3.2 Degrees of Freedom and Independent Coordinates

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The geometric configuration of a vibrating system can be completely determined by a set of

independent coordinates. This number of independent coordinates, for most systems, is termed the

number of DoFs of the system. For example, a particle freely moving on a plane requires two

independent coordinates to completely locate it (e.g., x and y Cartesian coordinates or r and u polar

coordinates); its motion has two DoF. A rigid body that is free to take any orientation in (threedimensional)

space needs six independent coordinates to completely define its position. For instance,

its centroid is positioned using three independent Cartesian coordinates ðx; y; zÞ: Any axis fixed in the

body and passing through its centroid can be oriented by two independent angles ðu; fÞ: The

orientation of the body about this body axis can be fixed by a third independent angle ðcÞ: Altogether,

six independent coordinates have been utilized; the system has six DoF.

Strictly speaking, the number of DoF is equal to the number of independent, incremental,

generalized coordinates that are needed to represent a general motion. In other words, it is the

number of incremental independent motions that are possible. For holonomic systems (i.e., systems

possessing holonomic constraints only), the number of independent incremental generalized

coordinates is equal to the number of independent generalized coordinates; hence, either definition

may be used for the number of DoF. If, on the other hand, the system has nonholonomic

3-2 Vibration and Shock Handbook

© 2005 by Taylor & Francis Group, LLC

constraints, the definition based on incremental coordinates should be used, because in these

systems the number of independent incremental coordinates is in general less than the number of

independent coordinates that are required to completely position the system.

3.2.1 Nonholonomic Constraints

Constraints of a system that cannot be represented by purely algebraic equations in its generalized

coordinates and time are termed nonholonomic constraints. For a nonholonomic system, more

coordinates than the number of DoF are required to completely define the position of the system. The

number of excess coordinates is equal to the number of nonalgebraic relations that define the

nonholonomic constraints in the system. Examples for nonholonomic systems are afforded by bodies

rolling on surfaces and bodies whose velocities are constrained in some manner.

Example 3.1

A good example for a nonholonomic system is provided by a sphere rolling, without slipping, on a

plane surface. In Figure 3.1, the point O denotes the center of the sphere at a given instant, and P is an

arbitrary point within the sphere. The instantaneous point of contact with the plane surface is denoted

by Q, so that the radius of the sphere is OQ ¼ a. This system requires five independent generalized

coordinates to position it. For example, the center O is fixed by the Cartesian coordinates x and y:

Since the sphere is free to roll along any arbitrary path on the plane and return to the starting point,

the line OP can assume any arbitrary orientation for any given position for the center O. This line can

be oriented by two independent coordinates u and f; defined as in Figure 3.1. Furthermore, since the

sphere is free to spin about the z-axis and is also free to roll on any trajectory (and return to its starting

point), it follows that the sphere can take any orientation about the line OP (for a specific location of

point O and line OP). This position can be oriented by the angle c: These five generalized coordinates

x; y; u; f; and c are independent. The corresponding incremental coordinates dx; dy; du; df; and dc

are, however, not independent, as a result of the constraint of rolling without slipping. It can be

shown that two independent differential equations can be written for this constraint, and that

consequently there exist only three independent incremental coordinates; the system actually has only

three DoF.

To establish the equations for the two nonholonomic constraints note that the incremental

displacements dx and dy of the center O about the instantaneous point of contact Q can be written

dx ¼ a db; dy ¼ 2a da

x

z

y

O θ

ψ

φ

β

α

P

Q

a

FIGURE 3.1 Rolling sphere on a plane (an example of a nonholonomic system).

Modal Analysis 3-3

© 2005 by Taylor & Francis Group, LLC

in which the rotations of a and b are taken as positive about the positive directions of x and y;

respectively (Figure 3.1). Next, we will express da and db in terms of the generalized coordinates.

Note that du is directed along the z direction and has no components along the x and y directions.

On the other hand, df has the components df cos u in the positive y direction and df sin u in the

negative x direction. Furthermore, the horizontal component of dc is dc sin f: This in turn has

the components ðdc sin fÞcos u and ðdc sin fÞsin u in the positive x and y directions, respectively.

It follows that

da ¼ 2df sin u þ dc sin f cos u

db ¼ df cos u þ dc sin f sin u

Consequently, the two nonholonomic constraint equations are

dx ¼ aðdf cos u þ dc sin f sin uÞ

dy ¼ aðdf sin u 2 dc sin f cos uÞ

Note that these are differential equations that cannot be directly integrated to give algebraic equations.

A particular choice for the three independent incremental coordinates associated with the three DoF

in the present system of a rolling sphere would be du; df; and dc: The incremental variables da; db;

and du will form another choice. The incremental variables dx; dy; and du will also form a possible

choice. Once three incremented displacements are chosen in this manner, the remaining two

incremental generalized coordinates are not independent and can be expressed in terms of these three

incremented variables using the constraint differential equations.

Example 3.2

A relatively simple example for a nonholonomic system is a single-dimensional rigid body (a straight

line) moving on a plane such that its velocity is always along the body axis. The idealized motion of a ship

in calm water is a practical situation representing such a system. This body needs three independent

coordinates to completely define all possible configurations that it can take. For example, the centroid of

the body can be fixed by two Cartesian coordinates x and y on the plane, and the orientation of the axis

through the centroid may be fixed by a single angle u: Note that, for a given location ðx; yÞ of the centroid,

any arbitrary orientation ðuÞ for the body axis is feasible, because, as in the previous example, any

arbitrary trajectory can be followed by this body and return the centroid to the starting point, but with a

different orientation of the axis of the body. Since the velocity is always directed along the body axis, a

nonholonomic constraint exists and it is expressed as

dy

dx ¼ tan u

It follows that there are only two independent incremental variables; the system has only two DoF.

Some useful definitions and properties that were discussed in this section are summarized in Box 3.1.

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