TECNOLOGIE PER LA RIABILITAZIONE. lezione # 7 dinamica del corpo rigido. A body. Rigid body kinematics in summary. What do we mean with kinetics?

Transcription

1 TENLGIE PER LA RIABILITAZINE AA Università degli Studi di Napoli "ederico II acoltà di Ingegneria orso di Laurea in Ingegneria Biomedica lezione # 7 dinamica del corpo rigido A body A body is a material system (matter) and may be modelled as follows: Its volume is small relative to the space within which it moves and we are only interested in its general position in space particle We have no interest in its deformations, we concentrate on its gross motion rigid body Prof. Aurelio appozzo Dipartimento di Scienze del Movimento Umano e dello Sport Laboratorio di Bioingegneria Istituto Universitario di Scienze Motorie - Roma dismus dinamica del corpo rigido 1 We are interested in its deformations and we try not to be simultaneously intrigued by its motion in space viscoelastic body dismus dinamica del corpo rigido 3 What do we mean with kinetics? The book reads as follows: Kinetics is the study of the relation existing between the forces acting on a body and its motion. Rigid body kinematics in summary A generic movement of a rigid body may be described as if it were made of a translation (linear displacement, velocity, acceleration) plus a rotation (angular displacement, velocity, acceleration) t 1 t2 This leads to the solution of two basic problems: predict the motion caused by given forces, Y forces Equations of motion kinematics determine the forces required to produce a given motion forces Equations of motion kinematics X Z dismus dinamica del corpo rigido 2 dismus dinamica del corpo rigido 4 1

2 Rigid body kinematics in summary A generic movement of a rigid body may be described as if it were made of a translation (linear displacement, velocity, acceleration) plus a rotation (angular displacement, velocity, acceleration) Rigid body kinematics in summary A generic movement of a rigid body may be described as if it were made of a translation (linear displacement, velocity, acceleration) plus a rotation (angular displacement, velocity, acceleration) t 1 t 1 t 2 Y Y X Z X Z dismus dinamica del corpo rigido 5 This is relevant to kinetics, because we shall look for a cause of movement variation with reference to translation and rotation separately. dismus dinamica del corpo rigido 7 Rigid body kinematics in summary A generic movement of a rigid body may be described as if it were made of a translation (linear displacement, velocity, acceleration) plus a rotation (angular displacement, velocity, acceleration) Y t 1 t2 The inertia principle it represents the answer to the following questions Why does a rigid body move as we see it to move? Because other bodies interact with it. What happens when a body interacts with another one? The characteristics of its motion change: it either goes faster or slower, or changes its direction (i.e., it accelerates). X Z dismus dinamica del corpo rigido 6 dismus dinamica del corpo rigido 8 2

3 The force We represent the interaction between two bodies using an abstract entity that we call force An external force both B and A are rigid bodies and interact through a single point A D E A D E B dismus dinamica del corpo rigido 9 dismus dinamica del corpo rigido 11 An external force both B and A are rigid bodies and interact through a single point Another external force is deformable and interacts with A through a line or an area A D E A D E B dismus dinamica del corpo rigido 10 dismus dinamica del corpo rigido 12 3

4 Another external force is deformable and interacts with A through a line or an area Internal forces A interacts with A through the cross sectional area A D E dismus dinamica del corpo rigido 13 dismus dinamica del corpo rigido 15 Internal forces A interacts with A through the cross sectional area The gravitational force The hearth acts on the body with a force applied at a point called centre of gravity (for all practical purposes coinciding with the centre of mass) that we call gravitational force or weight A A D E dismus dinamica del corpo rigido 14 dismus dinamica del corpo rigido 16 4

5 The gravitational force orces acting on a rigid body The hearth acts on the body with a force applied at a point called centre of gravity (for all practical purposes coinciding with the centre of mass) that we call gravitational force or weight two useful properties dismus dinamica del corpo rigido 17 dismus dinamica del corpo rigido 19 The free body diagram orces acting on a rigid body: two useful properties The principle of transmissibility: These three forces have exactly the same effect on the motion of the rigid body: the three forces are equivalent The question now is: what will the effect of this system of forces be on the motion of the body? dismus dinamica del corpo rigido 18 dismus dinamica del corpo rigido 20 5

