Short Answer:

A virtual displacement is a small imaginary change or movement of a system from its equilibrium position, consistent with the system’s constraints. It is not a real motion but an assumed infinitesimal movement used in theoretical analysis to study equilibrium conditions.

In simple words, virtual displacement means a very small assumed movement of a particle or body that satisfies all the geometrical and physical restrictions (constraints) of the system. It helps in applying the principle of virtual work to determine the equilibrium of mechanical systems.

Detailed Explanation :

Virtual Displacement

The concept of virtual displacement is an important part of analytical mechanics, particularly in the study of equilibrium and motion using the principle of virtual work. In mechanics, a body or system of particles is often subjected to several forces and constraints that restrict its motion.

A virtual displacement is an imagined or hypothetical infinitesimal change in the configuration (position) of the system that is consistent with the given constraints at a fixed instant of time. It is called virtual because it does not represent a real or actual movement occurring over time, but rather a mathematical assumption used for analysis.

In simple terms, a virtual displacement is a small possible movement that a body can make without violating the system’s restrictions.

Definition of Virtual Displacement

A virtual displacement can be defined as:

“A virtual displacement is an infinitesimal imaginary change in the position of a system consistent with its constraints at a given instant of time.”

The word infinitesimal means extremely small, and consistent with constraints means the displacement must follow the same limitations or boundary conditions that apply to the actual motion of the system.

For example:

• A bead moving on a smooth wire can only move along the wire. Any small movement (virtual displacement) must lie along the wire and not perpendicular to it.
• A pendulum can only move along the arc of a circle, not away from it; hence its virtual displacement is always tangential to the arc.

Mathematical Expression of Virtual Displacement

If a particle moves from position  to  under constraints, then a very small assumed change in its position is represented by δr, which denotes the virtual displacement.

If  is the position vector of a particle, then

This change  is not associated with any actual time interval (unlike real displacement, which occurs over time). It is only a possible small change that satisfies all the system’s constraints at that instant.

Characteristics of Virtual Displacement

1. Imaginary and Infinitesimal:
   Virtual displacement is not a real movement but a small assumed change used for analysis. It is extremely small (infinitesimal) and hypothetical in nature.
2. Consistent with Constraints:
   Every virtual displacement must satisfy all the constraints of the system. The system cannot move in directions that violate restrictions, such as rigid links, fixed supports, or surfaces.
3. Independent of Time:
   A virtual displacement does not occur over a period of time; it is considered at a particular instant. It has no relation to the velocity or acceleration of motion.
4. Possible Displacement:
   It represents one of the possible small displacements the system can make without violating its geometric conditions.
5. Used for Equilibrium Analysis:
   Virtual displacement is mainly used to apply the principle of virtual work, which helps determine whether a body or system is in equilibrium.

Difference Between Actual and Virtual Displacement

• Actual Displacement: It occurs in real time due to forces acting on a body, changing its position.
• Virtual Displacement: It is an imagined, infinitesimal movement at a fixed instant, used for theoretical analysis and does not depend on time or applied forces.

Example: When a pendulum swings, its movement is an actual displacement, but if we imagine a very small possible shift of the pendulum from its current position without actual motion, that is a virtual displacement.

Application of Virtual Displacement

The concept of virtual displacement is used in various areas of mechanics and engineering, especially in the principle of virtual work, which states:

“If a body is in equilibrium, the total virtual work done by all forces acting on the body during any virtual displacement is zero.”

This principle helps determine the equilibrium conditions of complex mechanical systems without directly calculating reaction forces.

Applications include:

1. Analysis of mechanical systems: Used in the study of equilibrium of structures, beams, and machines.
2. Dynamics and kinematics: Helps derive equations of motion for systems with constraints.
3. Robotics and mechanisms: Used to analyze motion and forces in linkages and jointed systems.
4. Structural engineering: Simplifies problems involving static equilibrium of frames or trusses.

Example to Understand Virtual Displacement

Consider a smooth inclined plane with a block resting on it. The block can only move up or down the plane because the surface prevents motion perpendicular to it.

If we imagine a very small movement  of the block along the plane, this small shift is the virtual displacement.
Even though the block may not actually move, this assumed displacement is used to test whether the forces (like weight, normal reaction, and friction) satisfy equilibrium conditions.

Importance of Virtual Displacement in Engineering Mechanics

1. It forms the basis of the principle of virtual work, which is essential for analyzing equilibrium without directly involving unknown reaction forces.
2. It helps in simplifying complex mechanical systems with multiple constraints.
3. It allows engineers to determine whether a structure or mechanism is stable and in equilibrium.
4. It is widely used in advanced mechanical system design, including robotics, mechanisms, and structural analysis.
5. It provides a clear theoretical understanding of how systems respond to forces under constraints.

Conclusion

A virtual displacement is an imaginary, infinitesimal change in the position of a system that is consistent with its constraints at a given instant. It does not occur in real time but is used as a mathematical concept to study the equilibrium of mechanical systems. The principle of virtual work, based on virtual displacement, helps engineers analyze complex systems easily and accurately. Thus, virtual displacement is a powerful tool in mechanical engineering and theoretical mechanics for understanding the behavior of constrained systems.