Tooth Trajectory Research Articles

Process geometry modeling with cutter runout for milling of curved surfaces

Prediction of cutting forces and machined surface error in peripheral milling of curved geometries is non-trivial due to varying workpiece curvature along tool path. The complexity in this case, arises due to continuously changing process geometry as workpiece curvature varies along tool path. In the presence of cutter runout, the situation is further complicated owing to changing radii of cutting points. The present work attempts to model process geometry in machining of curved geometries and in the presence of cutter runout. A mathematical model computing process geometry parameters which include cutter/workpiece engagements and instantaneous uncut chip thickness in the presence of cutter runout is presented. The developed model is more realistic as it accounts for interaction of cutting tooth trajectory with that of preceding teeth trajectories in computing process geometry. Computer simulation studies carried for this purpose has shown that it is essential to account for teeth trajectory interactions for accurate prediction of process geometry parameters. This aspect is further confirmed with machining experiments, which were conducted to validate this aspect. From the outcomes of present work, it is clearly seen that the computation of process geometry during machining of curved geometries and in presence of cutter runout is not straightforward and requires a systematic approach as presented in this paper.

Modelling of tooth trajectory and process geometry in peripheral milling of curved surfaces

The paper presents modelling of tooth trajectory and process geometry in peripheral milling of curved surfaces. The paper differs from previous work in this area, in two respects. Firstly it deals with milling of variable curvature geometries unlike zero and constant curvature geometries dealt in the past. Secondly true tooth trajectories are considered for modelling process geometry in milling of curved surfaces instead of simpler circular tooth trajectories. Mathematical expressions for, feed per tooth along cutter contact path, entry and exits angles of tooth, undeformed chip thickness and surface error are derived and effect of workpiece curvature on these variables is studied. As cutting forces depend on these process variables, physical experiments were also performed to study the effect of workpiece curvature on cutting forces. Process simulation experiments carried out show the need for modelling true tooth trajectories instead of circular tooth trajectories particularly for curved geometries. Results also show that using simpler constant curvature models to variable curvature geometries for the purpose of estimation of process geometry variables could be erroneous.

Theoretical modelling of cutting forces in helical end milling with cutter runout

A theoretical cutting force model for helical end milling with cutter runout is developed using a predictive machining theory, which predicts cutting forces from the input data of workpiece material properties, tool geometry and cutting conditions. In the model, a helical end milling cutter is discretized into a number of slices along the cutter axis to account for the helix angle effect. The cutting action for a tooth segment in the first slice is modelled as oblique cutting with end cutting edge effect and tool nose radius effect, whereas the cutting actions of other slices are modelled as oblique cutting without end cutting edge effect and tool nose radius effect. The influence of cutter runout on chip load is considered based on the true tooth trajectories. The total cutting force is the sum of the forces at all the cutting slices of the cutter. The model is verified with experimental milling tests.

A numerical study of the effects of cutter runout on milling process geometry based on true tooth trajectory

Cutter runout is a common phenomenon affecting the cutting performances in milling operations. To date, most of the milling process models considering cutter runout were established based on the circular tooth path approximation, which brought errors into the runout estimation. In this paper, a new approach is presented for modelling the milling process geometry with cutter runout based on the true tooth trajectory of cutter in milling. The mathematical relationship between the trajectories generated by successive cutter teeth with runout is analysed. The milling process geometrical parameters, including the instantaneous undeformed chip thickness, the entry and exit angles of a cutting tooth, and the ideal peripheral machined workpiece surface roughness, are modelled according to the true tooth trajectories. Numerical method is used to solve the derived transcendental equations. A simulation study of the effects of cutter runout on milling process geometry is conducted using the models. It was found that the change of cutter radius for a tooth relative to its preceding one is the most important factor in evaluating the effects of cutter runout.

Analytical modelling of the milling process geometry based on the true tooth trajectory

A new approach for modelling the milling process geometry is presented. The main geometrical parameters are modelled on the basis of the true tooth trajectory of a trochoid. A simulation study using the model has been conducted to investigate the effects of the feed, cutter diameter and number of teeth on the process geometry parameters.

