Jaap meijaard wrote:
>   To: bikengr@netnet.net
>       ruina@cornell.edu
>       a.l.schwab@tudelft.nl
>   Cc: mcoleman@cems.uvm.edu
>
> Nottingham, 23 October 2007
>
> Dear Jim, (and other bicycle dynamicists)
>
> See my comments to your previous mail. They are indicated by ***(JPM):
> As to the paper by Tarrant: this does not seem to be a good contribution to the
literature. I can't understand everything he does. For instance, in Fig.2, the trail
seems to be negative. Although the paper is full of equations, it seems to be more
of a qualitative nature.
>
> Best regards, Jaap Meijaard.
>
>
> Jaap: Thank you!
>
> I am without a full paper copy, so I was unable to make such a comparison. I
> noticed that the printed ERRATA were unchanged, but this tabular correction
> is very interesting [of course, in my silly quest to figure out when the
> book was published]. I also took my original copy of Bourlet, published by
> the same press, and looked at all the other books advertised in the final
> pages. Carvallo was not there....
>
> The remainder of this email is really to Mike Coleman, but of course I would
> appreciate if you have anything to add or corrections to make. I helpfully
> put *** next to two sentences below, in case you care to comment!
>
>
> Mike: Digging around in the University of Michigan pdf, I finally found the
> Monocycle stability calculation embedded in the Bicycle section.
>
> SUMMARY
> To hint at what you will see below, his equation is correct in form, but I
> am not clear (I don't even know if Carvallo is clear) whether he neglects
> the wheel mass or inertia in comparison to rider mass or inertia.
>
> ***(JPM): If you take the equations (1), (2) and (3) on pages 39-41, you
should obtain the correct equations. Some approximations have
been made in the equations at the bottom of page 41, valid for
small wheel mass. The inertia quantities should be interpreted
as for the complete unicycle. The most important term neglected
is -(B_1 - mu_1*R^2) s n cos(theta) in the equation (p).
> He does come up with the need for the appropriate inertia asymmetry, and the
> existence of a critical speed for both eigenvalues to be positive. But
> possibly he has some sign or coefficient errors in the contribution of the
> wheel (relative to the more massive rider).
>
>
> DETAILS
> With reference to the 'printed page numbers' (not the pdf page numbers) this
> is pages 116 -- 118, namely Section 78. It refers immediately to the
> equations for p, n, and s of Section 24, which are at the bottom of page 41.
>
> These quantities are defined on page 3: p is THETAprime (the lean rate), n
> is NUprime (rate of change of heading of the ground tangent line), and s is
> the axial spin rate.
>
> I don't pretend to follow exactly what he does on pages 117 and 118 [in
> part, because the equations of section 24 don't seem to be reproduced
> exactly in 78?]. But the conclusions are relatively clear, and analogous to
> our own:
>
> ***(JPM): on page 117, the terms with s' are left out, because the axial spin
rate is assumed to be constant.
> His equation (1') on page 117 is his eigenvalue equation. It is quadratic in
> LAMBDA, just like ours when fore/aft rider position is suppressed.
>
> At a zero lean angle THETA, in equation 1', the coefficient of LAMBDA^2
> reduces to AC-L^2, namely the determinant of the moment of inertia matrix
> with respect to the ground contact, as defined on page 37 with respect to
> axes given on page 1. [A is roll inertia, C is yaw inertia, and L is product
> of inertia.] Carvallo is a little casual later on in saying that these
> quantities include the contributions of the wheel (whereas for the wheel
> alone (a thin ring) we would see the subscript "1", such as B1, etc.)
>
> At any rate, his highest order term perfectly matches our "det(M)", and he
> notes that it is positive without further discussion.
>
> His zero order term is given as (2), and his first order term as (3).
> Naturally he wants them both to be positive for stability. These equations
> are simplified and specialized to zero lean and constant heading, in
> equations (2) and (3) at the top of page 118. Again, I would not be able to
> validate his steps, but simply take the results as he reports them.
>
> ***(JPM): for the more exact equations, an additional term in dP/dn (line 2
from the bottom) would appear: -(B_1-mu_1*R^2) s . This is the
moment of inertia of the wheel about the spin axis times the spin
rate, with a minus sign. For a ring, this is -mu_1 R^2 s .
>
> Results
> His quadratic coefficient is det(inertia matrix about contact)
> It agrees perfectly with our derivation.
>
> His first order coefficient is (product of inertia)(rolling speed)(m1R-mgh)
> I think the product of inertia is the negative of the tensor component of
> inertia; and mgh dominates m1*R (it refers either to the overall system
> mass, or least the rider mass)
>
> ***(JPM): L is the negative of the tensor component. Note x-axis is in
forward direction and z-axis is up. With corrected equations,
the mu_1 R would disappear. mu is total mass and h height of
centre of mass of the whole system. There is no sign error.
> But I think our coefficient p3 is (Ics)(R thetadot)(m1R + m2z)
> ***** IF I INTERPRET CORRECTLY, HIS PARENTHETICAL (m1R - mgh) SHOULD
> ACTUALLY BE -(m1R + mgh), IN OTHER WORDS HE MAY HAVE A SIGN ERROR IN THE m1R
> TERM. Such a sign error, if that is the case, would not alter his result,
> since the rider is generally far more massive and higher. His final
> conclusion for this coefficient is "L must be less than or equal to zero".
> He states this condition "is new and important; it shows that stability
> depends on the position of the rider on his cycle"
>
> Thus, even if he has some error in one sign of a small term, or has a mixup
> about whether he has already included the wheel mass in the system mass, his
> expression is very close to ours, and leads to the very same inertia
> restriction.
>
> His zero order coefficient is very similar to our quadratic expression in
> speed. His speed independent quantity -C MU g h matches ours exactly ***** if MU h
> represents the total moment of mass; otherwise it matches ours approximately
> by neglecting the contribution of the wheel.
>
> His m m1 R^3 h s^2 is also very close to our expression, but probably not
> exactly the same. ***** Where he has m*h, we have (2 m1 R + m2 z). He could
> be neglecting the wheel contribution; but if he includes it he would not
> have the factor of 2. Qualitatively his form is correct, but he would
> probably find a slightly different critical speed.
>
> ***(JPM): With the correction, a term (mu mu_1 s^2 R^4) should be added.
Remember, mu h should represent the total moment of mass. 
> OVERALL: qualitatively in the proper form, with possible differences from
> occasional neglect of the wheel relative to the rider; and possible errors
> (sign of the wheel contribution in one case; and neglect of a factor 2 on
> the wheel contribution in another case).
>
> ***(JPM): All differences are due to neglected terms in the equations at
the bottom of page 41.
> JMP
>
>
>
>