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# Statics ME 2560

WMU

GPA 3.52

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This 9 page Class Notes was uploaded by Mortimer Rutherford on Wednesday September 30, 2015. The Class Notes belongs to ME 2560 at Western Michigan University taught by Javier Sanchez in Fall. Since its upload, it has received 16 views. For similar materials see /class/216916/me-2560-western-michigan-university in Mechanical Engineering at Western Michigan University.

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Date Created: 09/30/15

CH VI ME2560 STATICS Equilibrium of a Rigid Body EQUILIBRIUM OF A RIGID BODY 61 CONDITIONS FOR RIGIDiBODY EQULIBRIUM As we know a rigid body is form by an in nite number of particles If we select arbitrarily one of the particles of a given rigid body in which we need to study the required conditions for its equilibrium we will see that such particle is subjected to internal and external forces The internal forces 7 are those caused by the interaction of the particle with the particles adjacent to it The external forces E are those caused by external effects such as gravitational electrical magnetic or contact forces Internal and external forces are schematically presented in figure 61 The condition required for the particle to be in equilibrium is that Figure 61 System of internal and external forces acting on a particle ofa rigid body Similar equations are obtained when applying the equilibrium equation to each of the other particles in the body If all these equations are added the result is 77 Z i 0 The summation of the internal forces will equal zero since the internal forces between particles within the body will occur in equal but opposite collinear pairs Newton s third law As a consequence only the summation of external forces remains in the equation for equilibrium such that 37 o If we consider summation of moments acting on the ith particle about an arbitrary point the equilibrium equation for this specific particle is written as Writing similar equations for all the particles forming the body the summation of moments for all the particles is 42 CH VI ME2560 STATICS Equilibrium of a Rigid Body erx izrjxfi20 The second term is zero since the internal forces occur in equal but opposite collinear pairs therefore the resultant moment of each pair of forces about point 0 is zero Thus ZMfm Then the equilibrium equations for a rigid body are Z 0 21 0 61 62 FREE BODY DIAGRAMS The correct determination of the forces and moments acting on a rigid body highly depends on the correct presentation of the free body diagram and since there are a number of different structural supports it is very important to know what class of forces those supports impose on a rigid body In general if a support prevents the translation of a body in a given direction then a force is developed on the body in that direction If a support prevents rotation then a couple moment is exerted on the body Figure 62 presents three of the most common supports Figure 62a shows a beam supported on one of its ends by a roller Since the support only prevents the movement in the vertical direction the roller can only exert force on the beam in the vertical direction In figure 62 b the beam is supported in a more restrictive manner using a pin In this case the pin prevents the translational movement in any direction39 therefore the resultant reaction on the beam will consist on a force with components in both directions Finally figure 62 c presents a fixed support This is the most restrictive support since prevents any kind of translational and rotational motion Thus the reaction on the beam will consist on a force with components in the vertical and horizontal directions and a couple moment that prevents rotation I pin fixedsupport a IOHCI F M F rwefif U E gt 39fa f Fy 3911 3 c Figure 62 Three of the most common supports 43 CH VI ME2560 STATICS Equilibrium of a Rigid Body Table 61 Two dimensional supports and their reactions Types of Connection Reaction Number of Unknowns One unknown The reaction is a tension force which acts away from the member in the direction of the ca e One unknown The reaction is a force which acts along the axis of the link weightless link One unknown The reaction is a force which acts 3 439 perpendicular to the 9 surface at the point of contact roller 4 3 One unknown The i reaction is a force 3 m k which acts F F perpendicular to the 191110 pin in slot con ned smwmb 51m 5 One unknown The reaction is a force which acts perpendicular to the surface at the point of contact maker One unknown The fag quot reaction is a force which acts 9 perpendicular to the F smaulh coma ing surface at the point of surfaae