Degrees of freedom

The number of degrees of freedom (DOF) is a number of inde­pen­dent para­me­ters that unam­bigu­ously define the state of a mechan­ical system.

Consider a plane hinge mech­a­nism ñcon­sisting of two similar paral­lel­o­grams that have two common fixed red hinges. The number of DOF of such a mech­a­nism obvi­ously equals two as paral­lel­o­grams can rotate inde­pen­dently and as para­me­ters one may take, for example, the angles counted off the hori­zontal direc­tion.

Does every mech­a­nism has a number of DOF «fixed»? Or there are mech­a­nisms whose number of DOF is vari­able? It turns out that there are…

The plane first hinge mech­a­nism with a vari­able number of DOF was invented by W. Wunder­lich in 1954. It contained two fixed hinges and 12 edges. We'll consider a simpler construc­tion with 9 edges created by a russian math­e­mati­cian Michael Kovalev.

Addto the paral­lel­o­grams «center­lines» and a short edge connecting their inter­sec­tion point with a fixed red hinge.

While the blue hinge remains on the central line that connects two initial fixed red hinges, the addi­tional edges don't change the number of DOF. The state is defined, for example, by two rota­tion angles counted off the hori­zontal direc­tion.

However, the blue hinge may leave the central line when the center­lines and the short edge lie on the same line. As soon as the blue hinge is off the central line, the state of the whole mech­a­nism is defined by a single para­meter! This para­meter can be, for example, the angle between the initial and the current loca­tion of the short edge.