Generalized Coordinates and Degrees of Freedom

In deriving the equations of motion for a system, one must start with selecting a set of coordinates for the problem. The set of coordinates or parameters, which uniquely describes the geometric position and/or orientation of body, or system of bodies, are called the generalized coordinates of the system. The minimum number of independent generalized necessary to completely describe the geometry is known as the number of degrees of freedom. For system there can be many valid choices for a set of coordinates and any set that can uniquely describe the geometry of the system is a suitable. This doesn’t mean, however, that any set of coordinates should be used. As you can guess, based on the geometry and type of information being examined, some sets of coordinates will allow for an easier solution with more physical insight than others. For example (Figure 1), one may solve the simple planar motion of swinging pendulum using Cartesian coordinates in x and y but the coordinate system that provides the clearest view of the motion is obviously polar coordinates. Therefore, even though we may appropriately select x and y as the set of generalized coordinates, the most convenient set is the angle θ.

 

Figure 1

 

 

Below in (Figure 2), a bar lies in the x-y plane. The bar is unconstrained and can be placed in any position or orientation.  Given the two end points A and B, four generalized coordinates,  and  are sufficient to completely describe the geometry of the bar.  However, if we know the position at one end of the bar and its rotation angle, θ, then we can completely describe the geometry. Therefore, the actual number of degrees of freedom for the bar is only three. As previously stated, it is the minimum number of generalized coordinates that determines the number of degrees of freedom and each coordinate must be independent. Here, the four coordinates,  and , are not independent. By specifying any 3, the 4^th coordinate can determined.

 

 

Figure 2

 

 

For systems which there are many possible sets of coordinates, it will always be possible to map one set of to another. In the above case, the mapping of the 4 generalized coordinates  to coordinates in  is the accomplished by the following relationship

 

                                        and  

 

Now, let’s pin one end of the bar at point A.

 

Figure 3

 

By pinning the end of the bar (Figure 3) such that  and  are constrained, the number of degrees of freedom is reduced by two. Now, the only coordinate necessary to describe the geometry of the system is the rotation angle θ. 

 

If we continued by pinned the other end of the bar at point B  (Figure 4) then the system would be completely constrained thus having zero remaining degrees of freedom. Note, instead of pinning the bar at point B, we could have just as easily fixed the angle θ  to completely constrain the bar.

 

 

Figure 4

 

Generally, the constraint of a pin reduces the number of degrees of freedom by two. In this instance, however, pinning the bar at B only reduced the system by one  for two reasons. First, you obviously cannot reduce the number of degree of freedom by more than it has. And second, the dependent relationship between the rotation angle and coordinates at B (given that there is already a pin at A) means that endpoint B really only represents a single degree of freedom, namely θ. Thus constraining B only reduced the number of degrees of freedom by one.

 

Now let’s take a final look at a multi-body system. Below in Figure 5, two masses are connected by rods. The first mass, m_1, is suspended by a rod which is also connected to a ball and socket joint at P. The second mass, m_2, is connected to the first mass suspended by an extensible rod and a ball and socket joint at m_1. Before selecting the generalized coordinates, let’s first determine the number of degrees of freedom.

Figure 5

 

For the first mass, m₁ is connected to an inextensible rod which is free to rotate in 3 directions. It can swing through a nutation angle and precession angle and but can also rotate about the long axis of the rod. The x,y,z position of m₁ is constrained by the rod and the ball and socket joint such that m₁ will always be a distance L from P. Since its position is constrained but it has the freedom to rotate in 3 directions, m₁ contributes 3 degrees of freedom to the complete system. 

 

The situation for the second mass, m₂, is almost identical to that of m_1. The second mass is attached to the first with a rod and socket joint, but in this case the rod is extensible. The extensibility of the rod adds an additional degree of freedom. M₂’s position is still constrained by the length of the rod, l, but l is now allowed to change. The extensible rod and the ball and socket joint give m₂ a total of 4 degrees of freedom. Summing these with those of m₁ we now have a total of 7 degrees of freedom for the entire system.

 

Now that we have a clear understanding of the number of degrees of freedom for this system and their origin, selecting the generalized coordinates becomes a much easier task. It would only make sense to select 3 rotation angles for m₁, 3 rotation angles for m₂, and one additional coordinate for the extensibility of the second rod, l. Therefore we select for the system the generalized coordinates .