MMET 275 502 MECHANICS FOR TECHNOLOGISTS Department of Engineering Technology and Industrial Distribution Texas A M University Spring 2023 MANUFACTURING MECHANICAL ENGINEERING TECHNOLOGY MANUFACTURING MECHANICAL ENGINEERING TECHNOLOGY MMET 1 REVIEW 2 Problem 1 Force System Resultants 30 points Problem 2 Principles of Moments 35 points Problem 3 Structural Analysis 35 points MANUFACTURING MECHANICAL ENGINEERING TECHNOLOGY MANUFACTURING MECHANICAL ENGINEERING TECHNOLOGY MMET 2 2 CHAPTER 4 FORCE SYSTEM RESULTANTS Objectives 1 Concept of the moment of a force in 2 D 3 D 2 Analyze a simple distributed loading 4 1 Moment of a Force Scalar Formation 4 2 Cross Product 4 3 Moment of Force Vector Formulation 4 4 Principle of Moments 4 9 Reduction of a Simple Distributed Loading MANUFACTURING MECHANICAL ENGINEERING TECHNOLOGY MANUFACTURING MECHANICAL ENGINEERING TECHNOLOGY MMET 3 4 3 MOMENT OF FORCE VECTOR FORM Moment of force F about point O can be expressed using cross product The magnitude of cross product Direction and sense are determined by right hand rule MANUFACTURING MECHANICAL ENGINEERING TECHNOLOGY MANUFACTURING MECHANICAL ENGINEERING TECHNOLOGY MMET 4 oMrF sinoMrF 4 3 MOMENT OF FORCE VECTOR FORM Principle of Transmissibility MANUFACTURING MECHANICAL ENGINEERING TECHNOLOGY MANUFACTURING MECHANICAL ENGINEERING TECHNOLOGY MMET 5 123oMrFrFrF 4 3 MOMENT OF FORCE VECTOR FORM Cartesian Vector Formulation Force expressed in Cartesian form MANUFACTURING MECHANICAL ENGINEERING TECHNOLOGY MANUFACTURING MECHANICAL ENGINEERING TECHNOLOGY MMET 6 OxyzxyzyzzyxzzxxyyxijkMrFrrrFFFrFrFirFrFjrFrFk EXAMPLE Two forces act on the rod Determine the resultant moment they create about the flange at O Express the result as a Cartesian vector MANUFACTURING MECHANICAL ENGINEERING TECHNOLOGY MANUFACTURING MECHANICAL ENGINEERING TECHNOLOGY MMET 7 PRACTICE MANUFACTURING MECHANICAL ENGINEERING TECHNOLOGY MANUFACTURING MECHANICAL ENGINEERING TECHNOLOGY MMET 8 EXAMPLES MANUFACTURING MECHANICAL ENGINEERING TECHNOLOGY MANUFACTURING MECHANICAL ENGINEERING TECHNOLOGY MMET 9 RFAwL 2000 22LLLRxwwxxdxwxdxLxFwLwL EXAMPLES MANUFACTURING MECHANICAL ENGINEERING TECHNOLOGY MANUFACTURING MECHANICAL ENGINEERING TECHNOLOGY MMET 10 2maxmaxmax0001 22LLLRxxFAwxdxwdxwwLLL 3maxmax000maxmax 3211322LLLRxxwwxdxwxxdxLLxLFwLwL EXAMPLE The granular material exerts the distributed loading on the beam Determine the magnitude and location of the equivalent resultant of this load MANUFACTURING MECHANICAL ENGINEERING TECHNOLOGY MANUFACTURING MECHANICAL ENGINEERING TECHNOLOGY MMET 11 EXAMPLE MANUFACTURING MECHANICAL ENGINEERING TECHNOLOGY MANUFACTURING MECHANICAL ENGINEERING TECHNOLOGY MMET 12 CHAPTER 5 OBJECTIVES Develop the EoE for a rigid body Concept of the FBD for a rigid body 5 1 Conditions for Rigid Body Equilibrium 5 2 Free Body Diagrams 5 3 Equations of Equilibrium 5 4 Two and Three Force Members 5 5 Free Body Diagrams 5 6 Equations of Equilibrium 5 7 Constraints and Statical Determinacy MANUFACTURING MECHANICAL ENGINEERING TECHNOLOGY MANUFACTURING MECHANICAL ENGINEERING TECHNOLOGY MMET 13 5 1 CONDITIONS FOR RIGID BODY EQUILIBRIUM Forces on a particle Forces acting on a particle are concurrent So rotation is not a concern Thus equilibrium is