Lagrange Approach Basilio Bona DAUIN – Politecnico di Torino Semester 1, 2016-17 B. Bona (DAUIN) Lagrange Semester 1, 2016-17 1 / 50
Introduction A multibody system is considered as a system in which the dynamic equations derive from a unifying principle . This principle is based on the fact that, in order to describe the motion of a system, it is sufficient to consider some scalar quantities. These quantities were in origin called vis viva and work function , today they are called kinetic energy and potential energy . Both are state functions , i.e., functions that map the value of the state vector into a scalar function. The concept of state will be defined later; for the moment we simply consider that the state corresponds to the two vectors of the generalized coordinates q ( t ) and of the generalized velocities ˙ q ( t ). B. Bona (DAUIN) Lagrange Semester 1, 2016-17 2 / 50
This general principle is called the Principle of Least Action . Let us consider the space Q of the generalized coordinates q ∈ Q , as sketched in Figure for a two-dimensional space Q . A particle starts its motion at time t 1 in Q 1 = q ( t 1 ) and ends it motion at time t 2 reaching the state Q 2 = q ( t 2 ) (or vice-versa, since time can be reversed). Assume that the motion keeps constant the total energy, i.e., the sum E = K + P of the kinetic energy K and the potential energy P that the particle has at time t 1 . B. Bona (DAUIN) Lagrange Semester 1, 2016-17 3 / 50
Q 1 and Q 2 are connected by a continuous path (trajectory) called true trajectory , that is unknown, since it is what we want to compute as the result of the dynamical equation analysis. If we choose at random different trajectories, with the only condition that the two boundary points remain the same ( perturbed trajectories ), the chance to obtain exactly the true trajectory will be very small. What characterizes the true trajectory with respect to all possible other perturbed trajectories? Euler contributed to the solution of this problem, but Lagrange developed a complete theory, that was later extended by Hamilton. B. Bona (DAUIN) Lagrange Semester 1, 2016-17 4 / 50
The true trajectory is the one that minimizes the integral of the vis-viva (i.e., twice the kinetic energy) of the entire motion between Q 1 and Q 2 . This integral is called action and has a constant and well defined value for each perturbed trajectory having constant total energy E ( E depends only on the initial state). The least action principle states that the nature “chooses”, among the infinite number of trajectories starting in q ( t 1 ) and ending in q ( t 2 ), the trajectory that minimizes the definite integral � t 2 K ∗ ( q ( t ) , ˙ S = q ( t )) d t t 1 of a particular state function K ∗ ( q ( t ) , ˙ q ( t )). The integral between the initial time t 1 and the final time t 2 must obey to the boundary constraints in the two time instants. B. Bona (DAUIN) Lagrange Semester 1, 2016-17 5 / 50
The scalar quantity S is the integral of a function and is called a functional . A functional is a mapping between a function and a real number; the function shall be considered as a whole, i.e., not a single particular value; in this sense a functional is often the integral of the function. The minimization of a functional is based on a particular mathematical technique, called calculus of variations . The conditions that guarantee the minimization of S provide a set of differential equations that contain the first and second time derivatives of the q i ( t ); this set of equations completely describes the system dynamics . B. Bona (DAUIN) Lagrange Semester 1, 2016-17 6 / 50
These differential equations specify the evolution of a physical quantity as the result of infinitesimal increments of time or position; summing up this infinitesimal variations we obtain the physical variables at every instant, knowing only their initial value and possibly some initial derivative: we can say that the motion has a local representation. The action characterizes the motion dynamics requiring only the knowledge of the states at the initial and final times; every intermediate value of the variables can be determined by the minimization of the action, that is a global , rather than a local, measure. The Lagrange approach is based on the definition of two scalar quantities, namely the total kinetic co-energy K ∗ and the total potential energy P associated with the body. The reason for using the term co-energy instead of the term energy , will be clarified later. B. Bona (DAUIN) Lagrange Semester 1, 2016-17 7 / 50
