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12.4 Example Calculations

Example calculations

The following example calculations illustrate the procedure for handling some typical problems.

Example 1: Equivalent Load P per Roller

Assumptions:

  • Linear guides type R 6
  • AC 6 cage with 8 rollers (RA = 8)
  • Load F = 350 N
  • Lever arm distance X = 120 mm
Example 1 mechanical diagram - single guideway configuration
For AC 6 type roller cage, the following applies:

Step 1: Calculate load-bearing length Kt

Kt = (RA – 1) · t = (8 – 1) · 9 = 63 mm

Step 2: Determine correction factor

Rtmin = 1 roller (according to chapter 5.1 AC 6 cage technical specifications)

C = 530 N (max. permissible load capacity per roller)

Step 3: Calculate equivalent load P per roller

P = (F · X) / (Kt · 2) · (1 / Rtmin)
P = (350 · 120) / (63 · 2) · (1 / 1)
P = 42,000 / 126 = 334 N

✓ P (334 N) is less than C (530 N), therefore this design is correct.

Example 2: Equivalent Load P per Roller (20 Rollers)

Assumptions:

  • Linear guides type R 6
  • AC 6 cage with 20 rollers (RA = 20)
  • Load F = 6,500 N (acting vertically on dual guideway system)
  • C = 530 N (max. permissible load capacity per roller)
Example 2 dual guideway configuration
Calculation Steps:

Step 1: Calculate number of load-bearing rollers

Since the load acts vertically, each guideway in the dual system bears half the load.

RT = RA/2 = 20/2 = 10 rollers

Step 2: Calculate equivalent load P per roller

P = F · (1/2) · (1/RT)
P = 6,500 · (1/2) · (1/10)
P = 3,250 / 10 = 325 N

✓ P (325 N) is less than C (530 N), therefore this design is correct. Safety factor = 530 / 325 ≈ 1.63

Example 3: Equivalent Load P per Ball

Assumptions:

  • Rigid cage structure
  • Linear guides type R 6
  • AK 6 cage with 12 balls (RA = 12)
  • Cage division t = 9 mm
  • Load F = 240 N
  • Lever arm distance X = 75 mm
  • C = 65 N (max. permissible load capacity per ball)
Example 3 ball cage configuration
Calculation Steps:

Step 1: Calculate load-bearing length Kt

Kt = (RA – 1) · t = (12 – 1) · 9 = 99 mm

Step 2: Determine correction factor Rtmin

According to the Rtmin calculation chart in chapter 12.3:

  • Structure type: Rigid structure (A)
  • X/Kt = 75/99 = 0.76 (less than 1)
  • From chart: Rtmin ≈ Rt/2
Rt = RA/2 = 12/2 = 6 balls
Rtmin = 6/2 = 3 balls

Step 3: Calculate equivalent load P per ball

P = (F/Kt) · (X/2) · (1/Rtmin)
P = (240/99) · (75/2) · (1/3)
P ≈ 30.3 N

✓ P (30.3 N) is less than C (65 N), therefore this design is correct. Safety factor = 65 / 30.3 ≈ 2.15

Example 4: Equivalent Load P for RNG Guideways

Assumptions:

  • RNG type linear guideway
  • KBN cage with 10 rollers (RA = 10)
  • Load F = 15,000 N
  • Lever arm distance X = 50 mm
  • Linear guideway center distance Q = 100 mm
Example 4 RNG guideway configuration
Calculation Steps:

Step 1: Calculate number of load-bearing rollers RT

RT = RA/2 = 10/2 = 5 rollers

Step 2: Calculate equivalent load P₁ from eccentric load

P₁ = F · (X/Q) · (1/RT)
P₁ = 15,000 · (50/100) · (1/5) = 1,500 N

Step 3: Calculate equivalent load P₂ from vertical load

P₂ = F/RA = 15,000/10 = 1,500 N

Step 4: Calculate total equivalent load P

P = P₁ + P₂ = 1,500 + 1,500 = 3,000 N

Step 5: Select appropriate KBN cage size

  • KBN 4: C = 850 N < 3,000 N ✗
  • KBN 6: C = 1,800 N < 3,000 N ✗
  • KBN 9: C = 3,900 N > 3,000 N ✓

✓ Select KBN 9 cage, C = 3,900 N > P = 3,000 N. Safety factor = 3,900 / 3,000 ≈ 1.30

KBN Cage Dimensions and Load Capacity

Type Size Roller Diameter Dw (mm) Division t (mm) Width w (mm) Load Capacity C per Roller (N)
KBN 4 4.5 6.5 approx. 4 850
6 6.5 8.5 approx. 5 1,800
9 9 12 approx. 7.5 3,900
12 12 15 approx. 9 6,500

Example 5: Equivalent Load P per Needle

Assumptions:

