[**The degree of freedom**](https://en.wikipedia.org/wiki/Degrees_of_freedom_(mechanics)) of a system is the number of independent scalar parameters (measurements) that are necessary and enough for definition its configuration in space and at each instance of time. In other words, degree of freedom of a system is the number of scalar parameters associated to the system configuration that can vary freely (independently). Note that degree of freedom is defined with respect to an arbitrary reference frame.
As your question is about a planar mechanism and because of simplicity, we limit our discussion in planar motion.
> Mechanisms can be divided into *planar mechanisms* and *spatial mechanisms*, according to the relative motion of the rigid bodies. In a *planar mechanisms*, all of the relative motions of the rigid bodies are in one plane or in parallel planes. If there is any relative motion that is not in the same plane or in parallel planes, the mechanism is called the *spatial mechanism*. In other words, planar mechanisms are essentially two dimensional while spatial mechanisms are three dimensional. ([Source]( https://www.cs.cmu.edu/~rapidproto/mechanisms/chpt3.html))
**Degree of freedom of a particle in a plane:** the number of independent scalar parameters (measurements) that are necessary and enough for definition an arbitrary particle position (configuration) with respect to a point (reference frame) in a plane is two. I.e. for planar motion, we have:
$$\mathrm{DOF}_{\textrm{particle}}=2$$
(I don’t know a proof for this. To be honest, I asked a [question]( https://math.stackexchange.com/questions/1873664/is-there-a-proof-for-this-or-we-should-accept-that) in math.stackexchange but they got angry, so I escaped!)
In figure below, I have chosen point $O$ as reference frame and used parameters $x$ and $y$ for definition of position of point $A$.
[![enter image description here][1]][1]
This means that only two parameters associated to the particle position can vary freely (independently). For example, if we use three parameters ($x$, $y$ and $\theta$ in figure below) for definition of particle position in a plane, all of those parameters cannot vary freely. In other words, for any arbitrary $(x,y)$, there is a position. But, that is not true for any arbitrary $x,y,\theta$.
[![enter image description here][2]][2]
As you cans see, point $A$ has been specified by $(x_1,y_1,\theta_1)$. There are infinite points that can be specified by $(x,y,\theta_1)$. I.e. both of $x$ and $y$ can vary simultaneously and freely. But no point is specified by $(x_2,y_2,\theta_1)$. If a point like $B$ is specified by $(x_2,y_2,\theta_2)$, then certainly $\theta_2=\tan^{-1}\frac{y_2}{x_2}$. I.e. all of $x$, $y$ and $\theta$ cannot vary simultaneously and freely.
**Degree of freedom of a rigid body in a plane:** the number of independent scalar parameters (measurements) that are necessary and enough for definition a rigid body configuration with respect to a point (reference frame) in a plane is three. I.e. for planar motion, we have:
$$\mathrm{DOF}_{\textrm{rigid body}}=3$$
Because we can define configuration of a rigid body in a plane by definition of position of an arbitrary point of it (by two parameters) and position of other points of it with respect to former point (by one parameter, because according to the definition of a rigid body, distance between any arbitrary two points of it is constant).
In figure below, I have chosen point $O$ as reference frame and used parameters $x$, $y$ for definition of position of point $A$ and parameter $\theta$ for definition of position of point $B$ with respect to point $A$.
[![enter image description here][3]][3]
This means that only three parameters associated to the rigid body configuration can vary freely (independently).
I am not expert in continuum mechanics, so I just discuss (a little bit) about simplified cases like linear springs and bars under axial loading.
**Degree of freedom of a linear spring (under axial loading) in a plane:** the number of independent scalar parameters (measurements) that are necessary and enough for definition a linear spring configuration with respect to a point (reference frame) in a plane and under axial loading is two. I.e. for planar motion, we have:
$$\mathrm{DOF}_{\textrm{spring}}=2$$
Because we can define configuration of a linear spring in a plane by definition of position of two arbitrary points of it (each point by one parameter due to axial loading) after loading (It is obvious that we need another parameter i.e. distance of those two arbitrary points when no force is exerted on the spring. But, since that parameter is constant, so it doesn’t increase degree of freedom of the spring).
