TPT March 2017

AR T I C L E

Advanced Machine Engineering

COMPLIANCE CALCULATIONS

Circumferential deformation [mm]

Angular deformation [rad]

Torque [Nm] 11.2985 11.2985 11.2985 11.2985

Radius [mm]

Compliance [rad/Nm]

Gear ratio [1]

Compliance ref to spindle axis Compl./Gear Ratio 2 [rad/Nm]

AXES

CL# Spindle

1.52E-04 2.54E-04 6.20E-04 8.89E-04

61.976 2.46E-06

2.18E-07 2.70E-07 1.05E-06 1.97E-06

1

2.17641E-07 1.44665E-07 1.63231E-08 3.1555E-09 3.81785E-07

C#3 C#2

83.16

3.05E-06

1.367 8.014

52.324 1.18E-05 40.005 2.22E-05

C#1 Input

24.966

TOTAL

TOTAL STIFFNESS [Nm/rad]

2.62E+06

Figure 4: Theoretical calculation of the gear train stiffness

Since 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.

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.

Advanced Machine & Engineering Rockford, IL, USA

Email: info@ame.com Website: www.ame.com

Figure 3: Compliance measurement

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MARCH 2017