Lagrangian approach The Lagrange method allows to define a set of Lagrange equations , that have some advantages with respect to the vector equations provided by the Newton-Euler approach. The approach provides n second-order scalar differential equations, directly expressed in the generalized coordinates ˙ q i ( t ) e q i ( t ). If holonomic constraint are present, the constraint forces do not appear in the equations . The kinetic co-energies and the potential energies are independent of the reference frame used to represent the body motion. The kinetic co-energies and the potential energies are additive scalars: in a multi-body system the total energies/co-energies are the sum of each energy/co-energy component. B. Bona (DAUIN) Lagrange Semester 1, 2016-17 8 / 50
Linear and angular momenta Linear momentum h L is the physical vector defined as the product of a body mass M for its linear velocity v (the center-of-mass velocity) h L ( t ) = M v ( t ) In non-relativistic mechanics the mass M of a body is constant (except for some particular cases, as rockets consuming fuel, etc.) Angular momentum h A (also called moment of momentum or rotational momentum) is the physical vector defined as the product of a body rotational inertia Γ for its rotational velocity ω h A ( t ) = Γ ( t ) ω ( t ) While the mass of a body is usually constant, the inertia matrix (or inertia tensor ) Γ may vary in time. B. Bona (DAUIN) Lagrange Semester 1, 2016-17 9 / 50
Kinetic energy and co-energy for single point-mass The mechanical kinetic energy associated to a point-mass m is defined as the work necessary to increase the linear or angular momentum from 0 to h , i.e., � h K ( h ) = 0 d W The infinitesimal work associated to the mass is given by d W = f · d x + τ · d α where the symbol · indicates the scalar product, and f is the resultant of the applied linear forces on the mass, d x is the infinitesimal linear displacement increment, τ is the resultant of the applied angular torques, and d α is the infinitesimal angular displacement increment. Moreover f = d h L τ = d h A d t d t B. Bona (DAUIN) Lagrange Semester 1, 2016-17 10 / 50
The resulting infinitesimal work is therefore the sum of two terms d W = d W L + d W A = d h L d t · d x + d h A d t · d α = v · d h L + ω · d h A and we can write � h K ( h ) = 0 v · d h L + ω · d h A The kinetic energy is a scalar state function associated to the particle states ( v , ω ) and ( h L , h A ). Another state function associated to the point-mass, called mechanical kinetic co-energy , is defined as � v K ∗ ( v ) = 0 h L · d v + h L · d ω As shown in Figure, between the mechanical energy and the co-energy a relation exists K ∗ ( v ) = h · v − K ( h ) (for notational simplicity, only the linear velocity is considered) B. Bona (DAUIN) Lagrange Semester 1, 2016-17 11 / 50
Figure: This relation is an example of the Legendre transformation B. Bona (DAUIN) Lagrange Semester 1, 2016-17 12 / 50
In particular, if the mass particle is moving at a velocity significantly smaller that the speed of light c , i.e., it is not a relativistic mass , the relation is h = m v with m constant, and the two “energies” become � h m h · d h = 1 1 2 m h · h = 1 2 m � h � 2 K ( h ) = 0 � v 0 m v · d v = 1 2 m v · v = 1 2 m � v � 2 K ∗ ( v ) = As one can see, in this case the kinetic energy and co-energy are the same since � h � 2 = m 2 � v � 2 This does not happen for relativistic masses where m = m ( v ( t )). B. Bona (DAUIN) Lagrange Semester 1, 2016-17 13 / 50
In an extended body composed by N masses m i the kinetic co-energy is the sum of the kinetic co-energy of each mass N K ∗ ( v ) = 1 ∑ m i v i · v i 2 i =1 B. Bona (DAUIN) Lagrange Semester 1, 2016-17 14 / 50
x 0 We consider the velocity v 0 i = ˙ i with respect to R 0 . Each velocity i in R 0 can be computed from the general relation 0 0 x 0 i ( t ) = ω 0 01 ( t ) × ρ 0 i ( t )+ R 0 x 1 i ( t )+ ˙ 1 ( t ) = ω 0 01 ( t ) × ρ 0 ( t )+ ˙ ˙ 1 ˙ d d 1 ( t ) where the term R 0 1 ˙ x i ( t ) is zero, since the point-masses are fixed with x 1 respect to the body-frame, i.e., ˙ i ( t ) = 0 . Now we consider a purely translatory motion and then a purely rotational motion. B. Bona (DAUIN) Lagrange Semester 1, 2016-17 15 / 50
Translational motion If the motion is purely translational all point masses have the same linear velocity v 0 with respect to R 0 0 x 0 i ( t ) = ˙ 1 ( t ) ≡ v 0 ( t ) ∀ i ˙ d then N K ∗ = 1 m i = 1 2 m tot v 0 · v 0 = 1 � 2 = 1 2 v 0 · v 0 � � v 0 � 2( v 0 ) T ( m tot I ) v 0 ∑ 2 m tot i =1 where the mass m tot is the total body mass. The kinetic co-energy is equivalent to that of one particle with total mass m tot with the translational velocity v 0 . The total mass m tot can be ideally concentrated in the body center-of-mass C , whose position is x c 1 x c m tot = ∑ m tot ∑ x i m i → x c = x i m i i i B. Bona (DAUIN) Lagrange Semester 1, 2016-17 16 / 50
Recommend
More recommend