6 orces acting on a rigid body: two useful properties orces acting on a rigid body: two useful properties The principle of transmissibility: These three forces have exactly the same effect on the motion of the rigid body: the three forces are equivalent The principle of transmissibility: These three forces have exactly the same effect on the motion of the rigid body: the three forces are equivalent oncurrent forces: The resultant force has exactly the same effect on the motion of the rigid body than the three forces acting simultaneously on the same particle or having the line of action passing through it. R = dismus dinamica del corpo rigido 21 dismus dinamica del corpo rigido 23 orces acting on a rigid body: two useful properties orces acting on a rigid body: two useful properties The principle of transmissibility: These three forces have exactly the same effect on the motion of the rigid body: the three forces are equivalent The principle of transmissibility: These three forces have exactly the same effect on the motion of the rigid body: the three forces are equivalent 1 2 R 3 oncurrent forces: The resultant force has exactly the same effect on the motion of the rigid body than the three forces acting simultaneously on the same particle or having the line of action passing through it. R = dismus dinamica del corpo rigido 22 dismus dinamica del corpo rigido 24 6

7 The effect of: The effect of forces (three special cases) ase 2: the effect of two forces having the same magnitude and line of action, but opposite direction (couple of forces) 1. a force the line of action of which passes through the M 2. a couple of forces 3. a force the line of action of which does not pass through the M is an angular acceleration α about the M dismus dinamica del corpo rigido 25 dismus dinamica del corpo rigido 27 ase 1: the effect of a force the line of action of which passes through the M ase 2: the effect of a couple of forces (limited to plane motion) a d is a linear acceleration identical for all points = m a may represent the resultant of n forces acting on the same point = I M α is an angular acceleration α about the M = d (couple or moment of the couple) where I M = dm r 2 (mass moment of inertia) dismus dinamica del corpo rigido 26 dismus dinamica del corpo rigido 28 7

8 The mass moment of inertia: definition about the centre of mass The mass moment of inertia (example) about an axis passing through the centre of mass I M = dm r 2 I M = dm r 2 dm 2 dm 1 r 1 r 2 r 3 dm 3 the mass moment of inertia depends on both the mass of the body and on its distribution relative to the reference axis A < B r i A greater couple is required to accelerate B than it is for A dm i dismus dinamica del corpo rigido 29 dismus dinamica del corpo rigido 31 The mass moment of inertia (example) ouples acting on a rigid body about an axis passing through the centre of mass I M = dm r 2 a useful property < A B A greater couple is required to accelerate B than it is for A dismus dinamica del corpo rigido 30 dismus dinamica del corpo rigido 32 8

9 1 3 Equivalent couples Equivalent couples 1 1 d 1 = 1 d 1 d 3 = 3 d 3 = I M α 3 = I M α dismus dinamica del corpo rigido 33 dismus dinamica del corpo rigido 35 Equivalent couples Equivalent couples d = 2 d 2 = 4 d 4 4 = I M α 4 d 4 = I M α dismus dinamica del corpo rigido 34 dismus dinamica del corpo rigido 36 9

10 Equivalent couples ase 3: The effect of a force the line of action of which does not pass through the M 4 = 4 d 4 4 d 4 = I M α these couples, that act in the same plane, are equivalent (i.e., they have the same effect on movement) They may be represented using this symbol: dismus dinamica del corpo rigido 37 dismus dinamica del corpo rigido 39 The effect of: The effect of forces: in summary ase 3: The effect of a force the line of action of which does not pass through the M 1. a force the line of action of which passes through the M 2. a couple of forces = m a d = I M α = M = d Appendix dismus dinamica del corpo rigido 38 dismus dinamica del corpo rigido 40 10

11 ase 3: The effect of a force The answer to the question (limited to the plane motion) the line of action of which does not pass through the M = m a M = I α (limited to the plane motion) dismus dinamica del corpo rigido 41 dismus dinamica del corpo rigido 43 The effect of forces: in summary The answer to the question (limited to the plane motion) The effect of: 1. a force the line of action of which passes through the M = m a 2. a couple of forces = I M α 3. a force the line of action of which does not pass through the M = m a = I M α dismus dinamica del corpo rigido 42 dismus dinamica del corpo rigido 44 11

