Questions: Principle of Virtual Work and Generalized Forces

Question 1

What is the key advantage of expressing the principle of virtual work in generalized coordinates rather than Cartesian coordinates?

• A. Generalized coordinates always produce simpler algebraic expressions regardless of the system
• B. Constraint forces automatically drop out because virtual displacements consistent with constraints do no work for them
• C. The equations of motion become linear, making them easier to solve numerically
• D. Generalized coordinates eliminate the need to know the kinetic energy of the system

Answer: (B)

The key advantage is that constraint forces (normal forces, pin reactions, roller reactions at frictionless contacts) do no virtual work for displacements that satisfy the constraints. When virtual displacements are defined to be compatible with the constraints, constraint forces are perpendicular to those displacements and drop out automatically. This means you never need to introduce or solve for constraint forces — only the applied forces (and inertia forces, in dynamics) appear in the equations. This is what makes the method so powerful for multi-degree-of-freedom systems.

Question 2

A slider-crank mechanism consists of a rotating crank, a connecting rod, and a piston sliding in a cylinder. How many generalized coordinates are needed to fully describe its configuration?

• A. Three — one for the crank angle, one for the rod angle, and one for the piston position
• B. One — a single angle (e.g., crank angle) determines the position of every other link
• C. Two — the crank angle and piston displacement are independently variable
• D. Six — three translational and three rotational degrees of freedom for each rigid body

Answer: (B)

A slider-crank mechanism has one degree of freedom: specifying the crank angle completely determines the position of the connecting rod and piston through the geometric constraints. Once the crank angle θ is known, the constraint equations give the rod angle and piston displacement uniquely. This is exactly the power of generalized coordinates: they encode the constraints, so only the true independent degrees of freedom appear. Six Cartesian coordinates per body (option D) would over-specify the system; the constraints reduce it to one.

Question 3

When the principle of virtual work is formulated in generalized coordinates, the constraint forces (e.g., normal forces at frictionless supports) appear in the generalized force Qᵢ and is expected to be included explicitly.

• T. True
• F. False

Answer: False

Constraint forces at frictionless contacts are perpendicular to the virtual displacements that satisfy the constraints, so they do zero virtual work. When computing the generalized force Qᵢ = ∂(δW)/∂(δqᵢ), only applied forces contribute — constraint forces have already been eliminated by the choice of admissible virtual displacements. This is the fundamental reason for using generalized coordinates: you never need to solve for constraint forces at all, greatly simplifying the equations of motion.

Question 4

If a generalized coordinate qᵢ is an angle (measured in radians), the corresponding generalized force Qᵢ must have units of torque (N·m).

• T. True
• F. False

Answer: True

The generalized force Qᵢ is defined so that Qᵢ δqᵢ has units of energy (Joules = N·m). If δqᵢ is a dimensionless angle in radians, then Qᵢ must have units of N·m (torque) so the product has units of energy. This unit analysis is not arbitrary — it reflects the physical content of the work-energy relationship expressed in generalized coordinates. If qᵢ were a length (meters), Qᵢ would be a force (N). The units of Qᵢ always depend on the units of the corresponding coordinate.

Question 5

Explain why constraint forces drop out when virtual work is expressed in generalized coordinates, and why this makes the method powerful for complex systems.

Model answer: Virtual displacements in generalized coordinates are defined to be admissible — they satisfy all geometric constraints. Constraint forces (pin reactions, normal forces at frictionless contacts) act perpendicular to the directions allowed by the constraints, so they do zero work for any admissible virtual displacement. Since they contribute nothing to the total virtual work, they never appear in the generalized force Qᵢ. This eliminates the need to solve for constraint forces, reducing an n-body system with many constraints to n decoupled equations of motion in the true degrees of freedom.

For a system with many rigid bodies and constraints, the Newtonian approach requires writing equations for every body and then solving a large system that includes all the internal constraint forces (tensions, normal forces, pin reactions). The virtual work method sidesteps this entirely — by choosing generalized coordinates that respect the constraints, you automatically eliminate those forces from the problem. For a robot arm with 6 joints, for example, this means 6 equations of motion in 6 generalized coordinates rather than equations for dozens of forces and reactions between links.