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ENGR 3340: Fundamentals of Statics and Dynamics Fundamentals of Statics and Dynamics - ENGR 3340 Professor: Dr. Omar E. Meza Castillo [email protected] http://facultad.bayamon.inter.edu/omeza Department of Mechanical Engineering of Statics and Dynamics ENGR 3340: Fundamentals Universidad Interamericana MSP21 - Bayamón Tentative Lectures Schedule Topic Lecture Equilibrium of a Particle in 3-D 7 2 of Statics and Dynamics ENGR 3340: Fundamentals Universidad Interamericana MSP21 - Bayamón One thing you learn in science is that there is no perfect answer, no perfect measure. A. O. Beckman Topic 7: Equilibrium of a Particle in 3-D Equilibrium 3 of Statics and Dynamics ENGR 3340: Fundamentals Universidad Interamericana MSP21 - Bayamón Objectives To introduce the concept of the free-body diagram for a particle To show how to solve particle equilibrium problems using the equations of equilibrium 4 of Statics and Dynamics ENGR 3340: Fundamentals Universidad Interamericana MSP21 - Bayamón Three-Dimensional Force System. Equilibrium F 0 Particle moves with constant velocity or remains at rest. Resolve the forces into their i, j, k components Fxi Fy j Fzk 0 Fx 0 Fy 0 Fz 0 Three equations Three unknowns Mathematical manipulation vs. Physical interpretation 5 of Statics and Dynamics ENGR 3340: Fundamentals Universidad Interamericana MSP21 - Bayamón Three-Dimensional Force System - Physical Interpretation Free Body Diagram • Establish the x, y, z axes in any suitable orientation. • Label all known and unknown force magnitudes and directions. • The sense of a force having an unknown magnitude can be assumed. Types of Forces Weight: Spring: W mg 6 F ks of Statics and Dynamics ENGR 3340: Fundamentals Universidad Interamericana MSP21 - Bayamón Three-Dimensional Force System - Mathematical Manipulation F Fu Position Vectors r uAC AC rAC r rAB rC rA (xC x A ) i (yC y A ) j (zC z A ) k Direction Cosines uF1 cos i cos j cos k cos 60 i cos 60 j cos135 k Projection angles (depends on each problem) uF2 cos 36.8 cos 30 i cos 36.8 sin 30 j sin 36.8 k 4 3 41 3 i j k 5 2 5 2 5 7 of Statics and Dynamics ENGR 3340: Fundamentals Universidad Interamericana MSP21 - Bayamón 8 of Statics and Dynamics ENGR 3340: Fundamentals Universidad Interamericana MSP21 - Bayamón Example 1 Given: A 600 N load is supported by three cords with the geometry as shown. Find: The tension in cords AB, AC and AD. Plan: 1) Draw a free body diagram of Point A. Let the unknown force magnitudes be FB, FC, FD . 2) Represent each force in the Cartesian vector form. 3) Apply equilibrium equations to solve for the three unknowns. 9 of Statics and Dynamics ENGR 3340: Fundamentals Universidad Interamericana MSP21 - Bayamón Solution 1 z FD FBD at A FC 2m 1m 2m A FB x 600 N FB = FB (sin 30 i + cos 30 j) N = {0.5 FB i + 0.866 FB j} N F C = – FC i N FD = FD (rAD /rAD) = FD { (1 i – 2 j + 2 k) / (12 + 22 + 22)½ } N = { 0.333 FD i – 0.667 FD j + 0.667 FD k } N 10 30˚ y of Statics and Dynamics ENGR 3340: Fundamentals Universidad Interamericana MSP21 - Bayamón Solution 1 z Now equate the respective i , j , k components to zero. FD FBD at A FC 2m Fx = 0.5 FB – FC + 0.333 FD = 0 1m Fy = 0.866 FB – 0.667 FD = 0 2m Fz = 0.667 FD – 600 = 0 A FB x 600 N Solving the three simultaneous equations yields FC = 646 N FD = 900 N FB = 693 N 11 30˚ y of Statics and Dynamics ENGR 3340: Fundamentals Universidad Interamericana MSP21 - Bayamón Homework5 http://facultad. bayamon.inter.edu/omeza/ Omar E. Meza Castillo Ph.D. 12 of Statics and Dynamics ENGR 3340: Fundamentals Universidad Interamericana MSP21 - Bayamón ¿Preguntas? Comentarios 13 of Statics and Dynamics ENGR 3340: Fundamentals Universidad Interamericana MSP21 - Bayamón GRACIAS 14