Ball Screw Selection and Calculations
ME EN 7960 – Precision Machine Design
Topic 4
ME EN 7960 – Precision Machine Design – Ball Screw Calculations
4-1
Ball Screw Selection Criteria
Maximum Rotational
Speed
Resonance (bending)
of threaded shaft
DN value
Applied Load
Thrust force
Required torque
Ball Screw
Selection
Ball Screw Life
Basic dynamic load
rating
Accuracy
Stiffness
ME EN 7960 – Precision Machine Design – Ball Screw Calculations
4-2
1
Based on Load
• A ball screw transforms rotational motion into
translational motion. As a result, the shaft is subject to
loads:
– Thrust force (the sum of all external forces such as machining
load, gravity, friction, inertial forces, etc.).
– Torque required to generate the thrust force.
ME EN 7960 – Precision Machine Design – Ball Screw Calculations
4-3
Driving Torque to Obtain Thrust
Fl
T= a
2πη
T:
Fa:
l:
η:
driving torque [Nm]
thrust force [N]
screw lead [m]
efficiency
Source: THK Co., Ltd.
ME EN 7960 – Precision Machine Design – Ball Screw Calculations
4-4
2
Required Thrust
• The thrust is the sum of all forces acting in the axial
direction.
Fa = FM + F f + Fi + Fg
FM:
Ff:
Fi:
Fg:
Machining force [N]
Frictional force [N]
Inertial force [N]
Gravitational force [N]
Source: THK Co., Ltd.
ME EN 7960 – Precision Machine Design – Ball Screw Calculations
4-5
Stresses from Applied Loads
σ axial =
Fa
πrtr2
τ torsional =
2T
πrtr3
The equivalent (Von Mises) stress:
2
2
σ eq = σ axial
+ 3τ torsional
→ σ eq =
4 Fa
12l 2
+
1
πd tr2
π 2 d tr2η 2
σmax:
Permissible stress [147 MPa]
ME EN 7960 – Precision Machine Design – Ball Screw Calculations
4-6
3
Graphic Solution
Von Mises Stress in Shaft (Fa = 2344N)
8
3 .10
1mm lead
2mm lead
4mm lead
5mm lead
8
Von Mises Stress [Pa]
2.5 .10
8
2 .10
8
1.5 .10
max
= 147 MPa
8
1 .10
7
5 .10
0
0.001
0.002
0.003
0.004
0.005
0.006
0.007
Root Diameter [m]
0.008
0.009
1mm lead -> dtr > 4.6mm
2mm lead -> dtr > 4.8mm
4mm lead -> dtr > 5.2mm
5mm lead -> dtr > 5.5mm
ME EN 7960 – Precision Machine Design – Ball Screw Calculations
4-7
Permissible Compressive Load
• Buckling Load
P1 =
λπ EI
2
lb2
P1:
lb:
E:
I:
λ:
Buckling load [N]
Distance between mounting positions [m]
Elastic modulus [Pa]
Second moment of inertia [m4]
Support factor
Fixed – free:
λ = 0.25
Fixed – supported:
λ = 2.0
Fixed – fixed:
λ = 4.0
ME EN 7960 – Precision Machine Design – Ball Screw Calculations
4-8
4
Fixed-Free Mount
Source: THK Co., Ltd.
Inexpensive but only applicable for short ball screws and/or slow speeds.
ME EN 7960 – Precision Machine Design – Ball Screw Calculations
4-9
Fixed-Supported Mount
Source: THK Co., Ltd.
Most commonly used mounting setup.
ME EN 7960 – Precision Machine Design – Ball Screw Calculations
4-10
5
Fixed-Fixed Mount
Source: THK Co., Ltd.
Overconstrained mounting setup for applications where high stiffness,
accuracy, and high shaft speed is required. Ball screw needs to be prestretched to avoid buckling in the case of thermal expansion.
ME EN 7960 – Precision Machine Design – Ball Screw Calculations
4-11
Basic Static Load Rating Coa
– When ball screws are subjected to excessive loads in static
condition (non rotating shaft), local permanent deformations are
caused between the track surface and the steel balls.
– When the amount of this permanent deformation exceeds a
certain degree, smooth movement will be impaired.
