Now that the general layout of the machine has been determined and the target force, speed, and power requirements for each axis have been calculated, the power transmission components for each axis can be designed. Lead screw length, diameter, and pitch will be selected to meet the requirements of each axis.

The HIWIN ballscrew catalog, like most bearing catalogs, contains plenty of engineering information on how to analyze your system and choose the right ball screw. This makes it a great design resource, even if you don’t buy from HIWIN! Besides looking at standard part sizes the main focus right now will be analyzing buckling and screw whip. Future analysis will examine other factors like stiffness.

## Screw Pitch Selection

From the previous analysis, it was determined that the maximum traverse rate when cutting would be about 300IPM. The NEMA 34 motors that I happen to have sitting in my garage can maintain the required 140W of power between about 300RPM and 1000RPM. Dividing the traverse rate by the motor speeds gives the range of acceptable screw pitches necessary to maintain the maximum traverse rate at peak power, in this case from **7.7 mm** to **25 mm**.

A similar analysis can be done on the Z axis. Recall that 60IPM was the highest projected speed. The NEMA 23 motors that I happen to have laying around my garage can easily maintain the required 40W of power from 250 RPM to 2000 RPM. This results in a range of acceptable screw pitches between** 0.8 mm** and **6.1 mm**.

Most of the cheap common ball screws available on the internet follow the DIN 69051 standard. This standard mostly uses 5mm and 10mm pitches, although it is not uncommon for manufacturers to ‘extend’ the product line to include 2 mm and 4 mm pitches for smaller diameter screws and 20mm pitches for larger diameter screws. Obviously 10mm is a good choice for the X and Y axes in this case (although 20 might allow for higher rapid speeds with lower acceleration rates). 4mm and 5mm screw pitches would both probably work fine for the Z axis, depending on what is more available in its diameter.

The great part about this selection technique, where all of the sizing is based on power instead of speed and torque, is that you end up with a range of acceptable gearing that all work the same because the motor is operating in its continuous power zone. Then, you can simply pick a close-by gear (or in this case, screw) out of a catalog and it will work. This cuts down on the number of times component selection and analysis needs to be iterated before the design is finalized.

## Screw Diameter Selection

The two cases driving a larger screw diameter are screw whip and buckling. Screw whip is the phenomenon where a screw tends to bow outwards when it is spun very fast due to the large centripetal forces. Buckling is the load case where a large length of the screw is under compression (even if not rotating) and may randomly bow outward. The HIWIN catalog has suggested analysis methods for these two cases when using the common fixed/free bearing support method.

For both load cases the length of the screw is an important input. The length of the screw is approximately equal to the desired range of motion of the axis, plus the spacing between bearings on that axis. Then, because screws are usually sold in even lengths of 100mm, rounded up to the next size. Here is the rough estimation of screw length for these analyses:

Axis | X | Y | Z |
---|---|---|---|

Desired Range of Motion (in) | 52 | 24 | 6 |

Estimated Bearing Spacing (in) | 8 | 12 | 8 |

Total Screw Length (mm) | 1524 | 914 | 356 |

Selected Screw Length (mm) | 1600 | 1000 | 400 |

### Screw Whip

The HIWIN catalog provides equations (below) for calculating the maximum permissive speed of a screw based on a selected diameter and screw length. However, in this case the maximum desired speed is already known and the minimum screw diameter needs to be computed.

The HIWIN equation can be reorganized to solve for the minimum root diameter of the screw:

\( d_r = \frac{N_p L_t^2}{0.8 \times 2.71 \times 10^8 \times M_f} \)The only input left to calculate is the maximum rapid speed. The rapid speed of each axis is limited only by the inertial loading (no cutting forces). Converting the inertial load into a motor torque using the selected screw pitch, allows you to look up the corresponding maximum motor speed at that output torque (with a safety factor) using the provided torque curve.

