Modeling-4

Modeling-4 - Transfer Functions For Systems With Gears...

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ransfer Functions For Systems With Gears Most rotational mechanical systems have gear trains associated Transfer Functions For Systems With Gears with them; especially those driven by motors Gears provide mechanical advantage to rotational systems The linearized interaction between two gears is depicted as follows: where an input gear with radius r 1 and N 1 teeth is rotated through angle θ 1 (t) due to torque, T 1 (t) . An output gear with radius r 2 and N 2 teeth responds by rotating through angle θ 2 (t) and elivering a torque ) delivering a torque T 2 (t) . 81 (c) 2010 Farrokh Sharifi
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ransfer Functions For Systems With Gears Let us find a relationship between θ and θ : Transfer Functions For Systems With Gears 1 2 as the gears turn, the distance traveled along each gear’s circumference is the same. Thus: by assuming the gears have negligible inertia and damping we can evelop lationship etween e put rque d e elivered or develop a relationship between the input torque, T 1 , and the delivered torque, T 2 , as follows: or Transfer function for angular displacement Transfer function for torque 82
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ransfer Functions For Systems With Gears Let’s see what happens to mechanical impedances when gears Transfer Functions For Systems With Gears are introduced to the system: the following system shows gears driving a rotational inertia, spring, d i and viscous damper: an equivalent representation can be derived by reflecting the input, T (t) , as an output: 1 i T )(N Since: (T 1 ) output =T 1 (t)(N 2 /N 1 ) 83 (c) 2010 Farrokh Sharifi
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ransfer Functions For Systems With Gears from this we write the equation of motion using mechanical impedance: Transfer Functions For Systems With Gears nverting ) to uivalent ) ields converting θ 2 (s) into an equivalent θ 1 (s) yields: after simplification: this suggest the following system: 84 (c) 2010 Farrokh Sharifi
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ransfer Functions For Systems With Gears from these results the following statement can be made: Rotational Transfer Functions For Systems With Gears mechanical impedances can be reflected through gear trains by multiplying the mechanical impedance by the ratio: where the impedance to be reflected is attached to the source shaft and is being reflected to the destination shaft the following example will make use of this concept of reflected impedance 85 (c) 2010 Farrokh Sharifi
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ransfer Functions For Systems With Gears Impedance Method : Transfer Functions For Systems With Gears Step 1: Redraw the free body diagram changing all time variables to equivalent Laplace variable θ (s), T (s),. .
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Modeling-4 - Transfer Functions For Systems With Gears...

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