高速微小孔振动钻削主轴系统动力学分析
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摘要
本文根据实际微小孔钻床的结构特点,将主轴系统简化为六个阶梯轴段。采用有限单元法和Timoshenko梁-轴理论,结合虚功原理和拉格朗日原理,在考虑了主轴系统的陀螺力矩、钻削轴向力、主轴轴承和比例阻尼等因素对主轴系统弯曲变形影响的基础上,建立了包含回转系统偏心的微小孔钻床主轴系统转子动力学模型,给出了主轴系统弯曲变形产生的正应力和剪切应力的计算式。在上述工作基础上,分别分析了普通钻削和振动钻削时,钻头悬伸长度、主轴轴承位置、轴承刚度和阻尼、轴向力恒定部分、轴向力幅度、轴向力频率和主轴转速等系统参数对轴段截面弯曲正应力、主轴系统共振特性和稳定性的影响。综合微小孔钻头的弯曲应力疲劳特性和主轴系统的稳定性,定义了评价振动钻削主轴系统转子动力学性能优劣的指标,给出了基于该指标的主轴系统合理参数。
Micro-holes have been employed widely in the fields of aviation, space, navigation, automobile, electronics, chemical industry, medicine, optical fiber communications and flow control, et al. High-speed drilling is an economic, practical and widely-used processing technology for the micro-holes. And the technology is much better than other processing methods in efficiency and precision. Furthermore, the novel technology of high-speed vibration-drilling for the micro-holes can increase the drilling quality, prolong the life of tool, and improve the processing performance of composite materials and some more special material. However, the problem of micro-drills breaking hinders development of the novel technology.
For the small stiffness of micro-drills and inevitable eccentricity of spindle system constructed by spindle, clamp and micro-drills, the micro-drills bends under the high-speed rotation of the spindle system and axial thrust of drilling. The bending causes the bending stresses in the cross-section of micro-drills, even result in the micro-drills breaking. The axial thrust of vibration-drilling varies in some period and amplitude. From dynamics, the spindle system of vibration-drilling can be modeled as a periodically time-varying parametrically excited system, and unbounded growth of a small disturbance can occur in the parametrically excited system, the parametric instability namely. If the improper parameters of vibration-drilling which could cause system resonance and even the parametric instability are employed, then the micro-drills must break.
From the structure of micro-drilling machine, the spindle system can be modeled and simplified as six segments of spindle A, shaft B, up-clamp C, down-clamp D, drill E and the drill with spiral groove F in this paper. Based on the precondition of equivalent bending stresses, the drill with spiral groove F can be equalized as a uniform shaft. The kinetic and potential energy of shaft element are obtained from finite element method and the theory of classic Timoshenko beam-shaft with shear deformation. With Lagrange principle, the mass, gyroscopic, stiffness and axial force induced stiffness matrices of shaft element are deduced. And centrifugal force from system eccentricity and supporting force from bearing are obtained from the principle of virtual work. As a result, from finite element method, rotor-dynamic model of spindle system of the micro-drilling machine is achieved with the relation of the shaft segments and the models of centrifugal force and supporting force from bearing. The rotor-dynamic model includes the effects of gyroscopic moment, axial thrust of drilling, bearing and centrifugal forces, and proportional damping on the spindle system bending.
From experimental data of drilling forces (axial thrust and torque) in traditional drilling, the fluctuation of the drilling forces is small compared with the mean drilling forces in normal drilling process. With the small fluctuation, the drilling forces in the traditional drilling can be considered as a constant one to facilitate the calculation in the study of dynamic stresses of micro-drills. Steady state response of the spindle system bending can be deduced from the rotor-dynamic model of spindle system for the traditional drilling, and then the bending normal and shear stresses are provided. As a result, there is a suitable clamping length of micro-drill with minimum stresses in the dangerous cross-section of the drill. Fixing the bearings at the ends of the spindle (increasing the bearings span) and employing bearings with large damping can reduce the dynamic stresses of the spindle system and improve the Processing Performance. When the spindle system rotates around resonance speed, the resonance should causes evident increase of dynamic stresses, and then the micro-drills break. The dynamic stresses increases with the axial thrust of drilling, and the micro-drills break could break when the axial thrust approaches the bulking loads of spindle system.
