Simple Harmonic Motion/Accelerating Pitch

[pbd1971]

Hi there, got a project I am working on at the moment, it requires me to manufacture a Timing Spiral on a 4th axis. The Spiral needs to have an accelerating pitch and has been specified as Simple Harmonic Motion.

Does anyone know what this means? from the investigations I have done so far I think it means the acceleration of the pitch needs to be "smooth"

I am thinking of writing a program that produces an incremental increase in the X axis for every 1 degree of angular rotation on the 4th axis.

For example, I need a pitch acceleration from 100mm to 125mm over 360 deg. Anyone any ideas of how to work it out?

Thanks

[Kiwi]

Is this a different project to the bottle screws?

Here is some code in increments of 10deg. over 112.5mm. Can supply increments of 1 deg if this is what you require.

G90 G01 X0 Y0 Z??
X2.8002 A10
X5.6177 A20
X8.4526 A30
X11.305 A40
X14.175 A50
X17.0628 A60
X19.9683 A70
X22.8918 A80
X25.8333 A90
X28.793 A100
X31.771 A110
X34.7673 A120
X37.7821 A130
X40.8156 A140
X43.8678 A150
X46.9388 A160
X50.0287 A170
X53.1378 A180
X56.266 A190
X59.4136 A200
X62.5805 A210
X65.7671 A220
X68.9733 A230
X72.1992 A240
X75.4451 A250
X78.7111 A260
X81.9972 A270
X85.3035 A280
X88.6303 A290
X91.9777 A300
X95.3456 A310
X98.7344 A320
X102.1441 A330
X105.5749 A340
X109.0268 A350
X112.5 A360

BTW. I have corrected an error and modified the program I provided for your other post.

[Geof]

You should Google Simple Harmonic Motion. It is a sine wave dependent type of motion. For instance if you have a pin on a wheel engaging in a slot on a bar that can slide every rotation pushes the bar forward then back and the motion of the bar is simple harmonic motion. I think this is called a Scotch Yoke but my memory may be faulty. As the wheel rotates the bar accelerates up to a maximum speed at the point where the pin is 90 degrees to the motion of the bar, then deccelerates to zero at the 180 degree point when the bar is at its maximum travel and then returns. The variation in speed of the bar plotted against the rotation of the wheel is a sine curve. A regular connecting rod moves a piston in a very close approximation to simple harmonic motion with the slight difference being caused by the fact that the connecting rod does not remain parallel to the motion of the piston through the entire cycle. The longer the rod the less the difference.

The way I would interpret your accelerating pitch spiral is that the change in pitch should start and end on a sine curve. This is opposed to my suggestions and example in your other thread where I had a constant pitch section then a (stepwise) increasing pitch section followed by constant pitcj again. Plotting my acceleration against the length of the spiral would give a flat line followed by a constant slope then another flat line.

What you need is a sine wave so your X increments have to vary in a sine wave fashion and I have to admit right at the moment I cannot dream up a math expression to do this.

As a side comment I will say this SHM spiral would have been fairly simple to cut in the old days of fully mechanical shape generation. To do this the X motion would be driven by a gear/crank/connecting rod or scotch yoke mechanism from the rotation of the spiral.

And now I am going to curse you because I will be thinking about this all through my golf game this morning. Just like I did last Monday about your other thread. Mind you I had a good game so maybe the distraction helps prevent me over-thinking my golf stroke and screwing things up.


EDIT:
Yes it is a Scotch Yoke, I just Googled. Have a look at this: http://en.wikipedia.org/wiki/Scotch_yoke

[pbd1971]

Thanks again guys, really appreciate the responses. To be honest, at the moment, I don't think my maths skills are up to the challenge so been thinking about paying for a couple of hours private tuition to get things fixed once and for all. I know that there is going to be variations of this component over the coming year, some of them are fairly straight forward, an angular rotation plotted against an incremental X movement on the drawing (these are obviously not SHM) but some will require me to produce a program when all I am given is the change in pitch over a length and within a certain number of degrees, using SHM. If I can get a formula for this I will be a happy bunny!

@ kiwi, did you use your "generator" to produce this code?

@Geof, hope the golf game went ok, at least you got someone to blame if it didn't

[Geof]

Getting the tutoring is a good idea. I did think about it while I was playing. Incidentally I had a great game. I am a novice having first swung a golf club on April 20th 2009 and I am totally non-competitive. I play with a friend and we get great laughs out of the horrible strokes we pull off. Anyway back to the math. It will be something like dividing the X distance into 180 increments, doing the calculation for X = X + something, then multiplying X by the sine of the incrment value. You should be able to find someone to help you develop a macro that will just need the start pitch, end pitch, the total length and the number of increments plugged in.

[Kiwi]

@ kiwi, did you use your "generator" to produce this code?

