Lecture Notes on D'Alembert's Principle and Applications

Introduction

• Principle of virtual work applies to static equilibrium systems.
• Extension to dynamic systems called D'Alembert's principle.

D'Alembert's Principle

• For a dynamical system, the rate of change of momentum of a particle is equal to the net force on it:
  • (\frac{dp}{dt} = F_{net})
• Can be split into:
  • Applied force
  • Constraint force
• Multiply both sides by virtual displacement (\delta r_i) and sum over all particles:
  • (\sum_{i} (F_{i_{applied}} - P_i) \cdot \delta r_i = 0) (Net work done by constraint forces is 0)

Example 1: Bead on a Wire

• Bead travels on a frictionless wire with coordinates (x,y).
• Constraint: (y = \tan(\alpha) \cdot x)
• Virtual displacements (\delta y = \tan(\alpha) \cdot \delta x)
• Forces:
  • Only applied force: gravitational force (F = -mg \hat{j})
• D'Alembert's Principle:
  • (m \cdot \ddot{r} - F \cdot \delta r = 0)
• Results in:
  • (m x \ddot{} = -g \sin(\alpha) \cos(\alpha))

Example 2: Wedge Block System

• Setup: Wedge with block sliding on it, both without friction.
• Constraints:
  • Wedge remains on the table.
  • Block does not leave the surface of the wedge.
• Position vectors for wedge and block:
  • (A: (X,Y)) and (B: (x,y))
• Constraint equations:
  • Wedge's y-coordinate fixed (set to 0).
  • Block's position described by (y = \tan(\alpha) (X - x))
• Applying D'Alembert's Principle:
  • Write forces for both wedge and block.
  • Resulting equations lead to:
    • Conservation of horizontal momentum in absence of net external force.

Example 3: Pendulum on a Moving Trolley

• Trolley moving horizontally, pendulum fixed to it.
• Known motion of the trolley governed by a function (f(t)).
• Length of pendulum remains fixed: (x - f(t)^2 + y^2 = L^2)
• Transformations: (x = L \sin(\theta), y = L \cos(\theta))
• Forces acting on the bob of the pendulum:
  • Only applied force: (-mg \hat{j})
• Actual vs virtual displacements:
  • Calculate dR accounting for the trolley's movement.
• Results in: (\theta) equation of motion relating to the movement of the trolley and gravitational force.

Conclusion

• D'Alembert's principle can be applied to derive equations of motion for dynamic systems.
• The complexity of these calculations highlights the need for a formal derivation method like Lagrange's equations for simplifying analysis.