The use of variable pitch or helix cutters is a known means to prevent chatter vibration during milling. In this article, an alternative method based on an improved semi-discretization method is proposed to predict the stability of variable pitch or variable helix milling. In order to consider the effect of distributed system delays attributed to helix variation, the average delays were calculated for each flute after the engaged cutting flutes were divided into a finite number of axial elements. Meanwhile, a straightforward integral force model, which can consider the piecewise continuous regions of the cutting that describe the helix angle is used to determine the cutting force. Through comparisons with prior works, timedomain simulations, and cutting tests, the proposed approach was verified. In addition, the method was applied to examine the effect of tool geometries on stability trends. Several phenomena for certain combinations of pitch and helix angles are shown and explained.
The use of variable pitch cutter is a known means to increase the stable limit depth of cut by disrupting the regenerative effect. In this paper, an improved semidiscretization algorithm is presented to predict the stability lobes for variable pitch cutters. Modeling efforts develop a straightforward analytical integral force model that can cover any case of piecewise continuous cutting regions regarding the helix angle. The proposed approach has been verified with the comparisons with prior works, time domain simulations, and cutting tests. In addition, the method is also applied to examine the effect of the tool geometries on the stability trends for variable pitch milling. Some new phenomena for certain combinations of parameters are shown and explained.
The micro-electro-mechanical system (MEMS) resonator developed based on surface processing technology usually changes the section shape either due to excessive etching or insufficient etching. In this paper, a section parameter is proposed to describe the microbeam changes in the upper and lower sections. The effect of section change on the mechanical properties is studied analytically and verified through numerical and finite element solutions. A doubly-clamped microbeam-based resonator, which is actuated by an electrode on one side, is investigated. The higher-order model is derived without neglecting the effects of neutral plane stretching and electrostatic nonlinearity. Further, the Galerkin method and Newton–Cotes method are used to reduce the complexity and order of the derived model. First of all, the influence of microbeam shape and gap variation on the static pull-in are studied. Then, the dynamic analysis of the system is investigated. The method of multiple scales (MMS) is applied to determine the response of the system for small amplitude vibrations. The relationship between the microbeam shape and the frequency response is discussed. Results show that the change of section and gap distance can make the vibration soften, harden, and so on. Furthermore, when the amplitude of vibration is large, the frequency response softening effect is weakened by the MMS. If the nonlinearity shows hardening-type behavior at the beginning, with the increase of the amplitude, the frequency response will shift from hardening to softening behavior. The large amplitude in-well motions are studied to investigate the transitions between hardening and softening behaviors. Finally, the finite element analysis using COMSOL software (COMSOL Inc., Stockholm, Sweden) is carried out to verify the theoretical results, and the two results are very close to each other in the stable region.
The dynamic equations of a four-degree-of-freedom micro gyroscope system were developed considering the nonlinearity of driving stiffness, the primary resonance, and the 1:1 internal resonance. Then, the perturbation analysis was carried out using the method of multiple scales. The influence of stiffness nonlinearity and system parameters on micro-gyro dynamic characteristics, output sensitivity, detection bandwidth, and working stability were discussed based on the analytic and numerical solutions of the dynamic equations. Through the singularity theory, the influence of system parameters on bifurcation behavior was analyzed. The results show that the amplitude jump and multi-stable solutions caused by the nonlinear hardening characteristics of the high robust two-degree-of-freedom drive-mode occur outside the detection bandwidth. In addition, the influence on the bandwidth was weak and the sensitivity of the bandwidth area was slightly reduced. Moreover, saturation existed in the response amplitude of the second drive-mode in spite of the primary resonance being completely tuned or detuned. As a result, although the electrostatic force amplitude was out of the unstable region and even took a larger value, the micro gyroscope obtained a larger stable output. Besides, nonlinearity will lead to energy transfer between various modes of multi-degree-of-freedom micro gyroscopes. That means the response amplitudes could change greatly due to the variation of the external environment even the system is under a constant excitation frequency. Therefore, increasing the stiffness coefficient of the micro beam and the electrostatic force amplitude can maintain the robustness of the system to environmental changes and avoid the occurrence of bifurcation.
Magnetic dipole theory has been widely and successfully used to explain the leakage magnetic field signals. Because the model parameter such as magnetic dipole density is not easy to quantify, magnetic dipole theory often needs normalizing in application, which is considered to be unsuitable for quantitatively analyzing the magnetic memory signals with the stress effect. In this paper, the theoretical model of magneto-mechanical coupling magnetic dipole is established, which is suitable for analyzing the stress effect on magnetic signals in magnetic memory testing method. Based on the ferromagnetic theory, the equivalent field under the combined action of the applied load and the magnetic field is determined. And then, the magneto-mechanical analytical model is obtained for the isotropic ferromagnetic material under the weak magnetic field based on the first-order magnetization approximation in the weak magnetization state. Under the assumptions of rectangular and V-shaped magnetic charge distribution for the two-dimensional magnetic signal problem, the theoretical analytical models of the magnetic memory signals from the smooth and cracked specimens, and the analytical models of the magnetic memory signal induced by the rectangular and V-shaped surface defect are established. Based on the analytical solution of the proposed magneto-mechanical magnetic dipole theory, the difference in signal between before and after the failure of the specimen, the signal from the rectangular and V-shaped defect, and other influencing factors and laws of the magnetic signal are analyzed in detail. In particular, the influence of stress, environmental magnetic field, defect morphology and size, lift-off effect, specimen size and other factors on magnetic memory signals can be described based on the analytical solution of magneto-mechanical magnetic dipole models proposed in this paper. The proposed analytical model of magneto-mechanical magnetic dipole in this paper is simple and easy to use, and the present research shows that the proposed analytical solution in this paper can explain some basic experimental phenomena and laws in magnetic memory testing experiments. In addition, the precise magneto-mechanical coupling quantitative model combined with the finite element analysis method is still needed for accurately analyzing the magnetic memory signals in experiment.
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