Milling force prediction using a dynamic shear length model

Modelling of cutting forces in milling is often needed in machining automation. In this paper, a new method for the determination of the cutting forces in face milling is presented, which applies a predictive machining theory originally developed for orthogonal cutting to milling operations, with a dynamic shear length model developed and incorporated. The proposed dynamic shear length model is developed based on the analysis for the true tooth trajectories of a milling cutter, taking into account of the characteristic wavy surface effects in milling. The prediction for the cutting forces is carried out at each step of the angular increment of cutter rotation from input data of fundamental workpiece material properties, tool geometry and cutting conditions. Cutting forces at a cutter tooth can be predicted once the shear angle, shear length, shear plane area, and the shear flow stress along the shear length have been determined. The milling force prediction using the dynamic shear length model is verified through milling experimental tests. The sensitivity of the difference between the static and dynamic shear length models with respect to the feed per tooth and the cutter diameter is discussed.

A new method for determining the undeformed chip thickness in milling

The determination of the instantaneous undeformed chip thickness is one of the key issues in the modelling of milling processes. The traditional chip thickness model, with an approximation form of h=f t sin θ , assumes the tooth path to be circular, and thus lacks accuracy. In this paper, a new approach for determining the undeformed chip thickness in milling with much higher accuracy is proposed. Through analysis of the true tooth trajectories, the true instantaneous undeformed chip thickness can be calculated by solving the deduced transcendental equation. Rather than using the cumbersome numerical method, a Taylor’s series is used to approximate the involved infinitesimal value in the transcendental equation, which results in an explicit expression for practical determination of the undeformed chip thickness with higher accuracy compared to the traditional model.

Development of a magnetic sensing device for tooth displacement under orthodontic forces.

We have developed a system for measuring tooth displacement from orthodontic force. Eight small magnetic sensors and a magnet are combined to measure three-dimensional displacement. Sensors, arranged cubically in the three planes of space, are placed in the mouth and fixed to the posterior teeth by a splint. A magnet is placed in the center of the eight sensors and attached to a front tooth that is subjected to orthodontic force. Sensors detect the magnet's movement as target tooth displacement. The system was designed to achieve displacement resolution of 1 microm. The mean percentage of measurement errors was determined to be less than 1% in a 600-cubic-microm volume from calibration. The system was tested clinically on human teeth. Although the oral environment, with high temperature and humidity, was not agreeable with the sensors, this system was stable and accurate enough for quantitative measurement of tooth displacement. The advantage of this system is the ability to detect tooth trajectories by decomposing displacement into translation and rotation and to determine the position of the center of rotation from these parameters.

A new method for qualitative and quantitative evaluation of tooth displacement under the application of orthodontic forces using magnetic sensors

The purpose of this study was to develop a new method for the quantitative and qualitative measurement of tooth displacement under load and to clarify the relationship between the force system acting on a tooth and a resultant movement. A combination of eight magnetic sensors and a magnet was employed to measure three-dimensional displacement. Two sets of sensors and a magnet were placed in front and at the back of a front tooth subjected to orthodontic forces. Magnets and sensors were fixed to the subject tooth and posterior teeth, respectively. Sensors detected displacements of two magnets extended from the subject tooth separately in real time. The tooth trajectories projected on the sagittal plane were studied. From displacements of two magnets, angle of tooth rotation, the position of center of rotation and the amount and direction of displacement of an arbitrary point on the tooth could be calculated with high accuracy. Also, the location of center of resistance could be determined. It was found that the center of rotation is moving substantially even with a small difference in the level of force application. Furthermore, a small amount of displacement of the center of resistance was observed with varying the force magnitude.

An application of magnet and magnetic sensor: measurement system for tooth movement

A measurement system for tooth displacement and rotation in the sagittal and frontal planes was designed and tested. Four small magnetic sensors were arranged at each corner of a rectangle on a plastic sheet. They were then placed in the mouth and fixed to the front teeth with a plastic splint. A powerful magnet made of a rare earth metal was fixed to the target tooth and placed at the center of the sensors. Movement of the magnet was detected by the four sensors as the tooth trajectory. This system was tested by measuring first molar movement in human subjects with a load generated by an orthodontic aid. This system was small enough to fit in a human oral cavity and did not interfere with the orthodontic aid. Although the oral environment with high humidity and high temperature was not agreeable for the sensors, resolution was estimated at better than 3 microns including the effects of system drift. System calibration was done extra orally and distortion was calculated at less than 6% in a square of 200 microns. The accuracy of this system was not affected by temperature due to the application in the oral cavity at a relatively constant temperature. The advantage of this system was not only the two-dimensional measurement but also rotational measurement in which a 0.05 degree resolution was calculated.