April 18, 2013

Degrees of freedom is the one of the most important concept in mechanics. This concept is widely used in robotics and kinematics. D.O.F means how many variables are required to determine position of a mechanism in space. In this video lecture we will understand how to predict degrees of freedom of a mechanism.

Consider the mechanism shown in first figure of Fig.4. Position of this 4 bar mechanism can be completely determined just by knowing angle or position of any one of the member. So degrees of freedom is one. Similarly degrees of freedom of the cam and follower mechanism is also one. But to determine position of the slider crank mechanism shown, we should know angle or displacement of at least 2 members. So here degrees of freedom is 2.

You have predicted D.O.F of some simple mechanisms from your intuition. But for a complex mechanism, such an approach may not work. So in coming sections we will see how we can predict D.O.F of a mechanism.

Consider the rigid body shown below, which is situated in space. It could have 3 translatory motions as shown. Also it could have 3 rotary motions as shown.In total we need 6 inputs to determine its position. So degrees of freedom of rigid body in space is 6.

If the body is in a plane it can have only 3 motions. 2 translational and 1 rotational. So degrees of freedom of a rigid body in a plane is 3.

A mechanism is a collection of rigid bodies or links, connected through pairs, provided one link is grounded. Consider the mechanism shown below.

If this system were not connected like this, then each link except the ground would have 3 degrees of freedom.

So total degrees of freedom, or mobility is 3(N-1). N represents total number of links. In this case N is 3. But when we connect it together through pairs, links will not have the same 3 degrees of freedom.
If joint between 2 links is having surface contact as shown below, then both the links will have same translatory motion, in X and Y directions.So for each such pairs, there will be a deduction of 2 mobility from total mobility. Where LP represents number of pairs with surface contacts. Such pairs are called lower pairs. In this case we have 2 lower pairs.

Now consider the joint which is having a line contact. If joint between 2 links is having line or point contact, both the link should have same translational motion along the common normal. However it could have different motion, in tangential direction. So for each such pairs, there will be deduction of 1 mobility from total mobility. This kind of pair is called higher pair. Here we have got 1 higher pair.

So this mechanism has got 1 degrees of freedom. Means, by knowing position of only one cam, we can completely determine this mechanism. The general equation to find out degrees of freedom of a planar mechanism is given below. This equation is also known as Kuthbach equation.

Here N represent total number of links in the mechanism. LP and HP represent number of lower pairs and higher pairs respectively.

Back to same old planar mechanisms.This mechanism is having 4 links, and 4 lower pairs. So you can predict from Kuthbach equation that mobility of the mechanism is 1.

Cam and follower is having 3 links, 2 lower pairs, and one higher pair.So mobility is again one.

This mechanism is having 5 links and 5 lower pairs. So mobility is 2.

If the mechanism is 3 dimensional in nature, you could easily derive an equation for mobility using the same concept. So equation for degrees of freedom would be as follows

Where Pn represents number of pairs which block ānā degrees of freedom. The main thing here will be determination of nature of pair. You can use this equation to predict D.O.F following 3 dimensional mechanism.

This article is written by Sabin Mathew, an IIT Delhi postgraduate in mechanical engineering. Sabin is passionate about understanding the physics behind complex technologies and explaining them in simple words. He is the founder of Learn Engineering educational platform. To know more about the author check this link