An evolutionary framework in modelling of multi-output characteristics of the bone drilling process

Abstract

For the improvement of economic and environmental performance of bone drilling operations, the optimum value of the spindle speed and feed rate must be known. This is because the input spindle speed is an important factor of energy consumption and with its approximate value known in optimization of output characteristics of bone drilling operations may result in significant energy savings. Optimization of multi-output characteristics of the bone drilling process is possible, if the relationships between the outputs and the inputs are known. Therefore, this study forms the strong basis for development of the models for the three output characteristics (maximum temperature, maximum force and maximum average surface roughness) for the bone drilling operation performed on the bovine bone. Experimental studies are conducted to measure these three outputs based on the spindle speed and feed rate. The validation of the formulated models is done based on the root-mean-square error, coefficient of determination, relative error, multi-objective error and mean absolute percentage error. The relationships between the three outputs and the inputs are further revealed by the 2-D analysis on the models. The findings from these relationships can be used for the predictive monitoring the bone drilling operation. The work concludes with discussion of environmental implications arising from the current study.

Appendix

$$\begin{aligned} &{\text{Maximum temperature}}_{\text{GP}} = 55.1011 + \left( {2.4727} \right)*\left( {{ \tan }\left( {{ \cos }\left( {x_{2} } \right)} \right)} \right) \\ & \quad + \left( { - 2.525} \right)*\left( {{ \cos }\left( {\left( {\left( {x_{1} } \right) + \left( {\left( {7.744684} \right)} \right)} \right) + \left( {x_{1} } \right)} \right)} \right) \\ & \quad + \left( { - 0.26165} \right)*\left( {{ \sin }\left( {{ \tanh }\left( {{ \sin }\left( {\left( {{ \cos }\left( {x_{2} } \right)} \right) + \left( {{ \cos }\left( {x_{2} } \right)} \right)} \right)} \right)} \right)} \right) + \left( { - 0.013513} \right) \\ & \quad *\left( {\left( {\left( {{ \sin }\left( {{ \cos }\left( {x_{2} } \right)} \right)} \right) + \left( {\left( {\left( {{ \cos }\left( {x_{1} } \right)} \right) + \left( {\left( {7.744684} \right)} \right)} \right) + \left( {x_{1} } \right)} \right)} \right)*\left( {{ \sin }\left( {\left( {x_{2} } \right) + \left( {{ \cos }\left( {x_{2} } \right)} \right)} \right)} \right)} \right) \\ & \quad + \left( {0.68269} \right)*\left( {{ \tanh }\left( {\left( {{ \tanh }\left( {{ \sin }\left( {x_{1} } \right)} \right)} \right) + \left( {{ \cos }\left( {\left( {x_{1} } \right) + \left( {x_{2} } \right)} \right)} \right)} \right)} \right) \\ & \quad + \left( { - 0.11163} \right)*\left( {\left( {{ \sin }\left( {\left( {x_{1} } \right) + \left( {{ \tan }\left( {{ \cos }\left( {x_{2} } \right)} \right)} \right)} \right)} \right) + \left( {x_{1} } \right)} \right) \\ \end{aligned}$$

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$$\begin{aligned} &{\text{Maximum thrust force}}_{\text{GP}} = - 26.7024 + \left( {0.35454} \right)*\left( {\left( {\tan \left( {x_{1} } \right)} \right)*\left( {\tanh \left( {x_{1} } \right)} \right)} \right) \\ & \quad + \left( {0.011571} \right)*\left( {\left( {\left( {\left( {x_{1} } \right)*\left( {\cos \left( {x_{2} } \right)} \right)} \right) + \left( {\left( {\tanh \left( {\sin \left( {\left( {8.500942} \right)} \right)} \right)} \right) - \left( {x_{2} } \right)} \right)} \right)*\left( {\cos \left( {x_{1} } \right)} \right)} \right) \\ & \quad + \left( { - 0.59937} \right)*\left( {\left( {\left( {8.500942} \right)} \right)*\left( {\cos \left( {x_{2} } \right)} \right)} \right) \\ & \quad + \left( {0.87393} \right)*\left( {x_{1} } \right) + \left( { - 0.0001998} \right)*\left( {\left( {x_{1} } \right)*\left( {\left( {\left( {\tanh \left( {x_{1} } \right)} \right) - \left( {\left( {x_{2} } \right) - \left( {x_{1} } \right)} \right)} \right)*\left( {\cos \left( {x_{1} } \right)} \right)} \right)} \right) \\ & \quad + \left( { - 0.1258} \right)*\left( {\tan \left( {\left( {x_{1} } \right)*\left( {\left( {x_{2} } \right)*\left( {\cos \left( {x_{1} } \right)} \right)} \right)} \right)} \right) \\ \end{aligned}$$

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$$\begin{aligned}& {\text{Maximum average surface roughness}} _{\text{GP}} = 1.2153 + \left( { - 0.016092} \right) \\ & \quad *\left( {\left( {\left( {\tan \left( {\tan \left( {\left( {x_{2} } \right) - \left( {\left( {5.304303} \right)} \right)} \right)} \right)} \right) - \left( {\cos \left( {\left( {\left( {x_{2} } \right) + \left( {\left( { - 5.107464} \right)} \right)} \right) + \left( {x_{2} } \right)} \right)} \right)} \right)} \right. \\ & \quad \left. { - \left( {\cos \left( {\left( {x_{2} } \right) - \left( {\left( {\left( {x_{2} } \right) - \left( {\left( {5.304303} \right)} \right)} \right)*\left( {\sin \left( {x_{1} } \right)} \right)} \right)} \right)} \right)} \right) \\ & \quad + \left( {0.36557} \right)*\left( {\sin \left( {\cos \left( {\left( {x_{2} } \right) - \left( {\left( {5.304303} \right)} \right)} \right)} \right)} \right) + \left( { - 0.00069953} \right) \\ & \quad *\left( {\left( {x_{1} } \right)*\left( {\tan \left( {x_{2} } \right)} \right)} \right) + \left( { - 3.6113e - 005} \right) \\ & \quad *\left( {\left( {\left( {x_{1} } \right) - \left( {\left( {x_{2} } \right) - \left( {\cos \left( {\left( {x_{2} } \right) - \left( {\left( {5.304303} \right)} \right)} \right)} \right)} \right)} \right) - \left( {\left( {5.304303} \right)} \right)} \right) \\ & \quad + \left( { - 0.0056038} \right)*\left( {\left( {x_{1} } \right)*\left( {\sin \left( {\cos \left( {\left( {x_{1} } \right) + \left( {\left( { - 1.501624} \right)} \right)} \right)} \right)} \right)} \right) + \left( {0.02242} \right)*\left( {\tan \left( {x_{2} } \right)} \right) \\ \end{aligned}$$

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