The axial thrust of vibration micro-drilling spindle system can be separated as a constant part and a periodically fluctuating part. As a periodically time-varying parametrically excited system, the rotor-dynamic model of vibration micro-drilling spindle system can be achieved. Analytical expression of steady state response of the system is deduced from the harmonic balance method in complex exponent form and the operation of block matrices, and the calculation method of bending normal and shear stresses are provided. The resonance condition of the system is achieved in eigenvalue problem from the theory of differential equation and vibration mechanics. Quadratic eigenvalue problem for characteristic exponent is expressed from harmonic balance method, the real part of the characteristic exponent is the natural frequency of system, and the image one can be used to determined parametric stability of the spindle system.
The bending stresses of E-F cross-section and resonance condition of the spindle system are investigated in this paper. The results show that the bending normal and shear stresses in the cross-section of E-F segment for vibration micro-drilling varies periodically with the fluctuation of axial thrust, but the variation is not synchronized due to the phase factors of damping. The normal stresses is 10 times larger than the shear stresses, thus the shear stresses can be neglected. The periodic fluctuation of bending normal stress should induce alternating stress in the vibration micro-drilling, and the alternating stress can result in fatigue damage of the micro-drills. And the fundamental factors of the fatigue damage can be measured by the maximum, minimum, mean and amplitude of the bending stress. Therefore, with the plot of single-factor and the gray image of double-factor, the maximum, minimum, mean and amplitude of the bending stress are investigated from the effects of system parameters, such as constant part, amplitude and frequency of axial thrust, rotating speed of spindle system, fixing location of bearings, clamping length of micro-drill, inertia and stiffness damping. As a result, the maximum, minimum and mean of the bending stress vary identically with the system parameters. The amplitude of axial thrust entirely increases the amplitude of bending stress, but the effect of constant part of axial thrust in not evident. When system resonates, the amplitude of axial thrust increases the maximum, minimum, mean and amplitude of the bending stress, and widens the range of resonance parameter causing large stress. The effects of frequency of axial thrust become obvious when system resonates. Effect of spindle speed on the difference between maximum and minimum of bending stresses is not evident. But compared with constant part, amplitude and frequency of the axial thrust, the spindle speed affects the maximum and minimum of bending stresses more evidently. With the increase of the span of spindle bearings and suspending length of micro-drills, the maximum, minimum and mean of the bending stress decrease. Furthermore, the span of spindle bearings and suspending length of micro-drills also can result in system resonance. The maximum, mean and amplitude of the bending stresses can be decreased by the inertial and stiffness damping of spindle system, thus large damping material should be selected to reduce the bending stresses. The minimum of bending stress decreases consistently with the inertial damping, but it increases firstly, and then decreases with the stiffness damping. And the reduction performance of the inertial damping is better than stiffness damping, but all of them can not cause system resonance. When system resonates, the amplitude of bending stresses should grow dramatically. The resonance of spindle system for vibration micro-drilling can be divided into two categories: the resonance of spindle speed and the resonance of axial thrust excitation. For the resonance of spindle speed, the maximum and minimum of the bending stress increase simultaneously, and the resonance obviously affects the maximum, minimum, mean and amplitude of the bending stresses. However, for the resonance of axial thrust excitation, the maximum bending stress increases, but the minimum one decreases correspondingly. The resonance induced by constant part, amplitude and frequency of axial thrust, fixing location of spindle bearings and suspending length of micro-drills all can be categorized as the resonance of axial thrust excitation.
As the periodically time-varying parametrically excited system, the spindle system of vibration micro-drilling can induce unbounded growth of a small disturbance. When the curves of natural frequencies overlapped with each other, there must be a characteristic exponent whose real part is greater than zero, and then the parametric instability should occur in the spindle system. However, the inertial and stiffness damping can reduce the real part of characteristic exponent to improve system stability. The parametric instability induced by the axial thrust of spindle system is determined by the constant part, amplitude and frequency of the thrust, but could not be influenced by the spindle speed. The constant part of axial thrust of drilling reduces the unstable frequency of axial thrust, and widens the range of parameters inducing system instability. The occurrence of parametric instability is determined by the frequency of axial thrust of drilling, and the amplitude of the axial thrust determines the extent of instability and the range of unstable frequency of axial thrust.
