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AR T I C L E
Advanced Machine Engineering
www.read-tpt.comMARCH 2017
87
Advanced Machine & Engineering
Rockford, IL, USA
Email:
info@ame.comWebsite:
www.ame.comSince the arc length (
S
) is a very small value compared
to the blade radius (
r
) it can be assumed that the linear
displacement and arc length (
S
) is the same and the following
equation can be used.
S=θ. r
where: S: Arc length (m),
θ: Angular displacement (rad)
The backlash for the gearbox is designed with a range of
0.030° to 0.047°. This is the total backlash of the gearbox
reflected to the spindle. The backlash was 0.035°, which is
within the expected total backlash range.
The measured values are shown in the graph. The X axis
expresses the gradual increase of the torque calculated by
multiplying the forces obtained by increasing the hydraulic
pressure in the cylinder with the radius of the blade where the
force is applied. The Y axis shows the angular displacement
of the carbide tooth on the blade, representing the actual wind
up of the gear train (in degrees).
The slope of this line is the compliance and the stiffness is
the reciprocal of this value. Any unevenness of such a graph
would show a problem within the gear train.
To make a sanity check the torsional compliance of the single
transmission shafts was analysed with FEA.The compliance
has been reflected to the blade spindle and compared to the
measurement.
To simplify the complexity of a system, simple models can be
created by reducing mechanical quantities such as stiffness,
inertia or damping to one shaft. The reduced system is
equivalent to the original system from an energy point of view.
COMPLIANCE CALCULATIONS
AXES
Torque
[Nm]
Circumferential
deformation
[mm]
Radius
[mm]
Angular
deformation
[rad]
Compliance
[rad/Nm]
Gear ratio
[1]
Compliance ref to spindle axis
Compl./Gear Ratio
2
[rad/Nm]
CL# Spindle
11.2985
1.52E-04
61.976 2.46E-06
2.18E-07
1
2.17641E-07
C#3
11.2985
2.54E-04
83.16
3.05E-06
2.70E-07
1.367
1.44665E-07
C#2
11.2985
6.20E-04
52.324 1.18E-05
1.05E-06
8.014
1.63231E-08
C#1 Input
11.2985
8.89E-04
40.005 2.22E-05
1.97E-06
24.966
3.1555E-09
TOTAL
3.81785E-07
TOTAL STIFFNESS
[Nm/rad]
2.62E+06
Figure 3: Compliance measurement
The potential energy
E
stored in a shaft with the torsional
stiffness
c
can be calculated with the acting torque
M
and the
twisting angle φ.
E
= 1
M
φ = 1
c
φ
2
2
2
Since the energy stored in a shaft has to be the same as the
reduced one:
E
= 1
c
an
φ
an
2
= 1
c
an, red
φ
ab
2
2
2
With
φ
an
/ φ
ab
=
i
being the gear ratio.
c
an, red
=
i
2
c
an
When reducing the stiffness to a slow running shaft such as
the saw blade spindle and |i|>1 then
c
an, red
>
c
an
The complete gear train is essentially a series connection of
shafts in which the total stiffness is
c
ges
1/(1/c1+1/c2+….). 1/c
is equal to the compliance, which can be easily added up and
expressed as the reciprocal at the end.
Conclusion
Compliance data could also help solving problems in the field.
If a head is acting up in the field or if the tool life suddenly
drops, a compliance test can easily be conducted at the
machine. The graph can be compared to the original graph
and the irregularity will give indications of the problems.
Every carbide saw has a certain compliance. This compliance
must be held low to obtain an acceptable tool life. However,
decreasing compliance will increase the machine cost by
making the machine stiffer. The secret is to find the golden
balance resulting in the most cost-efficient carbide saw.
Figure 4: Theoretical calculation of the gear train stiffness