Calculate Acceleration of drum due to falling object 2

[CraneEng87]

Hello,

I'm trying to work out the acceleration of a drum as a load is dropped and need to account for mass moment of inertia.

I've attached a free body diagram of the system.

The load is instantaneously dropped causing the drum to un-spool wire rope as the load falls to the ground.

Assume the following:

Efficiency of sheave system = 0.942
Load = 34002 LBF
Drum Mass Moment of Inertia = 24532.57 LBM-FT^2
Radius of drum to rope = 2.946 ft

Find Angular Acceleration of Drum:

I'm not sure how to relate these components together and working in the imperial system for a problem like this is killing me. I can convert to SI if needed.

I'd appreciate any help.

 

[hydtools]

The r/g appears in the block inertia moment terms. The block inertia moment form is m*v*r. The author chose to express m in terms of tension S. S = m*g or m = S/g Velocity v = w*r then m*v*r = (S/g)*w*r*r = S*w*r*(r/g)

After rearranging terms so that n final = n initial + (2*g*9.549*t)/(2r +4*g*J/(S*2r)) I am not sure how the drive system efficiency was figured in other than applied to show drive output torque to be 0.98*rope tension torque. The pulley efficiency may account for bearing friction so that rope tension between drum and pulley is 0.94 times the tension between the block and pulley.

Ted

 

[rb1957]

yeah, maybe what they're saying is if the lowering control system fails and an emergency brake is applied then the drum accelerates due to two loads ... the torque being applied by the cable and the gravitational acceleration of the mass ?

really ? ... doesn't the cable tension account for the mass of the block ? and it is the later term ...
r/g = 0.09, J/T = 8846/(140534*.9) = 0.07

another day in paradise, or is paradise one day closer ?

 

[hydtools]

There are two terms, momentum and impulse, determined by the mass of the block. The rotational momentum of the block and the impulse torque of the rope tension on the drum.

Ted

 

[hydtools]

Here is a textbook reference, Mechanics for Engineers, Beer and Johnston

Ted

 

[chicopee]

My solution came up with a winding drum angular velocity of .31 radian/sec^2 at the moment the load is released from an arresting position; the load will accelerate at .914 ft/sec^2. My calculations are based on the following: 1)adding a snatch block at the same elevation as the bottom rim of the drum to keep the cable at 45 degrees between the snatch block and the upper sheave 2) using sheave and snatch block efficiencies as given at .942 3) assuming the winding drum journal coefficient of friction at .01 which may be a little low if sloppy 3) drum weight at 5356.4 lbf by assuming drum as a homogeneous disk with I-.5MR^2 4) discounting rope weight added to the load and removed from the winding drum and related elasticity. Remember, sheave has to be able to take a load of nearly 35 Tons and snatch block rated for not less than 15 tons; also on earth, numerically, a mass of 1 lbm weights 1 lbf as a=32.2ft/sec^2 and gc=32.2 lbm-ft/lbf-sec^2 .

 

[hydtools]

chicopee, you changed the problem. The initial drum speed is 5.33rpm and there is no snatch block with your added efficiency and assumed a drum journal friction.

What is your point?

Ted

 

[chicopee]

An error in the statement "My solution came up with a winding drum angular velocity of .31 radian/sec^2..." it should be angular acceleration.

Hytools, I added the snatch block as you can better control the cable angular position to the upper sheave and minimizes the load line to the winding drum.. Installation of a snatch block makes more sense and construction riggers would have installed. Also, the original post asked for angular acceleration.
Looking at the sketch, it is faulty and probably should be changed to incorporate a snatch block.

 

[RolMec]

Hello craneeng87,

i believe the German calc is wrong. Pls. have a look at attached calc #1 (it's done metric ;-) ).
Considering your findings for the constants used in the equation & under metric units, the resulting unit doesn't come out correct too: see att. calc. #2.

I'd fear a system which goes overspeed 4 times just within a quarter of a second.

Regards
R.

 

[RolMec]

and here (corrected) calc. #1

 

[hydtools]

Rolmec, in your corrected should you not multiply by 60 rather than divide by 60 to change seconds to minutes? Omega is in the denominator that would put 60 in the numerator.

Ted

 

[rb1957]

I thought the german solution was dimensionally correct

another day in paradise, or is paradise one day closer ?

 

[hydtools]

The units of the German solution are correct.

units of 187.35 are m*rev/sec*min; 2*g*(60sec/min)/(2*pi rad/rev)
units of 39.24 are m/sec^2; 4*g
Not as noted in Rolmec's calc #1

Just for reference, if the weight were to free fall it would reach 2.45 m/sec in 0.25 sec. In the solution of the system the weight reaches 1.88 m/sec in 0.25 sec. Not quite free fall speed.

Ted

 

[RolMec]

Thanks hydtools, I was not sufficiently diligent when putting up my calc.
If the 60 changes into the numerator this would lead to a still higher rot. speed than already seen from OP --> my simplistic approach does not work here.
I'll try better next time.
Regards
R.

 

[RolMec]

Hello CraneEng87,

here's the derivation, now correct. German calc. confirmed ;-)
Hope this helps.

Regards
R.
;-) It pays to follow free body diagram rules.

 

[rb1957]

I thought the german calc buried a "g" in the J/T term, in the constant ...

another day in paradise, or is paradise one day closer ?

 

[CraneEng87]

Thanks to everyone who helped work out this equation.

RolMec's last calculation looks like he accounted for all of the terms in the original German calc and confirms the calculation is correct.

While the forum was tackling the German calc for me I was able to refine my boss's calc into a formula that made a little more sense and arrived at a closer answer than before. I'll attach it and see if everyone here thinks it is a reasonable substitution for the German calc or if the German calc is a drastically better way of doing this.

I didn't provide the variables for calculating "E".