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Constraints
If the motion of dynamics of a system is not allowed to extend freely in three
dimensional space. Then the system is said to be subjected to constraints.
When constraints apply to a system, then its coordinate don’t remain independent.
For Example:
1) Rigid Body: In a rigid body, the distance between two points remains unchanged.
2) The beads of an abacus are constrained to one dimensional motion by the
supporting wires.
3) Gas molecules with in a container are constrained to move only inside the container.
4) A particle placed on the surface of a solid sphere is subjected to move only on the
surface or in the region exterior to the surface.
Holonomic Constraints
If the conditions of constraints can be expressed in proper form of mathematical
equations
i-e , , , … … … , =
then the constraints imposed on the system are called Holonomic constraints. In above
equation , , are Generallized coordinates and the system is time dependent.
Example:
1) In a rigid body constraints are expressed by equation of the form
( − ) − =
Where Cij is known as the constraints.
2) A particle constrained to move along any curve or on a given surface is another example
of a Holonomic constraints.
+ = =
Where “a” is the radius of the sphere.
Non-Holonomic Constraints
If the conditions of constraints cannot be expressed in proper form of mathematical
equation then the system is said to subject to Non-holonomic constraints.
e.g A particle placed on the surface of a sphere is also example of non-holonomic constraints,
for it can be expressed as an equality
− ≥ 0
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Holonomic Constraints are further classified according to time dependence. There are two
types
Rehonomous Constraints:
If the constraints imposed on a system are changing (dependent) with time, then the
constraints are called Rehonomous constraints.
Scleronomous Constraints:
If the constraints imposed on a system are not changing (independent) with time, then
the constraints are called Scleronomous
Example:
A bead sliding on a rigid curved wire fixed in space is obviously subject to a
Scleronomous constraints.
If a wire is moving in prescribed fashion, then constraints are Rehonomous.
Difficulties introduced by Constrains
Constraints introduce two types of difficulties in solution of mechanical problem.
1) The coordinates are no longer independent, since they are connected to equation of
constraints.
2) The forces of constraints.
e.g the force that wire exerts on the bead or the wall on the gas particles.
Solution to Difficulty:
In case of Holonomic Constraints, the first difficulty is solved by introduction of generallized
coordinates.
A system of particles, free from constraints, has 3N independent coordinates or degree of
freedom. If there exists holonomic constraints, expressed by k, then we have 3N-k
independent coordinates and the system is said to have 3N-k degrees of freedom.
The elimination of the dependent coordinates can be expressed in another way, by
introducing 3N-k, independent variables , , … … . . in terms of which old coordinates
are , , … … are expressed by equations of the form
= ( , , … … . , )
= ( , , … … . , )
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Containing constrains in them implicitly. These are transformation equations and can be
considered as parametric representation of ( ) variables. These function must be invertible
which means inverse of these functions exists. Above equations combined with the k
equations of constraints can be inverted to obtain any as a function of the ( ) variable and
time.
Generallized coordinates are not divided
into group of three that can be associated to form a vector. In
double pendulum moving in a plane (two particles connected
by an in extensible light rod and suspended by a similar
rod fastened to one of the particle), satisfactory
generallized coordinates are two angle , .
Generallized coordinates, in the sense of coordinates other than Cartesian, are often useful in
system without constraints.
If the constraints is non-holonomic, the equation expressing the constraints cannot be
used to eliminate the
dependent coordinates.
Example: An object rolling
on a rough surface
without slipping.
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The coordinates used to determine the system will generally involve angular coordinate
to specify the orientation of the body plus the set of coordinates describing the location of the
point of contact on the surface. The constraint of “rolling” connects these two sets of
coordinates: they are not independent. A change in position of contact inevitably (without
doubt) means a change in its orientation. So we cannot reduce the number of coordinates, for
the rolling condition is not expressible as a equation between the coordinates in form of
, , , … … … , =
Consider a disk rolling on the horizontal xy-plane constrained to move so that the plane
of the disk is always vertical. The coordinates used to describe the motion might be x, y
coordinates of the centre of the disk, an angle of rotation ɸabout the axis of the disk, and an
angle Ө between the axis of the disk. As a result of constraint the velocity of the disk, v , has a
magnitude proportional to ɸ.
.
= ὠ = ɸ
.
− ɸ.
= 0
Where “a” is the radius of the disk, and its direction is perpendicular to the axis of the disk.
= . + .
.
= = (90 − Ө) = sin Ө
.
= = − (90 − Ө) = − Ө
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Force of constraints:
I f we apply the force on the particle (say A) then its motion is restricted by another
body (say B), the the force on the body A to the body B is called force of constraints.
Ideal Contraints:
If the work done by the force of constraint during virtual displacement is zero is called
Ideal constraints.
Virtual Displacement:
An imaginary small displacement at any instant of time is called Virtual displacement.
Coordinates , , , … … . ℎ
Virtual Work
Workdone by the force during the virtual displacement is called Virtual work.
Degree of freedom
An important characteristics of a given mechanical system is its number of degree of
freedom is the smallest number of coordinates required to specify completely the configuration
or state of the system.
Thus for a free particle the degree of freedom is 3.
Generallized coordiantes
Any set of parameter of quantity that can satisfy the configuration or state of the
system can be used as generallized coordinates and can be the quantities that can be observed
to change with the motion of the system.
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D’ Alembert’s Principle
The sum of work done by the applied force and reverse effective force is zero,
this is called D’ Alembert’s Principle
D’ Alembert’s Principle derived a method which was applicable for the system in
motion.
According to Newton’s 2nd
Law of motion
= .
− .
= 0
The above equation shows that the particle in the system will be in equilibrium under a
force which is equal to the actual force plus the reverse effective force(− .
).
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If a particle is constrained to move on a surface, then the force of constraint is
perpendicular to the surface while virtual displacement is tangent to it. Hence the work
done by constraints will be zero.
This relation is called D’ Alembert’s Principle.
Lagrange Equation of motion By D’ Alembert’s Principle
The equation of motion of the form
Ձ
Ձ . −
Ձ
Ձ
= 0
In a conservative field of force is called the Lagrangian Equation of motion by
D’Alembert’s Principle.
Proof:
In order to derive Lagrangian Equation of motion from D’Alembert’s Principle, we
use transformation equation
Now Virtual Displacement δriis
According to D’ Alembert’s Principle
Considering the 1st
term
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Where
is called Generallized Force.
2nd
part of equation ( ) is
Now, we find out the value of
Consider an equation
Differentiating Equation ( ) w.r.t
