Geometrical error improvement of Aramid honeycomb workpieces in robot-based triangular knife ultrasonic cutting process

Abstract

Due to the notable advantages of ultrasonic-assisted machining, this technology is being more and more adopted in different manufacturing processes. Aramid honeycomb structures are among materials for which ultrasonic-assisted machining has become an interesting alternative thanks to its effect in reduction of machining forces and less micro imperfections. A considerable number of research works have addressed micro machining defeats in ultrasonic application such as dust generation and fiber delamination in Aramid honeycomb workpieces. However, in this study, a great attention is dedicated to the geometrical errors observed in the workpieces in cutting process of Aramid honeycomb structures using ultrasonic technology. The effect of different elements involved in robotic-based ultrasonic-assisted cutting process of Aramid honeycomb structures on these error types has been studied. To proceed this goal, a machining force estimation model is experimentally developed using the machining forces captured during the execution of several cutting tests. An on-site measurement process for measuring the geometrical errors in resulted workpieces is presented. The compliance behavior of robot-tool system is simulated using an extended virtual joint method (VJM) approach. In this approach, the machining tool is considered an additional joint-link system attached to the 6-revolute joint industrial system. Simulation results based on this method applied on the case study showed that the portion of contribution of industrial robotic arm to the compliant errors of process is negligible compared with the one of ultrasonic cutting knife. An offline compensation algorithm is proposed based on the developed machining force and tool compliance behavior models. Experimental results supported the efficiency of the proposed compensation method in reducing the geometrical error by up to 95% based on the machining features.

Appendix: A computation of constants Fs, Fc and F0

To obtain a force predictive model, machining forces for 5 different chamfering angles (40:10:80) are registered in the sensor frame and further converted into the tool frame. Tables 4 and 5 summarize maximum values of each component as a function of chamfering angle in Kistler sensor and cutting tool frames respectively.

Table 4 Maximum chamfering force values defined in sensor frame

Table 5 Maximum chamfering force values defined in tool frame

To be able to predict the cutting forces in chamfering process of this honeycomb structure, the mathematical expression introduced in Eq. 1 is used. To find the constant values $F_{s_{i}}$, $F_{c_{i}}$ and $F_{0_{i}}$, where subscript i represents any arbitrary component defined in either coordinate systems, an error function E_i is introduced which evaluates the summation of squared errors of force prediction:

$$ E_{i}={\Sigma}_{j=1}^{N} (F_{s_{i}}\sin(\beta_{j})+F_{c_{i}}\cos(\beta_{j})+F_{0_{i}}-f_{j_{\exp}})^{2} $$

(26)

Where N is the number of data sets and $f_{j_{\exp }}$ is the j th experimental force value defined in Tables 4 and 5. To solve for concerned constants using least square method, the derivative of the error function with respect to each constant is equated to zero:

$$ \left\{ \begin{array}{rcl} \frac{\partial E_{i}}{\partial F_{s_{i}}}=0 &\Rightarrow& F_{s_{i}} {\Sigma} (\sin{\beta_{j}})^{2}+ F_{c_{i}} {\Sigma} (\sin{\beta_{j}}\cos{\beta_{j}})+F_{0_{i}} {\Sigma} (\sin{\beta_{j}}) \\ & & ={\Sigma} (f_{j_{\exp}}\times \sin{\beta_{j}}) \\ \frac{\partial E_{i}}{\partial F_{c_{i}}}=0 &\Rightarrow& F_{s_{i}} {\Sigma} (\sin{\beta_{j}}\cos{\beta_{j}})+ F_{c_{i}} {\Sigma} (\cos{\beta_{j}})^{2}+F_{0_{i}} {\Sigma} (\cos{\beta_{j}})\\ & & ={\Sigma} (f_{j_{\exp}}\times \cos{\beta_{j}}) \\ \frac{\partial E_{i}}{\partial F_{0_{i}}}=0 &\Rightarrow& F_{s_{i}} {\Sigma} (\sin{\beta_{j}})+ F_{c_{i}} {\Sigma} (\cos{\beta_{j}})+F_{0_{i}}\times N ={\Sigma} f_{j_{\exp}} \end{array}\right. $$

(27)

Where operator Σ has replaced ${\Sigma }_{j=1}^{N}$ to sake brevity. The set of equations expressed in Eq. 27 is then converted to the following final matrix notation:

$$ \left[\!\begin{array}{c} F_{s_{i}}\\ F_{c_{i}}\\ F_{0_{i}} \end{array}\! \right]= \left[\!\begin{array}{ccc} {\Sigma} \sin^{2}{\beta_{j}}& {\Sigma} \sin{\beta_{j}}\cos{\beta_{j}} & {\Sigma} \sin{\beta_{j}} \\ {\Sigma} \sin{\beta}\cos{\beta} & {\Sigma} \cos^{2}{\beta_{j}} & {\Sigma} \cos{\beta_{j}} \\ {\Sigma} \sin{\beta_{j}} & {\Sigma} \cos{\beta_{j}} & N \end{array}\!\right]^{-1}\times \left[\!\begin{array}{c} {\Sigma} f_{j}\sin{\beta_{j}}\\ {\Sigma} f_{j}\cos{\beta_{j}}\\ {\Sigma} f_{j} \end{array} \right] $$

(28)

Equation 28 is used to obtain the 6 sets of (F_s, F_c, F₀) corresponding to the 3 principal directions of the two mentioned coordinate systems (Tables 1 and 2).