ENGI 1313 Mechanics I

Transcription

1 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

2 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 S. Kenny, Ph.D., P.Eng.

3 Example Determine the force P needed to pull the 50-kg roller over the smooth step. Take θ = 60. α = S. Kenny, Ph.D., P.Eng.

4 Example (cont.) What XY-coordinate System be Established? y x α = S. Kenny, Ph.D., P.Eng.

5 Example (cont.) Establish FBD y θ x α = N B w = mg = (50 kg)(9.807 m/s 2 ) = 490 N N A S. Kenny, Ph.D., P.Eng.

6 Example (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 S. Kenny, Ph.D., P.Eng.

7 Example (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 S. Kenny, Ph.D., P.Eng.

8 Example (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 α = w = 490 N N A N B o x B = r sinφ = 0.6 m sin = m S. Kenny, Ph.D., P.Eng.

9 Example (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 S. Kenny, Ph.D., P.Eng.

10 Example (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 )( m) o o ( P cos 60 )( 0.5m) ( P sin60 )( ) = 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 = m S. Kenny, Ph.D., P.Eng.

11 Comprehension Quiz 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 S. Kenny, Ph.D., P.Eng.

12 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 S. Kenny, Ph.D., P.Eng.

13 Application to 3D Structures (cont.) Engineering Design Basic analysis Check more rigorous methods S. Kenny, Ph.D., P.Eng.

14 Application to 3D Structures (cont.) Design of Experimental Test Frame Axial Forces Lateral Loads Couple Forces For Bending S. Kenny, Ph.D., P.Eng.

15 3-D Structural Connections Ball and Socket Three orthogonal forces S. Kenny, Ph.D., P.Eng.

16 3-D Structural Connections (cont.) Single Journal Bearing Two forces and two couple moments Frictionless Circular shaft Orthogonal to longitudinal bearing axis S. Kenny, Ph.D., P.Eng.

17 3-D Structural Connections (cont.) Journal Bearing (cont.) Two or more (properly aligned) journal bearings will generate only support reaction forces S. Kenny, Ph.D., P.Eng.

18 3-D Structural Connections (cont.) Single Hinge Three orthogonal forces Two couple moments orthogonal to hinge axis S. Kenny, Ph.D., P.Eng.

19 3-D Structural Connections (cont.) Hinge Design Two or more (properly aligned) hinges will generate only support reaction forces S. Kenny, Ph.D., P.Eng.

20 Rigid Body Constraints What is the Common Characteristic? Statically determinate system S. Kenny, Ph.D., P.Eng.

21 Redundant Constraints Statically Indeterminate System Support reactions > equilibrium equations S. Kenny, Ph.D., P.Eng.

22 Improper Constraints Rigid Body Instability 2-D problem Concurrent reaction forces Intersects an out-of-plane axis S. Kenny, Ph.D., P.Eng.

23 Improper Constraints (cont.) Rigid Body Instability 3-D problem Support reactions intersect a common axis S. Kenny, Ph.D., P.Eng.

24 Improper Constraints (cont.) Rigid Body Instability Parallel reaction forces S. Kenny, Ph.D., P.Eng.

25 References Hibbeler (2007) mech_ S. Kenny, Ph.D., P.Eng.