Governing physics, pt II

In the last post we discussed cable tension resulting from acceleration and deceleration in both the arbor cables and power drum cable.  Now, lets discuss whether that acceleration and maximum velocity make sense considering winch capabilities and move time.

Since we defined our maximum linear velocity of the winch as 60 in/sec, we will accelerate up to that speed, coast there, then decelerate down to zero.  If you were to graph this as vertical velocity of the lift vs. time, it would look something like this…

Where the y-axis is velocity and the x-axis is time.  This is called a trapezoidal velocity profile (as opposed to a triangular velocity profile), and it’s the graph of the desired behavior of the lift during operation.  You can see from the graph that the 2 angled sides represent the acceleration and deceleration with respect to the maximum velocity. 

By drawing this shape and integrating over time (taking the area under the velocity curve), we can determine what the total move time should be given our parameters for acceleration and max velocity.  Here are the resulting equations for total move time, acceleration and deceleration time…

 Because we have the time in terms of move distance, acceleration and maximum velocity, we can change their values and observe how that change affects the move time.  We could also work backwards, specifying a move time and calculating the necessary acceleration.  This would be useful information as the applications of the elevator vary from show to show.

Governing physics, pt I

In order to get a better feel for the forces involved with accelerating and decelerating the maximum amount of weight, I’ve worked out the force balancing equations as well as created a velocity profile for the move.  The force balance equations will tell us what the cable tension will be given a load, counterweight and an acceleration, while the velocity profile will visually break the move down into pieces and calculate the timed acceleration and deceleration given a maximum velocity and travel distance.  However before we begin, certain parameters have to defined – namely the physical capabilities of the winch – and these parameter values will be the starting point of our calculations.  Here are some in no particular order:

Travel distance: 12 ft 4 in

Maximum linear velocity produced by the winch: 60 in/sec

Typical acceleration (dependent on the load): 30 in/sec

Must use a trapezoidal velocity profile! (this will be explained in pt. II)

To balance the forces and determine the tensions in both the arbor cables and the power drum cable, a sketch of the system was made and free-body diagrams were created.  Since we want our maximum load to be 454 kg (1000 lbs) and, just a guess, our lifting platform weighs 22 kg (50 lbs), then the total mass being lifted is 476 kg.  We will also assume that our counterweight will be approximately 50% of the total mass being lifted, or 227 kg (500 lbs) and use the gravitational constant for acceleration of 9.8 m/sec.  By drawing free-body diagrams, it is possible to balance the forces in each mass seperately but using a similar value for the cable tension connecting the mass (and we’ll only assume there is one cable connecting the weight/counterweight system right now).  This is because the tension in the cable is uniform throughout it’s length, so the tension acting on mass 1 (the weight) is the same as whats acting on mass 2 (the counterweight).  If we add the two equations the tension cancels out, allowing us to solve for the force required to accelerate the system to 30 in/sec.  You can check out the free-body diagrams and calculations I’ve done here…

Note the great thing about using variables in these types of calculations is that I can substitute any values into ‘m1’, ‘m2’, and ‘a’ to come up with the forces exerted on that specific system.  The next step would be to create an Excel spreadsheet that utilizes these equations, allowing us to change values at will and see how those changes affect the entire system.

Now that the tension in both cables are known (the arbor cables and the power drum cable), I can get a feel for how close we should come to a breaking strain.  The cables themselves will be 7 x 19, 1/4″ galvanized aircraft cable with a typical minimum breaking strain of 7000 lbs.  From the calculations above, it appears that to accelerate a 1000 lb load at 30 in/sec would put roughly 670 lbs of force on the wire rope attached to the power drum, and roughly 283 lbs of force onto each of the 4 arbor cables.  This means we’re currently working with a safety factor of about 10:1, which is a good thing considering we haven’t included any force from sheave friction or forces resulting from an emergency stop (e-stop).

In the next post we’ll talk about the velocity profile, total move time and motor limitations.

 

Concept wrap-up!

The ‘power drum’ concept is a go!  But first, a little background on the counter-weighted arbor systems that are used in fly systems of this sort.

The basic idea is to lessen the forces involved in lifting objects using a counterweight design.  This counterweight should ideally be 50% of the weight being lifted by the elevator, and because that weight can change drastically it makes sense to design a process that will allow the easy addition/subtraction of counterweight from the arbor.  The arbor houses the counterweights and acts as the object the lifting and lowering cables physically attach to.  Here’s a sketch of the proposed arbor – click on it to magnify…

The weights (which are steel slabs with u-shaped grooves cut into them) pile ontop of one another and are held in place by weight locks.  These locks could be anything from pipe clamps to rings with set screws and holding plates.  The lifting and lowering wires, as well as the wire control to the power drum, will be physically connected to the upper and lower arbor plates.  These plates will also be guided vertically to impede any rocking or swinging of the counterweight.

The power drum is what physically transmits power from the hand crank or electric winch to the arbor system.  It is the power drum wires labeled ‘PD’ in the drawing above that connect the weights to the power drum, and they are skewed to either side to allow the wire to wrap around the drum in the center, as the 2 wires spiral in toward eachother.

This is more easily seen than described…so heres a sketch of the power drum wire setup…The manufacturing constraints on fabricating the drum limit the diameter to 10 inches, so that’s the size of the drum I’ll design to.  The outer surface will also have grooves cut into it to allow for 1/4″ steel cable to lay securely around the drum face.

Brainstorming the hand/winch operations

When I last left this blog, I was beginning the brainstorming process for the design of a stage elevator based on certain specifications.  The main concern when operating this lift is safety, for both the operator and anyone (or thing) being lifted.  If the desire is to have the ability to operate the lift both by hand or using a standard endless loop electric winch, my first thought was to couple the system in a manner that would allow operation of the winch by hand if the electric winch ever failed.  The transition between hand and winch operation should be quick and easy, with the ability to switch in low lighting and relatively intuitive.  What if a circuit breaker blows?  What if the winch dies suddenly?  Of the many things that could cause winch failure, I wanted a system that would allow a stagehand to quickly switch to ‘hand’ mode and complete the lifting move (or lower to safety!).

The weight lifted should be counterbalanced using an arbor system.  I’ve sketched out a quick rendition of what my original thoughts were, and it involved a wheel and a ‘power’ drum around which lifting, lowering and winch wires would be attached.  When you wanted to operate the lift using the winch, everything was already in place: winch connected to power drum, drum connected to counterweights, counterweights connected to lifting platform.  Torque applied by the winch would transfer through to drum which then would raise or lower the counterweights and thus the platform itself.

If hand operation was necessary, the changeover would be simple.  Disengage the winch via a to-be-designed clutch (after brakes have been applied to stabilize the lift platform) and insert a wheel/hand crank into the end of the power drum.  Release the brakes with a foot-operated brake override and crank away, either up or down.

Here’s a sketch of the idea:

The interesting thing about this idea is that the wheel/crank is removable, so during winch operation the wheel can be tucked safely away from the action and is certain not to injure anyone passing by.  One of the main issues initially seems to be the fleet angle (ie. the angle of the wire from the power drum to a block…as the wire wraps around the drum this angle will change) and the clutch system.  Also, when the platform has been fully lifted, what holds it in place?  Are the brakes engaged?  Do you leave the winch on, providing a torque?  How about a solenoid-activated bolt system that locks the platform in place at stage height?  These are all questions that I am keeping in mind as I move forward…

A second idea is a modification in the manner the lift is hand operated.  Instead of a crank, the operator pulls a loop of rope directly attached to the counterweights.  Something like this:

However, this is just a rough draft.  More to come!!