Coa ≥ f s Fa
Coa:
f s:
Fa:
Basic static load rating [N, kgf, lbf]
Static safety factor
Load on shaft in axial direction [N, kgf, lbf]
Use conditions
fs (lower limit)
Normal operation
1.0 – 2.0
Operation with impacts and vibrations
2.0 – 3.0
ME EN 7960 – Precision Machine Design – Ball Screw Calculations
4-12
6
Permissible Speed
• When the speed of a ball screw increases, the ball screw
will approach its natural frequency, causing a resonance
and the operation will become impossible.
nc =
60λ2
2
2πlb
EI
ρA
15λ2 d tr
=
2πlb2
E
ρ
nc:
lb:
E:
I:
ρ:
A:
λ:
Critical speed [min-1]
Distance between supports [m]
Elastic modulus [Pa]
Second moment of inertia [m4]
Density [kg/m3]
Root cross sectional area [m2]
Support factor
Fixed – free:
λ = 1.875
Supported – supported:
λ = 3.142
Fixed – supported:
λ = 3.927
Fixed – fixed:
λ = 4.730
ME EN 7960 – Precision Machine Design – Ball Screw Calculations
4-13
Spindle Speed and DN Value
• Shaft speed
n=
va
l
n:
va:
l:
Revolutions per second [s-1]
axial speed [m/s]
lead [m]
D:
N:
Ball circle diameter [mm]
Revolutions per minute [min-1]
• DN Value. Unless specified
otherwise:
DN ≤ 70000
ME EN 7960 – Precision Machine Design – Ball Screw Calculations
4-14
7
Dynamic Load Rating Ca and Life
• The basic load rating Ca is the load in the shaft direction
with 90% of a group of the same ball screws operating
individually will reach a life of 106 (1 million) revolutions.
3
⎛ C ⎞
L = ⎜⎜ a ⎟⎟ ×106
⎝ f w Fa ⎠
L:
Ca:
fw:
Fa:
Rotation life [rev]
Basic dynamic load rating [N, kgf, lbf]
Load factor
Load in shaft direction [N, kgf, lbf]
Use conditions
fw
Smooth movement without impacts
1.0 – 1.2
Normal movements
1.2 – 1.5
Movement with impacts and vibrations
1.5 – 2.5
ME EN 7960 – Precision Machine Design – Ball Screw Calculations
4-15
Example
•
•
•
•
•
•
•
•
Mass of axis: 350kg
Maximum velocity: 20m/min
Acceleration time: 0.05s
Bearing friction factor: 0.003
Machining force: 500N
Length of work piece: 500mm
Length of travel at maximum speed: 100mm
Orientation of axis: horizontal
ME EN 7960 – Precision Machine Design – Ball Screw Calculations
4-16
8
Machining Profile of Making a Slot
Machining speed
Maximum speed
v
Work piece
Variable speed
vmax
v0 = 0
v1 = vm
ax
v2 = vm
ax
v3 = vm
v4 = vm
vm
x
v5 = 0
v6 = −vm
-vm
v7 = −vm
v8 = −vm
ax
vr = −vm
ax
v1 0 = 0
-vmax
ME EN 7960 – Precision Machine Design – Ball Screw Calculations
4-17
Load Profile for Making a Slot
0 1
2 3
4 5 6
7 8
910
V,F
vmax
v0 = 0
Fa1 = Fi + F f = 2344N
v1 = vmax
Fa 2 = F f = 10 N
v2 = vmax
v3 = vm
t v4 = vm
vm
v5 = 0
-vm
v6 = −vm
v7 = −vm
v8 = −vmax
vr = −vmax
v10 = 0
Fa 3 = − Fi + F f i = −2324N
Fa 4 = Fm + F f = 510 N
Fa 5 = F f = 10 N
Fa 6 = − F f = −10 N
Fa 7 = − Fm − F f = −510 N
Fa 8 = − Fi − F f = −2344N
Fa 9 = − F f = −10 N
Fa10 = Fi − F f = 2324N
-vmax
1
2
3
4
5 6
7
8
9
10
ME EN 7960 – Precision Machine Design – Ball Screw Calculations
4-18
9
Running Lengths Depending on Usage
Running distance during acceleration:
laccl =
(v1 + v2 )taccl
2
(v1 + v2 )tdecl
Running distance during deceleration: ldecl =
2
ME EN 7960 – Precision Machine Design – Ball Screw Calculations
4-19
Mean Axial Force
Determine mean axial load in the positive direction by collecting all individual,
positive axial loads.