Yes, but after reading Geof's reply it looks like the path should return back to the start.
I now think you need a path which is a circle rotated/tipped so the ellipse view is 25mm wide for 100mm pitch.
Here is some code in increments of 10 for 180deg with the pitch starting at zero to 100 and back to zero.
The second 180deg needs the X decreasing the same values.
G90 G01 X0 Y0 Z37.5
X0.601 A10
X1.3222 A20
X2.1876 A30
X3.2261 A40
X4.4724 A50
X5.9678 A60
X7.7624 A70
X9.9158 A80
X12.5 A90
X15.3876 A100
X17.6976 A110
X19.5456 A120
X21.0241 A130
X22.2068 A140
X23.153 A150
X23.91 A160
X24.5155 A170
X25 A180

[Kiwi]

I've had a bit more of a play and produced some code which I think is 'Simple Harmonic Motion' with a pitch of 100mm
This is a single plane circle in increments of 10deg.

G90 G01
X0 A0
X0.3798 A10
X1.5077 A20
X3.3494 A30
X5.8489 A40
X8.9303 A50
X12.5 A60
X16.4495 A70
X20.6588 A80
X25 A90
X29.3412 A100
X33.5505 A110
X37.5 A120
X41.0697 A130
X44.1511 A140
X46.6506 A150
X48.4923 A160
X49.6202 A170
X50 A180
X49.6202 A190
X48.4923 A200
X46.6506 A210
X44.1511 A220
X41.0697 A230
X37.5 A240
X33.5505 A250
X29.3412 A260
X25 A270
X20.6588 A280
X16.4495 A290
X12.5 A300
X8.9303 A310
X5.8489 A320
X3.3494 A330
X1.5077 A340
X0.3798 A350
X0 A360

This is the XYZ code for the above.

G90 G01
X0 Y0 Z35.3553
X0.3798 Y-4.3412 Z34.8182
X1.5077 Y-8.5505 Z33.2232
X3.3494 Y-12.5 Z30.6186
X5.8489 Y-16.0697 Z27.0838
X8.9303 Y-19.1511 Z22.726
X12.5 Y-21.6506 Z17.6777
X16.4495 Y-23.4923 Z12.0922
X20.6588 Y-24.6202 Z6.1394
X25 Y-25 Z0
X29.3412 Y-24.6202 Z-6.1394
X33.5505 Y-23.4923 Z-12.0922
X37.5 Y-21.6506 Z-17.6777
X41.0697 Y-19.1511 Z-22.726
X44.1511 Y-16.0697 Z-27.0838
X46.6506 Y-12.5 Z-30.6186
X48.4923 Y-8.5505 Z-33.2232
X49.6202 Y-4.3412 Z-34.8182
X50 Y0 Z-35.3553
X49.6202 Y4.3412 Z-34.8182
X48.4923 Y8.5505 Z-33.2232
X46.6506 Y12.5 Z-30.6186
X44.1511 Y16.0697 Z-27.0838
X41.0697 Y19.1511 Z-22.726
X37.5 Y21.6506 Z-17.6777
X33.5505 Y23.4923 Z-12.0922
X29.3412 Y24.6202 Z-6.1394
X25 Y25 Z0
X20.6588 Y24.6202 Z6.1394
X16.4495 Y23.4923 Z12.0922
X12.5 Y21.6506 Z17.6777
X8.9303 Y19.1511 Z22.726
X5.8489 Y16.0697 Z27.0838
X3.3494 Y12.5 Z30.6186
X1.5077 Y8.5505 Z33.2232
X0.3798 Y4.3412 Z34.8182
X0 Y0 Z35.3553

My other code was 0 to 100mm pitch following the path of a helix.

[Kiwi]

I'm now thinking that the circle should be true-round looking along the X axis.
I still think the XA code is right.

#Addition, Updated code
G90 G01
X0 Y0 Z35.3553
X0.3798 Y-6.1394 Z34.8182
X1.5077 Y-12.0922 Z33.2232
X3.3494 Y-17.6777 Z30.6186
X5.8489 Y-22.726 Z27.0838
X8.9303 Y-27.0838 Z22.726
X12.5 Y-30.6186 Z17.6777
X16.4495 Y-33.2232 Z12.0922
X20.6588 Y-34.8182 Z6.1394
X25 Y-35.3553 Z0
X29.3412 Y-34.8182 Z-6.1394
X33.5505 Y-33.2232 Z-12.0922
X37.5 Y-30.6186 Z-17.6777
X41.0697 Y-27.0838 Z-22.726
X44.1511 Y-22.726 Z-27.0838
X46.6506 Y-17.6777 Z-30.6186
X48.4923 Y-12.0922 Z-33.2232
X49.6202 Y-6.1394 Z-34.8182
X50 Y0 Z-35.3553
X49.6202 Y6.1394 Z-34.8182
X48.4923 Y12.0922 Z-33.2232
X46.6506 Y17.6777 Z-30.6186
X44.1511 Y22.726 Z-27.0838
X41.0697 Y27.0838 Z-22.726
X37.5 Y30.6186 Z-17.6777
X33.5505 Y33.2232 Z-12.0922
X29.3412 Y34.8182 Z-6.1394
X25 Y35.3553 Z0
X20.6588 Y34.8182 Z6.1394
X16.4495 Y33.2232 Z12.0922
X12.5 Y30.6186 Z17.6777
X8.9303 Y27.0838 Z22.726
X5.8489 Y22.726 Z27.0838
X3.3494 Y17.6777 Z30.6186
X1.5077 Y12.0922 Z33.2232
X0.3798 Y6.1394 Z34.8182
X0 Y0 Z35.3553

[Andre' B]

Here is my take.
Assuming you have macro B on the machine.
The following is a hacked up version of the code I have used to make an ellipse.
Basicly just replaced the Y value with the A at the current angle.
Back plot in NCPlot is just a line but when animated you can see it speeding up and slowing down as it should.
The step angle can be made as small as needed for the finish but very small values will slow down the machine.

