Degrees of Freedom and Equilibrium Equation (1)

Here’s the first article I’ve created for you as part of the MechaDevs Blog. In it, I’d like to introduce the basics of the wonderful branch of mechanics called statics and explain some concepts we’ll need for further discussions on mechanical and strength calculations for beams, frames, and trusses.

The article, in its three parts, will answer questions related to what degrees of freedom are, what equilibrium equations are, what geometric rigidity/stability is, and why we calculate support reactions. Understanding the information presented here would surely allow students to avoid many different accidents (and drastically reduce the number of funny videos on YouTube…).

In the first part of the article, we will cover the following topics:

• Statics as a part of mechanics
• Degrees of freedom
• Static equilibrium equations

Statics – Mechanics of the Resting Body (Well, Motionless at Least)…

As you probably already know, mechanics is a part of physics that deals with the motion of points, bodies, and all sorts of other things that are supposed to move (but don’t… or vice versa). Since we analyze their motion, kinematics and dynamics deal with this, but we should also consider situations where a given object doesn’t move. The most important aspect in this context is the conditions that must be met for the body to remain at rest. What needs to be fulfilled for an object not to move? The description of these conditions, as well as everything related to the mechanical analysis of immobile elements, is precisely what statics deals with.

But before we dive into statics itself, let’s go back to preschool, where we learned about the linear motion of a point. In the theory presented, we talked about the position of a point (described, for example, by the variable x), its velocity (V_x), which determines the change in position, and perhaps someone also mentioned the point’s acceleration (a_x), which describes the change in velocity.

When discussing acceleration, you probably heard some typical kinematic jargon that, oddly enough, sticks quite well in memory – terms like uniform motion or uniformly accelerated motion. However, you may have missed the name of a rather unknown scientist – Mr. Isaac Newton, who, thanks to a certain accident (as legend has it), laid down very sensible foundations for today’s discussions on dynamics.

You’re probably wondering why I’m so insistently referring to dynamics – well, it’s in Newton’s laws of motion that we find the answer to our pressing question… It’s the first law of Newton that explains when we can expect no movement. Paraphrasing:

If no force acts on an object, or if the forces acting on it are balanced, the object remains at rest or continues to move at a constant velocity (if it had an initial velocity).

So, we now have a basis – since it’s practically impossible for no forces to act (something is always pulling on us somewhere), to remain at rest, the forces must balance out – they must cancel each other out properly. In the preschool example of a body moving in a straight line, the condition for this can be very easily described mathematically:

∑F_{ix} =0

In other words – the sum of all forces acting in the direction of the body’s movement must equal zero. However, this is a very simple case, and I think we can raise the bar a little.

Degrees of Freedom – What Moves and How…

In your academic career, you’ll most often encounter the mechanical analysis of planar systems – like various types of plates. By plates, we mean systems whose geometry can be drawn on a piece of paper (without using 3D projection methods) and are loaded by forces that also act in the plane of that paper. In other words – everything happens on a flat surface (welcome to the world of “flatlanders”). Examples of such plates include parts of beams, flat frames, trusses, and so on.

When analyzing planar systems, it’s worth considering how these objects can move on the mentioned paper. Most often, when I ask students this question, the answer is “up, down, and sideways…”. In engineering terms, this would be vertically and horizontally, but unfortunately, that’s not all! Such a system can also rotate around its own axis, which will be perpendicular to our paper. Combining these three types of motion gives us unlimited possibilities for the movement of a planar object (plate) on a plane:

At this point, we should introduce a new concept – degrees of freedom.

In the context we are analyzing, degrees of freedom refer to the number of independent types (or directions) of motion that a given body can perform.

So, a plate on a plane – attention – has 3 (in words: three) degrees of freedom. It might be worth adding that, in this case, we are analyzing the plate as a rigid element (a deformable object would have an infinite number of degrees of freedom – but that’s a topic for another time…). Similarly, going back to the example of a point on a line – such a point had one degree of freedom. Completing the picture – in our three-dimensional space, according to classical mechanics, a solid body has 6 (in words: six) degrees of freedom (three translations in the directions of the coordinate axes and three rotations around those axes).

Static Equilibrium Equations – Balance is Key…

So, let’s return to the issue of the plate at rest. If a point on a line had one degree of freedom and needed to meet one condition to remain motionless, then how many and what conditions must the plate with three degrees of freedom meet to stay still?

I should give you a moment to think about this and respond.

Unfortunately, a blog post limits our interaction a bit… oh well… let’s assume that by reading this rambling, you’ve had enough time to consider it. Yes – the answer is – such a system must meet three conditions to remain motionless – one for each degree of freedom.

These conditions can also be nicely described mathematically:

∑F_{ix} =0,      ∑F_{iy}=0,     ∑M_{iz} =0

In simpler terms – the sum of all forces (or their components) acting along the x-axis must equal zero (preventing horizontal motion), as must the forces along the y-axis (preventing vertical motion) and the moments about the z-axis (preventing rotation in the plane of the paper). And yes, rotational motion of a body is caused by moments acting on it. Here, a moment refers to the moment of a force, meaning the action of forces at a certain distance from a point.

The conditions described above are called static equilibrium conditions, and their expanded forms, into which we substitute the actual forces acting on the system, are called static equilibrium equations (or simply equilibrium equations). If the forces acting on the system satisfy the equilibrium conditions, the system has a chance of remaining at rest (or moving at a constant velocity – you remember that, right ????). And what if the conditions aren’t met? Once again, we turn to Mr. Newton and his second law of motion:

If the forces acting on a body are not balanced, the body will accelerate in proportion to the imbalance and inversely proportional to its mass.

 In such a case, we’re dealing with a completely different branch of mechanics – dynamics (when we’re trying to figure out why something is moving).

In the next part of the article, we’ll discuss geometric stability, where we’ll try to determine how to ensure that a body satisfies the conditions of equilibrium…