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

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1 Analtical Elements of Mechanisms Introduction. Degrees of Freedom he number of degrees of freedom (DOF) of a sstem is equal to the number of independent parameters (measurements) that are needed to uniquel define its position in space at an instant of time. he number of DOF is defined with respect to a reference frame. Figure. shows a rigid bod () ling in a plane. he rigid bod is assumed to be incapable of deformation, and the distance between two particles on the rigid bod is constant at an time. If this rigid bod alwas remains in the plane, three parameters (three DOF) are required to completel define its position: two linear coordinates (, ) to define the position of an one point on the rigid bod, and one angular coordinate θ to define the angle of the bod with respect to the aes. he minimum number of measurements needed to define its position are shown in the figure as,, and θ. A rigid bod in a plane then has three degrees of freedom. Note that the particular parameters chosen to define its position are not unique. An alternative set of three parameters could be used. here is an infinit of sets of parameters possible, but in this case there must alwas be three parameters per set, such as two lengths and an angle, to define the position because a rigid bod in plane motion has three DOF. Si parameters are needed to define the position of a free rigid bod in a three-dimensional (3-D) space. One possible set of parameters that could be used are three lengths (,, z), plus three angles (θ,θ,θ z ). An free rigid bod in three-dimensional space has si degrees of freedom.. Motion A rigid bod free to move in a reference frame will, in the general case, have comple motion, which is simultaneousl a combination of rotation and

2 Analtical Elements of Mechanisms Analtical Elements of Mechanisms Z rigid bod () Y θ X Figure.. in planar motion with three DOF: translation along the ais, translation along the ais, and rotation, θ, about the z. translation. For simplicit, onl the two-dimensional (-D) or planar case will be presented. For planar motion the following terms will be defined (see Fig..). Pure rotation is that in which the bod possesses one point (center of rotation) that has no motion with respect to a fied reference frame; see Fig... All other points on the bod describe arcs about that center. Pure translation is that in which all points on the bod describe parallel paths; see Fig... Comple motion is that which ehibits a simultaneous combination of rotation and translation; see Fig..(c). With general plane motion, points on the bod will travel nonparallel paths, and there will be, at ever instant, a center of rotation, which will continuousl change location. ranslation and rotation represent independent motions of the bod. Each can eist without the other. For a -D coordinate sstem, as shown in Fig.., the and terms represent the translation components of motion, and the θ term represents the rotation component..3 Links and Joints Linkages are basic elements of all mechanisms. Linkages are made up of s and s. A, sometimes known as an element or a member, is an (assumed) rigid bod that possesses s. Nodes are defined as points at which s can be attached. A connected to its neighboring elements b s s is an element of degree s. A of degree is also called unar, as in Fig..3; that of degree, binar, as in Fig..3; that of degree 3, ternar, as in Fig..3(c); and so on.

3 Analtical Elements of Mechanisms Introduction 3 rotation θ rectilinear translation curvilinear translation general plane motion (c) Figure.. in motion: pure rotation, pure translation, and (c) general motion. A is a connection between two or more s (at their s). A allows some relative motion between the connected s. Joints are also called kinematic pairs. he number of independent coordinates that uniquel determine the relative position of two constrained s is termed the degree of freedom of a given. Alternativel the term class is introduced. A kinematic pair is of the jth class if it diminishes the relative motion of ed bodies b j

4 Analtical Elements of Mechanisms 4 Analtical Elements of Mechanisms Link Schematic representation (c) Figure.3. pes of s: unar, binar, and (c) ternar elements. degrees of freedom; that is, j scalar constraint conditions correspond to the given kinematic pair. It follows that such a has (6 j) independent coordinates. he number of degrees of freedom is the fundamental characteristic quantit of s. One of the s of a sstem is usuall considered to be the reference, and the position of other s is determined in relation to this reference bod. If the reference is stationar, the term frame or ground is used. he coordinates in the definition of degree of freedom can be linear or angular. Also, the coordinates used can be absolute (measured with regard to the frame) or relative. Figures.4.9 show eamples of s commonl found in mechanisms. Figures.4 and.4 show two forms of a planar, one degree of freedom, namel a rotating pin and a translating slider

