Delay, Parametric Excitation, and the Nonlinear Dynamics of Cutting Processes

Abstract

It is a rule of thumb that time delay tends to destabilize any dynamical system. This is not true, however, in the case of delayed oscillators, which serve as mechanical models for several surprising physical phenomena. Parametric excitation of oscillatory systems also exhibits stability properties sometimes defying our physical sense. The combination of the two effects leads to challenging tasks when nonlinear dynamic behaviors in these systems are to be predicted or explained as well. This paper gives a brief historical review of the development of stability analysis in these systems, induced by newer and newer models in several fields of engineering. Local and global nonlinear behavior is also discussed in the case of the most typical parametrically excited delayed oscillator, a recent model of cutting applied to the study of high-speed milling processes.

2:1 Resonance in the delayed nonlinear Mathieu equation

“…Further evidence of the existence of the degenerate Hopf in the original Equation (1) was obtained by numerically integrating Equation (1) in the neighborhood of T = 2π for a variety of other parameters. Since a limit cycle in the slow flow corresponds to a quasiperiodic motion in the original equation, we searched for quasiperiodic motions in Equation (1). No quasiperiodic motions were observed.…”

“…Equation (1) is a model for high-speed milling: "High-speed milling is a kind of parametrically interrupted cutting as opposed to the self-interrupted cutting arising in unstable turning processes." [1]. See, e.g., [2,3].…”

“…Various special cases of Equation (1) have been studied by others, depending on which parameters are zero. In the case that b and c are zero, we have the linear Mathieu equation, the stability chart for which is well known [4], see Fig.…”

Effect of Nominal Chip Thickness on Stability of Interrupted Turning

“…In the above studies [1][2][3][4][5][6][7][8][9][10][11][12][13][14], the cutting force models are linear, without considering nonlinearities in terms of chip thickness. Given this, powerlaw function for cutting force was used to analyse chatter by Hanna and Tobias [15] and Stépán et al [16] and Yang et al [17]. Recently, instead of approximating the cutting forces as a power-law function of the third-degree Taylor series (in terms of chip thickness), Moradi et al proposed an extended nonlinear model of cutting force which is expressed as a completed third-order polynomial function of the chip thickness [18,19].…”

“…It is worth noting that, in the above studies [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19], dynamic cutting forces are only the function of dynamic chip thickness, and cutting force versus unit of dynamic chip thickness must be unique. However as shown in Figure 1, cutting forces versus the same dynamic chip thickness (Δℎ 1 = Δℎ 2 ) are unequal at different nominal chip thickness (Δ 1 ̸ = Δ 2 ), due to the fact that the nominal chip thickness is neglected.…”