The 7th Degree of Freedom

The entire idea that there are only six possible degrees of freedom is based on the flawed concept that there is any such thing as a rigid body. If we concede the relevance of elasticity as a player in kinematic arena it opens Pandora’s Box on constraint.

If elasticity is accepted as a seventh degree of freedom, how would the result manifest itself in the real world? Obviously its effects will be in a direction that is opposite, i.e. at 180°, to the force that generates the deflection! Elasticity isn’t actually a seventh degree of freedom, but the deflections caused by any applied force does result in that degree of freedom. Here is where the fun begins.

If there are two vector forces instead of one, are there then eight degrees of freedom, or is there a single deflection that is the resultant of the multitude of forces? From a functional standpoint this is obviously the case. With all of this distortion of the six degrees of freedom and the rigid body concept that we all know really doesn’t ever exist, where are we on constraint?

When does over-constraint occur? As an example of this constraint dilemma, let’s look at a long cylindrical bar. If we support it at four tangent points that have an included angle of ninety degrees, the cylinder can only roll around and translate along the X axis. When held horizontally, there will be various patterns of sag (elastic deformation) due to the force of gravity.

The pattern of these sags will be determined by the longitudinal position of the four points of contact and the long cylindrical bar. If we place the four points near the ends of the cylinder, there will be a gradual bow down toward the center of the earth. Now comes the question of constraint, or rather over constraint. If we support the center of the long cylindrical bar with another cylinder, placed at right angles to the long cylinder, is this over constraint?

If we accept elasticity as a seventh degree of freedom, this would not be over constraint. The long cylindrical bar will obviously be elastic in all three hundred and sixty degrees, around its longitudinal axis! If we supported the center of the bowed cylinder with two 90° points of contact, would this be over-constraint? I contend that it would not be, due to the 360º nature of the of elasticity for the long cylindrical bar.

If we are working with a very long cylinder such as a scale bar, where there would be multiple bows between supports, would a multitude of two point supports be over-constraint? I would contend again that it would not be over-constraint. If we try to go back to the good old six degrees of freedom with “rigid bodies,” it wouldn’t describe what we know to be the real bubblegum world that we all live in.

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