机床非线性颤振的描述函数分析

本文采用描述函数方法分析机床的非线性颤振.以描述函数刻划切削过程复杂的非线性特性,并将非线性振动分析所依赖的复平面推广到三维空间进行稳定性分析。从而求出颤振振幅和颤振频率。该法不仅可揭示切削过程等效刚度系数和等效阻尼系数的变化,而且还可考察各切削参数对颤振频率和振幅的影响.结果表明,描述函数方法是分析机床颤振这个极其复杂的非线性系统的一个有效的手段.     

机床非线性颤振的分段线性化及其数学模型

通过引入非线性调制环节N(S),本文实现了非线性颤振的分段线性化,建立了统一的切削动力学系统模型。这一模型使线性颤振理论、非线性颤振理论趋于统一,并使超声振动切削抑制颤振的机理从理论上得到解释。     

Experimental investigation on using the piezoelectric and electromagnetic vibration absorbers in milling

The paper presents an active control system that counteracts the development of chatter vibration. The vibration amplitude depends on the dynamic properties of the machine tool, cutting tool and work-piece. In the paper we analyze the case when the loss of machining stability is caused by the work-piece. The proposed active control system employs electromagnet or piezoelectric actuator to suppress vibration during milling. The active control introduces damping into the system, thereby raising the critical depth of cut and reducing forced vibration amplitude. It enables stable cutting under a much wider range of cutting parameters that for the uncontrolled system. Cutting tests are performed on JAFO FYN-50 machine with mill DIN 845 B-25 K-N HSS to demonstrate an effectiveness of the proposed systems.     

Delay equations with fluctuating delay related to the regenerative chatter

The mathematical models representing machine tool chatter dynamics have been cast as differential equations with delay. In this paper, non-linear delay differential equations with periodic delays which model the machine tool chatter with continuously modulated spindle speed are studied. The explicit time-dependent delay terms, due to spindle speed modulation, are replaced by state-dependent delay terms by augmenting the original equations. The augmented system of equations is autonomous and has two pairs of pure imaginary eigenvalues without resonance. The reduced bifurcation equation is obtained by making use of Lyapunov-Schmidt Reduction method. By using the reduced bifurcation equations, the periodic solutions are determined to analyze the tool motion. Analytical results show both modest increase of stability and existence of periodic solutions near the new stability boundary.     

Design and Modeling for Chatter Control

Boring bars for single-point turning on a lathe are particularly susceptible to chatter and have been the subject of numerous studies. Chatter is, in general, caused by instability. Clearly, the cutting process can be limited to regions of known stable operation. However, this severely constrains the machine-tool operation and causes a decrease in productivity. The more aggressive approach is to attack the stability problem directly through application of vibration control. Here, we demonstrate a new biaxial vibration control system (VPI Smart Tool) for boring bars. We present the experimentally determined modal properties of the VPI Smart Tool and demonstrate how these properties may be used to develop models suitable for chatter stability analysis, simulation, and development of feedback compensation. A phenomenological chatter model that captures much of the rich dynamic character observed during experiments is presented. We introduce the notion that the mean cutting force changes direction as the width of cut increases due to the finite nose radius of the tool. This phenomenon is used to explain the progression from chatter that is dominated by motions normal to the machined surface at small widths of cut to chatter that is dominated by motions tangential to the machined surface at large widths of cut. We show experimental evidence to support our assertion that a biaxial actuation scheme is necessary to combat the tendency of the tool to chatter in both directions. We then present some preliminary theoretical results concerning the persistence of subcritical instability as we expand consideration to high-speed machining.     

Chatter mitigation using the nonlinear tuned vibration absorber

A passive vibration absorber, termed the nonlinear tuned vibration absorber (NLTVA), is designed for the suppression of chatter vibrations. Unlike most passive vibration absorbers proposed in the literature for suppressing machine tool vibrations, the NLTVA comprises both a linear and a nonlinear restoring force. Its linear characteristics are tuned in order to optimize the stability properties of the machining operation, while its nonlinear properties are chosen in order to control the bifurcation behavior of the system and guarantee robustness of stable operation. In this study, the NLTVA is applied to turning machining.     

