modulated_transf

# modulated_transf - NONLINEAR MECHANICAL SYSTEMS(MECHANISMS...

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NONLINEAR MECHANICAL SYSTEMS (MECHANISMS) The analogy between dynamic behavior in different energy domains can be useful. Closer inspection reveals that the analogy is not complete. One key distinction of mechanical systems is the role of kinematics — the geometry of motion E XAMPLE : automobile internal combustion engine. reciprocating translational motion of a piston is converted to crankshaft rotation by a crank and slider mechanism. x 1 θ x 2 x Mod. Sim. Dyn. Sys. Nonlinear Mechanics Intro. page 1

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Consider torque applied to rotate the crankshaft. Inertia and friction opposing crankshaft rotation depend on position. if θ 0° or θ 180° small crankshaft rotation does not move the piston — inertia is small if θ ±90° crankshaft rotation is nearly proportional to piston translation — inertia is large How should this phenomenon be modeled? Mod. Sim. Dyn. Sys. Nonlinear Mechanics Intro. page 2
CAUTION: A POSITION - MODULATED INERTIA WOULD VIOLATE ENERGY CONSERVATION ! I τ ω θ S e S HOULD THIS BE SOME KIND OF MULTIPORT ? unclear the angle that "modulates" inertia also modulates stored kinetic energy that change of stored energy should be associated with a power port ... but that angle is already associated with a power port how can a separate port be justified? Mod. Sim. Dyn. Sys. Nonlinear Mechanics Intro. page 3

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E XAMINE THE MECHANISM KINEMATICS The mechanism imposes a relation between crankshaft angle and piston position x = l 2 2 – l 1 2 sin 2 ( θ ) + l 1 cos( θ ) where l 1 = crank length l 2 = connecting rod length check: θ = 0° x = l 2 + l 1 θ = ±90° x = l 2 2 – l 1 2 θ = 180° x = l 2 – l 1 OK Mod. Sim. Dyn. Sys. Nonlinear Mechanics Intro. page 4
The kinematic relation x = x( θ ) may be regarded as a transformation between coordinates. An energetically consistent description of the mechanism inertia includes a modulated transformer . For example, if the inertia of the crank and connecting rod are neglected, and the piston mass is assumed to be the dominant inertia, a bond graph of that model is 1 I τ ω θ MTF 1 x . . θ

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modulated_transf - NONLINEAR MECHANICAL SYSTEMS(MECHANISMS...

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