Ball Screw Selection

Metric_1801-1886

3/1/06

10:29

Page 1804

Selection of Ball Screws 1

’Technical Calculations»

I Reference Formulas

I Example of Selection of Ball Screws (For X-Axis of Orthogonal Robot) Conditions of Use KWork&Table Weight KMaximum Stroke KThreading Speed KAcceleration Constant KPositioning Precision KRepeat Accuracy KLife Span KDirect Acting Guide Coefficient of Friction KDriving Motor KDuty Cycle Model Diagram V a/s

Work

W=50(kg) Smax.=720(mm) Vmax.=1000(mm/s) t=0.15(s) Ú0.1/720(mm) Ú0.01(mm) Lh=30000 hours O=0.02 Nmax.=3000(r/min)

240a(Stroke)

Table

Coupling

Starting Motor

(

P m=

720a 1000a/s

Distance Between Supports (Critical Speed:Fixed-SupportR2)

6Buckling Load(Pk)Derived with Euler's Equations of Motion

1Mean Axial Load (Pm)2Mean Revolution Frequency(Nm) (t 1 + t 2 + t 3 =100%) Revolution Frequency Operating Time Ratio Axial Load P1daN(Max.) N 1 r/min t1 % P2daN(Normal) N 2 r/min t2 % P3daN(Min.) N 3 r/min t3 %

) 1

P 1 N 1 t 1+ P 2 N 2 t 2+ P 3 N 3 t 3 3 (N)EEEEEEEEEEE 1 N 1 t 1+ N 2 t 2 + N 3 t 3 3

3

3

N 1 t 1+ N 2 t 2+ N 3 t 3 ( min-1 )EEEEEEEEEEEEE 2 N m= t1 + t2 + t3 If little difference is obtained between the maximum(P1) and

P k=

(N )EEEEEEEEEEEEEEEEEEEE 6 R2 Where; Pk :Load at Buckling Moment(daN) R :Distance between Points of Buckling Load (mm) E : Young's Modulus(2.06M104daN/mm2) I : Min. Geometrical Moment of Across Root Thread Area(mm4) Q I= d 4 64 d : Thread Root Diameter(mm) n : Coefficient Determined by Method of Screw Support Support-Support Fixed-Support Fixed-Fixed Fixed-Free

minimum(P3)axial l oads, o r if the load change is almost linear, an approximated value can be obtained with the following formula. 0.15 0.09 0.15 0.51

0.15

0.57

0.15 0.53

P m—

s

0.9 0.9M3=2.7

Distance Between Load Acting Points (Buckling Load:Fixed-FixedR1)

1.4 4.1(1 Cycle)

1.Setting Lead(L)

3.Allowable Buckling Load and Critical Speed

Set lead based on maximum motor revolutions and threading speed. Use the following formula :

Check overall length of thread(L), critical speed(Nc), and buckling load(Pk)as follows.

VmaxM60

=20(mm) Nmax 2.Calculating Basic Dynamic Load Rating L˘

L= Max. Stroke+Nut Length+Margin+Dimensions of Both Ends

Examines the required basic dynamic load rating and the Allowable Revolution Frequency (DN Value) Calculation of axial load for each operating pattern A In Acceleration Vmax Acceleration(Å)= M 10 -3 =6.7(m/s 2 ) t Axial Load(Pa)=(WCÅ+OCWCg)M10–1=34.5 (daN) (g:Gravitational Acceleration 9.8m/s2)

=720+62+60+78=920(mm) Check allowable axial load in terms of buckling load. Assuming the distance between load acting points R1=820, the following is obtained from formulas 6and 7on P.1086. P k =722(daN)

B At Constant Speed Axial Load (Pb)=OCWCgM10 =1 (daN) –1

This satisfies the conditions of use.

C In Deceleration Axial Load (Pc)= (WCÅ–OCWCg)M10–1=32.5 (daN) Operating Time(s)Per 1 Cycle for Each Operating Pattern(s) A

Operating Pattern Operating Time

B

0.60

0.84

C

Total Operating Time

0.60

To calculate critical speed, assuming the distance between supportsR2=790 the following is obtained from the formula5(Fixed-Support)on P.1086. Nc=3024(r/min)

2.04 This satisfies the conditions of use.

A

B

C

Axial Load

345N

10N

325N

Number of Revolutions

1500r/min

3000r/min

1500r/min

According to the tolerance values for lead accuracy(P.1086),

Oprating Time Ratio

29.4%

41.2%

29.4%

the class of positioning Precision Ú0.1/720mm is

Nm=2118(r/min)

Calculation of the required basic dynamic load rating (C) The actual life span in running(Lho), which excludes downtime, is as follows:

(

L ho =30000

)

2.04

=14927 (hrs)

4.1

4.Design Precision Investigate the degree of Precision and axial play

C5 (Accumulated typical lead errors=0.035 fluctuation=0.025) Axial play is max. 0.005 based on repeated positioning Precision ofÚ0.01 of the Selection of Ball Screws and Support Units 5. 5.Results From the calculations made, the best selection of the ball screw is Catalog No.

