(Spring 2003)


Results: Here are some of my preliminary results on the analysis of the dynamic behavior of a bicycle. For the analysis we used our software system SPACAR. This software has been developed at Delft University of Technology and is capable of doing dynamic analysis of flexible multibody systems. The method is based upon the Finite Element Method where the expressions for the generalized strains can be used as constraint equations to model partly rigid systems. Pure rolling is be modeled by zero generalized slips. The equations of motion are expressed in terms of independent generalized coordinates this to facilitate the numerical integration of the equations of motion.

The model of the bicycle is an ordinary Dutch city bike, like this one:

Dutch Bicycle

This is a sketch of the SPACAR model, with all the element and node numbers:

Model Sketch

Finally the input file, which contains the complete definition of the bike, looks like this:

* bike3
pinbody  1  1 2 3
pinbody  2  1 2 4
pinbody  3  1 2 5
hinge    4  2 6      1 0 0
pinbody  5  5 6 7
hinge    6  2 8      0.36 0 0.9
pinbody  7  3 8 9
pinbody  8  3 8 10
hinge    9  8 11     0 -1 0
wheel   10  10 11 12 0 -1 0
hinge   11   2 13    0 -1 0
wheel   12   4 13 14 0 -1 0
hinge   13  2 15     0 1  0
hinge   14  15 16   -1 0  0
hinge   15  16 18    0 0 -1
pinbody 16  17 18 14
x  1  0.50 0 -0.60
x  3  0.90 0 -0.90
x  4  0    0 -0.35
x  5  0.25 0 -1.00
x  7  0.30 0 -1.20
x  9  1.00 0 -0.70
x 10  1.20 0 -0.35
x 12  1.20 0  0
x 14  0    0  0
x 17  0    0  0
fix 17  1 2 3
fix 18  1 2 3 4
line 6 1
rlse 9 1
enhc 10 4  9 1
enhc 10 5 15 1
inpute 11 1
enhc 12 4 16 1
enhc 12 5 16 2
rlse 13 1
line 14 1
rlse 15 1
rlse 16 1 2 3
mass  1 16
mass  2  2 0 0 8 0 2
mass  4  2
* mass  6 10 0 0 10 0 2
mass  6 6 0 -4 10 0 6
mass  7 80
mass  9  1
mass 10  2
mass 11  0.12 0 0 0.24 0 0.12
mass 13  0.12 0 0 0.24 0 0.12
* dependent should be calculated from m*g
* take g=10
force  1  0 0 160
force  4  0 0  20
force  7  0 0 800
force  9  0 0  10
force 10  0 0  20
ed 11 1 30
epskin 1e-3
epsint 1e-3
epsind 1e-3
timestep 1 0.001
hmax 0.01
end
eof

In a first analysis we look at the steady motion of the upright bicycle with a rigid rider, hands free, and pure rolling. These are the root-loci from the linearized equations of motion with the forward speed v as a parameter.

You can take a closer look at the original Figure Bike3xrl.pdf.

In order to get an idea about the stability of this upright motion look at the bottom-left figure, the forward speed v versus the Real part of the eigenvalues l. Now its customary to have the parameter, here the forward speed v, on the abscissa and the Real part of the eigenvalue l on the ordinate. The stability diagram then looks like this:

You can take a closer look at the original Figure Bike3xRev.pdf, the dots are horizontally equidistant at 0.1 m/s.

We see that at a forward speed v of less then 0.9 m/s the bike simple falls over, 4 real eigenvalues l with 2 positive ones. We call this the capsize mode. At a speed of 0.9 m/s two real eigenvalues become identical and start forming a conjugated pair after which we have an unstable oscillatory motion, the so-called weave motion. This weave motion is an oscillatory motion in which the bicycle sways about the headed direction. At about 4.1 m/s this weave becomes stable. But then at about 5.7 m/s the previously stable capsize becomes marginally unstable. So at high speed, v>5.7 m/s, we have an unstable capsize mode but the timescale is so long, l=0.2 1/s or t=5 s, that in practice you can easily correct this mode. Now look at the bottom-right part of the previous figure, the 3D depiction of the root loci as a function of the forward speed, and identify the different modes at increasing speed v.

In a second full nonlinear analysis we look at the motion of the bike by means of a forward dynamic analysis of the perturbed upright motion. The perturbation is a small lateral velocity of 0.1 m/s for the whole bike to start the unstable motion, if present. The results are visualized by a number of VRML (Virtual Reality Modeling Language) files at different initial forward speeds. You can view these VRML files in internet browsers like Firefox, Opera or Internet Explorer with the plugin Freeware from Cosmo. For the bike to start moving you must click on the red frame of the bike. If you want to see the path of the rear and front wheel, then you can click on one of the wheels. You can change your viewpoint: look in the Viewpoint List located below left. A very nice one is the one called 'Camera', which is a moving camera with stable horizon (as if you were riding along on the rear passenger seat).

Ok, so now for the VRML movies:

bike3v000.wrl at v=0 m/s, unstable capsize.
bike3v175.wrl at v=1.75 m/s, unstable weave.
bike3v350.wrl at v=3.50 m/s, unstable weave.
bike3v368.wrl at v=3.68 m/s, stable weave in a curve! (a nice nonlinear result)
bike3v490.wrl at v=4.90 m/s, a stable weave.
bike3v630.wrl at v=6.30 m/s, an unstable capsize.

Note that obtaining a speed of 36 km/h and above is no problem in Ithaca, although I myself do not dare to go that fast.