Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu

1. Engineering 36 Chp5: MechEquilibrium Bruce Mayer, PE Licensed Electrical & Mechanical EngineerBMayer@ChabotCollege.edu

2. ReCall Equilibrium Conditions • Rigid Body in Static Equilibrium Characterized by • Balanced external forces and moments • Will impart no Tendency toward Translational or Rotational motion to the body • The NECESSARY and SUFFICIENT condition for the static equilibrium of a body are that the RESULTANT Force and Couple from all external forces form a system equivalent to zero

3. Equilibrium cont. • Rigid Body Equilibrium Mathematically • Resolving into Rectangular Components the Resultant Forces & Moments Leads to an Equivalent Definition of Rigid Body Equilibrium

4. 2D Planar System • In 2D systems it is assumed that • The System Geometry resides completely the XY Plane • There is NO Tendency to • Translate in the Z-Direction • Rotate about the X or Y Axes • These Conditions Simplify The Equilibrium Equations

5. 2D Planar System: • No Z-Translation → NO Z-Directed Force: • No X or Y Rotation → NO X or Y Applied Moments

6. 2D Planar System: • If r Lies in the XY Plane, then rz = 0. With Fz = 0 the rxF Determinant: • So in this case M due to rxF is confined to the Z-Direction:

7. Example  Crane problem • Solution Plan • Create a free-body diagram for the crane • Determine Rcns at B by solving the equation for the sum of the moments of all forces about Pin-A • Note there will be no Moment contribution from the unknown reactions at A. • Determine the reactions at A by solving the equations for the sum of all horizontal force components and all vertical force components. • A fixed crane has a mass of 1000 kg and is used to lift a 2400 kg crate. It is held in place by a pin at A and a rocker at B. The center of gravity of the crane is located at G. • Determine the components of the reactions at A and B.

8. Example  Crane problem • Reaction Analysis • PIN at A • X & Y Reactions • ROCKER at B • NORMAL Reaction Only • +X Rcn in This Case • Solution Plan cont. • Check the values obtained for the reactions by verifying that the moments about B of all forces Sum to zero.

9. Example  Crane problem • Draw the Free BodyDiagram (FBD) • Determine the reactions at Pt-A by solving the eqns for the sum of all horizontal & vertical forces • Determine Pt-B Rcn by solving the equation for the sum of the moments of all forces about Pt-A.

10. Example  Coal Car • Solution Plan • Create a free-body diagram for the car with the coord system ALIGNED WITH the CABLE • Determine the reactions at the wheels by solving equations for the sum of moments about points above each axle, in the LoA of the Pull Cable • Determine the cable tension by solving the equation for the sum of force components parallel to the cable • A loaded coal car is at rest on an inclined track. The car has a gross weight of 5500 lb, for the car and its load as applied at G. The car is held in position by the cable. • Determine the TENSION in the cable and the REACTION at each pair of wheels.

11. Example  Coal Car • Create Free Body Dia. • Determine the reactions at the wheels • Determine cable tension

12. Example  Cable Braced Beam • Solution Plan • Create a free-body diagram for the frame and cable. • Solve 3 equilibrium equations for the reaction force components and Moment at E. • Reaction Analysis • The Support at E is a CANTILEVER; can Resist • Planar Forces • Planar Moment • Also • The frame supports part of the roof of a small building. The tension in the cable is 150 kN. • Determine the reaction at the fixed end E.

13. Triangle Trig Review SohCahToa

14. Example  Cable Braced Beam • Create a free-body diagram for the frame and cable sin cos • Solve 3 equilibrium eqns for the reaction force components and couple

15. SIX scalar equations are required to express the conditions for the equilibrium of a rigid body in the general THREE dimensional case Rigid Body Equilibrium in 3D • These equations can be solved for NO MORE than 6 unknowns • The Unknowns generally represent REACTIONS at Supports or Connections.

16. The scalar equations are often conveniently obtained by applying the vector forms of the conditions for equilibrium Rigid Body Equilibrium in 3D • Solve the Above Eqns with the Usual Techniques; e.g., Determinant Operations • A Clever Choice for the Pivot Point, O, Can Eliminate from the Calculation as Many as 3 Unknowns (a PoC somewhere)

17. Example  3D Ball-n-Socket • Solution Plan • Create a free-body diagram of the Sign • Apply the conditions for static equilibrium to develop equations for the unknown reactions. • Reaction Analysis • Ball-n-Socket at A can Resist Translation in 3D • Can NOT Resist TWIST (moment) in Any Direction • Cables Pull on Their Geometric Axis • A sign of Uniform Density weighs 270 lb and is supported by a BALL-AND-SOCKET joint at A and by two Cables. • Determine the tension in each cable and the reaction at A.

18. 3D Ball-n-Socket Prob cont. • Create a free-body diagram of the Sign • The x-Axis rotation Means that the Sign is Only Partially Constrained. Mathematically • But If the Sign is NOT Swinging (i.e., it’s hanging STILL) The Following Condition MustExist • With The Absence of the Mx Constraint This Problem Generates only 5 Unknowns • MX = 0 by X-Axis P.O.A.for W, TEC & TBC • Note: The Sign can SWING about the x-Axis (Wind Accommodation?)

19. 3D Ball-n-Socket Prob cont.2 • State in Component-Form the Two Cable-Tension Vectors

20. 3D Ball-n-Socket Prob cont.3 • Apply the conditions for static equilibrium to develop equations for the unknown reactions. • Solving 5 Eqns in 5 Unkwns

21. >> r = [5 -3 -7] r = 5 -3 -7  >> F = [-13 8 11] F = -13 8 11  >> rxF = cross(r,F) rxF = 23 36 1 WhiteBoard Work Solve 3DProblem byMechanics& MATLAB • Determine the tensions in the cables and the components of reaction acting on the smooth collar at G necessary to hold the 2000 N sign in equilibrium. The sign weight may concentrated at the center of gravity.

25. Engineering 36 Appendix Bruce Mayer, PE Registered Electrical & Mechanical EngineerBMayer@ChabotCollege.edu