Insert the work factor(fw)=1.2 into the formula of deformation given in 3on P.1806,

(

C=

60L ho N m 10 6

) 1 3

The result is BSS1520. Next, look at the DN values(P.18064)as the Allowable Revolution Frequency.

1805

BSS1520-950-FC18

MPmMfw=370(daN)

and select a fitting ball screw from P.1807~.

For a DmN¯70000, DmN=15.8M3000=47400 This is within the Allowable range: therefore, proceed to the following investigation using this size of ball screw.

7Allowable Axial Load(P)for Buckling Load

3Life Span Hours(hrs) Lh =

106 60Nm

(

) 3

C Pmfw

(hrs)EEEEEEEEEEEEEE 3

Where; Lh :Life Span Hours(hrs) C :Basic Dynamic Load Rating (daN) Pm :Mean Axial Load (daN) Nm :Mean Revolution Frequency(r/min) fw :Work Factor f w =1.0~1.2 Impactless Run Normal Run f w =1.2~1.5 Run with Impact f w =1.5~2.0 The basic dynamic load rating that satisfies the set lifespan hours is expressed by the following formula.

(

C=

60LhNm 10

6

This section provides a guide for selecting ball screw frictional characteristics and the driving motor.

I Friction and Efficiency Ball screw efficiency ˝ can be expressed in the following formulas; wherein O i s the coefficient of friction and ı i s the screw's lead angle. Variabls are determined through analysis of a dynamic model. K When rotational force is converted into axial force(forward action)

) 1 3

P=ÅP(N ) EEEEEEEEEEEEEEEEEEEEEEEE 7 k Where; Pk :Buckling Load (daN) Å: Safety Factor(Å=0.5) Depending on the required safety level, a higher safety factor may be needed.

Driving Torque

P mfw(N) ˝=

Setting life span hours longer than what is actually necessary only requires a larger ball screw, but also increases the price. In general, the following standards are used for life span hours: Machine Tools:20000hrs Automatic Control Equipment:15000hrs Industrial Machinery:10000hrs Measuring Instruments:15000hrs

DmN¯70000 (Precision Ball Screws) DmN¯50000 (Rolled Ball Screws)

EEEEEEEEEEEEEE 4

Where;

The best support unit is Catalog No.BSW12.

Nm:Max. Revolution Frequency(r/min)

Ball Dia. 1.5875 2.3812 3.175 4.7625 6.35

A Value 0.3 0.6 0.8 1.0 1.8

5Critical Speed(Nc) N c =f a

n= 1 n= 2 n= 4 n=0.25

(N)

Dm:Thread Outer Diameter(mm)-A Value

Operating Pattern

Pm=25(daN)

3

60 2

EI

2Q R2

ÇA

M102 (min-1) EEEEEEEEE 5

Where; R :Distance Between Supports(mm) fa :Safety Factor(0.8) E :Young's Modulus(2.06M104daN/mm2) I :Min. Geometrical Moment of Across Root Thread Area(mm4) Q I= d4 64 d : Thread Root Diameter(mm) Ç: Specific Gravity(7.8M10–6kg/mm3) A : Root Thread Section Area(mm2) Q A= d2 4  : Coefficient Determined by Method of Screw Support Support-Support Fixed-Support Fixed-Fixed Fixed-Free

=Q =3.927 =4.730 =1.875

1 -O tan ı 1 +O/tanı

EEEEEEEEEEEEEEEEEEE 1

K When axial force is converted into rotational force(reverse action) ˝' =

4Allowable Revolution Frequency(DN)

Load Conditions for a Lead of 20

Calculating the mean axial load(Pm) and the mean turns(Nm)by load conditions (P.18061, 2),

2P 1 + P 3

nQ2 EI

1 -O/tanı 1 +O tanı

EEEEEEEEEEEEEEEEEEE 2

I Load Torque The load torque(constant speed driving torque)required in drive source design(motors, etc.)is calculated as follows. KForward Action Torque required when converting rotational force into axial force PL T= (NCcm)EEEEEEEEEEEEEEEEEE 3 2Q ˝ Where; T : Load Torque(daNCcm) (daN) P : External Axial Load (cm) L : Ball Screw Lead ˝ : Ball Screw Efficiency(0.9) KReverse Action External axial load when converting axial force into rotational force 2Q T P= (N )EEEEEEEEEEEEEEEEEEEE 4 L ˝' Where; P :External Axial Load (daN) T

:Load Torque(daNCcm)

L

:Ball Screw Lead (cm)

˝ ':Ball Screw Efficiency(0.9) KFriction Torque Caused by Preloading This is a torque generated by preloading. As external loads increase, the preload of the nut is released and therefore the friction torque by preloading also decreases.

1806