524
9 Variational Principles in Mechanics and Their Applications
9.2 Principle of Virtual Displacement
9.2.1 Principle of Virtual Displacement for System
of Particles
The principle of virtual displacement is also called the principle of virtual work
or virtual displacement principle. Galileo began to apply this principle as far back
as the 17th century. In 1717 John Bernoulli first realized the universal significance of
the principle and its effect on solving the problem of statics. The virtual displacement
principle of particle system can be expressed as: If a particle system with the steady
and ideal constraints is in equilibrium state at a certain configuration, then
the sufficient and necessary conditions maintaining the balance is: The sum of
elementary works done by all positive forces acting on the particle system on
any virtual displacement is zero. Its mathematical expression is
δW =
n
i=1
F i · δr i = 0
(9.2.1)
or it is written in the form of components as
δW =
n
i=1
(F i x δx + F iy δy + F iz δz) = 0
(9.2.2)
Equations (9.2.1) and (9.2.2) is called the equation of virtual work.
Proof First prove the necessity of condition of the principle.
Let the composite force of active forces acting on the particle m i be F i , the
composite force of reactive forces of constraint be N i . Since the particle system is
in equilibrium state in a configuration, there will be ¨
r i = 0, according to Newton’s
second law, there will be
F i + N i = 0 (i = 1, 2, . . . , n)
Thus for any virtual displacement the expression
n
i=1
(F i + N i ) · δr i =
n
i=1
F i · δr i +
n
i=1
N i · δr i = 0
holds, because the particle system is an ideal constraint system, there is
n
i=1
N i · δr i = 0
9 Variational Principles in Mechanics and Their Applications
9.2 Principle of Virtual Displacement
9.2.1 Principle of Virtual Displacement for System
of Particles
The principle of virtual displacement is also called the principle of virtual work
or virtual displacement principle. Galileo began to apply this principle as far back
as the 17th century. In 1717 John Bernoulli first realized the universal significance of
the principle and its effect on solving the problem of statics. The virtual displacement
principle of particle system can be expressed as: If a particle system with the steady
and ideal constraints is in equilibrium state at a certain configuration, then
the sufficient and necessary conditions maintaining the balance is: The sum of
elementary works done by all positive forces acting on the particle system on
any virtual displacement is zero. Its mathematical expression is
δW =
n
i=1
F i · δr i = 0
(9.2.1)
or it is written in the form of components as
δW =
n
i=1
(F i x δx + F iy δy + F iz δz) = 0
(9.2.2)
Equations (9.2.1) and (9.2.2) is called the equation of virtual work.
Proof First prove the necessity of condition of the principle.
Let the composite force of active forces acting on the particle m i be F i , the
composite force of reactive forces of constraint be N i . Since the particle system is
in equilibrium state in a configuration, there will be ¨
r i = 0, according to Newton’s
second law, there will be
F i + N i = 0 (i = 1, 2, . . . , n)
Thus for any virtual displacement the expression
n
i=1
(F i + N i ) · δr i =
n
i=1
F i · δr i +
n
i=1
N i · δr i = 0
holds, because the particle system is an ideal constraint system, there is
n
i=1
N i · δr i = 0