![]() ![]() These axes do need to be perpendicular to one another, but they do not necessarily have to be horizontal or vertical. Next you will need to choose the x, y, and z axes. It is also useful to label all forces, key dimensions, and angles. This is done by removing everything but the body and drawing in all forces acting on the body. The first step in equilibrium analysis is drawing a free body diagram. In the free body diagram, provide values for any of the know magnitudes or directions for the force vectors and provide variable names for any unknowns (either magnitudes or directions). This diagram should show all the known and unknown force vectors acting on the body. The first step in finding the equilibrium equations is to draw a free body diagram of the body being analyzed. Once we have written out the equilibrium equations, we can solve theĮquations for any unknown forces. We do this by summing up all the x components of the force vectors and setting them equal to zero in our first equation, and summing up all the y components of the force vectors and setting them equal to zero in our second equation. For two dimensional problems, we will split our one vector equation down into two scalar equations. In order to solve for any unknowns in our sum of forces equation, we actually need to turn the one vector equation into a set of scalar equations. ![]() This is the basis for equilibrium analysis for a particle. If we know that the body is not accelerating then we know that the sum of the forces acting on that body must be equal to zero. If a body is in static equilibrium, then by definition that body is not accelerating. The number of unknowns that you will be able to solve for will again be the number of equations that you have.Equilibrium Analysis for Concurrent Force Systems Once you have your equilibrium equations, you can solve these formulas for unknowns. All moments will be about the z axis for two dimensional problems, though moments can be about x, y and z axes for three dimensional problems. To write out the moment equations simply sum the moments exerted by each force (adding in pure moments shown in the diagram) about the given point and the given axis (x, y, or z) and set that sum equal to zero. ![]() Remember that any force vector that travels through a given point will exert no moment about that point. Any point should work, but it is usually advantageous to choose a point that will decrease the number of unknowns in the equation. To do this you will need to choose a point to take the moments about. Next you will need to come up with the the moment equations. Your first equation will be the sum of the magnitudes of the components in the x direction being equal to zero, the second equation will be the sum of the magnitudes of the components in the y direction being equal to zero, and the third (if you have a 3D problem) will be the sum of the magnitudes in the z direction being equal to zero. Once you have chosen axes, you need to break down all of the force vectors into components along the x, y and z directions (see the vectors page in Appendix 1 page for more details on this process). If you choose coordinate axes that line up with some of your force vectors you will simplify later analysis. ![]() Next you will need to choose the x, y, z axes. In the free body diagram, provide values for any of the known magnitudes, directions, and points of application for the force vectors and provide variable names for any unknowns (either magnitudes, directions, or distances). This diagram should show all the force vectors acting on the body. \Īs with particles, the first step in finding the equilibrium equations is to draw a free body diagram of the body being analyzed. The sum of each of these will be equal to zero.įor a two dimensional problem, we break our one vector force equation into two scalar component equations. This means that a rigid body in a two dimensional problem has three possible equilibrium equations that is, the sum of force components in the x and y directions, and the moments about the z axis. In a two dimensional problem, the body can only have clockwise or counter clockwise rotation (corresponding to rotations about the z axis). This means that our vector equation needs to be broken down into scalar components before we can solve the equilibrium equations. The addition of moments (as opposed to particles where we only looked at the forces) adds another set of possible equilibrium equations, allowing us to solve for more unknowns as compared to particle problems. For an rigid body in static equilibrium, that is a non-deformable body where forces are not concurrent, the sum of both the forces and the moments acting on the body must be equal to zero. ![]()
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