# Ch1 - Chapter 1 1 1 1.1 INTRODUCTION Degrees of freedom The...

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Chapter 1 1 1 INTRODUCTION 1.1 Degrees of freedom The number of degrees of freedom (DOF) of a system is equal to the number of independent parameters (measurements) that are needed to uniquely define its position in space at any instant of time. The number of DOF is defined with respect to a reference frame. Figure1.1 shows a rigid body (RB) lying in a plane. The rigid body is assumed to be incapable of deformation and the distance between two particles on the RB is constant at any time. If this rigid body will always remain in the plane, three parameters (three DOF) are required to completely define its position: two linear coordinates ( x, y ) to define the position of any one point on the RB, and one angular coordinate θ to define the angle of the body with respect to the axes. The minimum number of measurements needed to define its position are shown in the figure as x, y, and θ . A rigid body in a plane then has three DOF. Note that the particular parameters chosen to define its position arenot unique. Any alternate set of three parameters could be used. There is an infinity of sets of parameters possible, but in this case there must be three parameters per set, such as two lengths

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Chapter 1 2 and an angle, to define the position because a rigid body in plane motion has three DOF. Six parameters are needed to define the position of a free rigid body in a three-dimensional (3-D) space. One possible set of parameters which could be used are three lengths, ( x, y, z ), plus three angles ( θ x y , θ z ). Any free rigid body in three-dimensional space has six degrees of freedom. 1.2 Motion 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 the planar motion the following terms will be defined, Fig. 1.2: pure rotation in which the body possesses one point (center of rotation) which has no motion with respect to a “fixed” reference frame, Fig. 1.2(a). All other points on the body describe arcs about that center. pure translation in which all points on the body describe parallel paths, Fig. 1.2(b). complex motion which exhibits a simultaneous combination of rotation and

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Chapter 1 3 translation, Fig. 1.2(c). 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. Translation and rotation represent independent motions of the body. Each can exist without the other. For a 2-D coordinate system, as shown in Fig. 1.1, the x and y terms represent the translation components of motion, and the θ term represents the rotation component. 1.3 Links and joints Linkages are the basic elements of all mechanisms. Linkages are made up of links and joints. A link or an element or a member is an (assumed) rigid body which possesses nodes. Nodes are defined as points at which links can be attached. A link connected to its neighbouring elements by s nodes is an element of

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