[![enter image description here][4]][4]
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> I just want to ask how do we actually approach such a problem?
We want to determine the configuration of whole of the system at each time. The system is formed of various elements (particles, rigid bodies and non-rigid bodies). We should determine the configuration of all elements. For this purpose, we need some parameters (measurements) and references (we can do all measurements with respect to one reference, but most times, it is easier that we use more references).
> How to identify which kinematic degrees of freedom are relevant?
This question is a little vague. Because any parameter (measurement) that is associated to the system is relevant. Obviously an irrelevant parameter is a parameter which doesn’t measure anything related to the system. I guess you mean *” How to identify which kinematic degrees of freedom are suitable?”* It depends on the problem. For example, consider to the following case.
[![enter image description here][5]][5]
The block is moving on the incline surface. It is a rigid body. So, it should have three degrees of freedom. But it experiences translation only. Hence, we can substitute it by a particle with same mass in its center of mass. Now, it should have two degrees of freedom. But it is moving on a straight line. So, it has only one degree of freedom.
Now, we can choose **any arbitrary parameter associated with block’s position** as degree of freedom e.g. $x_1$, $x_2$, $x_3$, $x_4$, etc. But a suitable choice will make calculations easier.
> How to identify which kinematic degrees of freedom are independent?
Obviously this question is valid for $DOF\ge 2$. I don’t know a mathematical answer for this question. But I want to explain another method for recognizing independency of parameters (measurements). A set of non-constant parameters associated to the system motion, are dependent on each other if we can calculate one or more of them by **measuring** the others **without writing equations of motion of the system**. I.e. if we can calculate one or some parameters by considering to the **geometry** of the system, they are dependent in our current discussion. But, if for expressing the parameters as functions of each other, we have no other way except writing equations of motion (i.e. kinematic constraints must be applied), then those parameters are independent. I think some simple examples are useful here.
Example 1: particle motion on a straight line.
In figure below, there is a block which is moving on a straight line. As it was mentioned before, the block has one degree of freedom and we can choose any arbitrary parameter (measurement) associated with position of its center of mass. So, if we choose more than one parameter (measurement) associated with its position, they certainly will be dependent on each other. Let’s see how.
[![enter image description here][6]][6]
We have chosen two parameters ($x_1(t)=\overline{O_1C}(t)$ and $x_2(t)=\overline{O_2C}(t)$) for determining its position at each moment. According to the geometry, we always have:
$$ x_2(t)=\overline{O_2C}(t)=\sqrt{\overline{O_1O_2}^2+\left(\overline{O_1C}(t)\right)^2-2\overline{O_1O_2}\cdot\overline{O_1C}(t)\cos\theta}$$
And as $\overline{O_1O_2}$ and $\theta$ are constant, we can calculate $x_2(t)=\overline{O_2C}(t)$ by measuring $x_1(t)=\overline{O_1C}(t)$ and vice versa. So, they are dependent of each other.
Example 2: particle motion on a plane.
In figure below there is a particle moving on a plane. As it was mentioned, it has two degrees of freedom. Hence, if we choose more than two parameters (measurements) associated with its position, they certainly won’t be independent of each other. This time, we choose one parameter (measurement) and we will see that it is not enough for specifying the position (configuration) of the particle.
[![enter image description here][7]][7]
We have chosen parameter $x(t)=\overline{OP}(t)$ for specifying the position of the particle at each time. As you can see, there are infinite number of points (on the circle) that have the same $x(t)=\overline{OP}(t)$ (remember that $x(t)$ is a scalar). So we cannot specify particle position by using the only measurement of $x(t)$.
