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Mechanical Systems • Translation Point mass concept P(t) = F(t)*v(t) Newton’s Laws & Free-body diagrams • Rotation Rigid body concept P(t) = T(t)*w(t) Newton’s laws & Free-body diagrams • Transducer devices and effects 1 Mechanical translation • Newton’s Laws Every body persists in a state of uniform motion, except insofar as it may be compelled by force to change that state. The time rate of change of momentum is equal to the force producing it. To every action there is an equal and opposite reaction. (Principia Philosophiae, 1686, Isaac Newton) 2 Quantities and SI Units • “F-L-T” system • Define F: force (Newton [N]) Define L: length (meter [m]) Define T: time (second [s]) Derive v: velocity, m: mass “M-L-T” system Define M: mass (kilogram [kg]) Define L: length (meter [m]) Define T: time (second [s]) Derive v: velocity, F: force 3 Physical effects and engineered components • • • • Inertia effect - rigid body with mass Compliance (stiffness) effect - spring Dissipation (friction) effect - damper System boundary conditions: motion conditions – velocity specified force conditions - drivers and loads 4 Translational inertia • Physical effect: *dV • Engineered device: rigid body “mass” • Standard schematic icon (stylized • • picture) Standard multiport representation Standard icon equations 5 Inertia in translation: standard forms v 1 F1 F2 m m * v F1 F 2 I F2 F1 F3 1 0 F1 F 2 F 3 6 Compliance (stiffness) • • • • • Physical effects: =E* Engineered devices: spring Standard schematic icon Standard multiport representation Standard icon equations 7 Standard translation icons 0 C F k * ( L Lfree) dL v1 v 2 dt 8 Dissipation (resistance) • • • • • Physical effects Engineered devices: damper Standard schematic icons Standard multiport representation Standard icon equations 9 Standard translation icons 0 F b * (v1 v2 ) R 10 Free-body diagrams • Purpose: Develop a systematic method for generating the equations of a mechanical system. • Setup method: Separate the mechanical schematic into standard components and effects (icons); generate the equation(s) for each icon. • Standard form of equations: the composite of all component equations is the initial system set; select a reduced set of key variables (generalized coordinates); reduce the initial equation set to a set in these variables. 11 Multiport modeling of mechanical translation • Multiport representations of the standard • • icons: focus on power ports Equations for the standard icons Multiport modeling using the free-body approach 12 Multiport modeling of translation based on free-body diagrams • Identify each mass point and rigid connector. • • Define an inertial velocity for each. Use a standard multiport component to represent each mass point • Write the standard equation for each component. 13 Suitcase example Vo 14 Bobsled example See the file Bobsled report.pdf for a study of this model. 15