B-2-3 Permissible Rotational
Speed
* Please consult NSK if the maximum rotational
speed exceeds the criteria of maximum
rotational speed on page B50, even both the
d·n value are in range of the allowable limit.
B-2-3.1 Critical Speed of the Screw Shaft
Calculate the critical speed which is the
matching value of the ball screw rotational
speed and the natural frequency of the screw
shaft. The 80% of the critical speed is defined as
the permissible rotational speed.
Calculate the critical speed of the screw shaft
whether you use shaft rotation or nut rotation.
Critical speed varies by the nut traveling
position. Please consult NSK for detailed
calculation.
If using a ball screw exceeding the critical
speed, it is necessary to increase the natural
frequency by using an intermediate support,
etc. If using with nut rotation, it is possible to
operate exceeding critical speed by installing
a vibration energy absorbing system (optional,
vibration control damper: patented by NSK) to
the screw shaft. (Refer to "Nut rotatable drive
ND Series" on page B541.)
B47
j125
j100
6
on critical speed nc as follows, taking in account
j80
j63
4
"B-2-4 Supporting Conditions for Calculation of
Buckling Load and Critical Speed" on page B51.
Fig. 3.1 shows the permissible rotational speeds
against critical speed for each shaft diameter.
2
nC = α ×
=f
60λ
2
2πL
√
E•I•g
γ•A
··· 7)
dr
7
–1
× 10 (min )
L2
In this formula:
α : Safety factor (α = 0.8)
5
E : Elastic modulus (E = 2.06 × 10 MPa)
I : Moment of inertia of area of screw shaft
4
I = π dr (mm4)
··· 3)
64
dr : Screw shaft root diameter (mm) (See the
dimension table.)
g : Acceleration of gravity ( = 9.8 × 103 mm/s2)
γ : Specific weight (γ = 7.65 × 10-5 N/mm3)
A : Cross section area of the screw shaft root
2
diameter (mm )
2
π
×dr (mm2)
A=
··· 5)
4
L : Unsupported length (mm) (See Figs. 4.1, and 4.2
"Supporting conditions of screw shaft and ball
nut" on page B51)
f, λ : Factors determined by the supporting condition
2
j50
j40
j32
103
8
j25
j20
j16
6
j12
j10
4
j8
2
102
Fixed-Simple support 2
10
Fixed-Fixed support
2
2
4
Fixed-Free support
Simple-Simple support
Shaft end mounting
4
8
4
102
6
6
8
6
2
8
10
102
103
8
2
3
2
4
2
4
6
4
8
4
103
6
6
6
8
2
104
8
4
8
10
103
2
4
6
Rotational speed (min–1)
Fig. 3.1 Permissible rotational speeds vs. critical speeds
Table 3.1 Coefficients of critical speed
Supporting condition
f
λ
Fixed - Simple support
15.1
3.927
Fixed - Fixed support
21.9
4.730
Fixed support - Free
3.4
1.875
Simple - Simple support
9.7
π
B48
Ball Screw
critical speed of screw shaft rotation and the
Calculate the permissible rotational speed based
Unsupported length (mm)
Permissible rotational speed is determined by
the feeding speed and ball screw lead. When
selecting a ball screw, it is important to know
the permissible rotational speed.
It is necessary to calculate two items below, and
whichever smaller is the permissible rotational
speed.
The lower of the following two factors, d·n
and critical speed, will determine the overall
permissible rotational speed of the ball screw.
• Critical speed which is the resonance vibration
of the shaft.
• d·n value which is involved in damaging the
ball recirculation components.
104
8
B-2-3.2 d·n Value
<<Calculation example of permissible rotational speed to the critical speed>>
Calculate the permissible rotational speed to the critical speed under
conditions in Fig. 3.2.
<Use conditions>
Nut model: DFT4010-5
Supporting condition is Fixed - Simple support (From the supporting
condition (ii) in Fig. 4.1 "Supporting conditions of screw shaft and ball
nut" on page B51.)
Unsupported length
L = 2 000 mm
Screw shaft root diameter dr = 34.4 mm (from the dimension table)
<Calculation>
An increase of ball orbital speed increases the
Notes: 1. Special measure must be taken for
collision impact of balls to ball recirculation
high-speed specification products.
Please consult NSK.
parts, and thus resulting in damage to them. For
this reason, the permissible rotational speed is
2. Please consult NSK if the maximum
also limited by the d·n value (d, shaft diameter
rotational speed or the d·n value
in millimeters; n, rotational speed per minutes).
exceed the values on the table below,
Table 3.2 shows the allowable d·n value and the
even both the critical speed of screw
maximum rotational speed of ball screws.
shaft and the d·n value are in ranges
of the allowable limit.
Supporting condition is Fixed-Simple support, from Table 3.1 on page B47
λ = 3.927
f =15.1
By formula 7) on page B47, permissible rotational speed to critical
speed is
Table 3.2 Criteria of allowable d·n value and maximum rotational speed
Ball screw recirculation system, Series/Type
Standard ball screw Ball screw for transfer equipment R series
Standard nut ball
screws
Nut model number:
DFT4010-5
High-speed
Criterion of permissible
rotational speed [min−1]
50 000 or less
−
3 000
End-deflector type
180 000 or less
−
5 000
Return tube type
70 000 or less
100 000 or less
3 000
Deflector(bridge) type
84 000 or less
100 000 or less
3 000
3 000
End cap type
80 000 or less
100 000 or less
HMD type for high-speed machine tools
160 000 or less
−
4 000
HMS type for high-speed machine tools
160 000 or less
−
5 000
HMC type for high-speed machine tools 100 000 or less, 135 000 or less
−
3 750
BSL type for miniature lathes
−
4 000
−
3 225
*1
(180 000 or less)
*1
HTF-SRC type for high-load drives 140 000 or less, 160 000 or less
Applicationoriented ball screws HTF-SRD type for high-load drives
120 000 or less
HTF type for high-load drives
Fixed support
L = 2 000
Simple support
Fig. 3.2 Calculation example of permissible rotational speed to the critical speed
B49
−
50 000 or less, 70 000 or less*1 100 000 or less
Ball Screw
nc = f d2r × 107 =15.1 × 34.4 2 × 107 = 1 298.6 (min–1)
L
2 000
nc = 1 290 min–1 or under
Allowable d·n value
Standard
2 400
3 125
VSS type for contaminated environment
150 000 or less
−
3 000
ND series nut-rotatable ball screws
70 000 or less
100 000 or less
3 000
™ series for robots
70 000 or less
−
3 000
R series for transfer equipment
50 000 or less
−
3 000
*1) Please refer to the explanation of each ball screw for which two allowable d·n values are listed
· HMC type for high-speed machine tools: page B501
· HTF-SRC type for high-load drives: page B511
· HTF type for high-load drives: page B519
B50