# Modeling a Milling Dynamometer as a 3 DOF Dynamic System by Stiffness Identification
## 1 Introduction  
The most important parameters that has to be known in any machining are the cutting forces that develop in all three directions of a spatial orthogonal system. In the case of milling, the force values for all three directions are essential for analyzing, controlling, and optimizing the milling process. Taking into account the dynamics of the milling process and machining conditions, determining these values is often difficult. The determination of the force values can be achieved either theoretically, based on computational relationships, or experimentally, by measuring them directly. The analytical relationships that provide the values of the forces that appear in the milling process are based on the use of experimentally obtained coefficients. In essence, the experimental component is fundamental in the determination of cutting forces. The direct determination of cutting forces using measurements can be achieved by means of specialized devices called dynamometers. Over time, different types of dynamometers have been developed that use strain gauges for deformation measurements [1; 2; 3; 4; 5; 6], or rotating dynamometers [7; 8] have been applied as integrated systems on tool holders for the milling process. The most frequently used dynamometers are made with strain gauge or piezoelectric sensors, but there are some other indirect methods such as those from the feed drivers of spindle or from machine tool vibrations. The direct methods are mainly used to measure, with high precision, the forces, while the indirect methods are used in process monitoring applications. In practice, the best result can be obtained by using a flexural mechanism [9; 10; 11; 12; 13]. Flexural mechanisms offer some real advantages such as a simple design and assembly,low cost of manufacturing, monolithic structure, linear relationship between the applied force and displacement for small distortions, and continuous and smooth displacement characteristics. A dynamometer that is used to determine milling forces consists of two flat plates linked together by four octagonal rings in a symmetrical structure. The workpiece is placed on the upper plate. The bottom plate is fixed to the table of the milling machine. Elastic rings are subjected to forces that are monitored and summed accordingly. The forces that develop along the three directions can be obtained in this way. Generally, dynamometers can be divided, from the point of view of their production, into two classes: commercial devices, produced by renowned companies in the field and used for general industrial processes, and laboratory devices, designed and manufactured for certain experiments, which assume certain particularities that must be taken into account. In the case of commercial dynamometers, they are made of a set of piezoelectric sensors that are arranged to measure the machining forces and/or torque. The laboratory dynamometers are made with either strain gauges or piezoelectric sensors. Due their high sensitivity and rigidity, piezoelectric ones are preferred. The usage of strain gauges is based on Hooke's law, which takes into consideration the relations between the elastic strain, stress, and associated force. The dynamometers with strain gauges involve some disadvantages, such as the following:  
* They have a complicated measurement structure: the strain gauges must be connected in an electrical circuit (measuring bridge), which must be balanced and calibrated;
* The deformation-electrical resistance relationship is more nonlinear;
* They are fragile and difficult to mount on curved surfaces;
* The maximum deflection that can be measured is an order of magnitude smaller (typically, less than 0.01 m/m);
* They have temperature sensitivity. In the case of using of piezoelectric crystals, a very good compromise is obtained between the stiffness and sensitivity, and as a result, piezoelectric dynamometers are widely used.