Statics of Particles - College of Engineering | University ...



Statics of Particles (3D)Forces in 3DMost real-world problems are formulated for bodies and forces not arranged on a single plane.Hence, for quantitative analysis we need mathematical representation of location and orientation for bodies, magnitude and direction of forces, and means to formulate conditions imposed on them.We will use 3D vectors for the representation. However, there are multiple mutually equivalent ways to represent vector itself and one should select the most efficient way based on the problem.Operations on forces in 3D are performed, almost exclusively, by use of formal vector algebraForce Acting on a Line Between Two points376316742846800If a force F is acting along the line ?? with magnitude ?, then:What are rectangular components of F?Any vector can be defined as the magnitude times the unit vector in the same direction:F=F*uF=F*uAB Unit vector in the direction of ??: uAB=ABABNeed to find the vector and vector magnitude of ABAB=xB-xAi+yB-yAj+zB-zAkAB=xB-xA2+yB-yA2+zB-zA2?Expanding to force components:F=F*uAB=FAB*AB=FxB-xA2+yB-yA2+zB-zA2?xB-xAi+yB-yAj+zB-zAkForces in 3D, Example 1:Example 2:Resultant of the sum of forces in 3D:R=F=Fxi+Fyj+Fzk=Fxi+Fyj+FzkComponents of the resultant:Rx=Fx,??Ry=Fy,???Rz=FzMagnitude of the resultant: Rx=Fx,??Ry=Fy,???Rz=FzDirection cosines: cosα=RxR;cosβ=RyR;cosγ=RzRExample:Particle Equilibrium (3D)Similarly to a 2D situation, the EE are expanded to 3D:F=Fxi+Fyj+Fzk=0Or, if separated into components:Fx=0,Fy=0,Fz=0Note: Three equations can be used to find 3 unknowns at most.Analysis of Equilibrium follows the same procedure as in 2D.Draw FBD to include all acting forces.Apply equations of equilibrium.In case of 3D there are three equations per particle. Hence, to solve them, there should be maximum 3 unknowns per particle.In 3D elimination technique is almost never used. Instead, use formal vector algebra.Possible unknowns: Force components / magnitudes / angles; some distances. Examples: 1 force – 3 components, or magnitude and 2 angles; 3 forces: magnitudes; etc.Example 1:Example 2:Example 3:Example 4:Example 5: ................
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