Compliance is defined as the measure of the ability of a mechanical system to respond to an applied vibrating force, expressed as the reciprocal of the system stiffness. In short, it measures the weakness of the system. In a carbide saw, the most critical component subject to torsional and lateral vibration of the saw blade, is the gearbox, commonly called the head. When the blade tooth first contacts the material, the reaction force 'winds up' the gear train. First the backlash is removed and then the additional loading will increase the torsional displacement. If there is any backlash in the feed mechanism, it will also act the same way as the power train backlash.
The saw blade and its mounting shaft have relatively little inertia. During the time the backlash is being removed, the blade tooth momentarily pauses in its rotation while the motor continues at its full speed. When the backlash is eliminated, the blade comes up to speed almost instantly. The speed may momentarily be even higher if the compliance is high and the cutting tooth 'springs' forward. If this happens when the tooth exits the material, the backlash will open up again and the process repeats until some teeth will stay in the cut. This exciting frequency measured in Hz could become critical when its frequency matches a natural frequency to result in resonance. As more teeth are engaged, the torque of the gear train will increase but the fluctuating load is only caused by one tooth engaging and disengaging the cut. This fluctuation of the wind-up of the gear train is very damaging to the carbide teeth and reduces the tool life.
Figure 1. Locked input schaft on the gearbox to prevent rotation
The compliance can be measured statically. In this case, we measured a head mounted on our AMSAW pivot saw. A rigid steel bar was clamped with a “c” clamp to the flanged bushing of the motor shaft. The steel bar at the toothed pulley was locked between (2) screws to prevent the pulley from turning. The dial indicator on the pulley measured any small movement Fig 1. On the blade side of the head a steel bar was locked between the tooth gullet and the blade lift hole, and a hydraulic cylinder was used to apply a gradual force to put a torque load on the gear train. The displacement value between a fixed point of the head and the tooth of the saw blade was measured with a dial indicator Fig. 2.
Figure 2. Hydraulic cylinder applied tangential force on the blade
and the displacement was measured with a dial indicator
Figure3. Compliance Measurement
The torque calculated by the relationship:
During the test a dial indicator was used and a linear displacement obtained. 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 analyzed with FEA. The compliance has been reflected to the blade spindle and compared to the measurement.
Figure 4. Theoretical calculation of the gear train stiffness
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.
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