Differential screw for fine adjustments of precision equipment

Differential screw components

A differential screw is a mechanism that provides very fine motions of machine parts. There are several forms of its configuration.

The picture shows one common form of the differential screws. There are 3 main components as follows:

• Spindle (differential screw) -- The spindle has two different threads on the same axis.
• Base -- It's the base of the whole mechanism which has one threaded hole.
• Nut -- It has one threaded hole with sliding joint. This is the end mover where we will get fine motion. This part may be connected to other machine components to provide precise motion.

How it works

Different threads on the same spindle / distance between marks

The spindle has two different thread sizes. In this example, the larger one (thread A) has M12 coarse thread which has a pitch of 1.75 mm. Another thread (B) is M10 coarse thread which has a pitch of 1.5 mm.

Pitches (leads) of the differential screw

The pitch (or lead) of a screw is distance the screw advances when it turns one revolution. Therefore, when the handle turns one revolution, thread A rotates one revolution and moves in a distance equal to the pitch of thread A (1.75 mm). Since thread B is on the same spindle, it also moves together with thread A (1.75 mm) and also rotates one revolution. However, thread B connects to the nut which is unable to rotate. So, the nut retracts a distance equal to the pitch of thread B which is 1.5 mm. Hence, the motion of the nut is the advance distance of thread A minus the retracted distance. It is the difference between the pitch of threads. This is why it is called the differential screw.

Differential screw displacement formula

where:

• ΔS_nut = travelling distance of the nut (mm)
• L_A = pitch of thread A (mm)
• L_B = pitch of thread B (mm)
• Δθ_screw = number of turns of the screw (rev)

From this example, we have ΔS_nut = 1.75 - 1.5 = 0.25 mm. That means the nut travels 0.25 mm per each turn of the spindle. As shown in the above picture, the distance between 2 marks is 2.5 mm. Then we need to turn the spindle 10 revolutions so that the nut will travel 2.5 mm.

As we can see from the formula, if we need the nut to move 0.1 mm per one turn of the spindle, we need to select the different screw threads. Since we know that the standard metric coarse threads have the following values:

• M5, pitch = 0.8 mm
• M4, pitch = 0.7 mm

The difference between pitches is 0.8 - 0.7 = 0.1 mm which is as per the requirement. So, thread A will be M5 and thread B is M4 and we will get 0.1 mm per turn.

Watch the following video to see how it moves. We use Unigraphics NX4 motion simulation to show all motions.

Reference:

• Machines & Mechanisms Third Edition by David H. Myszka

A pull rod for position adjustment of a cam-driven mechanism

Kinematic diagram of a cam-driven mechanism

A cam-driven mechanism is commonly used in most production machines since all motions and timings can be controlled. Not only the displacement is controlled, but also the velocity and acceleration as well as jerk can be controlled. Cam-driven mechanism allows overlapping motion between machine parts since the positions of all relevant parts can be determined from the timing diagram which is desirable for high speed application.

A simple cam-driven mechanism consist of the following parts as shown in the kinematic diagram:

• Cam: for motion generation (displacement, velocity, acceleration and timing).
• Cam follower: rolling part mounted on a lever.
• Spring (not shown): to keep contact between cam surface and cam follower.
• Lever: to transfer continuous cam rotation to swinging motion.
• Pull rod: to transfer the motion from the lever to the slider.
• Slider: end equipment (processing equipment)

In this post, we're going to focus on the pull rod (also known as push rod or tie rod) which is one of the common parts for most machines. The pull rod allows position adjustment of its connected parts since its length can be adjusted. Normally, the pull rod consists of the following parts:

• Pull rod
• Rod end bearing RH thread
• Rod end bearing LH thread
• Nut RH thread
• Nut LH thread

Pull rod with both female rod end bearings

The pull rod usually made of a hexagonal post. The mechanic can use a wrench on the hexagonal part to tighten or loosen the pull rod from the rod end bearings. Rod end bearings must have RH thread on one side and LH thread on the other side otherwise the distance between the rod ends will remain the same.

