What is constraints and classification of constraints

In today’s article What is constraints and classification of constraints, we are going to discuss the basic Introduction of constraints and constraint motion. We will also going to discuss the basic properties of constraints force, classification of constraints, and principal of virtual work etc in detail.


A constrained motion is a motion which cannot proceed arbitrarily in any manner. The limitation on the motion of a system are called constraints. And motion is said to be constraint motion. The constraints are always related to force which restricts the motion of a system. Hence these force are known as constraint force.

Example 1. Let us consider the motion of a simple pendulum confined to move in the vertical plane. We would need only two co-ordinates (Cartesian coordinate x,y and polar coordinate r and θ with respect to the point of suspension O, as origin) to locate the position of the bob bob in motion. However, motion of bob is not free but takes place under a constraint that the distance l of the bob is to remains same from O all the time. This condition imposed by the constraint can be expressed in the form of an equation either x and y or r and 0,

x² + y² = l²

Or, r = l   (1)

In plane Polar coordinates the equation looks simpler. Again one coordinate 0, in polar coordinates, would suffice to describe the motion. Note that we have utilised equation (1) to reduce the number of coordinate, which otherwise would have been two.

What is Constraints and classification of constraints

Basic property of constraints force

  1. They are Elastic in nature and appear at the surface of contact.
  2. They are so strong that they allowed the body under consideration to a slidely from a prescribed path.
  3. The sole effect of constant force to keep constraint relation satisfied.

Classification of constraints

On the basis of time, they can be devided into two parts i.e. time dependent and time independent.

1. Holonomic constraints-

The constraints limit that motion of a system and the number of independent coordinates needed to describe the motion is reduced.

Suppose the constraints are present in the system of n particles. If the constraints are expressed in form of equation of term,

F( r1, r2, r3…t) =0

2. Non-Holomonic constraints

The constraints which are not expressed in the form of equation (1) is known as non- holonomic constraints.

3. Scleronomic constraints

A constraints is said to be scleronomic constraint if constraint relation do not explicitly dependent on time.

4. Rheonomic constraint

A constraint is said to be rheonomic if dependent on time.

5. Conservative constraints

In case of conservative total energy of the system is conserved during the constant motion and constant force do not do any work.

6. Dissipative constraints

In dissipative constraints the constant force do work and the total energy (mechanical energy) is not conserved.

Generalised co-ordinates

The generalised co-ordinate are as a set of independence co-ordinates sufficient in number to describe completely the state of configuration of a dynamical system. They are denoted as-

q1 , q2 , q3…..qk…qn

n = total number of generalised coordinate.

Consider a system n particles. The co-ordinate of ith particle.

xi = xi (q1, q2,…..qk….qn,t)

yi = yi (q1, q2,…..qk….qn,t)

zi = zi ( q1, q2,…..qk….qn,t)

In the position vector general form

ri = ri (q1, q2,…..qk….qn,t)

Above equation are called transformation equation.

Principle of virtual work

Any Imaginary arbitrary instantaneous change in the position vector of the particle of the system is called virtual displacement. This is displacement of position co-ordinate only and does not involve variation of time.

δri = δri (q1 , q2,….qn)        (a)

Suppose the system is in equilibrium than the total force of any particle is zero.

ƒi = 0.     i = 1,2,3…..N

Where i shows the state.

The virtual work of the force Fi in the virtual displacement δri will also be zero.

δwi = Fi . δri = 0            (b)

Similarly, the sum of virtual work for all the particle must vanish.

δw = ∑ˆN (i=1) Fi.δri = 0     (c)

The result represent the principle of virtual work which state that the work done is zero. In the case of an arbitrary virtual displacement of a system for a position of equilibrium.

Fi = fiˆa + fi

Where Fi is total force

fiˆa is applied force

And fi is constraint force.

Hence, this is total force fi on the ith particle.

By equation (c) we can write,

δw = ∑ˆN (i=1) (fiˆa + fi) δri =0

= ∑ˆN (i=1) (fiˆaδri )+ ∑ˆN (i=1) (fi δri) = 0

= ∑ˆN (i=1)fiˆaδri = 0

We restricts ourselves to the system where the virtual work to force of constraint is zero. In case of rigid body then,

δw =  ∑ˆN (i=1)fiˆaδri = 0

It mean for equilibrium of a system the virtual work applied is zero. We see that the principle of virtual work deals with a static of system of a particle.


In this article we have discussed about constraint motion, its properties, types of constraints and principal of virtual work in easiest way possible. And now we are able to clarify all the doubts related to constraints.

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