Combining the bending fatigue properties of micro-drills and stability of spindle system, a factor is defined to indicate the rotor-dynamic performance of spindle system of vibration micro-drilling. Using the factor, some suitable parameters with good rotor-dynamic performance are listed.
Micro-holes have been employed widely in the fields of aviation, space, navigation, automobile, electronics, chemical industry, medicine, optical fiber communications and flow control, et al. High-speed drilling is an economic, practical and widely-used processing technology for the micro-holes. And the technology is much better than other processing methods in efficiency and precision. Furthermore, the novel technology of high-speed vibration-drilling for the micro-holes can increase the drilling quality, prolong the life of tool, and improve the processing performance of composite materials and some more special material. However, the problem of micro-drills breaking hinders development of the novel technology.
For the small stiffness of micro-drills and inevitable eccentricity of spindle system constructed by spindle, clamp and micro-drills, the micro-drills bends under the high-speed rotation of the spindle system and axial thrust of drilling. The bending causes the bending stresses in the cross-section of micro-drills, even result in the micro-drills breaking. The axial thrust of vibration-drilling varies in some period and amplitude. From dynamics, the spindle system of vibration-drilling can be modeled as a periodically time-varying parametrically excited system, and unbounded growth of a small disturbance can occur in the parametrically excited system, the parametric instability namely. If the improper parameters of vibration-drilling which could cause system resonance and even the parametric instability are employed, then the micro-drills must break.
From the structure of micro-drilling machine, the spindle system can be modeled and simplified as six segments of spindle A, shaft B, up-clamp C, down-clamp D, drill E and the drill with spiral groove F in this paper. Based on the precondition of equivalent bending stresses, the drill with spiral groove F can be equalized as a uniform shaft. The kinetic and potential energy of shaft element are obtained from finite element method and the theory of classic Timoshenko beam-shaft with shear deformation. With Lagrange principle, the mass, gyroscopic, stiffness and axial force induced stiffness matrices of shaft element are deduced. And centrifugal force from system eccentricity and supporting force from bearing are obtained from the principle of virtual work. As a result, from finite element method, rotor-dynamic model of spindle system of the micro-drilling machine is achieved with the relation of the shaft segments and the models of centrifugal force and supporting force from bearing. The rotor-dynamic model includes the effects of gyroscopic moment, axial thrust of drilling, bearing and centrifugal forces, and proportional damping on the spindle system bending.
From experimental data of drilling forces (axial thrust and torque) in traditional drilling, the fluctuation of the drilling forces is small compared with the mean drilling forces in normal drilling process. With the small fluctuation, the drilling forces in the traditional drilling can be considered as a constant one to facilitate the calculation in the study of dynamic stresses of micro-drills. Steady state response of the spindle system bending can be deduced from the rotor-dynamic model of spindle system for the traditional drilling, and then the bending normal and shear stresses are provided. As a result, there is a suitable clamping length of micro-drill with minimum stresses in the dangerous cross-section of the drill. Fixing the bearings at the ends of the spindle (increasing the bearings span) and employing bearings with large damping can reduce the dynamic stresses of the spindle system and improve the Processing Performance. When the spindle system rotates around resonance speed, the resonance should causes evident increase of dynamic stresses, and then the micro-drills break. The dynamic stresses increases with the axial thrust of drilling, and the micro-drills break could break when the axial thrust approaches the bulking loads of spindle system.
The axial thrust of vibration micro-drilling spindle system can be separated as a constant part and a periodically fluctuating part. As a periodically time-varying parametrically excited system, the rotor-dynamic model of vibration micro-drilling spindle system can be achieved. Analytical expression of steady state response of the system is deduced from the harmonic balance method in complex exponent form and the operation of block matrices, and the calculation method of bending normal and shear stresses are provided. The resonance condition of the system is achieved in eigenvalue problem from the theory of differential equation and vibration mechanics. Quadratic eigenvalue problem for characteristic exponent is expressed from harmonic balance method, the real part of the characteristic exponent is the natural frequency of system, and the image one can be used to determined parametric stability of the spindle system.