12 The answer to the question (limited to the plane motion) The equations of motion (limited to the plane motion) R R Given a rigid body which undergoes the action of n forces (free body diagram), it is true that R = m a M = I M α where R = = M M dismus dinamica del corpo rigido 45 dismus dinamica del corpo rigido 47 The answer to the question (limited to the plane motion) The static case R R Given a rigid body which undergoes the action of n forces (free body diagram), if what is the effect of this system of forces on the motion of the body? R = m a M = I α where R = = M M dismus dinamica del corpo rigido 46 R = 0 = 0 where R = = then, the rigid body is in a static equilibrium M M that is = 0 dismus dinamica del corpo rigido 48 M M = 0 12

13 A special case: rigid body with a fixed axis This body is constrained by a cylindrical hinge which we do not remove. Thus, the forces exchanged through that hinge are not made explicit. A special case: rigid body with a fixed axis This body is constrained by a cylindrical hinge which we do not remove. Thus, the forces exchanged through that hinge are not made explicit R = I α where = M dismus dinamica del corpo rigido 49 dismus dinamica del corpo rigido 51 A special case: rigid body with a fixed axis This body is constrained by a cylindrical hinge which we do not remove. Thus, the forces exchanged through that hinge are not made explicit. Moment of inertia of rigid body relative to an arbitrary point 2 3 dm i R r i 4 R = = M as demonstrated in appendix G I = Σ dm r 2 dismus dinamica del corpo rigido 50 dismus dinamica del corpo rigido 52 13

14 A special case: rigid body with a fixed axis In the static case: In Summary (limited to the planar case) Given a rigid diagram with a fixed axis (limited to the planar case), 2 3 The equation of motion is: M = I R 4 = 0I αx and where thus M= 0 = M and the condition that grant for static equilibrium is: M = 0 dismus dinamica del corpo rigido 53 dismus dinamica del corpo rigido 55 In Summary (limited to the planar case) Given a free body diagram, The equations of motion are: = ma M M = I M M fine della lezione # 9 and the conditions that grant for static equilibrium are: = 0 M M = 0 dismus dinamica del corpo rigido 54 dismus dinamica del corpo rigido 56 14

15 TENLGIE PER LA RIABILITAZINE AA A force is equivalent to the same force passing through an arbitrary given point plus a couple Università degli Studi di Napoli "ederico II acoltà di Ingegneria orso di Laurea in Ingegneria Biomedica Appendix Equivalent force systems P Prof. Aurelio appozzo Dipartimento di Scienze del Movimento Umano e dello Sport Laboratorio di Bioingegneria Istituto Universitario di Scienze Motorie - Roma dismus dinamica del corpo rigido 57 By adding the two primed forces nothing changes, the motion of the body is unaffected dismus dinamica del corpo rigido 59 A force is equivalent to the same force passing through an arbitrary given point plus a couple A force is equivalent to the same force passing through an arbitrary given point plus a couple d P P P = d We simply change names and form a couple thus, a force, the line of action of which passes through point P, and a couple are now acting on the body dismus dinamica del corpo rigido 58 dismus dinamica del corpo rigido 60 15

16 A force is equivalent to the same force passing through an arbitrary given point plus a couple Moving a force to a given point P any given force is equivalent to the same force the line of action of which is made to pass through point P plus a couple the moment of which is equal to the moment of the force with respect to that point d = M = d It is evident that point P may be the centre of mass of the body = d d P P dismus dinamica del corpo rigido 61 dismus dinamica del corpo rigido 63 The moment of a force The general case the moment of a force with respect to a point P is defined as having a magnitude equal to M = d d 4 P dismus dinamica del corpo rigido 62 dismus dinamica del corpo rigido 64 16

17 The general case The general case dismus dinamica del corpo rigido 65 dismus dinamica del corpo rigido 67 The general case The general case dismus dinamica del corpo rigido 66 dismus dinamica del corpo rigido 68 17

18 The general case R R = = P M dismus dinamica del corpo rigido 69 The end of Appendix dismus dinamica del corpo rigido 70 18