Example of pull rods on the machine

Adjustment of the pull rod length usually happens when both sides of the rod end bearings are already connected to other parts in the machine (in this example, it is connected to the lever and the slider already). To adjust the length, no need to disconnect the rod end bearing, first we have to loosen both RH and LH nuts so that the pull rod can be turned. Then turn the pull rod in either direction and its length will change. By doing this, we can then adjust the position of the connected parts which, in the case, is the slider. After the slider is at the desired position, tighten both nuts.

The male rod end bearings version is also available. We can use the same nuts, but the hexagonal post will have threaded holes instead (see the following picture).

Pull rod with both male rod end bearings

The increment of the pull rod length (distance between both rod end bearings) is determined by the pitch of the thread on the rod end bearings. For this example, the M10 thread has a pitch of 1.5 mm. One turn of the pull rod will change the distance of each rod end bearing by 1.5 mm. Therefore, the increment is 2 times the pitch (2 x 1.5 = 3 mm/turn).

The pull rod length is increased or decreased according to the following directions.

Pull rod length extension and retraction according to the turning direction

The following is the animated picture showing how the slider position can be adjusted by turning the pull rod.

Animated picture of pull rod length adjustment

Watch the following video for how the cam driven-mechanism works and where the pull rod is used in the system. The simulation uses NX4 motion simulation module.

Example of rotary indexer sizing calculation for table plate drive application (3/3)

Step 4 : Rotary Indexer Model Selection

Minimum Follower Wheel Pitch Diameter (Credit : Sankyo)

• Recommended Size of Rotary Indexer can be estimated. By calculate Radius of Gyration divide by Follower Wheel Pitch Radius, this value should be less than 5 (this value may vary depend on difference manufacturer)

Recommended Follower Wheel Pitch Radius of Rotary Indexer

• Minimum Follower Wheel Pitch Radius, [PR_min] = 427/5 = 85.424 ≈ 85.4 mm
• Then, Minimum Follower Wheel Pitch Diameter, [PD_min] ≈ 85.4 x 2 ≈ 171 mm

• Recommended Size of Rotary Indexer also can be estimated by another method. By calculate Table Diameter divide by center distance between input & output shaft, this value should be less than 7 (this value may vary depend on difference manufacturer)

Distance Between Input & Output Shaft

Recommended Center Distance between Input & Output Shaft of Rotary Indexer

• Then, Minimum Distance Between Input & Output Shaft, [CD_min] = 1000/7 = 142.857 ≈ 143 mm
• Total Output Torque for Indexer Selection, [T_{t select}] = T_t x SF = 307.271 x 1.2 = 368.726 ≈ 369 N.m @ Input Shaft(camshaft) Speed 60 rpm

Thus, Select Indexer that Dynamic Rated Output Torque [T_op] is more than 369 N.m at input shaft speed more than 60 rpm, 8 stops, Indexing Angle 270 degree, MS motion curve, minimum center distance between input & output shaft 143 mm and minimum follower wheel pitch diameter 171 mm

Step 5 : Gear & Motor Selection
Find the maximum torque on gear output shaft at working shaft speed

• In this case, gear is directly connect to the rotary indexer. Then the maximum torque on gear output shaft is equal to the camshaft torque of rotary indexer
• The camshaft torque of rotary indexer[T_c] is determined by following formula:

Rotary Indexer Cam Shaft Torque

• Internal Indexer Inertia Torque, [T_oi] = 2.943 N.m (from rotary indexer manufacturer data)
• Camshaft Friction Torque, [T_x] = 16.677 N.m (from rotary indexer manufacturer data)
• Maximum Camshaft Torque Coefficient, [Q_m] = 0.987 (This is standard value, find the Qm for Modified Sine Cam Curve from Cam Curve Characteristic Table)
• Then, Camshaft Torque, [T_c] = 87.552 N.m @ camshaft speed 60 rpm (This is the torque at input shaft of rotary indexer which is equal to torque at output shaft of gear)

Next, calculate the equivalent camshaft torque for gear selection [T_{ce select}] by considering of operating condition & safety factor