Modeling and active control of chatter vibrations of piezoelectric stacked machining arm in micro-turning process

Self-excited vibrations known as chatter are considered as the most detrimental issue in micro-turning processes. Occurring unpredictably, they adversely affect the tool life, productivity rate and surface quality of the machining processes. In this paper, a novel machining arm is modeled as a piezoelectric stacked rod which is subjected to a chatter force in the orthogonal micro-turning process. Due to the fact that machining processes are affected by various sources of uncertainties, H_∞ robust control approach is used to suppress the chatter vibrations of the machining arm in the presence of tool wear and dynamic model parameter variations. Also, input control force of the system is provided by exciting the input voltage of piezoelectric layers of the rod. In order to be certain that the designed controller succeeds in suppressing vibrations of the effective structural modes, behavior of the first three modes of vibrations are considered in the final response of the machining arm. In the following, performance of the robust H_∞ controller is compared with a modified PID controller. Simulation results show that the H_∞ controller improves the robustness and performance of the system against uncertainties. The PID controller extends the stability region of the sharp tool and fails to achieve this purpose for the worn tool although its performance is acceptable in suppressing chatter vibrations.

     

Estimating Critical Hopf Bifurcation Parameters for a Second-Order Delay Differential Equation with Application to Machine Tool Chatter

Nonlinear time delay differential equations are well known to havearisen in models in physiology, biology and population dynamics. Theyhave also arisen in models of metal cutting processes. Machine toolchatter, from a process called regenerative chatter, has been identifiedas self-sustained oscillations for nonlinear delay differentialequations. The actual chatter occurs when the machine tool shifts from astable fixed point to a limit cycle and has been identified as arealized Hopf bifurcation. This paper demonstrates first that a class ofnonlinear delay differential equations used to model regenerativechatter satisfies the Hopf conditions. It then gives a precisecharacterization of the critical eigenvalues on the stability boundaryand continues with a complete development of the Hopf parameter, theperiod of the bifurcating solution and associated Floquet exponents.Several cases are simulated in order to show the Hopf bifurcationoccurring at the stability boundary. A discussion of a method ofintegrating delay differential equations is also given.     

Modeling of the phase lag causing fluidelastic instability in a parallel triangular tube array

Fluidelastic instability is considered a critical flow induced vibration mechanism in tube and shell heat exchangers. It is believed that a finite time lag between tube vibration and fluid response is essential to predict the phenomenon. However, the physical nature of this time lag is not fully understood. This paper presents a fundamental study of this time delay using a parallel triangular tube array with a pitch ratio of 1.54. A computational fluid dynamics (CFD) model was developed and validated experimentally in an attempt to investigate the interaction between tube vibrations and flow perturbations at lower reduced velocities U_r=1–6 and Reynolds numbers Re=2000–12 000. The numerical predictions of the phase lag are in reasonable agreement with the experimental measurements for the range of reduced velocities U_g/fd=6–7. It was found that there are two propagation mechanisms; the first is associated with the acoustic wave propagation at low reduced velocities, U_r<2, and the second mechanism for higher reduced velocities is associated with the vorticity shedding and convection. An empirical model of the two mechanisms is developed and the phase lag predictions are in reasonable agreement with the experimental and numerical measurements. The developed phase lag model is then coupled with the semi-analytical model of Lever and Weaver to predict the fluidelastic stability threshold. Improved predictions of the stability boundaries for the parallel triangular array were achieved. In addition, the present study has explained why fluidelastic instability does not occur below some threshold reduced velocity.     

Forced vibration analysis of the milling process with structural nonlinearity,internal resonance,tool wear and process damping effects

In this paper, forced vibration analysis of an extended dynamic model of the milling process is investigated, in the presence of internal resonance. Regenerative chatter, structural nonlinearity, tool wear and process damping effects are included in the proposed model. Taking into account the average and first order expansion of Fourier series for cutting force components; their closed form expressions are derived. Moreover, in the presence of large vibration amplitudes, the loss of contact effect is included in this model. Analytical approximate response of the nonlinear system is constructed through the multiple-scales approach. Dynamics of the system is studied for two cases of primary and super-harmonic resonance, associated with the internal resonance. Under steady state motion, the effects of structural nonlinearity, cutting force coefficients, tool wear length and process damping are investigated on the frequency response functions of the system. In addition, existence of multiple solutions, jump phenomenon and energy transfer between vibration modes are presented and compared for tow cases of primary and super-harmonic resonances.     