Example 3: what does the phrase of *” without writing equations of motion of the system”* mean?
Imagine a particle projectile motion in a plane (there is no friction). We know that it has two degrees of freedom. As you can see in the figure below, we have chosen the parameters $x$ and $y$ as its degrees of freedom.
[![enter image description here][8]][8]
As you can see in the figure, we cannot calculate $x$ by measuring $y$ and vice versa (there are infinite number of $x$ for a $y$ and vice versa). But, if we write equations of motion of the particle, i.e. if we apply the kinematic constraints; we will have:
$$y=\large{\frac{-g(x-x_0)^2}{2v_0^2\cos^2\theta}}+(x-x_0)\tan\theta+y_0=f(x)$$
[![enter image description here][9]][9]
Therefore, we can determine $y$ by measuring $x$. But as it was mentioned before this doesn’t reduce kinematic degrees of freedom of the problem (because we used kinematic itself).
> Now how to visualize this?
This question has been answered in previous explanations.
> How do I methodically come to this conclusion?
First step: assumptions
1. Displacements are small. I.e. the vertical springs and rod always are vertical and the horizontal spring always is horizontal.
2. The blocks, pulleys and rod are rigid bodies.
3. The springs are linear.
4. Geometry of the system is known.
Second step: recognizing and choice
[![enter image description here][10]][10]
The spring $S_1$ has a fixed point. So, it has 1 degree of freedom. I choose the position of the other end of it as a degree of freedom of the system ($x_1$). This parameter (measurement) $x_1$ can specify configuration of the block $C$. So, there is no need to a new parameter associated to the block $C$. Also, $x_1$ determines configuration of a point of the spring $S_2$. So, $S_2$ needs one else parameter to be specified. I choose the position of the other end of it, i.e. parameter $x_2$. The pulley $O_2$ has a fixed point and parameter $x_2$ specify position of another point of that with respect to its fixed point. So, there is no need to other parameter associated to the configuration of the pulley $O_2$. As configuration of the pulley $O_2$ has been determined, so the spring $S_3$ has a specified point. So, $S_3$ needs one else parameter to be specified. I choose the position of the other end of it, i.e. parameter $x_3$. Parameter $x_3$ determines the position of the block $B$, too. As configuration of the pulley $O_2$ has been determined, so the spring $S_4$ has a specified point. So, $S_4$ needs one else parameter to be specified. I choose the position of the other end of it, i.e. parameter $x_4$. The pulley $O_1$ has a fixed point and parameter $x_4$ specify position of another point of that with respect to its fixed point. So, there is no need to other parameter associated to the configuration of the pulley $O_1$. As configuration of the pulley $O_1$ has been determined, so the only$^1$ degree of freedom of the rod is removed i.e. it has already been specified. Just like the rod, position of the block $A$ can be determined by $x_4$. Finally, the spring $S_5$ has one fixed point and one specified point. So, there is no need to other parameter associated to it. Thus, the system has 4 degrees of freedom and one possible combinations of its degrees of freedom is $x_1$, $x_2$, $x_3$ and $x_4$.
<sub>
$^1$ The rod $R$ has one degree of freedom just like the block in example 1.
</sub>
> is there any other combination possible?
Yes, there is. There are infinite number of possible combinations and I have already chosen one other of them in the previous question.
[1]: https://i.sstatic.net/4K4jP.jpg
[2]: https://i.sstatic.net/GYP32.jpg
[3]: https://i.sstatic.net/onJ5w.jpg
[4]: https://i.sstatic.net/3LyI5.jpg
[5]: https://i.sstatic.net/JnVZd.jpg
[6]: https://i.sstatic.net/DWfrY.jpg
[7]: https://i.sstatic.net/KKS7G.jpg
[8]: https://i.sstatic.net/6YFfv.jpg
[9]: https://i.sstatic.net/qNaql.jpg
[10]: https://i.sstatic.net/1Hog2.jpg