Nonlinear dynamic analysis of flexible workpiece and tool in turning operation with delay and internal resonance
Introduction
In manufacturing science, the sources of self excited vibration or chatter still is a research of interest to have precise and stable turning operation as its presence leads to the poor surface finish, loss of energy, workpiece dimensional inaccuracy and reduction of tool life due to large amplitude of vibration. Apart from regenerative effect, mode coupling effect [1], frictional [2], thermo-mechanical effect came out as main contributory factors for the onset of chatter. Starting from the work of Taylor [3] and Tobias [4] till today, several linear and nonlinear models are studied on the chatter in turning process which are briefly described below.
Hanna and Tobias [5,6], Grabec [7] generated a nonlinear theory of regenerative chatter taking into account a single degree of freedom (SDOF) structure with nonlinear stiffness and cutting force. The analyses of these models reveal the chaotic behavior for the intensive cutting. Lin and Weng [8] presented a nonlinear orthogonal cutting force model considering the effective geometry of cutting tool due to wavy surface on the workpiece. Going one step ahead, in Ref. [9], the stiffness nonlinearities and nonlinear time delay terms are included in the model of [8] which yields chaotic oscillations for some cutting parameters. In paper of Moradi et al. [10], a SDOF model of turning process considering quadratic and cubic structural nonlinearities and contact force due to the presence of tool flank wear between the workpiece and the tool, is investigated for primary, subharmonic and superharmonic resonance conditions. A nonlinear dynamic model of orthogonal cutting is presented in Nosyreva et al. [11] where effects of the cutting velocity on the mean friction coefficient are accounted for. Both strong and weak nonlinear dynamic models of cutting processes are investigated in Hwang et al. [12] using numerical method and the method of multiple scales. The literature review in Wiercigroch et al. [13] shows that, cutting operation like turning itself is a complex and highly nonlinear phenomenon which includes nonlinearity because of complex interrelation in between its dynamics and thermo-dynamics apart from structural nonlinearity, force nonlinearity [14], multiple delays etc. It is observed that the perturbation methods like method of multiple scales are very much efficient to investigate nonlinear cutting process which may include weak structural or force nonlinearity. In Refs. [[15], [16], [17], [18]], higher order MMS are used in the presence of large delay to study the stability, bifurcation diagram and time response.
Most of the previously studied research works concentrate on either the tool or the workpiece vibration to study the stability of turning operation. It is assumed that either the tool or the workpiece is sufficiently stiffer compared to the other one which means that either the tool or the workpiece will have more motion than the other one. Urbikain et al. [19,20] and Ozturk et al. [21] proposed a model to avoid the chatter considering only the tool dynamics. Quyang and Wang [22] suggested to consider the three directional moving cutting force and deflection dependent axial feed force's induced bending moment on the study of dynamic behavior of rotating beam like workpiece for turning operation. Chanda et al. [23] plotted the stability diagram only for thin cylindrical workpiece turning operation considering three directional cutting force using semi-discretization method. The bifurcation study is performed in Kim et al. [24] considering only the tool motion in feed and cutting force direction. In case of parallel turning, the dynamics of tool is tuned in feed direction only to increase the stability in the study of [21] where it is assumed that each tool has SDOF motion. But, in actual case, there may be comparable stiffness difference; thus tool and workpiece both have a prominent individual motion. So the effective chip thickness variation becomes a combined effect of tool and workpiece motion. Keeping such a consideration, the stability of the turning process is studied in Chen and Tsao [25]. Experimental investigation of regenerative chatter, induced by phase lag of the undulation due to deflection of workpiece is made in Kato and Marui [26]. Srinivas and Kotaiah [27] studied the stability in turning using a nonlinear force feed dynamic model by considering three-dimensional cutting tool geometry and relative motion of the workpiece and the tool where each has two-degree of freedom.
In overall, it is observed that till now, most of the studies are performed considering either the workpiece or the tool vibration. But, in many occasions, the tool and the workpiece both may have a prominent individual motion which will contribute to the cutting force in terms of effective chip thickness because of relative motion and hence a 2-DOF model becomes more realistic and important. Though in Refs. [25,27,28], stability study of turning operation are found considering tool and workpiece interaction, no study is performed for internal resonance condition due to the tool and the workpiece interaction. In some cases, it may happen that internal resonance condition appears during turning operation when any of the natural frequencies of the workpiece may come closer to any of the natural frequencies of the tool.
Hence, in the present work, a unique case is studied when internal resonance and primary resonance may occur due to cross-coupling of two degree of freedom system in terms of effective chip thickness and harmonic force on workpiece. Thus, present study will be very interesting to see the tool and workpiece dynamics with the consideration of regenerative effect for the internal and primary resonance conditions. Apart from that, stiffness nonlinearity is considered in equation of motion. The instability regions are investigated for different system parameters like chip width (depth of cut), speed, delay, cutting force coefficient, workpiece to tool stiffness ratio, workpiece diameter to wall thickness ratio. The stability of the systems are studied for steady state solution for internal resonance only and for internal resonance along with primary resonance condition.
In the present study, method of multiple scales (MMS) is used as it gives reduced equations which can be easily used to obtain the steady state frequency response curves. It may be noted that one has to spend several hours to obtain the frequency response curve by using numerical methods for the system's equation of motion as one will obtain a single point on the frequency response curve at a time. Also, it requires a huge memory space to store all the time response data for time domain based numerical methods (e.g. 4th or 5th order Runge Kutta method). Hence, this approach is not generally followed in the analysis of nonlinear systems and perturbation analysis [[29], [30], [31], [32]] is usually used to find the frequency response curves. Moreover, use of symbolic software (e.g., Matlab, Mathematica) reduces the time of deriving the required equations for MMS. Here, it is to be noted that time response is used to validate few points on instability curves which are obtained for a wide range of system parameters. In addition, the solution of the reduced single DOF system from present equation, in case of no resonance condition is validated comparing the existing solution in Ref. [16] in results and discussion section.
Section snippets
Description of the physical and mathematical model
As previously discussed, in many available research works [6,7,10,18,[33], [34], [35]], the turning operation is modeled as a single degree of freedom system considering either the workpiece or the tool motion as it is assumed that either the workpiece or the tool is enough stiffer than the other one. However, in many cases, it happens that the tool and the workpiece dynamic characteristic is comparative. Thus, the tool and the workpiece's relative motion contributes to the chatter via cutting
Results and discussions
Eliminating the nonlinear stiffness terms and coupling terms from Eq. (5), (6) of tool and workpiece, the reduced single degree of freedom (SDOF) system equation can be written as
This equation is same as that considered by Das and Chatterjee [16]. The solution using the present standard MMS for this equation is validated by comparing the time response solution with the approach mentioned in Ref. [16] for the slow flow modulation or large delay system where,
Conclusions
In the present work, the governing nonlinear equations of motion of the workpiece and the tool are developed for the turning operation considering both the workpiece and the tool to be flexible. Here, the regenerative effect due to delay effect and internal resonance are considered. The harmonic force is also assumed to be present on the workpiece during turning. The governing equations are solved using the 2nd order method of multiple scales. In the absence of the structural nonlinearity and
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