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.