• Calculated Operation Factor, [f] = 1.8 (Calculate from type of load(steady or shock load), operating hours per day, frequency of starts/stops, ambient temperature, type of lubrication or others factor. Please check gear manufacturer information. In this case use 1.8)
• Then, T_{ce select} = T_ce x f x SF = 87.552 x 1.8 x 1.2 = 189.113 N.m @ Output Shaft Speed 60 rpm

Calculate Gear Input Shaft Torque

• Gear Ratio, [i_g] = 10.33 (Check from gear manufacturer info.)
• Gear Input Shaft Speed, [N_g] = N_rpm x i_g = 60 x 10.33 = 619.8 ≈ 620 rpm
• Gear Running Efficiency, [Eff_g] = 92% (Check from gear manufacturer info.)
• Gear Input Shaft Friction Torque, [T_xg] = 0.9 N.m (Check from gear manufacturer info.)
• The Gear Input Shaft Torque[T_g] is determined by following formula:

Gear Input Shaft Torque

• Gear Input Shaft Torque, [T_g] = 10.113 N.m @ Input Shaft Speed 620 rpm

Calculate Motor Torque

• Motor Revolution per Minute, [N_motor] = 1730 rpm (from motor manufacturer data)
• Gear Ratio Required, [i_req] = N_motor / N_rpm = 1730 / 60 = 28.833
• Pulley Speed Ratio, [i_pulley] = I_req / I_g = 28.833 / 10.33 = 2.791 ≈ 2.8 (←if very close to 1, motor can be mounted directly to gear, then set Eff_pulley = 100% & T_xp = 0

Table Drive Application with Pulley (Credit : Sankyo)

• Pulley Running Efficiency, [Eff_pulley] = 90% (estimated)
• Pulley Shaft Friction Torque, [T_xp] = 2 N.m (estimated)
• The Motor Torque [T_motor] is determined by following formula:

Motor Torque

• Then, Motor Torque, [T_motor] = 10.003/(2.791 x 0.9) + 2 = 6.026 N.m @ Speed 1730 rpm

Calculate Motor Power

• The Peak Motor Power [P_{motor peak}] is determined by following formula:

• Then, The Peak Motor Power [P_{motor peak}] = 2 x 3.1416 x 1730 x 6.026 / 60 = 1091.617 Watt ≈ 1.092 Kw

Calculate Motor Power for Motor Selection

• Power for Motor Selection, [P_{motor sel}] = P_{motor peak} x SF = 1.092 x 1.2 = 1.310 Kw

Then,  select gear that maximum continuous output torque is more than 189 N.m at output shaft speed more than 60 rpm, gear ratio 10.33
Motor power 1.31 Kw at speed 1730 rpm
Ratio of pulley between gear and motor 2.8

• Example of Rotary Indexer Sizing Calculation for Table Plate Drive Application Part 1
• Example of Rotary Indexer Sizing Calculation for Table Plate Drive Application Part 2

Example of rotary indexer sizing calculation for table plate drive application (2/3)

Calculations

Rotary Indexer with Dial Plate Driving Application Torque Diagram

Step 1 : Moment of Inertia Calculations

• How to calculate the Moment of Inertia?
• Dial Plate Inertia, [I_d] = 1/2 x 43.040 x ((1000/1000)/2)^2 = 5.380 Kg.m²

Dial Plate Inertia

• Stations Inertia, [I_st] = 160 x ((450/1000)/2)^2 = 32.400 Kg.m²

Stations Inertia

• Additional Part Inertia, [I_{add part}] = 8 x ((300/1000)/2)^2 = 0.720 Kg.m²

Additional Parts Inertia

• If there are others group of additional parts, do the same method.
• Total Moment of Inertia, [I_total or ∑MK²] = 5.380 + 32.400 + 0.720 = 38.500 Kg.m²

Total Inertia

• GD² = 38.500 x 4 x 9.81 = 1510.739 N.m²

GD Square

• Total Moving Mass, [∑M] = 43.040 + 120 + 8 = 211.040 Kg.