The bending stresses of E-F cross-section and resonance condition of the spindle system are investigated in this paper. The results show that the bending normal and shear stresses in the cross-section of E-F segment for vibration micro-drilling varies periodically with the fluctuation of axial thrust, but the variation is not synchronized due to the phase factors of damping. The normal stresses is 10 times larger than the shear stresses, thus the shear stresses can be neglected. The periodic fluctuation of bending normal stress should induce alternating stress in the vibration micro-drilling, and the alternating stress can result in fatigue damage of the micro-drills. And the fundamental factors of the fatigue damage can be measured by the maximum, minimum, mean and amplitude of the bending stress. Therefore, with the plot of single-factor and the gray image of double-factor, the maximum, minimum, mean and amplitude of the bending stress are investigated from the effects of system parameters, such as constant part, amplitude and frequency of axial thrust, rotating speed of spindle system, fixing location of bearings, clamping length of micro-drill, inertia and stiffness damping. As a result, the maximum, minimum and mean of the bending stress vary identically with the system parameters. The amplitude of axial thrust entirely increases the amplitude of bending stress, but the effect of constant part of axial thrust in not evident. When system resonates, the amplitude of axial thrust increases the maximum, minimum, mean and amplitude of the bending stress, and widens the range of resonance parameter causing large stress. The effects of frequency of axial thrust become obvious when system resonates. Effect of spindle speed on the difference between maximum and minimum of bending stresses is not evident. But compared with constant part, amplitude and frequency of the axial thrust, the spindle speed affects the maximum and minimum of bending stresses more evidently. With the increase of the span of spindle bearings and suspending length of micro-drills, the maximum, minimum and mean of the bending stress decrease. Furthermore, the span of spindle bearings and suspending length of micro-drills also can result in system resonance. The maximum, mean and amplitude of the bending stresses can be decreased by the inertial and stiffness damping of spindle system, thus large damping material should be selected to reduce the bending stresses. The minimum of bending stress decreases consistently with the inertial damping, but it increases firstly, and then decreases with the stiffness damping. And the reduction performance of the inertial damping is better than stiffness damping, but all of them can not cause system resonance. When system resonates, the amplitude of bending stresses should grow dramatically. The resonance of spindle system for vibration micro-drilling can be divided into two categories: the resonance of spindle speed and the resonance of axial thrust excitation. For the resonance of spindle speed, the maximum and minimum of the bending stress increase simultaneously, and the resonance obviously affects the maximum, minimum, mean and amplitude of the bending stresses. However, for the resonance of axial thrust excitation, the maximum bending stress increases, but the minimum one decreases correspondingly. The resonance induced by constant part, amplitude and frequency of axial thrust, fixing location of spindle bearings and suspending length of micro-drills all can be categorized as the resonance of axial thrust excitation.
As the periodically time-varying parametrically excited system, the spindle system of vibration micro-drilling can induce unbounded growth of a small disturbance. When the curves of natural frequencies overlapped with each other, there must be a characteristic exponent whose real part is greater than zero, and then the parametric instability should occur in the spindle system. However, the inertial and stiffness damping can reduce the real part of characteristic exponent to improve system stability. The parametric instability induced by the axial thrust of spindle system is determined by the constant part, amplitude and frequency of the thrust, but could not be influenced by the spindle speed. The constant part of axial thrust of drilling reduces the unstable frequency of axial thrust, and widens the range of parameters inducing system instability. The occurrence of parametric instability is determined by the frequency of axial thrust of drilling, and the amplitude of the axial thrust determines the extent of instability and the range of unstable frequency of axial thrust.
Combining the bending fatigue properties of micro-drills and stability of spindle system, a factor is defined to indicate the rotor-dynamic performance of spindle system of vibration micro-drilling. Using the factor, some suitable parameters with good rotor-dynamic performance are listed.
引文
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