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Acceleration resistance in crane design

  1. Apr 9, 2017 #1
    Hi everybody
    Does anyone familiar with this equation?

    Acceleration resistance,
    = {(load MOI/mech. efficiency) + (motor MOI) x rpm^2} / {3.65 x 10^5 x acceleration time}
    = {(kgm^2/ƞ) + (kgm^2) x rpm^2} / {3.65 x 10^5 x second}

    = final value unit is in kilowatt (kW)

    It used to calculate the acceleration power for trolley and gantry in crane design. I am looking for the origin/principal of this equation... thanks guys
     
  2. jcsd
  3. Apr 9, 2017 #2

    jack action

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    First, I would guess your equation should read:

    {(load MOI/mech. efficiency) + (motor MOI)} x rpm^2 / {3.65 x 10^5 x acceleration time}

    It comes from the definition of power ##P = T\omega##, where ##T## is the torque and ##\omega## is the angular velocity (i.e. rpm).

    The torque, if converted entirely to acceleration, is ##T = I\alpha##, where ##I## is the mass moment of inertia and ##\alpha## is the angular acceleration.

    The angular acceleration is ##\alpha= \frac{\omega}{t}##, where ##t## is the time.

    So, ##P = T\omega = I\alpha \omega = I\frac{\omega}{t}\omega = \frac{I\omega^2}{t}##.

    Thus ##I## = (load MOI/mech. efficiency) + (motor MOI), ##\omega## = rpm/2 and ##t## = acceleration time.

    The constant 3.65 x 10^5 = ##2^2 \times 1000 \frac{W}{kW} \div \left(\frac{\pi}{30}\frac{rpm}{\frac{rad}{s}}\right)^2##. So it is for unit conversions. The ##2^2## is from ##\omega## = rpm/2, because the acceleration time is measured from 0 to rpm, so we used the average rpm during the process, i.e. rpm/2.
     
  4. Apr 9, 2017 #3
    Dear sir, thank you for your explanation. it is very helpful. I appreciate it so much.
    just wonder, what came out from the (kgm^2 x rpm^2) / t... to be specific, what unit it produce that enable it to be convert to kW?
     
  5. Apr 10, 2017 #4

    jack action

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    The basic SI units of your formula are:
    [tex]\frac{kg.m^2 \left(\frac{rad}{s}\right)^2}{s}= \frac{kg.m^2}{s^3}[/tex]
    The basic SI units that define the Watts are:
    [tex]W = \frac{J}{s} = \frac{N.m}{s} = \frac{\left(kg\frac{m}{s^2}\right)m}{s} = \frac{kg.m^2}{s^3}[/tex]
    So you can see that they have equivalent dimensions, i.e. ##\frac{mass \times length^2}{time^3}##
     
  6. Apr 10, 2017 #5
    Dear Jack Action
    You mentioning the 3.65 x 10^5 is constant for unit conversions to kW. i am unable to figure out what unit to be converted to kW. can you help me? thx
     
  7. Apr 11, 2017 #6

    jack action

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    The units in your formula are all basic SI units (kg, m, s), except for rpm. Once you converted rpm to rad/s (see post #2), you have all SI basic units. In the end, the final SI unit is kg.m2/s3. In post #4, I showed that 1 kg.m2/s3 = 1 W. Then in post #2, W is converted to kW.
     
  8. Apr 19, 2017 #7
    Thanks Jack
     
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