9.2 Principle of Virtual Displacement
525
Therefore, there is
n
i=1
F i · δr i = 0
The necessity of condition of the principle has been proved.
Then prove the sufficiency of condition of the principle, using the reduction
to absurdity. Assuming that the expression (9.2.1) is valid for an arbitrary virtual
displacement, consequently the original equilibrium state of the system under the
action of the active force has been broken, then at least one particle m j under the action
of the active force F j and reactive force of constraint N j produces the acceleration
a j
a j = ¨
r j =
F j + N j
m j
= 0
Since the motion starts from static, in the time element dt it will produce a real
displacement dr j which is the same as the direction of a j , so at least for the particle
there is
(F j + N j ) · dr j > 0
For the whole system there is
n
i=1
F i · dr i > 0
For the steady system, the real displacement dr i is one of the possible forms of
virtual displacement δr i , thus when taking δr i = dr i , there is
n
i=1
F i · δr i > 0
This is in contradiction with the assumption condition (9.2.1), that is to say, as
long as the expression (9.2.1) holds, the case in which the balance is broken could not
happen. Conversely, as long as the expression (founded 9.2.1) is valid, the system
will maintain equilibrium state, thus the sufficiency of condition of the principle.
Quod erat demonstrandum.
As the virtual displacement principle of a foundation for the variational principle
of mechanics, it is necessary to emphasize its variational characteristics. The virtual
displacement principle points out that the location of the system balance refers to
such a position that the system may have all the positions (corresponding to various
virtual works), the sum of the virtual works done by the active forces at this time
525
Therefore, there is
n
i=1
F i · δr i = 0
The necessity of condition of the principle has been proved.
Then prove the sufficiency of condition of the principle, using the reduction
to absurdity. Assuming that the expression (9.2.1) is valid for an arbitrary virtual
displacement, consequently the original equilibrium state of the system under the
action of the active force has been broken, then at least one particle m j under the action
of the active force F j and reactive force of constraint N j produces the acceleration
a j
a j = ¨
r j =
F j + N j
m j
= 0
Since the motion starts from static, in the time element dt it will produce a real
displacement dr j which is the same as the direction of a j , so at least for the particle
there is
(F j + N j ) · dr j > 0
For the whole system there is
n
i=1
F i · dr i > 0
For the steady system, the real displacement dr i is one of the possible forms of
virtual displacement δr i , thus when taking δr i = dr i , there is
n
i=1
F i · δr i > 0
This is in contradiction with the assumption condition (9.2.1), that is to say, as
long as the expression (9.2.1) holds, the case in which the balance is broken could not
happen. Conversely, as long as the expression (founded 9.2.1) is valid, the system
will maintain equilibrium state, thus the sufficiency of condition of the principle.
Quod erat demonstrandum.
As the virtual displacement principle of a foundation for the variational principle
of mechanics, it is necessary to emphasize its variational characteristics. The virtual
displacement principle points out that the location of the system balance refers to
such a position that the system may have all the positions (corresponding to various
virtual works), the sum of the virtual works done by the active forces at this time