Hello, I am a Mechanical Engineering student and my project is to design a self stable bicycle. The bicycle will be run by a brushless dc motor and I want to get the eigenvalues as a function of forward speed to determine the range of speeds the bicycle is self stable. My question is when all of the terms of the linearized equation of motion of the bicycle are calculated , how can the eigenvalues of the equation be calculated?

Thank you for your time, Christofors Miltiadous — Preceding unsigned comment added by 31.205.38.184 (talk) 17:58, 6 February 2019 (UTC)[reply]

First, in case you haven't already done this, you need to rewrite the equations in matrix form: ${\displaystyle M\mathbf {\ddot {q)) +C\mathbf {\dot {q)) +K\mathbf {q} =\mathbf {f} }$ where where M = M0, C = C1*v, and K = K0 + K2*v^2

Then there are at least four different methods of calculating eigenvalues in MATLAB directly from the matrix coefficients, similar to the ones provided in the Benchmark Paper:

1. eig(A,B), to calculate the generalized eigenvalues λ such that Ax = λBx where A = [-C -K; eye(2) zeros(2)] and B = [M zeros(2); zeros(2) eye(2)];
2. eig(C), to calculate the eigenvalues λ such that Cx = λx where C = [-M0^-1*C1*v, -M0^-1*(g*K0+v2*K2); eye(2), zeros(2)]
3. roots(D) of the characteristic equation; and
4. polyeig().

The first three methods produce eigenvalues that differ from those published in the Benchmark Paper by 1e-13 to 1e-16, usually 1e-14, depending on the particular method, the particular eigenmode, and the forward speed. The fourth, polyeig(), merely considers the first two methods as special cases.

I provide more details in a 2011 ASME paper titled "Benchmarking Bicycle and Motorcycle Equations of Motion"

Alternatively, you could just use JBike6, which calculates the eigenvalues for you.

If none of this helps, look me up at UWM and email me at uwm.edu -AndrewDressel (talk) 19:31, 6 February 2019 (UTC)[reply]