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ForceAnalysisNew - Force Analysis of Machinery I...

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Force Analysis of Machinery I. Introduction: In a dynamic analysis, we create equations that relate force and motion of a body (as in ME 233) or in our case a mechanism or machine. These are called equations of motion . There are 2 directions to these problems: the Forward Dynamics problem, where the motion is given and the forces are to be determined, and the Inverse Dynamics problem, where the driving force is given and the resulting motion is to be found. 1) Forward kinematics: We will call our method Kinetostatics look for dynamic equilibrium at a specific position and time, a snapshot of the mechanism. The equations will look like: With m , a known, F unknown, or for a mechanism: The eq’s for force are linear and solved using linear algebra a m F = [ ] { } { } a m F C =
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Review first derivatives (Velocity) of vectors 1. Inverse dynamics: Called the time-response problem. This solves the motion of a mechanism given the input driving force. Force example, the time history of the flight of an arrow leaving a bow. For these problems, we write equations of motion (which are now differential eqations of motion) that might look like: And solve motion as a function of time through numerical integration. For ex. F x k x c x m = + + ( 29 t x x x t x x x m x k x c F x + = + = - - = 0 0 ,
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Summary Forward Dynamics: Kinetostatics Given motion, find required driving force and all bearing reactions Inverse Dynamics: Time response Given input force, solve output motion as a function of time.
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Motivational slide 1. Performing force analysis of a mechanism draws on all your modeling skills: 1. Mechanism modeling, position, velocity and acceleration analysis, force analysis 2. Goals from this section: 1. Learn how to carry out a force analysis (kinetostatics) 2. Apply to several examples in class and HW. 3. Create a computer model (Matlab) for force analysis and apply to a specific problem
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Review of Dynamics 1. Newtonian mechanics: 1. Conservation of momentum 2. Force = rate of change of momentum 3. Action/reaction 2. Corollary, Euler’s Equation: 1. Torque = rate of change of angular momentum ( 29 v m dt d F = 0 0 = = a F 21 12 F F = ( 29 ϖ ϖ α ϖ I I I dt d T × + = =
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Review of Dynamics 1. Dynamic forces: 1. Linear acceleration: Acceleration of pt. P: Sum forces on particle P: And integrate to solve: Result: ** ( 29 i i g i g p r r a r dt d a a × × + × + = + = βϖ ϖ α 2 2 dm a dF p = ( 29 = × × + × + = × × + × + = = = dm dm dm dm dm dm d g ext i i g ext i i g p ext a F r r a F r r a a F F ϖ ϖ α ϖ ϖ α g ext a F m =
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Review of Dynamics 2. Dynamic Moments: 1. General layout shown in figure:
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Review of Dynamics 2.
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