ME101 (Division III) webpage Lecture Slides available on http://www.iitg.ernet.in/kd/me101.htm Also available on: http://shilloi.iitg.ernet.in/~kd/me101.htm
Equivalent Systems: Resultants Equilibrium Equilibrium of a body is a condition in which the resultants of all forces acting on the body is zero. Condition studied in Statics When the resultant of all forces on a body is not zero, acceleration of the body is obtained by equating the force resultant to the product of the mass and acceleration of the body. Condition studied in Dynamics 2
Equivalent Systems: Resultants Vector Approach: Principle of Transmissibility can be used Magnitude and direction of the resultant force R is obtained by forming the force polygon where the forces are added head to tail in any sequence 3
Resultant of Concurrent Forces Two dimensional plane 4
Equivalent Force and Couple Two dimensional plane Three dimensional space 5
Resultant of Two Couples M = M 1 + M 2 6
Resultant of Force System:: 3D Three dimensional space Equivalent force-couple systems for each force Resultant equivalent force-couple system 7
Wrench Action Coincidence of resultant R and M axes Positive Wrench Negative Wrench 8
Wrench Action Wrench action or Screw Driver action Positive Wrench 9
Replacement of force system by wrench Resolve M into components Resolve M 2 into a couple Resultant Wrench 10
Example on Equivalent System Solution: For the beam, reduce the system of forces shown to (a) an equivalent forcecouple system at A, (b) an equivalent force couple system at B, and (c) a single force or resultant a) Compute the resultant force for the forces shown and the resultant couple for the moments of the forces about A. b) Find an equivalent force-couple system at B based on the force-couple system at A. c) Determine the point of application for the resultant force such that its moment about A is equal to the resultant couple at A. 11
Example on Equivalent System SOLUTION: a) Compute the resultant force and the resultant couple at A. R F 150 N j 600 N j 100 N j 250 N R 600 N j M R A j r F 1.6 i 600 j 2.8 i 100 j 4.8 i 250 j M R A 1880 Nmk 12
Example on Equivalent System b) Find an equivalent force-couple system at B based on the force-couple system at A. The force is unchanged by the movement of the force-couple system from A to B. R 600 N j The couple at B is equal to the moment about B of the force-couple system found at A. M R B M R A r B A R 1880 N mk 4.8 mi 600 N 1880 N mk 2880 N mk M R B j 1000 Nmk 13
Example on Equivalent System d R c) F R = F 1 + F 2 + F 3 + F 4 R = 150-600+100-250 = -650 N F R d = F 1 d 1 + F 2 d 2 + F 3 d 3 + F 4 d 4 d = 3.13 m 14
Example on resultant of a force system Determine the resultant of the system of parallel forces acting on the plate. Solve with a vector approach. 15
Example on resultant of a force system 16
Example on Wrench Replace the two forces and the negative wrench by a single force R applied at A and the corresponding couple M. 17
Example on Wrench 18
Example on Wrench 19
Example on Wrench 20
Rigid Body Equilibrium A rigid body will remain in equilibrium provided sum of all the external forces acting on the body is equal to zero, and Sum of the moments of the external forces about a point is equal to zero y z x 21
Rigid Body Equilibrium Free-Body Diagrams Space Diagram: A sketch showing the physical conditions of the problem. Free-Body Diagram: A sketch showing only the forces on the selected particle. 22
Rigid Body Equilibrium Support Reactions Prevention of Translation or Rotation of a body Restraints 23
Rigid Body Equilibrium Various Supports 2-D Force Systems 24