OBRABOTKAMETALLOV MATERIAL SCIENCE Том 23 № 3 2021 EQUIPMEN . INSTRUM TS Vol. 5 No. 4 2023 Dimensional Analysis Second set of twenty experiments as depicted in table 2 are performed to calibrate a theoretically developed flank wear and power consumption models for UVAHT. Experiments were performed varying the cutting speed, frequency, and amplitude of vibrations as depicted in table 2 and at constant feed and depth of cut of 0.085 mm/rev and 0.4 mm, respectively. The Buckingham Pi Theorem, named after the physicist Edgar Buckingham, is a fundamental principle in dimensional analysis. It states that when a physical problem involves “n” variables and “m” fundamental dimensions (such as length, time, mass, etc.), the problem can be expressed using n–m dimensionless parameters i.e. (Pi terms). The Pi terms are constructed as products of the original variables raised to appropriate powers such that the resulting expression is dimensionless [18–20]. The process of determining the Pi terms involves finding dimensionally independent groups of variables that describe the physical phenomena in the problem. According to Buckingham Pi theorem, the equation linking all the variables will have (n – m) dimensionless groups if the problem has “n” variables and those variables comprise “m” fundamental dimensions (for instance, M, L, and T): π1 = f (π2, π3, …….π n–m). The resulting equation takes the following form: the groups should not be dependent on one another, and no group should be established by adding the powers of other groups together. This approach has the benefit of being easier to use than the simultaneous equation method for determining the values of the indices (the exponent values of the variables). There are two prerequisites for using this approach to solve the equation. Each of the fundamental dimensions should be represented by one of the “m” variables at a minimum. One of the variables in a recurrent set should not be able to be formed into a dimensionless group. A dimensionless group of variables known as a repeating set. Selection of Dimensionless Parameters The selection of dimensionless parameters (Pi terms) involves identifying dimensionally independent groups of variables. These groups are chosen based on the underlying physics of the problem. The goal is to capture the significant interactions and relationships between the variables that govern the behavior of the system. In the context of conventional as well as ultrasonic vibration-assisted hard turning (UVAHT) of AISI 52100 steel, several process variables play a crucial role in influencing the machining performance. This process involves identifying the fundamental dimensions (length [L], time [T], mass [M], etc.) and determining the number of dimensionless parameters (Pi terms) required to describe the behavior of the system. By examining the relevant process variables and its corresponding units, one can establish the relationships between the variables and form dimensionless groups. Modelling of Power Consumption (Pc ) After conducting the dimensional analysis, dimensionless groups were formulated to represent the relationships between the relevant process variables. These dimensionless groups provide valuable insights into the interactions between ultrasonic vibration parameters and conventional turning parameters during the UVAHT process. Power consumption is depending on four parameters namely: material removal rate (MRR), density of the material (ρ), vibrational amplitude (A), and frequency of vibration (F). Now by selecting M (mass), L (length), and T (time) as the basic dimensions, the dimensions of the foregoing quantities would then be (table 3). Furthermore, Pc = φ (MRR, ρ, A, F)
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