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Rigid Body Dynamics Rigid Body Dynamics If the aim of kinematics is to \describe" the body motion, the aim of dynamics is to \explain" it; the history of mechanics shows that the passage from description to explanation requires the introduction of a new physical entity, that of mass or, in alternative, that of force. The fundamental law of mechanics, due to Newton, is analytically expressed as a vectorial equation between force, mass and acceleration of a translating rigid mass 2 3 2 3 fx ax 4 5 4 5 f = ma or else fy = m ay fz az [ ] T where the linear acceleration a = ax ay az is a kinematic quantity, obtained ad the time derivative of the body linear velocity. This law is true in principle only for a single point-mass particle with mass m, where the applied force f and the acceleration a can exchange the role of cause and effect: if a force f is applied to the particle, the particle accelerates with a linear acceleration equal to a, and conversely, if a particle has a linear acceleration a, the particle is subject to a force f proportional to its mass. [ ] [ ] T T If we jointly know the two vectors f = fx fy fz and a = ax ay az , we can compute the mass from one or any of the following scalar relations as f f f m = x = y = z ax ay az Though, in this last case, we are applying a circular argument, since no definitions of \acceleration" or \force" exist that are independent from the measurement of a mass; we are therefore compelled to use some trick, as clearly illustrated in [?]. If a single point mass is connected to others to form a rigid body, the Newton law is still valid, given that we observe some precautions. Every point-mass shall be isolated and we must consider and deduce the forces applied on it by the other masses; that is, we must introduce the constraint forces in addition to the external forces applied on the body. It is important to notice that the vector equations are dependent on the representa- tion used to characterize its components, and change changing the reference frame 1 2 used to represent the force and acceleration vectors, although the masses, at least in a non-relativistic motion state, do not vary changing the reference frame. In this textbook we will use an analytical approach, as defined in [?], instead of the vectorial one due to Newton. We consider the multibody system as a system in which the dynamical equations derive from a unifying principle that implicitly includes and generates these equations. This principle is based on the fact that, in order to describe the motion of a multi- body system is sufficient to consider and use in a proper way some suitable scalar quantities; these were in origin called by Leibnitz vis viva and work function, nowa- days take the name of kinetic energy and potential energy. They are an example of the so-called state functions, since to each value of the state vector1 This general principle takes the name of principle of least action; it can be roughly described in the following way. Let us consider the space Q of the generalized coordinates q 2 Q, as sketched in Figure 8 for a two-dimensional space Q; let us assume that a particle starts its motion at time t1 in the state Q1 = q(t1) and ends it motion at time t2 having reached the state Q2 = q(t2). Let us further assume that the motion keeps constant the sum E = C + P of the kinetic energy C and the potential energy P that the particle has at time t1. Given the continuity of motion, the two points Q1 and Q2 are connected by a continuous path (or trajectory), as the one represented by a continuous line in Figure 8; this trajectory is called the true trajectory, and at least in principle, it is unknown, since it is exactly what we want to compute as the result of the dynamical equation analysis. If we had chosen at random a different trajectory, with the only condition that the two boundary point remain fixed, and called this trajectory a perturbed trajectory, the chance to obtain exactly the true trajectory would have been minimal. Now we ask what is that characterize the true trajectory with respect to all possible other perturbed trajectories. Euler was the first mathematician to contribute to the solution of this problem, but it was Lagrange that developed a complete theory, that was later extended by Hamilton: the true trajectory is the one that minimizes the integral of the so-called vis-viva (i.e., two times the kinetic energy) of the entire motion between Q1 and Q2. This integral is called action and has a constant and well defined value for each perturbed trajectory at constant E. The least action principle states that the nature \chooses", among the infinite number of trajectories starting in q(t1) and ending in q(t2), the trajectory that 1The concept of state will be defined in Section XXX; for the moment we simply consider that the state corresponds to the two vectors q(t) and q_ (t). 3 Figure 1: True and perturbed trajectories in the configuration space Q. minimizes the definite integral S of a particular state function C∗(q(t); q_ (t))2 that depends on the generalized coordinates q(t) and the generalized velocities, where q_ (t) Z t2 S = C∗(q(t); q_ (t))dt (1) t1 In order to apply this principle it is therefore necessary to compute the trajectory in the space Q that minimizes S. The integral in (1) is computed between the initial time t1 and the final time t2 and must obey to the boundary constraints active at these two times. The minimization of a functional3 is based on a particular mathematical technique, called calculus of variations, whose illustration goes beyond the scope of the present text. The interested reader can find additional treatment in [?], Appendix D. FARE APPENDICE? The conditions that assure the attainment of the minimum of S provide a set of differential equations that contain the first and second time derivatives of the qi(t); this set completely describes the system motion dynamics. As a final comment, we can say that the least action principle makes use of a concept - the action - that overturns the usual representation of the physical system dynamics 2We will see later what this state function C∗ is. 3A functional is a mapping between a function and a real number; the function shall be con- sidered as a whole, i.e., not a single particular value; in this sense a functional is often the integral of the function, as in the above equation for S. 4 by means of a set of differential equations. With some approximation we can say that the 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. On the contrary, 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 local, measure. In any case the implementation of the principle of least action produces as a result a set of differential equations, different from these obtained with the local representation. In the following Sections we will illustrate how to obtain these equations. 0.1 Point Mass System Yo introduce the Lagrangian approach and compare it with the Newtonian approach we must define some general physical entities Let us start to consider a sate of N point masses mi, with i = 1; ··· ;N), as schemat- ically represented inn Figure 2. These systems are called multi-point systems. Figure 2: A system of N point masses mi. The position r i and the velocity v i are represented in a generic reference frame Rb. Now we consider a pure rotation motion around a point O, that, for simplicity, is the origin of the reference frame Rb; all vectors will be represented in this frame. If b necessary we indicate the vectors as r i , otherwise with r i. [ ] T The position of the generic mass mi is defined by the vector r i = xi yi zi . If the rotation velocity of the system is given by the vector !, every mass will acquire 0.1. POINT MASS SYSTEM 5 a linear velocity v i, as the result of the rotation, whose value is v i(t) = !(t) × r i(t) = S(!(t))r i(t) (2) 4 We define the linear momentum pi(t) as the product between the mass mi and the velocity v i(t) pi(t) = miv i(t) In this Chapter, the symbol p indicates the linear momentum an not the cartesian pose of a body or the position of a point, as in the preceding Chapters. 0.1.1 Moment of a Force We briefly recall that, given a point mass located in a point P represented by the vector r in the reference frame R, and a force f applied to it, the moment of the force with respect to a point O is given by the cross product dv dp r × f = r × m = r × (3) dt dt where the terms on the right side arise from the Newton equation.
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