introduction

introduction - Chapter 1 Introduction 1.1 Degrees of...

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Chapter 1 Introduction 1.1 Degrees of Freedom and Motion The number of degrees of freedom (DOF) of a mechanical system is equal to the number of independent parameters (measurements) that are needed to uniquely de- ±ne its position in space at any instant of time. The number of DOF is de±ned with respect to a reference frame. Figure 1.1 shows a rigid body (RB) lying in a plane. The distance between two particles on the rigid body is constant at any time. If this rigid body always remains in the plane, three parameters (three DOF) are required to completely de±ne its position: two linear coordinates ( x , y ) to de±ne the position of any one point on the rigid body, and one angular coordinate θ to de±ne the angle of the body with respect to the axes. The minimum number of measurements needed to de±ne its position are shown in the ±gure as x , y , and θ . A rigid body in a plane then has three degrees of freedom. The particular parameters chosen to de±ne its position are not unique. Any alternative set of three parameters could be used. There is an in±nity of sets of parameters possible, but in this case there must always be three parameters per set, such as two lengths and an angle, to de±ne the position because a rigid body in plane motion has three DOF. Six parameters are needed to de±ne the position of a free rigid body in a three- dimensional (3-D) space. One possible set of parameters that could be used are Fig. 1.1 Rigid body in planar motion with three DOF: translation along the x -axis, translation along the y -axis, and rotation, θ , about the z -axis θ x y z RB 1

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2 1 Introduction three lengths, ( x , y , z ), plus three angles ( θ x , θ y , θ z ). Any free rigid body in three- dimensional space has six degrees of freedom. A rigid body free to move in a reference frame will, in the general case, have complex motion, which is simultaneously a combination of rotation and translation. For simplicity, only the two-dimensional (2-D) or planar case will be presented. For planar motion the following terms will be de±ned, Fig. 1.2: (b) (a) pure curvilinear translation pure rectilinear translation pure rotation general plane motion (c) pure rotation pure rectilinear translation pure curvilinear translation general plane motion R R θ Fig. 1.2 Rigid body in motion: (a) pure rotation, (b) pure translation, and (c) general motion
1.2 Kinematic Pairs 3 1. pure rotation in which the body possesses one point (center of rotation) that has no motion with respect to a “Fxed” reference frame, ±ig. 1.2a. All other points on the body describe arcs about that center; 2. pure translation in which all points on the body describe parallel paths, ±ig. 1.2b; 3. complex or general plane motion that exhibits a simultaneous combination of rotation and translation, ±ig. 1.2c. With general plane motion, points on the body will travel non-parallel paths, and there will be, at every instant, a center of rotation, which will continuously change location.

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This note was uploaded on 10/28/2009 for the course MECH 320 taught by Professor D.a during the Spring '09 term at American University of Beirut.

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introduction - Chapter 1 Introduction 1.1 Degrees of...

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