  • Linear guideway type N/O 2025
  • SHW 15 cage with length K = 194 mm
  • Cage end width w = 2.9 mm
  • Needle division t = 4 mm
  • Load F = 5,000 N
  • Lever arm distance X = 280 mm
  • Guideway center distance Q = 75 mm
  • C = 750 N (max. permissible load capacity per needle)
Example 5 needle guideway configuration
Calculation Steps:

Step 1: Calculate total number of needles RA

RA = [(K - 2w + 1)/t] · 2
RA = [(194 - 5.8 + 1)/4] · 2 ≈ 96 needles

Step 2: Calculate number of load-bearing needles Rt

Rt = RA/2 = 96/2 = 48 needles

Step 3: Calculate equivalent load P per needle

P = F · (X/Q) · (1/Rt)
P = 5,000 · (280/75) · (1/48) ≈ 388.3 N

✓ P (388.3 N) is less than C (750 N), therefore this design is correct. Safety factor = 750 / 388.3 ≈ 1.93

Example 6: Equivalent Load P per Roller (Longitudinal and Lateral)

Assumptions:

  • Rigid structure
  • Linear guides type R 12
  • Cage type AC 12, length K = 400 mm
  • F = 2,000 N
  • X = 500 mm
  • X1 = 200 mm
  • Q = 100 mm
  • C = 2,500 N (see chapter 5.1 AC 12 cage technical specifications)
Example 6 R 12 guideway configuration
For the roller cage AC 12 the following applies:

Calculate load-bearing length Kt

Kt = K - 2w = 400 - 22 = 378 mm

Calculate total number of rollers RA

RA = Kt/t + 1 = 378/18 + 1 = 22 rollers

Calculate number of load-bearing rollers Rt

Rt = RA/2 = 22/2 = 11 rollers
Calculation for P per roller:

Load laterally

PQ = (F · X1)/Q · (1/RTQ)
PQ = (2,000 · 200)/100 · (1/11) = 364 N

Load longitudinally

PL = (F · X)/(Kt · 2) · (1/RTL)
PL = (2,000 · 500)/(378 · 2) · (1/2) = 662 N

Total equivalent load

P = PQ + PL = 364 + 662 = 1,026 N

✓ P is smaller than C. The design is correct.

Example 7: Equivalent Load P for Recirculating Unit

Assumptions:

  • Recirculating unit type SR 6-100
  • Linear guideway type R 6
  • Number of recirculating units Rt = 2 (dual configuration)
  • Load F = 6,000 N
  • C = 2,150 N (max. permissible load capacity per recirculating unit)
Example 7 recirculating unit configuration
Calculation Steps:

Step 1: Calculate single-side load

Fside = F/2 = 6,000/2 = 3,000 N

Step 2: Calculate equivalent load P per recirculating unit

P = Fside/Rt = 3,000/2 = 1,500 N

✓ P (1,500 N) is less than C (2,150 N), therefore this design is correct. Safety factor = 2,150 / 1,500 ≈ 1.43

Example 8: Moment Load M Longitudinally and Laterally

Assumptions:

  • Recirculating unit type SR 6-150
  • Linear guideway type RD 6
  • Load F = 2,000 N
  • Lever arm distance X = 45 mm
  • ML = 112 Nm (permissible moment load longitudinally and laterally)
Example 8 moment load diagram
Calculation Steps:

Step 1: Convert lever arm distance to meters

X = 45 mm = 0.045 m

Step 2: Calculate actual moment load M

M = F · X = 2,000 N · 0.045 m = 90 Nm

Step 3: Verify moment load is within permissible range

  • Actual moment load: M = 90 Nm
  • Permissible moment load: ML = 112 Nm
  • M < ML ✓

✓ Moment load M (90 Nm) is less than permissible load ML (112 Nm), therefore this design is correct. Safety factor = 112 / 90 ≈ 1.24

Example 9: Equivalent Loads PL and PQ

Assumptions:

  • Recirculating unit type NRT 26 111 (top)
  • Recirculating unit type NRT 19 077 (bottom and side)
  • C = 98,000 N (top recirculating unit)
  • C = 43,000 N (bottom and side recirculating units)
  • K = 700 mm
  • Kt = 450 mm
  • Rtmin = 0.5
  • F = 83,000 N
  • X = 500 mm
  • Y = 100 mm
Example 9 recirculating unit configuration
Calculation Steps:

Load longitudinally

PL = (F · X)/(K · 2) · (1/Rtmin)
PL = (83,000 · 500)/(700 · 2) · (1/0.5) = 59,286 N

Load laterally

PQ = (F · Y)/(Kt · 2) · (1/Rtmin)
PQ = (83,000 · 100)/450 · (1/0.5) = 36,889 N
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