(CHANGE AS NEEDED)
(CHANGE AS NEEDED)
#100= 2    (RADIUS)
#101= 1.0  (Y SCALE, 1.0 FOR A CIRCLE)
           (OTHER VALUES MAKE AN ELLIPSE)
#102= 1    (STEP ANGLE)
(----------------------)


#103= 0    (CURRENT ANGLE)
G1X[#100*COS[0]]Y[#101*#100*SIN[0]]

WHILE [#103 LT 360] DO1
(USE THE FOLLOWING LINE TO DO A CIRLE IN X,Y PLANE)
(G1X[#100*COS[#103]]Y[#101*#100*SIN[#103]]F10.0)
(OR)
(USE THE FOLLOWING LINE TO DO THE SIMPLE HARMONIC MOTION)
G1X[#100*COS[#103]] A[#103]F10.0

#103=#103+#102
END1
(G1X[#100*COS[360]]Y[#101*#100*SIN[360]])
G1X[#100*COS[#103]] A[#103]F10.0
G0X0.0Y0.0

[Geof]

I think I have created confusion by mentioning crank pin rotation and 180 degrees with inadequate explanation. So I will try to be adequate based on what seems to me to be the way in which Simple Harmonic Motion could apply to the change in pitch. And I hope I am not totally out to lunch in all this.

It is not the motion along the screw that is Simple Harmonic Motion. The SHM refers to the way in which the change in pitch occurs. Changing the pitch from 4tpi to 5tpi over the length is fairly simple in a linear manner and how smooth it is depends on how many steps are used but the change between each step is the same. If it is changed in 100 steps then the first step is 4.01tpi, then 4.02tpi, etc. If a graph is drawn for the change in pitch versus distance along the screw it would be a straight line and the rate at which the pitch is changing is constant all the way. Note this is the change in pitch between each step, not the actual pitch at each step.

For the rate of change to act in Simple Harmonic Motion it has to start out very slowly and gradually increase to a maximum at the mid point then reduce again to fade away at the end. If this change in pitch is drawn on a graph it will be a line that starts out almost flat, curves up and over a peak and then curves back down to flat again at the end. The curve in this graph is related to a sine curve, it is not just a combination of simple circular, constant radius, curves. Note I say it is "related", the actual shape of the curve is the sine curve squared.

To get the changes in pitch at each step for this curve it is necessary to multiply the average change, that is the .01 per step for 100 steps by a number that is obtained from the sine of an angle. This is where a rotation of 180 degrees comes, it is just a calculation tool and 180 degrees is used because the sine of an angle is zero at zero degrees, reaches a maximum of 1 at 90 degrees and then goes back to zero at 180 degrees. When 100 steps are used each step corresponds to 1.8 degrees so part of the calculation for the first step is:

sin1.8 = 0.03141

0.03141^2 = 0.00099

0.00099 x 0.01 = 0.0000099

At the 25th step the corresponding angle is 45 degrees which gives 0.7071^2 x 0.01 = 0.005

The 50th step is 90 degrees which gives 1^2 x 0.01 = 0.01

75 steps is 135 degrees which is the same as 45 degrees at 0.005

But there is a complication/fault in this calculation. The result obtained for the SHM rate of change never goes above 0.01 so the average rate of change will be less than 0.01 per step. There has to be a correction factor which I think would be 2. This would make the change at the first step 2 x 0.0000099 = 0.00002 (close enough). At the 25th step mark it would be 0.01 and midway at 50 steps it would be 0.02.

And this is as far as I can go. I don't know how to set this type of calculation up into a macro so that you can just enter the starting tpi, the finish tpi, the total length to be covered during the change and the number of steps in the change.

Now I will go back to playing golf. It hurts my head less.


EDIT I did make a mistake in calculating the size of the steps. The change in pitch is from 4tpi to 5tpi which is 1 inch. Divide this into 100 steps and the step size is 0.01 and I had used 0.05. I have corrected the numbers above.

[pbd1971]

This is getting more in depth than I anticipated I am just in the process of getting my new printer/scanner up and running. Once I finish this, I will attempt to upload the drawing which should make things clearer.........picture painting a thousand words and all that

But thanks to everyone for the help so far, really appreciate it.

[Kiwi]

Do these pictures show what is required?
The slot around the cylinder is like an ellipse on a single plane.
When flattened the slot forms the shape of sine wave.