A two degree-of-freedom model of a caster produces a fourth-order system that almost certainly is capable of being unstable under certain conditions. The analysis of a fourth-order system in literal coefficients is, however, a formidable job so a third-order caster model will be developed first. Allowing the pivot to have flexibility will do this if the slip angle is constrained to vanish. This is particularly logical for casters using nearly rigid wheels instead of pneumatic tires. In such cases, an equivalent cornering coefficient would be very large indeed. In many practical cases it would be logical to assume that the wheel would have almost no slip angle until it started to slide.
Fig. 8.4shows a top view of the caster. Because of the similarities between the equations of motion for vertical and inclined axis casters, the simpler case of a vertical axis caster will be analyzed. The new feature is the introduction of a spring with constant k at the pivot. This spring actually represents the flexibility of the structure supporting the wheel in the pivot bearing or, if the spring constant is zero, an amount of lateral play in the pivot or wheel bearings.
This analysis essentially recapitulates the analysis of the two degree- of-freedom trailer model in Chapter 5which began with a fourth-order dynamic model using Lagrange’s equations and then reduced the system to third order by assuming a very large cornering coefficient. In this case, however, Newton’s laws and the imposition of a zero slip angle assumption will lead directly to a third-order equation of motion.
Observing that the lateral acceleration of the center of mass is given approximately by the expression, x¨+ah¨, Newton’s laws can now be written
assuming that h is small enough that the small angle approximations can be used.
mð ¨x þ a ¨hÞ ¼ kx F ð8:22Þ
mj2 ¨h ¼ akx Fb ð8:23Þ
Now instead of relating the lateral force to the slip angle, we simply assume that the slip angle or, more directly, that the lateral velocity at the wheel- road contact point is zero.
x:þ Uh þ ða þ bÞh:¼ 0 ð8:24Þ
The lateral force can be eliminated from Eqs. (8.22) and (8.23) and, after differentiating the result, Eq. (8.24) can be used to eliminate x:and x¨. The result is a third-order equation of motion,
mðj2þ b2Þh:::þ mbU ¨h þ kða þ bÞ2
h:þ kða þ bÞUh ¼ 0 ð8:25Þ
The first check for stability is to examine the coefficients that would appear in the characteristic equation to see if any could be negative. Obviously, this happens if U is negative or if b is negative. One should really check to see if Eq. (8.25) correctly yields the equation for the caster moving in reverse when U is simply given a negative value. In any case, it is in the nature of casters to turn around when moved backward so the system should be unstable for backward motion.
It is highly unlikely that one could construct a caster with a center of mass behind the wheel center so a negative b is unlikely to be a cause of instability.
On the other hand, even if all coefficients of a third-order character- istic equation are positive, Routh’s criterion provides another criterion, which must be met for the system to be stable. When this criterion, Eq. (3.24), is applied to the coefficients in Eq. (8.25), a number of factors can be canceled out and the criterion for stability in this case finally reduces to a very simple inequality.
ab> j2 ð8:26Þ
The surprise is that this criterion does not involve the spring constant k. This means that in addition to flexibility in the caster structure, play in the pivot, which could be interpreted as a spring of zero spring constant, could be a cause of case instability if the criterion above were not satisfied.
A. Introduction of a Damping Moment
So far the model has shown the possibility of instability but there is no critical speed. This means that a caster is predicted to be either stable for all speeds however high or unstable even at extremely low speeds. This does not seem realistic. A potentially unstable caster is certainly stable if the speed is reduced sufficiently.
This can be explained by the lack of any friction in the model. Although the friction in the pivot is almost certainly better described by some sort of ‘‘dry’’ friction than by viscous friction, we will introduce a damping moment proportional to h:in order to keep the model linear. Let the damping moment about the pivot axis be given by the equation
Md¼ ch :
ð8:27Þ When this moment is inserted into Eq. (8.23), and the three equations, Eqs. (8.22)–(8.24), are combined into a single equation, the only change in the final equation of motion for h, Eq. (8.25), is in the coefficient of h¨ which becomes (c+mbU) instead of just mbU. This removes one possible cause of instability at low speeds. Even if b were to be negative, at a sufficiently low speed, the coefficient would be positive and the system would not neces- sarily be unstable.
In addition, the Routh stability criterion changes from Eq. (8.26) to cða þ bÞ
mU þ ab > j
2 ð8:28Þ
This implies that all casters are stable at sufficiently low speeds if viscous friction is included in the model as the stability criterion of Eq. (8.28) is always satisfied if U is small enough.
If the caster has the potential for instability,
ab< j2 ð8:29Þ
then there is a critical speed above which the system becomes unstable. Ucrit¼
cða þ bÞ
mðj2 abÞ ð8:30Þ
This model demonstrates that casters can exhibit unstable behavior for certain combinations of parameters and that friction may stabilize the motion of a caster at least up to a critical speed. The conclusion applies to both vertical axis and inclined axis casters.
V. A VERTICAL AXIS CASTER WITH PIVOT FLEXIBILITY