University:
Massachusetts Institute of Technology
Course:
2.032 | Dynamics
Academic year:

2023
Views:

274

Pages:

2
Author:

Uriel Fox

Session 13 Variational Approach to Dynamics (Lagrangian Mechanics, Energy Principle) Scalar (work-energy based) (as opposed to vector-based) Frame invariant Renders equations of motion without (in most cases) getting the constraint reaction forces involved Reaction forces are not immediately available (disadvantage) INGREDIENTS Generalized Coordinates x1, x2, ..., xn F: active force R: constraint force Often it is possible to select a smaller set of coordinates (it necessarily position) that uniquely determines the position of the system and adequately accounts for (one of the) constraints. q1, q2, ..., qn generalized coordinates x = x(q1, ..., qn) y = y(q1, ..., qn) z = z(q1, ..., qn) NOTE: q_i is a complete set of coordinates but not necessarily independent. (ii) Constraints: Scalar relations that limit possible motions of the system. f_i(q1, q2, ..., qn, t) = 0, j = 1, ..., m Types of Constraints: Holonomic/Scleronomic: Independent of time Non-holonomic/Rheonomic: Time-dependent Example 1. Moving double pendulum: f = (x₁ - x₂)² + (y₁ - y₂)² - l₁² - l₂² = 0 Holonomic, Scleronomic Constraint 2. Pendulum with Oscillating Support: x - x₀cosωt = 0 y - l cosθ = 0 Holonomic, Rheonomic Constraint II Analytical Mechanics Ingredients: Generalized Coordinates: q = {q₁, q₂, ..., qₙ} (Complete, not necessarily independent) Constraints f₁(q₁, q₂, ..., qₙ, t) = 0 Holonomic/Scleronomic Non-holonomic/Rheonomic #DOF = #DOF - #Constraints Example 3. 2D Rolling Disc: yₚ = 0 vₚ = 0 ⇒ ẋ + Rφ̇ = 0 Strictly Speaking (4) gives 2 non-holonomic, scleronomic constraints But f gives (xₚ - Rcosθ)² + (yₚ - Rsinθ)² - R² = 0 integrated non-holonomic constraint (semi-holonomic) yₚ + y_c = R

Session 13 Notes: Generalized Coordinates, Constraints, Virtual Displacements

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