satisfied by no translation Forces on a rigid body Forces need not be concurrent moment 0 Hence rigid body rotation is a concern For equilibrium the net force net moment about any arbitrary point O must be 0 no translation no rotation MANUFACTURING MECHANICAL ENGINEERING TECHNOLOGY MANUFACTURING MECHANICAL ENGINEERING TECHNOLOGY MMET 14 0F 0F 0M SUPPORT REACTIONS General rule 1 If a support prevents translation of a body in a given direction then a force is developed on the body in the opposite direction 2 If rotation is prevented then a moment is exerted on the body in the opposite direction MANUFACTURING MECHANICAL ENGINEERING TECHNOLOGY MANUFACTURING MECHANICAL ENGINEERING TECHNOLOGY MMET 15 5 2 FREE BODY DIAGRAMS Procedure for Drawing a FBD 1 Draw Outlined Shape Imagine body to be isolated or cut free from its constraints Draw outline shape 2 Identify your Coordinate System MANUFACTURING MECHANICAL ENGINEERING TECHNOLOGY MANUFACTURING MECHANICAL ENGINEERING TECHNOLOGY MMET 16 5 2 FREE BODY DIAGRAMS 3 Show All Forces and Couple Moments Identify all external forces and couple moments that act on the body 4 Identify Each Loading and Give Dimensions Indicate dimensions for calculation of forces Known forces and couple moments should be properly labeled with their magnitudes and directions MANUFACTURING MECHANICAL ENGINEERING TECHNOLOGY MANUFACTURING MECHANICAL ENGINEERING TECHNOLOGY MMET 17 5 4 TWO AND THREE FORCE MEMBERS Simplify some problems by recognizing members that are subjected to only 2 or 3 forces Two Force Members When a member is subject to no couple moments and forces are applied at only two points on a member the member is called a two force member MANUFACTURING MECHANICAL ENGINEERING TECHNOLOGY MANUFACTURING MECHANICAL ENGINEERING TECHNOLOGY MMET 18 5 4 TWO AND THREE FORCE MEMBERS Three Force Members When subjected to three forces the forces are concurrent or parallel This simplifies the equilibrium analysis of some rigid bodies since the directions of the resultant forces at A and B are known MANUFACTURING MECHANICAL ENGINEERING TECHNOLOGY MANUFACTURING MECHANICAL ENGINEERING TECHNOLOGY MMET 19 5 5 EQUILIBRIUM IN THREE DIMENSIONS FBD IMPORTANT NOTE A single bearing or hinge can prevent rotation by providing a resistive couple moment However it is usually preferred to use two or more properly aligned bearings or hinges Thus in these cases only force reactions are generated and there are no moment reactions created MANUFACTURING MECHANICAL ENGINEERING TECHNOLOGY MANUFACTURING MECHANICAL ENGINEERING TECHNOLOGY MMET 20 5 6 EQUATIONS OF EQUILIBRIUM Vector Equations of Equilibrium body in vector form F 0 MO 0 For two conditions for equilibrium of a rigid If all external forces and couple moments are expressed in Cartesian vector form F Fxi Fyj Fzk 0 MO Mxi Myj Mzk 0 MANUFACTURING MECHANICAL ENGINEERING TECHNOLOGY MANUFACTURING MECHANICAL ENGINEERING TECHNOLOGY MMET 21 EQUILIBRIUM OF A RIGID BODY IN THREE DIMENSIONS Six scalar equations are required to express the conditions for the equilibrium of a rigid body in the general three dimensional case These equations can be solved for no more than 6 unknowns which generally represent reactions at supports or connections The scalar equations