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Generalized Dynamics: Kinematics and Kinetics 357

11.3 Holonomic and Nonholonomic Constraints
The constraint equations of the examples of the foregoing section might be described as
being positional or geometrical. That is, they involve the relative positions of, or distances
between, points of a mechanical system. They are developed from position vectors of the
system. They do not involve velocities, accelerations, or derivatives of system coordinates.
Such positional or geometrical constraints are said to be holonomic, and the associated
mechanical system is said to be a holonomic system. If, however, a system has constraint
equations that involve velocities, accelerations, or derivatives of system coordinates, the
constraint equations are said to be nonholonomic, or kinematic, and the mechanical system
is said to be a nonholonomic system.
Specifically, suppose a constraint equation has the form:

r (
,
fq t) = 0 (11.3.1)
where the q (r = 1,…, n) are coordinates of the system (n is the number of degrees of
r
freedom of the unrestrained system). Such a constraint is said to be holonomic (or geometric).
Alternatively, suppose a constraint equation has the form:

r (
, ˙
f qqq ˙˙ ,… ) = 0 (11.3.2)
t ,
,
r
r
Such a constraint is said to be nonholonomic (or kinematic).
If a nonholonomic constraint equation does not involve second-order or higher-order
derivatives of the coordinates, and if it is a linear function of the first derivatives, then it
is said to be a simple nonholonomic constraint.
As might be expected, holonomic systems are easier to study and analyze than non-
holonomic systems. The reason is that holonomic systems produce algebraic constraint
equations, whereas nonholonomic systems produce differential constraint equations.
Indeed, expressions of the form of Eq. (11.3.2) are generally nonintegrable and as a con-
sequence cannot be solved in terms of elementary functions. Fortunately, the vast majority
of mechanical systems of interest and importance in machine dynamics can be modeled
as holonomic systems.
To illustrate these concepts, consider again the rolling circular disk (or “rolling coin”)
discussed in Sections 4.12 and 8.13 and as shown in Figure 11.3.1. Recall from Eqs. (4.12.3)
and (8.13.1) that the velocity v of the mass center G of the disk D in the inertia reference
G
frame R was found to be:

v = ( ψφ ˙ n ) θ 1 − rθ ˙ n 2 (11.3.3)
G
r ˙
+ sin
where, as before, θ, φ, and ψ are the angles shown in Figure 11.3.1. Also, from Eqs. (4.12.2)
and (8.13.3) the angular velocity ωω ωω of D in R is:

)
˙
φ
φ
˙
˙
˙
ωω= θn +( ψ + sinθ n − cosθn (11.3.4)
1 2 3