ENGI 1313 Mechanics I Lecture 25: Equilibrium of a Rigid Body Shawn Kenny, Ph.D., P.Eng. Assistant Professor Faculty of Engineering and Applied Science Memorial University of Newfoundland spkenny@engr.mun.ca
Lecture Objective to illustrate application of 2D equations of equilibrium for a rigid body to examine concepts for analyzing equilibrium of a rigid body in 3D 2 2007 S. Kenny, Ph.D., P.Eng.
Example 25-01 Determine the force P needed to pull the 50-kg roller over the smooth step. Take θ = 60. α = 3 2007 S. Kenny, Ph.D., P.Eng.
Example 25-01 (cont.) What XY-coordinate System be Established? y x α = 4 2007 S. Kenny, Ph.D., P.Eng.
Example 25-01 (cont.) Establish FBD y θ x α = N B w = mg = (50 kg)(9.807 m/s 2 ) = 490 N N A 5 2007 S. Kenny, Ph.D., P.Eng.
Example 25-01 (cont.) Determine Force Angles Roller self-weight y α 70 x α = 20 y θ x α = 20 α = N B w = 490 N N A 6 2007 S. Kenny, Ph.D., P.Eng.
Example 25-01 (cont.) Determine Force Angles Normal reaction force at A y x y θ x 90 α = N B N A w = 490 N N A 7 2007 S. Kenny, Ph.D., P.Eng.
Example 25-01 (cont.) Determine Force Angles Normal reaction force at B y y x θ x y B = (0.6 m 0.1 m) = 0.5 m φ = acos 1 0.5m 0.6m r = 0.6 m = N B o 33.56 α = w = 490 N N A N B o x B = r sinφ = 0.6 m sin 33.56 = 0.3317 m 8 2007 S. Kenny, Ph.D., P.Eng.
Example 25-01 (cont.) Draw FBD w = 490 N α = 20 y P θ = 60 x y x φ N B α = N B N A = 0 N w = 490 N N A 9 2007 S. Kenny, Ph.D., P.Eng.
Example 25-01 (cont.) What Equilibrium Equation should be Used to Find P? ΣM B = 0 ( w sinα )( y B ) + ( w cosα )( xb ) + ( P cosθ )( y ) ( P sinθ )( x ) = 0 B o o ( 490N sin 20 )( 0.5m) + ( 490N cos 20 )( 0.3317 m) o o ( P cos 60 )( 0.5m) ( P sin60 )( 0.3317 ) = 0 P = 6.35kN B K w = 490 N α = 20 y y B = 0.5 m + K φ N A P θ = 60 N B x x B = 0.3317 m 10 2007 S. Kenny, Ph.D., P.Eng.
Comprehension Quiz 25-01 If a support prevents rotation of a body about an axis, then the support exerts a on the body about that axis. A) Couple moment B) Force C) Both A and B D) None of the above. Answer: A 11 2007 S. Kenny, Ph.D., P.Eng.
3-D Equilibrium Basic Equations + F x = 0 + = 0 M x = 0 F y M y = 0 + F z = 0 M z = 0 Moment equations can also be determined about any point on the rigid body. Typically the point selected is where the most unknown forces are applied. This procedure helps to simplify the solution. 12 2007 S. Kenny, Ph.D., P.Eng.
Application to 3D Structures (cont.) Engineering Design Basic analysis Check more rigorous methods 13 2007 S. Kenny, Ph.D., P.Eng.
Application to 3D Structures (cont.) Design of Experimental Test Frame Axial Forces Lateral Loads Couple Forces For Bending 14 2007 S. Kenny, Ph.D., P.Eng.
3-D Structural Connections Ball and Socket Three orthogonal forces 15 2007 S. Kenny, Ph.D., P.Eng.
3-D Structural Connections (cont.) Single Journal Bearing Two forces and two couple moments Frictionless Circular shaft Orthogonal to longitudinal bearing axis 16 2007 S. Kenny, Ph.D., P.Eng.
3-D Structural Connections (cont.) Journal Bearing (cont.) Two or more (properly aligned) journal bearings will generate only support reaction forces 17 2007 S. Kenny, Ph.D., P.Eng.
3-D Structural Connections (cont.) Single Hinge Three orthogonal forces Two couple moments orthogonal to hinge axis 18 2007 S. Kenny, Ph.D., P.Eng.
3-D Structural Connections (cont.) Hinge Design Two or more (properly aligned) hinges will generate only support reaction forces 19 2007 S. Kenny, Ph.D., P.Eng.
Rigid Body Constraints What is the Common Characteristic? Statically determinate system 20 2007 S. Kenny, Ph.D., P.Eng.
Redundant Constraints Statically Indeterminate System Support reactions > equilibrium equations 21 2007 S. Kenny, Ph.D., P.Eng.
Improper Constraints Rigid Body Instability 2-D problem Concurrent reaction forces Intersects an out-of-plane axis 22 2007 S. Kenny, Ph.D., P.Eng.
Improper Constraints (cont.) Rigid Body Instability 3-D problem Support reactions intersect a common axis 23 2007 S. Kenny, Ph.D., P.Eng.
Improper Constraints (cont.) Rigid Body Instability Parallel reaction forces 24 2007 S. Kenny, Ph.D., P.Eng.
References Hibbeler (2007) http://wps.prenhall.com/esm_hibbeler_eng mech_1 25 2007 S. Kenny, Ph.D., P.Eng.