently obtuse, making it hard to understand what they are talking about.

 

[hilbert2]

I wrote a linear algebra code to solve the eigenfrequencies of a four-mass system, as in the image

but with all masses and spring constants different random numbers all taken from a Gaussian distribution with a mean value of $1.0$ dimensionless units and standard deviation of $0.02$ dimensionless units. The four normal mode frequencies and the histograms describing their distribution for 2000 instances of this random system are in these figures:

To find out whether the percentual standard deviation of any of these is smaller than that of the others, for same randomness in the masses and spring constants, I'd have to calculate this for even more instances of the random system and try to fit a Gaussian function in the histograms.

Edit: Just to be sure, I have to mention that the values calculated here are actually proportional to $\omega^2$, and not the unsquared frequencies $\omega$. I just set up the equation of motion with a matrix that would have 2:s on the diagonal and -1:s on the first off-diagonals in the case of zero standard deviation in the masses and spring constants, and then found the eigenvalues of this matrix.

 

[tech99]

If the individual springs are closely coupled than they behave as one spring with one combined mass. If the springs are slightly over coupled there is the possibility of oscillation at two frequencies. If the springs and masses are connected in tandem, forming a transmission line, the number of standing waves can be increased, and the total energy stored can be increased, corresponding to higher Q and less dependence on the driving power. In all cases, however, the temperature dependence is unaltered.

 

[hilbert2]

If the individual springs are closely coupled than they behave as one spring with one combined mass.

I guess you mean a situation where all the springs except those fixed to the endpoints of the system have really high $k$ so that the distances between neighboring masses remain almost constant. That is one special case where I could find out how randomness in the masses and springs affects the normal mode frequencies.

 

[hilbert2]

I may be resurrecting this thread from two months ago a bit late, but I just wanted to add the results I calculated for this (in case someone is looking for this topic in Google search). I calculated the fractional standard deviations (std. dev. divided by average value) of the normal mode freqs. for linear oscillator chains with 3 to 8 oscillators. All masses were exactly $m_i = 1$ in all simulations, and the spring contants were $k_i =1$ in average and their fractional standard deviation varied from $0.005$ to $0.100$. The calculation was made for $6000$ systems like this, with the force constants picked from that random Gaussian distribution.

The standard deviations of the $\omega$:s are shown here as a function of the std. devs. of the spring constants in these figures.

The eigenfrequencies clearly become more accurate when making the chain longer. If there is an odd number of frequencies, the "middle" frequency has smaller uncertainty than the others.

The fractional std. dev. of the highest eigenfrequency, when the frac. std. dev. of the force constants is $0.005$, is shown in the next image as a function of the number of oscillators.

There don't seem to be any publications about this, except one where they calculated the effect of this kind of randomness on the behavior of the Fermi-Pasta-Ulam oscillator chain: https://www.math.ucla.edu/~mason/papers/nelson-fput.pdf