contact 6 44 CH VI ME2560 STATICS Table 61 Cont Equilibrium of a Rigid Body 7 member pin camecred to collar on smooth rad One unknown The reaction is a force which acts perpendicular to the rod 8 Two unknowns The reactions are two components of force or the magnitude and direction 45 of the resultant force Note that Q5 and 0 are not necessarily equal usually not unless the rod shown is a link as in 2 member xed cannected m collar on smooth will Two unknowns The reactions are the couple moment and the force which acts perpendicular to the rod 10 J M fixed support Three unknowns The reactions are the couple moment and the two force components or the couple moment and the magnitude and direction 45 of the resultant force 45 CH VI ME2560 STATICS Equilibrium of a Rigid Body External and Internal Forces Since a rigid body is fonned by an infinite number of particles both external and internal loadings may act on it However when preparing a free body diagram the internal forces are not presented in the diagram since these forces always occur in opposite collinear pairs haVing a net effect of zero on the body Weight and Center of Gravity When a body is subjected to a gravitational field each of its particles has a specified weight The sum of all the indiVidual weights represents the weight of the body W and the location where this weight W acts corresponds to the center of graVity Homework No 61 5 9 5 10 63 EQUATIONS OF EQUILIBRIUM The conditions sufficient and necessary to ensure equilibrium in a rigid body are summation of forces equal to zero and summation of moments equal to zero In a coplanar system these equations are reduced to ZFX0 ZFy0 2M0 0 72 64 TWO AND THREE FORCE lVIElVIBERS TwoiForce Members When a member is subjected to no couple moments and forces are applied only at two points on the member the element is called twoiforce member Figure 63 shows some typical examples of two force members 13 F3 44 39 F2 72 F1 gt 7 i FR 39 FA FA A B FB7FA Two force members FD FA b Tuoforce member Figure 63 Two force members frequently found 46 CH VI MIE2560 STATICS Equrhbnum ofa Rrgrd B ody nteeeFutse Members If a member rs subjected to nnly three forces then rt rs Pm Irkymhcx U m mettom Frgure o 4 Three force member Hum zwnrk Nn 52 5719 5730 542 549 5757 5762 o 5 EQUILIBRIUM 1N THREE DLMZENSIONS Srmuarly as m the case of equrhbnum m coplanar systems the rst step m three the proper reaeuons at the supports The reaeuve forces and eo p1e moments aetrng at vanous types of supports and eonneeuons are shown m table 62 As m the two 64m mmem 15 developed when the rotation afthe attached member 15 prevented CH VI ME2560 STATICS Equilibrium of a Rigid Body Table 62 Three d1mens1onal supports and their reactlons Types of Connection Reaction Number of Unknowns 1 K One unknown The reaction is a force which F r a quot1 acts away from the member in the known direction of the cable 1 I i 3 One unknown The reaction is a force which 3 acts perpendicular to the surface at the point of i x A contact month surface support 3 i One unknown The reaction is a force which acts perpendicular to the surface at the point of contact roller 4 Three unknowns The reactions are three rectangular force components ball and socket 5 M1 C1 Four unknowns The reactions are two force and two couplemoment components which act perpendicular to the shaft singlejnumal bearing 48 CH VI Equilibrium of a Rigid Body ME2560 STATICS 6 single mmm bearing square shaft Five unknowns The reactions are two force and three couplemoment components 7 in 31 vt lmslr hearing 23 FT Five unknowns The reactions are three force 5 a and two couplemoment components 8 sing mm pin Five unknowns The reactions are three force and two couplemoment components 9 Five unknowns The reactions are three force and two couplemoment components 10 fixed suppom Six unknowns The reactions are three force and three couplemoment components 49 CH VI ME2560 STATICS Equilibrium of a Rigid Body EQUATIONS OF EQUILIBRIUM The conditions for equilibrium if a rigid body subjected to a three dimensional force system require that both the resultant force and the resultant couple moment acting on the body be equal to zero The two equations for equilibrium in a rigid body can be expressed in vector form as 2130 2M0 0 63 In these equations 2F is the vector sum of all the external acting on the body and 2M is the sum of the couple moments and the moments of all the forces about any point 0 located either on or off the body The two equations presented in Eq 63 can be alternatively presented as Z 2115 Zij ZFZIE 0 2M0 ZMJZMyjZMZIE0 64 Or also as ZFX0 ZFy0 ZFZ0 ZMX0 ZMy0 ZMZ0 65 Homework No 63 53 5766 5781 5783 5786 5789 5790 50

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