Total Moving Mass

• Radius of Gyration, [K] = (I_total/∑M)^1/2 = 427.118 mm

Radius of Gyration

Step 2 : Maximum Angular Acceleration Calculation

Cam Curve Characteristic Value

• From the table, find the Dimensionless Maximum Angular Acceleration for Modified Sine Cam Curve, [A_m] = 5.53
• Then, the Maximum Angular Acceleration, [a_m] = 0.785 / 0.75^2 x 5.53 = 7.721 rad/sec²

Maximum Angular Acceleration

^{Step 3 : Required Torque Calculations}

Torque on Rotary Indexer

• Inertia Torque at Index Shaft, [T_i] = 38.500 x 7.721 = 297.271 N.m

• External Friction Torque to Indexer, [T_f] = 10 N.m (Such as outboard support bearings, assume to 10 N.m)
• Work Torque, [T_w] = 0 N.m (Indexer is doing work such as lifting parts, assume to 0)
• Total Output Torque, [T_t] = 297.271 + 10 + 0 = 307.271 N.m

Total Torque at Output Shaft

• Example of Rotary Indexer Sizing Calculation for Table Plate Drive Application Part 1
• Example of Rotary Indexer Sizing Calculation for Table Plate Drive Application Part 3

Example of rotary indexer sizing calculation for table plate drive application (1/3)

Indexing System Information

Physical Properties Info.

Rotary Indexer with Dial Table Driving Application

Rotary Indexer with Dial Table Driving Application

Dial Table Info.

• Diameter, [D_dial] = 1000 mm
• Thickness, [Y] = 20 mm
• Material = Aluminum (Density 0.00274 g/mm³)

Mass of dial table

• Then Table Mass, [M_dial] = 43.040 Kg.

Station Info.

• Number of Stations, [N_stations] = 8 stations
• Mass per Station, [M_station] = 20 Kg/staion
• Then Total Station Mass, [SM_stations] = 8 x 20 = 160 Kg
• Rotating Radius to Station Center, [R_st] = 450 mm (If the station is large or complex shape, a radius of gyration of each station should be calculated.)

Additional Part Info. (The same part at the same rotation radius)

• Number of parts, [N_{add part}] = 4 parts
• Part Mass, [M_{add part}] = 2 Kg/part
• Total Additional Part Mass, [SM_addpart] = 4 x 2 = 8 Kg
• Rotating Radius to Part Center, [R_{add part}] = 300 mm (If the part is large or complex shape, a radius of gyration of each part should be calculated.)
• If there are others group of additional parts, do the same method.

Movement Info.

Rotary Indexer Movement Term

• Number of Stops, [S] = 8 stops

Displacement per 1 index

• Dial Plate Displacement per Index, [h_m] = 2 x 3.1416 / 8 = 0.785 rad
• Indexer Input Shaft Cycle, [T] = 1 sec (Assume to 1 second, this can be adjusted later)

Indexing Rate

• Indexing Rate, [N_rpm] = 60/1 = 60 rpm
• Indexing Period, [B_m] = 270 deg (Assume to 270 degree, should be taken as long as possible for smoother movement. Please check to the manufacturer information)

Indexing Time

• Indexing Time, [t_m] = 270/360 x 60/60 = 0.750 sec

Dwell Time

• Dwell Time, [t_dwell] = 1 - 0.750 = 0.250 sec
• Required Dwell Time, [t_{dwell required}] = 10 sec (This is actual required dwell time according to the working process.)
• In case of t_{dwell required} > t_dwell  we can use clutch and brake to extend the dwell period, but if t_{dwell required} < t_dwell that mean we can reduce t_dwell to be the same as t_{dwell required}
• Reducing t_{dwell required} can be consider in 2 cases. Reducing indexer input shaft cycle [T] or Increasing indexing period [B_m].
• This case,   t_{dwell required} > t_dwell

Motor stop time

• Input Shaft Stop Time, [t_stop] = 9.750 sec

Machine cycle time

• Machine Cycle Time, [Tmachine] = T + t_stop = 1 + 9.750 = 10.750 sec
• Machine Speed, [Speedmachine] = 3600/10.750 x 8/8 = 334.884 UPH

Machine Speed

Displacement VS Indexing Angle

• Cam Curve : Modified Sine (assumed, please check from rotary indexer manufacturer data)
• Safety Factor, [SF] = 1.2
• Operating Condition : 24 Hrs running, Steady load, Starts/stops 2 times per hour, Ambient temperature 30 C, Without cooling fan

• Example of Rotary Indexer Sizing Calculation for Table Plate Drive Application Part 2
• Example of Rotary Indexer Sizing Calculation for Table Plate Drive Application Part 3

Rotary indexer sizing calculation for table plate drive application

In this post we will explain how to calculate the size of rotary indexer, gear & motor for dial plate table application.