\(InertialLoad = Mass \times Acceleration\)\(TargetTorquePerMotor = SafetyFactor \times \frac{InertialLoad \times ScrewPitch}{2 \pi NumberOfMotors}\)\(SpeedOfAxis = SpeedOfMotor \times ScrewPitch \)Axis | X | Y | Z |
---|---|---|---|

Inertial Load (N) | 350 | 850 | 400 |

Number of Motors | 1 | 2 | 1 |

Screw Pitch (mm) | 10 | 10 | 4 |

Torque Required per Motor (Ncm) | 56 | 68 | 25 |

Safety Factor | 2 | 2 | 2 |

Target Torque per Motor (Nm) | 112 | 136 | 50 |

Max Rapid Speed at Motor (RPM) | 1125 | 975 | 1800 |

Max Rapid Speed of Axis (IPM) | 443 | 384 | 283 |

For the support style being used (fixed at the motor end, supported at the other end) the mounting factor is 0.689. Plugging all of this into the reorganized HIWIN equation from above equation yields the minimum required root diameter for each axis due to screw whip. It is no surprise that the X axis is the most heavily impacted due to the squared factor of screw length.

Axis | X | Y | Z |
---|---|---|---|

Screw Length (mm) | 1600 | 1000 | 400 |

Max Rapid Motor Speed (RPM) | 1125 | 975 | 1800 |

Minimum Rapid Root Diameter (mm) | 19.3 | 6.5 | 1.9 |

Max Cutting Motor Speed (RPM) | 768 | 768 | 381 |

Minimum Cutting Root Diameter (mm) | 13.2 | 5.1 | 0.4 |

What these two results show are the minimum root diameter needed to prevent the screw diameter form being the limiting factor for rapid or wood cutting use cases. Smaller root diameters can be selected, but then the max speed of the axis will need to be limited in software accordingly, and motor performance would be left unused.

### Buckling

The HIWIN catalog also provides equations for calculating the maximum acceptable compressive force before buckling occurs.

Similarly to the screw whip equation, this equation can be reversed to solve for a target screw diameter given a known axial compressive load.

\(d_r = \frac{2 \times F_p L_t^2}{40720 \times N_f}\)For the support style being used (fixed at the motor end, fixed at the ball nut) the mounting factor used is 1.0. The expected load is simply the maximum torque of the motor (probably at standstill) applied through the pitch of the screw.

\(Force = \frac{2 \pi Torque}{Pitch}\)Axis | X | Y | Z |
---|---|---|---|

Maximum Torque (Ncm) | 850 | 850 | 200 |

Screw Pitch (mm) | 10 | 10 | 4 |

Maximum Force (N) | 4000 | 4000 | 2400 |

Screw Length (mm) | 1600 | 1000 | 400 |

Minimum Root Diameter (mm) | 16.2 | 12.8 | 7.1 |

The X axis ended up with a similar minimum diameter as the screw whip case (again, the squared length term was quite significant), but the Y and Z axes were significantly more limited by buckling than screw whip.

One trick to reduce the buckling loads is to limit the amount of torque produced by the motor at low speeds without reducing the total power output at higher speeds. Most stepper motor drivers target a maximum specified current, which is roughly proportional to torque at low speeds. However, past a certain speed motors become **voltage limited** not **current limited**, so an artificial current limit is not relevant at those high speeds.

The tradeoff when limiting current artificially across all speed ranges (the ‘simple’ way) is that the beginning of the ‘constant power’ regime is pushed up to a slightly higher motor speed. In the case of my specific motor, limiting the motor to 75% current barely impacts the beginning of the constant power zone, but a 50% limitation pushes the beginning of that zone past the 300RPM point used in the screw pitch selection (although not by much).

Losing the low-speed high torque capability of the motors is not actually a downside for this design because the limiting factor was always high speed cutting. In fact, the high torque at low traverse rates really only hurts the design because it is an extreme load case that must be protected against! For a CNC router the ideal motor curve would be constant torque at low RPMs, then sharply transition into a constant power zone (similar to the 50% current version of my NEMA 34 motor curve).

### Selection

Recalculating the minimum root diameter due to buckling with various current limited torques and comparing to the screw whip limitation provides a great chart for selecting the desired screw diameter.

