SLA 3D Printer Project Log 5: 200 Step Program

I've recently obtained a Raspberry Pi (and instantly overclocked it - reached 900MHz CPU, 500MHz GPU and 600MHz RAM with an overvolt of 8), as skeptical as I was at first, this £25 Set-Top Box with pretensions to being a Linux PC is giving me some real inroads in this project.

I've already got it to run a NEMA 17 Stepper Motor through an EasyDriver V4.4, and my coding prowess is as extensive as the Waterloo and City Line; I did this via WiringPi, a library for the Pi that enables you to use the GPIO Pins without using Python or typing sudo (super-user do) ahead of every command.

The commands entered were:

• gpio mode 1 pwm - this assigns Pin 1 to PWM operation
• gpio pwm-bal - this enables Balanced PWM, which is SoC default
• gpio pwmr 100 - this sets the range to 100, or 0.1ms per unit
• gpio pwm 1 - this sets the duty cycle to 1, or 0.1ms out of 10ms

Keep in mind it is possible I am hideously wrong at this stage, I don't own an oscilloscope and with that in mind I am a cretin.  I frequently used PicoScope in Electronics class in College so that'll be my first port of call.  I believe I also need to vary the frequency to change the speed, not the duty cycle; maybe that'll need a dedicated board...

In other news I have finished my Projector Mount for the SLA 3D Printer, images below:

Frontal View, the width of the frame (sans handles) is 580mm.

Isometric View.

Rear view, showing the main frame.

A side-on shot showing the guts of the traverser.

I'm slightly doubtful about the 8mm steel rods I'm using to support a Projector which may weigh up to 5kg, so let's do a few Beam Bending Equations :P

Let's assume that the projector is positioned dead centre of the supporting rods in both axes, and that it weighs 5kg, it's centre of mass is spot-on in the middle of the Traverse Plate and 100mm above it (to account for an early, heavy and highly bulky DLP projector).  These are the crucial dimensions (for moment calculations):

• Horizontal Rods are 560mm long, so 280mm is the moment arm
• Vertical Rods are 565.5mm long, gives a moment arm of 282.75mm
• There is 62mm from the Horizontal Rods to the Traverse Plate
• This falls to 13mm with respect to the Vertical Rods
• On the Traverse Plate there is 170mm between the centres of the bearings (for the vertical axis)
• 317mm is the distance between the Horizontal Bearings

In order to find the deflection (which will cause uneven projection distance and hence ruin our prints), we need to find the moment arms and hence the force experienced by the beams at their various points; we know the Mass of the Projector is 5kg, this multiplied by 9.81m/s² gives us 49.05N of force, which for the sake of simplicity we will disregard for the vertical axis for now, and focus on it's effect on the horizontal axis.

Mooching around on the internet found me these highly useful formulae:

Thanks to Andy Ruina of Cornell University for this PDF so I didn't have to format this pile of symbols

The formula at the end is of greatest interest to us, and the symbols are as follows:

• P = the deflecting force:  what we are hunting for
• l = the length of the beam in question
• E = Young's (Elastic) Modulus = ~200GPa for Mild Steel
• I = Second Moment of Area, details below

There are many formulas for the Second Moment of Area, as it's highly dependent on the cross-section of the beam in question:  we'll be using (Pi*r⁴)/4 since that corresponds to a solid cylinder.  Our cylinders are 4mm in radius so (3.14*0.004⁴)/4 = 2.011x10^-10 m⁴ , there being two rods so double that to 4.021x10^-10 m⁴

With this information gathered, we can deduce that the denominator of this fraction is 48*200,000,000,000*4.021x10^-10 = 3860.39 GPa.m⁴ (Dimensions will be useful later); the numerator will be 49.05*0.560³ = 8.61N.m³, therefore we get a δ_max of 0.00223m or 2.23mm down in the centre.  Keep in mind this doesn't take moments into account nor does it account for the bearing spacings.

Wow, this really is looking like 200 steps!  I didn't expect to go back into my university course this soon, but it did highlight the fact that I may need to review my reliance of Reprap-derived mechanisms especially since I don't want my threaded rods to take on an undue amount of mechanical strain.  I'll end it there for today and continue with this lark at some other point, just because I'm tired of entering HTML mode to put in ^{(Superscript text here)} constantly.  Time for a kip...