Non-linear analysis and quench control of chatter in plunge grinding

This paper aims at mitigating regenerative chatter in plunge grinding. To begin with, a dynamic model is proposed to investigate grinding dynamics, where eigenvalue and bifurcation analyses are adopted, respectively, for prediction of grinding stability and chatter. Generally, it is found that most grinding chatter is incurred by subcritical Hopf bifurcation. Compared with supercritical instability, the subcritical generates coexistence of stable and unstable grinding in the stable region and increases chatter amplitude in the chatter region. To avoid these adverse effects of the subcritical instability, bifurcation control is employed, where the cubic non-linearity of the relative velocity between grinding wheel and workpiece is used as feedback. With the increase of feedback gain, the subcritical instability is transformed to be supercritical not only locally but also globally. Finally, the conditionally stable region is completely removed and the chatter amplitude is decreased. After that, to further reduce the chatter amplitude, quench control is used as well. More specifically, an external sinusoid excitation is applied on the wheel to quench the existing grinding chatter, replacing the large-amplitude chatter by a small-amplitude forced vibration. Through the method of multiple scales, the condition for quenching the chatter is obtained.     

Cross-recurrence plot quantification analysis of input and output signals for the detection of chatter in turning

Metal cutting is a complex nonlinear dynamical process. Analysis of signals from turning operation shows that the machining exhibits a low-dimensional chaos. The self-excited vibration caused by the regenerative effect, usually called chatter, can be created during machining by increasing one cutting parameter, while keeping all other cutting parameters constant. A cross-recurrence plot (CRP) enables the study of synchronisation or time differences in two time series. CRP-based methodology is used to find the point of transition from normal cutting to chatter cutting. In this method, two signals, one input signal (power to the lathe motor) and one output signal (cutting tool vibration), are recorded simultaneously at a constant sampling rate during cutting. A time series is generated from the recorded values, and cross-recurrence plot is prepared. This CRP can be quantified using Cross-Recurrence Quantification Analysis (CRQA). Abrupt variation in the CRQA parameters indicates the onset of chatter vibration. The results are verified using permutation entropy (PE) to detect the onset of chatter from the time series. The present study ascertains that this CRP-based methodology is capable of recognising the transition from regular cutting to the chatter cutting irrespective of the machining parameters or work piece material.     

含非线性摩擦系数的单自由度颤振系统全局动力学行为数值研究

摩擦系数模型取更具普适性的Stribeck非线性模型,基于事件驱动理论,利用C与Matlab联合仿真的方法开发了干摩擦颤振问题的快速求解程序。给出改进的胞映射算法,对含非线性摩擦的单自由度摩擦颤振系统的演化过程及其全局性态进行数值分析和研究,得到系统在任意的初始状态下的响应特性、系统收敛域的数值计算分析结果。分析结果表...     

Design of a tuned vibration absorber for a slender hollow cylindrical structure

Abstract

In this paper, details of the design work for a tuned vibration absorber to be used on a hollow cylindrical structure is presented. The vibration problem is of resonant type and the tuned vibration absorber is designed to suppress the displacement vibration response of the free end of the slender hollow structure dominated by the contribution of its lowest transverse vibration modes. The structure is modeled using a commercial finite element software. Finite element model of the structure is verified using experimentally obtained frequency response functions and modal parameters. Effective parameters of the tuned vibration absorber design are then determined based on finite element analysis simulations of the vibration suppression performance of the tuned vibration absorber as it is used on the structure. Details of the tuned vibration absorber design are determined and a prototype is fabricated. Prototype tuned vibration absorber is then characterized experimentally both as a standalone system and also as it is used on the main structure. Vibration reduction performance of the physical prototype of the tuned vibration absorber is also compared with its vibration reduction performance estimated from finite element analysis simulations so that the analysis based design process can be validated.

Communicated by Dumitru Caruntu.     

Stability analysis for milling process

In this article, the stability of a milling process is studied by using a semi-discretization method. The model of the workpiece–tool system includes loss-of-contact effects between the workpiece and the tool and time-delay effects associated with the chip-thickness variation. In addition, feed-rate effects are also considered. The governing system of equations is a non-autonomous, delay-differential system with time-periodic coefficients. Stability of periodic orbits of this system is studied to predict the onset of chatter and numerical evidence is provided for period-doubling bifurcations and secondary Hopf bifurcations. Stability charts generated using the semi-discretization method are found to compare well with the corresponding results obtained through time-domain simulations.     

