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9 Variational Principles in Mechanics and Their Applications
9.1.3 Actual Displacement and Virtual Displacement
An unfree system moves under an initial condition, the radius vectors r i of various
particles in the system satisfy the differential equation of dynamic and initial condition, as well as satisfy all the constraint equations, this kind of motion is called the
actual motion or true motion. The actual motion means really happened motion.
The variation of each particle in the particle system in its actual motion in space
position is called the actual displacement or real displacement, it is called the
displacement for short. In other words, any change of the configuration of a system
is called the displacement. A real displacement takes time. The displacement of an
object can be divided into two kinds, one is after the displacement occurs, that each
point in the object still remains the relative position of the initial state unchanged,
namely the distance between arbitrary two points in the object does not change, this
displacement of the object is called the rigid (body) displacement. The other is
when the displacement occurs, the object changes its relative position of each point
in space, namely changes its size and shape, this phenomenon is called the deformation. This displacement is caused by the deformation of the object. In general,
the above two kinds of displacements often exist at the same time, and the size and
direction are different respectively, but elastic mechanics is more interested in the
latter displacement, because this displacement relates to the stress within the object.
The material or object under the action of external force produces deformation,
which can fully recover its original shape after removing the external force, and the
stress and strain have a single value corresponding relationship, this property is called
elasticity. The object composed of elastic material is called the elastic body, elastic
system or system of elasticity. When an elastic body deforms, the change of position
of each point in space is called the displacement of elastic body. The displacement
of elastic body needs to satisfy the deformation compatibility conditions and the
geometrical boundary conditions. The vector connecting elastic body before and
after deformation at some point is called the displacement vector at the point.
The real motion trajectory of a particle in a particle system passing through the
fixed endpoints is called the actual path, natural path, proper path or true path.
Other possible motion trajectories having a common starting point and end point with
the true path, and satisfying the constraint conditions are called the nearby path,
neighboring path or varied path. The true path in addition to satisfy the constraint
conditions must also satisfy the fundamental law of dynamics. Due to the uniqueness
of real motion system, which determines the real displacement of each particle in a
certain time period is unique. A real displacement can be either finite value, also can
be infinitely small magnitude.
The imaginary arbitrary infinitesimal displacement that a particle or particle
system in the given time satisfies the constraint equation and allowed by the constraint
is called the virtual displacement of the particle or particle system at the moment.
In other words, an assumed any tiny displacement that a particle or particle system
under the condition of constraint allowance has nothing to do with the time is
called the virtual displacement. The virtual displacement is assumed, it does not
9 Variational Principles in Mechanics and Their Applications
9.1.3 Actual Displacement and Virtual Displacement
An unfree system moves under an initial condition, the radius vectors r i of various
particles in the system satisfy the differential equation of dynamic and initial condition, as well as satisfy all the constraint equations, this kind of motion is called the
actual motion or true motion. The actual motion means really happened motion.
The variation of each particle in the particle system in its actual motion in space
position is called the actual displacement or real displacement, it is called the
displacement for short. In other words, any change of the configuration of a system
is called the displacement. A real displacement takes time. The displacement of an
object can be divided into two kinds, one is after the displacement occurs, that each
point in the object still remains the relative position of the initial state unchanged,
namely the distance between arbitrary two points in the object does not change, this
displacement of the object is called the rigid (body) displacement. The other is
when the displacement occurs, the object changes its relative position of each point
in space, namely changes its size and shape, this phenomenon is called the deformation. This displacement is caused by the deformation of the object. In general,
the above two kinds of displacements often exist at the same time, and the size and
direction are different respectively, but elastic mechanics is more interested in the
latter displacement, because this displacement relates to the stress within the object.
The material or object under the action of external force produces deformation,
which can fully recover its original shape after removing the external force, and the
stress and strain have a single value corresponding relationship, this property is called
elasticity. The object composed of elastic material is called the elastic body, elastic
system or system of elasticity. When an elastic body deforms, the change of position
of each point in space is called the displacement of elastic body. The displacement
of elastic body needs to satisfy the deformation compatibility conditions and the
geometrical boundary conditions. The vector connecting elastic body before and
after deformation at some point is called the displacement vector at the point.
The real motion trajectory of a particle in a particle system passing through the
fixed endpoints is called the actual path, natural path, proper path or true path.
Other possible motion trajectories having a common starting point and end point with
the true path, and satisfying the constraint conditions are called the nearby path,
neighboring path or varied path. The true path in addition to satisfy the constraint
conditions must also satisfy the fundamental law of dynamics. Due to the uniqueness
of real motion system, which determines the real displacement of each particle in a
certain time period is unique. A real displacement can be either finite value, also can
be infinitely small magnitude.
The imaginary arbitrary infinitesimal displacement that a particle or particle
system in the given time satisfies the constraint equation and allowed by the constraint
is called the virtual displacement of the particle or particle system at the moment.
In other words, an assumed any tiny displacement that a particle or particle system
under the condition of constraint allowance has nothing to do with the time is
called the virtual displacement. The virtual displacement is assumed, it does not