2. Fundamentals: Structural analysis 0Contents2 Fundamentals: Structural analysis 12.1 Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Links and Joints . . . . . . . . . . . . . . . . . . . . . . . . . 32.3 Degrees of Freedom . . . . . . . . . . . . . . . . . . . . . . . . 72.4 Planar Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . 112.5 Dyads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.6 Independent Contours . . . . . . . . . . . . . . . . . . . . . . 122.7 Decomposition of Kinematic Chains . . . . . . . . . . . . . . . 132.8 Linkage Transformation . . . . . . . . . . . . . . . . . . . . . 162.9 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.10 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192. Fundamentals: Structural analysis 12 Fundamentals: Structural analysis2.1 MotionThe number of degrees of freedom (DOF) of a system is equal to the number ofindependent parameters (measurements) that are needed to uniquely defineits position in space at any instant of time. The number of DOF is definedwith respect to a reference frame.Figure 2.1 shows a rigid body (RB) lying in a plane. The rigid body is as-sumed to be incapable of deformation and the distance between two particleson the rigid body is constant at any time. If this rigid body always remainsin the plane, three parameters (three DOF) are required to completely defineits position: two linear coordinates (x, y) to define the position of any onepoint on the rigid body, and one angular coordinate θ to define the angle ofthe body with respect to the axes. The minimum number of measurementsneeded to define its position are shown in the figure as x, y, and θ. A rigidbody in a plane then has three degrees of freedom. Note that the particularparameters chosen to define its position are not unique. Any alternative setof three parameters could be used. There is an infinity of sets of parameterspossible, but in this case there must always be three parameters per set, suchas two lengths and an angle, to define the position because a rigid body inplane motion has three DOF.Six parameters are needed to define the position of a free rigid body in athree-dimensional (3-D) space. One possible set of parameters which couldbe used are three lengths, (x, y, z), plus three angles (θx, θy, θz). Any freerigid 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 andtranslation. For simplicity, only the two-dimensional (2-D) or planar casewill be presented. For planar motion the following terms will be defined,Fig. 2.2:• pure rotation in which the body possesses one point (center of rotation)which has no motion with respect to a “fixed” reference frame [Fig. 2.2(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. 2.2(b)].• complex or general plane motion which exhibits a simultaneous combinationof rotation and translation [Fig. 2.2(c)]. With general plane motion, points2. Fundamentals: Structural analysis 2on the body will travel nonparallel 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 inFig. 2.1, the x and y terms represent the translation components of motion,and the θ term represents the rotation component.Figure 2.1xyθXYZ(b)θ(a)pure curvilinear translationpure rectilinear translationpure rotationgeneral plane motion(c)Figure 2.2pure rotationpure rectilinear translationpure curvilinear translationgeneral plane motion2. Fundamentals: Structural analysis 32.2 Links and JointsLinkages are basic elements of all mechanisms. Linkages are made up oflinks and joints. A link, sometimes known as an element or a member, isan (assumed) rigid body which possesses nodes. Nodes are defined as pointsat which links can be attached. A link connected to its neighboring ele-ments by s nodes is an element of degree s. A link of degree 1 is also calledunary [Fig. 2.3(a)], of degree 2, binary [Fig. 2.3(b)], and of degree 3, ternary[Fig. 2.3(c)], etc.A joint is a connection between two or more links (at their nodes). Ajoint allows some relative motion between the connected links. Joints arealso called kinematic pairs.The number of independent coordinates that uniquely determine the rel-ative position of two constrained links is termed degree of freedom of a givenjoint. Alternatively the term degree of constraint is introduced. A kine-matic pair has the degree of constraint equal to j if it diminishes the relativemotion of linked bodies by j degrees of freedom; i.e. j scalar constraint con-ditions correspond to the given kinematic pair. It follows that such a jointhas (6 − j) independent coordinates. The number of degrees of freedom isthe fundamental characteristic quantity of joints. One of the links of a sys-tem is usually considered to be the reference link, and the position of otherRBs is determined in relation to this reference body. If the reference link isstationary, the term frame or ground is used.The coordinates in the definition of degree of freedom can be linear orangular. Also the coordinates used can be absolute (measured with regardto the frame) or relative. Figures 2.4-2.9 show examples of joints commonlyfound in mechanisms. Figures 2.4(a) and 2.4(b) show two forms of a planar,one degree of freedom joint, namely a rotating pin joint and a translatingslider joint. These are both typically referred to as full joints and has 5degrees of constraint. The pin joint allows one rotational (R) DOF, and theslider joint allows one translational (T) DOF between the joined links. Theseare both special cases of another common, one degree of freedom joint, thescrew and nut [Fig. 2.5(a)]. Motion of either the nut or the screw relative tothe other results in helical motion. If the helix angle is made zero [Fig. 2.5(b)],the nut rotates without advancing and it becomes a pin joint. If the helixangle is made 90◦, the nut will translate along the axis of the screw, and itbecomes a slider joint.Figure 2.6 shows examples of two degrees of freedom joints, which simul-(a)(b)(c)Figure 2.3Schematic