Fa ,mean + = 3
∑F l
∑l
3
ai + i +
i+
i
i
Determine mean axial load in the negative direction by collecting all individual,
negative axial loads.
Fa , mean − = 3
∑F
∑l
3
ai − i −
i−
l
i
i
Determine mean axial load:
Fa ,mean =
Fa ,mean + + Fa ,mean −
2
ME EN 7960 – Precision Machine Design – Ball Screw Calculations
4-20
10
Load Profile Based on Utilization
Mean axial force:
Fa ,mean = 3
3
3
Fm3lm + Funi
luni + Faccl
laccl
lb
Fm:
Funi:
Faccl:
lm:
luni:
laccl:
lb:
Machining force
Force at constant velocity (not machining)
Maximum force during acceleration and deceleration
Total travel per cycle during machining
Total travel per cycle at constant velocity
Total travel per cycle during acceleration and deceleration
Length of ball screw
Total travel length:
lb = lm + luni + laccl
Travel length:
lm = qm lb
luni = quni lb
laccl = qaccl lb
ME EN 7960 – Precision Machine Design – Ball Screw Calculations
4-21
Load Profile Based on Utilization (contd.)
qm + quni + qaccl = 1
Utilization:
qm:
quni:
qaccl:
Percentage per cycle spent machining (typically 0.5 – 0.9)
Percentage per cycle spent at constant velocity (typically 0.05 – 0.45)
Percentage per cycle spent during acceleration and deceleration (typically 0.05 – 0.1)
ME EN 7960 – Precision Machine Design – Ball Screw Calculations
4-22
11
Dynamic Load Rating Ca and Life
• When the rotation life L has been obtained, the life time
can be obtained according to the following formula if the
stroke length and the operation frequency are constant:
Rotation life [rev]
Life time [hr]
mean rotational speed [min-1]
L:
Lh:
nm :
L
Lh =
60nm
∑ (n l )
=
∑l
i i
nm
i
i
i
ni:
l i:
rotational speed at phase i [min-1]
distance traveled at phase i [m]
ME EN 7960 – Precision Machine Design – Ball Screw Calculations
4-23
Axial Rigidity
ME EN 7960 – Precision Machine Design – Ball Screw Calculations
4-24
12
Axial Rigidity
Fixed-free
Fixed-supported
1 1
1
1
1
= +
+ +
k ks k N kB kH
Fixed-fixed
1 1
1
1
1
= +
+
+
k k s k N k B1 + k B 2 k H 1 + k H 2
k:
k s:
kN:
kB:
kH:
Axial rigidity of linear motion system [N/m]
Axial rigidity of screw shaft [N/m]
Axial rigidity of nut [N/m]
Axial rigidity of support bearing [N/m]
Axial rigidity of support housing [N/m]
ME EN 7960 – Precision Machine Design – Ball Screw Calculations
4-25
Ball Screw Selection Procedure
• Axial rigidity of shaft
Fixed-free and fixed-supported:
ks =
AE
L
Fixed-fixed:
AEL
ab
4 AE
k s ,min =
L
ks =
ME EN 7960 – Precision Machine Design – Ball Screw Calculations
4-26
13
Ball Screw Accuracy
• Applicable if used in combination with rotary encoders.
Source: THK Co., Ltd.
ME EN 7960 – Precision Machine Design – Ball Screw Calculations
4-27
Overall Error
• The overall error is the summation of a number of individual errors:
–
–
–
–
Non-uniformity of threaded shaft.
Resolution/accuracy of rotary encoder.
Axial compliance of ball screw assembly.
Torsional compliance of threaded shaft.
δa = δS +
δa:
δs:
l:
Nrot:
Fa:
32Tl l
F
l
+ a + 4b⋅
N rot k axial πd tr G 2π
overall linear error [m]
ball screw non-linearity [m]
lead [m]
resolution of rotary encoder
[pulses/rev]
thrust force onto system [N]
kaxial:
Τ:
lb:
dtr:
G:
overall linear stiffness [N/m]
applied torque [Nm]
ball screw length [m]
root diameter of shaft [m]
shear modulus [Pa]
ME EN 7960 – Precision Machine Design – Ball Screw Calculations
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