Axis | X | Y | Z |
---|---|---|---|

Screw Whip, Rapid Motion (mm) | 19.3 | 6.5 | 1.9 |

Screw Whip, Wood Cutting (mm) | 13.2 | 5.1 | 0.4 |

Buckling, 100% Current (mm) | 16.2 | 12.8 | 7.1 |

Buckling, 75% Current (mm) | 15.1 | 11.9 | 6.6 |

Buckling, 50% Current (mm) | 13.6 | 10.8 | 6.0 |

Recall that all of this analysis was done based on the ‘root diameter’, which is the innermost part of the curved channel in the screw that the ball bearings contact (smaller than the ‘nominal’ diameter).

A selection of nominal and root diameters for various relevant screw sizes is listed below.

Nominal Diameter (mm) | Root Diameter (mm) |
---|---|

8 | 6.15 |

10 | 7.95 |

12 | 9.79 |

16 | 12.32 |

20 | 16.74 |

25 | 21.32 |

The X axis is a near perfect fit for a 20 mm screw, even at 100% current. The rapid speed will need to be reduced slightly, but the X axis already had more capability than the Y axis because of the lower inertial load per motor. Matching the y axis max rapid speed is likely within the capabilities of the 20 mm screw.

The Y axis can sneak just into the 16 mm screw’s capability by running at 75% current. Considering there are two of these screws in the system the cost savings of going down a size will definitely pay off!

Pushing the Z axis down to an 8 mm screw by limiting it to 50% current doesn’t seem worth the pain to me, especially when the cost difference between 8, 10, and 12 mm screws at only 400 mm long is pretty small. I will target a 12 mm diameter Z axis because it appears to be the most widely available.

## Design Confirmation

The previous analyses were done ‘in reverse’ – calculating the perfect, minimum pitch and diameter for known loads – in order to identify the proper parts from the catalog. Now, after selecting the exact dimensions from the catalog the calculations can be done ‘forward’ (as presented in the catalog’s engineering guidance) to find out how much margin the design has. Here is a review of the selected axis configuration:

Axis | X | Y | Z |
---|---|---|---|

Length (mm) | 1600 | 1000 | 400 |

Pitch (mm) | 10 | 10 | 4 |

Nominal Diameter (mm) | 20 | 16 | 12 |

Root Diameter (mm) | 16.74 | 12.32 | 9.79 |

Motor Configuration | 1 x NEMA 34 | 2 x NEMA 34 (75%) | 1 x NEMA 23 |

Recall that the maximum permissible motor speed due to screw whip is a function of the root diameter and length of the screw (see the above equation from HIWIN). In this case the margin of safety for screw whip is calculated as the difference between the target rapid speed of the motor and the max permissible speed of the motor, as a percentage (beyond 100%) of the target rapid motor speed.

\(MaxMotorSpeed = 0.8 \times 2.71 \times 10^8 \times \frac{0.689 RootDiameter}{Length^2}\)\(MarginOfSafety_{ScrewWhip} = \frac{MaxPermissibleSpeed – TargetRapidSpeed}{TargetRapidSpeed}\)Axis | X | Y | Z |
---|---|---|---|

Target Rapid Tool Speed (IPM) | 384 | 384 | 283 |

Target Rapid Motor Speed (RPM) | 975 | 975 | 1800 |

Max Permissible Speed From Screw Whip (RPM) | 977 | 1840 | 9139 |

Margin of Safety for Screw Whip | 0.2% | 89% | 408% |

The buckling strength of a screw is also a function of the screw length and diameter. In this case the margin of safety for buckling is calculated as the difference between the max expected load (adjusted for current limiting) and the max allowable load, as a percentage (beyond 100%) of the max expected load.

\(MaxAllowableLoad = 40720 \frac{1.0 RootDiameter^4}{Length^2}\)\(MarginOfSafety_{Buckling} = \frac{MaxAllowableLoad – MaxExpectedLoad}{MaxExpectedLoad}\)Axis | X | Y | Z |
---|---|---|---|

Max Torque Output per Motor (Ncm) | 850 | 638 | 205 |

Max Expected Load per Screw (N) | 5341 | 4006 | 3216 |

Max Allowable Compressive Load per Screw (N) | 6120 | 4597 | 11456 |

Margin of Safety for Buckling | 15% | 15% | 256% |

All analysis cases have positive margin! Although the X axis barely passes the screw whip case, and leaves some motor performance on the table. Now that the motors and screws have been picked out, the only analysis work remaining before detailed design can start is bearing selection and layout.