5 Analtical Elements of Mechanisms Introduction 5 pe of full 0 Schematic representation Figure.4. One degree of freedom, full (fifth class): pin and slider.. hese are both tpicall referred to as full s and are of the fifth class. he pin allows one rotational () DOF, and the slider allows one translational () DOF between the joined s. hese are both special cases of another common, one degree of freedom, the screw and nut, shown in Fig..5. Motion of either the nut or the screw relative to the other results in helical motion. If the heli angle is made zero, Fig..5, the nut rotates without advancing and it becomes a pin. If the heli angle is made 90, the nut will translate along the ais of the screw, and it becomes a slider. Figure.6 shows eamples of two degrees of freedom s, which simultaneousl allow two independent, relative motions, namel translation () and rotation (), between the joined s. A two degrees of freedom is usuall referred to as a half- and is of the fourth class. A half- is sometimes also called a roll slide because it allows both rotation (rolling) and translation (sliding).

6 Analtical Elements of Mechanisms 6 Analtical Elements of Mechanisms pe of full Schematic representation p p z p α A C θ O r A α rθ C πr p α = heli angle p = pitch Figure.5. Screw and nut ; helical motion. A jostick, ball-and-socket, or sphere, shown in Fig..7, is an eample of a three degrees of freedom (third class), which allows three independent angular motions between the two s that are joined. his ball would tpicall be used in a 3-D mechanism, one eample being the ball s used in automotive suspension sstems. A plane, Fig..7, is also an eample of a three degrees of freedom, which allows two translations and one rotation. Note that in order to visualize the degree of freedom of a in a mechanism, it is helpful to mentall disconnect the two s that create the from the rest of the mechanism. It is easier to see how man degrees of

7 Analtical Elements of Mechanisms Introduction 7 half cam follower half half (c) (d) gear X gear gear half half gear X (e) Figure.6. wo degrees of freedom, half- (fourth class): general, clinder, (c) roll slide disk, (d) cam-follower, and (e) gear. freedoms the two joined s have with respect to one another. Figure.8 shows an eample of a second class (clinder on plane), and Fig..9 represents a first class (sphere on plane). he tpe of contact between the elements can be point (P), curve (C), or surface (S). he term lower was coined b euleau to describe s with

8 Analtical Elements of Mechanisms z O Schematic representation z Figure.7. hree degrees of freedom (third class): ball and socket, and plane. z Figure.8. Four degrees of freedom (second class) clinder on a plane. z Figure.9. Five degrees of freedom (first class) sphere on a plane. 8

9 Analtical Elements of Mechanisms Introduction 9 surface contact. He used the term higher to describe s with point or curve contact. he main practical advantage of lower s over higher s is their better abilit to trap lubricant between their enveloping surfaces. his is especiall true for the rotating pin. A closed is a that is kept together or closed b its geometr. A pin in a hole and a slider in a two-sided slot are forms of closed s. A force closed, such as a pin in a half-bearing or a slider on a surface, requires some eternal force to keep it together or closed. his force could be supplied b gravit, b a spring, or b some eternal means. In ages, closed s are usuall preferred, and the are eas to accomplish. For cam-follower sstems, force closure is often preferred. he order of a is defined as the number of s joined minus one. he simplest combination of two s has order one and it is a single, shown in Fig..0. As additional s are placed on the same, the order is increased on a one for one basis, as shown in Fig..0. Joint order of order one (one pin ) of order two (two pin s) Figure.0. Order of a : of order one, and of order two (multiple s).

10 Analtical Elements of Mechanisms 0 Analtical Elements of Mechanisms 3 A contour D C 3 A C z 0 0 ground ground Figure.. Kinematic chains: closed and open kinematic chains. has significance in the proper determination of overall degrees of freedom for an assembl. odies ed b s form a kinematic chain. Simple kinematic chains are shown in Fig... A contour or loop is a configuration described b a polgon, as shown in Fig... he presence of loops in a mechanical structure can be used to define the following tpes of chains. Closed kinematic chains have one or more loops so that each and each is contained in at least one of the loops, as shown in Fig... A closed kinematic chain has no open attachment point. Open kinematic chains contain no loops, as shown in Fig... A common eample of an open kinematic chain is an industrial robot. Mied kinematic chains are a combination of closed and open kinematic chains. Another classification is also useful. Simple chains contain onl binar elements. Comple chains contain at least one element of degree 3 or higher. A mechanism is defined as a kinematic chain in which at least one has been grounded or attached to the frame, as shown in Figs.. and.. According to euleau s definition, a machine is a collection of mechanisms