Investigation of the onset of bioconvection in a suspension of oxytactic microorganisms subjected to high-frequency vertical vibration

This paper investigates the effect of vertical vibration on the stability of a dilute suspension of oxytactic microorganisms in a shallow horizontal fluid layer. For the case of high-frequency vibration, an averaging method is utilized to derive the equations describing the mean flow by decomposing the solutions of governing equations into two components: one that varies slowly with time, and a second that varies rapidly with time. Linear stability analysis is used to investigate the stability of the obtained averaged equations. It is predicted that high-frequency, low-amplitude vertical vibration has a stabilizing effect on a suspension of oxytactic microorganisms confined in a shallow horizontal layer. PACS 47.27 Te; 68.60 Dv     

Self-Excited Vibration of a Lap Spring Clutch Used in Electrophotography Fuser

The self-excited chatter vibration of a spring clutch used in an electrophotography fuser was investigated. From the results of theoretical and experimental investigations, the following points were deduced: (1) the vibration is induced by an asymmetric friction force due to asymmetric contact at the slipping surface; (2) Lissajous figures of the vibration were circular and the vibration was forward-whirling; and (3) calculated results based on the present model qualitatively agreed with experimental observations. Several methods of suppressing vibration were proposed. The spring clutch should be designed such that it has the following characteristics: (a) sufficient damping; (b) low friction coefficient; (c) sufficient gap during the slipping period and perfect fastening when transferring torque; and (d) small static offset displacement. These measures have been used to overcome the abnormal chatter vibration problem.     

Chaotic vibrations in high-speed milling

A large number of literatures are devoted to the stability of the milling process and various control methods for chatter suppression. But chaotic dynamics beyond the stable region has not been considered extensively. Moreover, modeling issues for chaotic motion need more challenge for accurate prediction of its complex dynamical behavior. This paper presents a detailed two-degree-of-freedom mechanics based model for the study of chaotic vibrations in milling. Segmental multiple regenerative effect that is the principle feature of nonlinear vibrations in milling processes besides two state dependent time delays has been considered. Exact geometrical formulation of multiple regenerative effects by considering simultaneously different numbers of delayed tool positions over the cutting zone is presented for the first time. Phase portrait, bifurcation diagram, largest Lyapunov exponent, and surface profile were calculated for a given machine tool and workpiece parameters. The simulation results show positive values of the largest Lyapunov exponent corresponding to the existence of chaos in high-speed milling operations. Also, investigation of the machined surface of the workpiece formed by the helical mill demonstrates an irregular pattern on the surface.     

Bifurcation analyses on the chatter vibrations of a turning process with state-dependent delay

Bifurcation analyses are performed using the methods of multiple scales and harmonic balance in order to investigate the chatter vibration characteristics of a nonlinear turning system with state-dependent delay. In this study, the tool of the turning system is modeled as a two degrees of freedom oscillator and both the nonlinear stiffness and the nonlinear cutting force are considered. Prior to performing the bifurcation analyses, the nonlinear cutting force is appropriately expanded using a Taylor series, and an eigenanalysis is performed on the linearized system to obtain the linear stability boundaries. The bifurcation analyses are then performed and examined to explain the effect of state-dependent delay on the small- and large-amplitude behavior of the turning system. Analytic results derived from this study are validated through direct comparison with numerical results.     

A novel nonlinear contact stiffness model of concrete–steel joint based on the fractal contact theory

The heavy-duty machine tool is usually assumed in the concrete foundation, in which the machine tool-foundation joints have a critical effect on the working accuracy and life of heavy-duty machine tool. This paper proposed a novel contact stiffness model of concrete–steel joint based on the fractal theory. The topography of contact surface exist in concrete–steel joint has a fractal feature and can be described by fractal parameters. Asperities are considered as elastic, plastic deformation in micro-scale. However, the asperities of concrete surface will be crushed when the stress is larger than their yield limit. Then, the force balance of contact surfaces will be broken. Here, an iteration model is proposed to describe the contact state of concrete–steel joint. Because the contact asperities cover a very small proportion (less than 1%), the load on crushed asperities is assumed carried by other no contact asperities. This process will be repeated again and again until the crushed asperities are not being produced under external load. After that, the real contact area, contact stiffness of the concrete–steel joint can be calculated by integrating the asperities of contact surfaces. Nonlinear relationships between contact stiffness and load, fractal roughness parameter G, fractal dimension D can be revealed based on the presented model. An experimental setup with concrete–steel test-specimens is designed to validate the proposed model. Results indicate that the theoretical vibration mode shapes agree well with the experimental variation mode shapes. The errors between theoretical and experimental natural frequencies are less than 4.18%. The presented model can be used to predict the contact stiffness of concrete–steel joint, which will provide a theoretical basis for optimizing the connection characteristic of machine tool-concrete foundation.