Dr. Wang Xingbo Fall ， 2005 Mathematical & Mechanical Method in Mechanical Engineering

1.Generalized coordinates, generalized forces and constraints 2.Generalized Newton’s 2nd Law 3.Generalized Equation of Motion 4.Lagrange’s Equations (Scalar Potential Case) Mathematical & Mechanical Method in Mechanical Engineering Introduction to Lagrangian Mechanics

In two-dimensions the positions of a point can be specified either by its rectangular coordinates (x, y) or by its polar coordinates. There are other possibilities such as confocal conical coordinates that might be less familiar. In three dimensions there are the options of rectangular coordinates (x, y, z), or cylindrical coordinates (ρ, , z) or spherical coordinates (r, θ,  ) (see Fig. 9.1) – or again there may be others that may be of use for specialized purposes (inclined coordinates in crystallography, for example, or a more general curvilinear coordinate system). Mathematical & Mechanical Method in Mechanical Engineering Coordinates

In many cases, other more generalized coordinates are needed to describe a dynamic system Any that is used to describe the dynamic state of a system is called generalized coordinate. Mathematical & Mechanical Method in Mechanical Engineering Generalized coordinates

In many systems, the particles may not be free In many systems, the particles may not be free to wander anywhere at will; The particle ’ s motion may be constrained to lie within a submanifold of the full configuration space Mathematical & Mechanical Method in Mechanical Engineering Constraints A constraint that can be described by an equation relating the coordinates (and perhaps also the time) is called a holonomic constraint.

The total number of generalized coordinates minus the number of holonomic constraints contributes the degrees of freedom of a system. If a system of N particles is subject to k holonomic constraints, then the degree of freedom is 3N-k Mathematical & Mechanical Method in Mechanical Engineering Degrees of freedom

Generalized dynamics — the derivation of differential equations (equations of motion) for the time evolution of the generalized coordinates Mathematical & Mechanical Method in Mechanical Engineering Generalized Newton’s 2nd Law the goal of generalized dynamics is to find universal forms of the equations of motion

Position of each of the N particles making up the system be given as a function of the n generalized coordinates q by r k = x k (q, t), k = 1,...,N. δq the virtual work done on the system in displacing it by an arbitrary infinitesimal amount δq at fixed time t is given by Mathematical & Mechanical Method in Mechanical Engineering Generalized dynamics

Calculate the virtual work in terms of the displacements of the N particles assumed to make up the system and the forces F k acting on them Mathematical & Mechanical Method in Mechanical Engineering Generalized dynamics

Generalized force Mathematical & Mechanical Method in Mechanical Engineering Generalized dynamics

When there are holonomic constraints on the system we decompose the forces acting on the particles into what we shall call explicit forces and forces of constraint. Mathematical & mechanical Method in Mechanical Engineering Generalized forces By forces of constraint,, mean those imposed on the particles by the rigid rods, joints, sliding planes etc The explicit force on each particle,, is the vector sum of any externally imposed forces, such as those due to an external gravitational or electric field, plus any interaction forces between particles such as those due to elastic springs coupling point masses, or to electrostatic attractions between charged particles

Mathematical & mechanical Method in Mechanical Engineering Properties of Cross Product The force of constraint is N

Mathematical & mechanical Method in Mechanical Engineering Generalized Equation of Motion

Mathematical & mechanical Method in Mechanical Engineering Generalized Equation of Motion

Mathematical & mechanical Method in Mechanical Engineering Generalized Equation of Motion

Mathematical & mechanical Method in Mechanical Engineering Generalized Newton’s second law

Mathematical & mechanical Method in Mechanical Engineering Example 9.1 Let us check in Cartesian system a motion that we can recover Newton’s equations of motion as a special case when q = {x, y, z}

Mathematical & mechanical Method in Mechanical Engineering Example 9.2

Mathematical & mechanical Method in Mechanical Engineering Example 9.2 L=T-V

In many problems in physics the forces F k are derivable from a potential, V (r 1, r 2, · · ·, r N ). The classical N-body problem the particles are assumed to interact pairwise via a two-body interaction potential V k,l (r k, r l ) ≡ U k,l (|r k - r l |) such that the force on particle k due to particle l is given by Mathematical & mechanical Method in Mechanical Engineering Lagrange’s Equations (Scalar Potential Case)

Mathematical & mechanical Method in Mechanical Engineering Lagrange’s Equations

Mathematical & mechanical Method in Mechanical Engineering Example of fields L ≡ T - V

Physical paths in configuration space are those for which the action integral is stationary against all infinitesimal variations that keep the endpoints fixed Mathematical & mechanical Method in Mechanical Engineering Hamilton’s Principle

Class is Over! See you next time! Mathematical & Mechanical Method in Mechanical Engineering