Constrained Motion: Definition, Types, and Examples

When we talk about constrained motion in mechanics, we’re essentially referring to the limitations placed on an object’s movement, its position, or sometimes both at once. In machines, whenever two components come into contact, they form what we call a pair.

Now, this pairing gets a special name a kinematic pair if the movement between those two elements is either wholly or partially restricted.

In other words, the way these parts can move relative to each other isn’t entirely free; there’s some control or boundary set by the nature of their connection.

What is a Constrained Motion?

Constrained motion happens when an object doesn’t have the freedom to move however it wants instead, something limits its path or behavior.

Picture a ball rolling inside a curved track, or a block sliding on a table that’s moving upward, or even an object that always stays in contact with a wedge that’s accelerating. In each case, the object can’t just do its own thing; its motion is being guided or held back in some way.

More formally, whenever a particle is made to move along a specific path because of outside forces, we call the limitations on its movement or position constraints. These constraints shape how a whole system of particles can behave, setting the rules for what’s possible.

The forces that actually keep an object in line with these rules are known as constraint forces. They’re responsible for making sure the object sticks to the allowed motion, rather than wandering off in some random direction.

Constraint forces, in short, keep an object from moving freely. When we talk about constrained motion, we mean movement that follows a required pattern one that isn’t totally arbitrary, but instead is determined by the constraints we’ve set.

A simple example: if we want the blades of a fan to move in a circle and they do just that, their motion is constrained. The constraints, and the forces behind them, are making sure the fan spins

The simplest example of such motion are:

1. A particle moving down an inclined plane, whether the surface is rough or smooth: Think of a particle placed on a slope. Regardless of whether the surface offers friction or not, the particle’s movement is directly influenced by gravity and the nature of the incline. The forces at play may change a bit depending on the texture, but the basic principle remains the same: gravity pulls it downward, and the incline determines the direction and rate at which it slides.
2. A particle sliding down a curved path due to gravity: Here, imagine a small object or particle let loose on a curved track. Gravity takes over and guides the particle along the curve, following the path’s unique shape. The way the particle picks up speed or changes direction depends entirely on how the curve bends and dips.
3. A particle attached to a string and set in motion: Now picture tying a string to a particle and swinging it or moving it in a controlled way. The string acts as a constraint, making sure the particle follows a specific route. How it moves whether it’s in a circle, a spiral, or any other path—is determined by the string’s length and the force applied.
4. Forcing straight-line motion on a car that naturally follows a curve: Sometimes, you might want a car to travel in a straight line, but it’s designed (or positioned) to follow a curved path. If you intervene and control the steering so the car moves straight instead, you’re imposing what’s called a constrained motion. The car’s natural tendency is altered by an external force to fit a different, specific path.
5. Bicycling as an example of input and output in motion: A familiar example is riding a bicycle. When you pedal, you’re creating a turning force (a couple) with your legs. That force doesn’t just disappear it’s transferred through the bike’s mechanisms to make the rear wheel spin. Your input the act of pedaling translates into the output: the rotation of the back wheel and, ultimately, forward movement.

Types of Constrained Motion

There are three types of Constrained Motions:

• Completely Constrained Motion
• Partially or Successfully constrained motion
• Incompletely Constrained Motion

#1. Completely Constrained Motion.

A motion is described as completely constrained when the movement between two components is restricted to a single, specific direction, no matter which way a force is applied.

To put it simply, the parts can only move in one direction relative to each other, regardless of how you try to push or pull them.

A good everyday example of this is a rectangular shaft that turns inside a matching rectangular hole the shape itself prevents the shaft from moving in any way except the one allowed.

Another classic example comes from steam engines: think about the piston moving inside its cylinder.

Here, the piston can only slide back and forth (that is, it reciprocates) within the cylinder, and this happens no matter how the crank attached to it moves. The design of the parts makes sure the motion stays locked to just that direction.

Examples of Completely constrained motions:

• The motion of a square bar in a square hole
• The motion of a shaft with a collar at each end in a circular hole,
• A piston in the cylinder of an IC engine.

#2. Partially or Successfully constrained motion.

A partially or successfully constrained motion refers to a situation where an object is technically able to move in more than one direction if no outside force is acting on it. However, once an external force comes into play, the movement becomes limited to just one specific direction.

This is the key reason why this kind of motion gets labeled as “partially constrained” or “successfully constrained.”

Let’s consider a shaft supported by a footstep bearing as an example. If you place a compressive load on the shaft, you might notice that it can either rotate within the bearing or even slide upwards. When there’s nothing stopping that upward movement, we’re dealing with what’s called incompletely constrained motion.

But as soon as a load is applied in such a way that it blocks the shaft from moving axially upwards so it can only rotate this changes the situation. Now, the shaft’s movement is deliberately restricted by that external force, making it a clear example of successfully constrained motion.

Examples of Successfully constrained motions:

• The motion of an I.C. engine valve, these are kept on their seat by a spring
• The piston reciprocating inside an engine cylinder
• Shaft in a footstep bearing

#3. Incompletely Constrained Motion.

Sometimes, the movement between two connected parts isn’t limited to just one direction. When that happens, we call it incompletely constrained motion. In these cases, if the direction of the force applied changes, the way the two parts move relative to each other can also shift.

A classic example is when you have a round shaft or a circular strip fitted into a round hole. Here, the shaft isn’t restricted to only one type of motion it can both slide back and forth and rotate within the hole.

What’s interesting is that these two movements don’t depend on each other at all; the shaft can spin without sliding, or slide without spinning. This lack of connection between the two motions is what makes it a textbook case of incompletely constrained motion.

Examples of incompletely constrained motions:

• In a circular hole, the circular shaft can either rotate or slide in the hole.

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