Rotary Indexer with Table Plate Driving Application

There are 2 things we need to know for the calculation

1. Physical properties info.
2. Movement info.

Physical properties information of movement part is data we need to use for calculate the total moment of inertia of the moving system. It consists of dial plate diameter, thickness, density for calculate mass, the number of stations, station mass, station radius to rotation center, etc.

Movement information is data we need for find the maximum angular acceleration of the moving system. It consists of number of stops, RPM of rotary indexer input shaft, indexing angle, dwell time, type of cam curve, etc.

After we know the value of the total moment of inertia and the maximum angular acceleration of the moving system, we can easily calculate the required torque for driving the system. Then we can select the rotary indexer that match the required torque.

For selection of gear, we can use this required torque at output shaft of indexer to calculate the required torque of input shaft of the indexer by using maximum camshaft torque coefficient, internal indexer inertia torque, camshaft torque from indexer manufacturer data, number of stops and indexing angle. When we know the torque required at indexer camshaft(which is output shaft torque of gear), we can select the gear.

For the motor selection, we have calculate the gear input shaft torque by converting gear output shaft torque with gear ratio. Then use this torque for calculate the motor power required.

Example of Rotary Indexer Sizing Calculation for Table Plate Drive Application

Indexers with single dwell (1-dwell) VS double dwell (2-dwell)

To configure a rotary indexer, there are several parameters to decide such as number of stops (S), indexing angle (b_m), maximum output torque, etc. In this post, we will show one more parameter which may affect the design if misunderstood. It's a number of dwell. In most cases, we use 1-dwell (single dwell) for the application. The indexer performs single index within the designated indexing angle and wait (dwell) until cycle complete.

However, for some indexer models, there will be the option to select 2-dwell cam type (double dwell, double indexes). Or there may be only 2-dwell version especially for the models which have large number of stops and long indexing angle e.g. S=20, b_m = 210 deg.

2-dwell indexer will perform differently from 1-dwell indexer though they're both having the same S and b_m. The 1-dwell indexer will index only one once per each full revolution of the input cam shaft. But the 2-dwell indexer will index 2 times and also stop 2 times per each revolution of the input cam shaft as can be seen in the following displacement diagram.

Displacement diagram of both 1-dwell indexer and 2-dwell indexer

The 2-dwell indexer divides displacement into 2 halves. The first half take half indexing angle (b_m/2) to rotate the output shaft to next station. The dwell angle is also divided by 2. It has 2 times displacement compared with the displacement of the 1-dwell cam as can be seen in the red line on the above chart.

In this post, we use indexers with following parameters for comparison.
They both have ...

• Number of stops, S = 12 stops. So the displacement (h_m) becomes h_m = 360/12 = 30 deg.
• Indexing angle, b_m = 210 deg.
• Same input shaft speed (w)

1-dwell and 2-dwell rotary indexers with S = 12 and b_m = 210 deg.

According to the explanation of the 2-dwell cam motion, the second indexer will perform 2 indexes and 2 dwells per 1 turn of the input shaft. The actual indexing angle will become b_m/2 = 210/2 = 105 deg. Therefore, the 2-dwell indexer takes 105 degrees to complete the first index with output shaft displacement (h_m) of 30 deg. Then it waits (dwell) until the input shaft angle reaches 180 deg. and it restart the next indexing from 180 deg. to 180 + 105 = 285 deg. After that, it waits until the input shaft complete its turn. Then the next cycle starts...

Here is the animation of how both indexers move.

How 1-dwell indexer and 2-dwell indexer move

Watch the following video for the animation made with Unigraphics NX4 motion simulation.