A Method for Stability Analysis of Periodic Delay Differential Equations with Multiple Time-Periodic Delays

Delay differential equations (DDEs) are widely utilized as the mathematical models in engineering fields. In this paper, a method is proposed to analyze the stability characteristics of periodic DDEs with multiple time-periodic delays. Stability charts are produced for two typical examples of time-periodic DDEs about milling chatter, including the variable-spindle speed milling system with one-time-periodic delay and variable pitch cutter milling system with multiple delays. The simulations show that the results gained by the proposed method are in close agreement with those existing in the past literature. This indicates the effectiveness of our method in terms of time-periodic DDEs with multiple time-periodic delays. Moreover, for milling processes, the proposed method further provides a generalized algorithm, which possesses a good capability to predict the stability lobes for milling operations with variable pitch cutter or variable-spindle speed

Chaos Suppression of an Electrically Actuated Microresonator Based on Fractional-Order Nonsingular Fast Terminal Sliding Mode Control

This paper focuses on chaos suppression strategy of a microresonator actuated by two symmetrical electrodes. Dynamic behavior of this system under the case where the origin is the only stable equilibrium is investigated first. Numerical simulations reveal that system may exhibit chaotic motion under certain excitation conditions. Then, bifurcation diagrams versus amplitude or frequency of AC excitation are drawn to grasp system dynamics nearby its natural frequency. Results show that the vibration is complex and may exhibit period-doubling bifurcation, chaotic motion, or dynamic pull-in instability. For the suppression of chaos, a novel control algorithm, based on an integer-order nonsingular fast terminal sliding mode and a fractional-order switching law, is proposed. Fractional Lyapunov Stability Theorem is used to guarantee the asymptotic stability of the system. Finally, numerical results with both fractional-order and integer-order control laws show that our proposed control law is effective in controlling chaos with system uncertainties and external disturbances

Investigation on tribological performance of LIPSS-structured nano-crystalline diamond films

The aim of present work was to investigate the tribological behavior of CVD nano-crystalline diamond (NCD) films with laser-induced periodic surface structures (LIPSS). The LIPSS textures were fabricated in ambient air using a Yb-doped photonic crystal fiber amplifier femtosecond (fs) laser with pulse duration of 200 fs and central wavelength of 1040 nm. Two types of LIPSS texture, namely continuously distributed LIPSS (CDL) texture and a complex texture formed of evenly spaced LIPSS stripes (ESLS), were obtained with the above processing conditions. The tribological behavior of as-fabricated LIPSS textures was evaluated via ball-on-disc tests in dry reciprocating sliding motion, and ZrO2 ceramic balls were used as counterparts. In addition, there were two types of reciprocating direction involved in the frictional tests, namely the direction parallel/vertical to the orientation of LIPSS ripples. The results indicate that as the LIPSS can work as reservoir for wear debris and reduce the contact area during the friction process, the LIPSS textured NCD surface showed obviously reduced friction coefficient in comparison with the conventional NCD surface, which was reduced from 0.42 to 0.09. Besides, for the CDL surface, the friction coefficient as the LIPSS ripples vertical to the reciprocating direction was higher than that as the LIPSS ripples parallel to the reciprocating direction. In the case of ESLS surface, however, the direction of the LIPSS ripples had little influence on the friction coefficient

RESEARCH ON VARIABLE INERTIA CHARACTERISTICS OF RECIPROCATING CRANKSHAFT SYSTEM TAKEN CONNECTING ROD RATIO INTO CONSIDERATION

Variation of connecting rod ratio in reciprocating crankshaft system can affect the value of system equivalent moment of inertia. However,current traditional variable inertia formula does not consider this important factor,which induces certain errors during torsional vibration analysis. This paper aims to investigate the effects of connecting rod ratio on equivalent moment of inertia and torsional vibration. First,a modified variable inertia formula concerning connecting rod ratio is deduced and its accuracy is compared with traditional one. Then,torsional dynamic equation is established and the method of multiple scales is used to derive the average equation under primary resonance condition. The effects of connecting rod ratio on stable dynamics of the system are investigated. Moreover,traditional and modified variable inertia formulas are compared with exact variable inertia through a detailed single-cylinder crankshaft system. And the shortage of traditional one is pointed out. Finally,numerical results on system vibration characteristic based on different variable inertia formulas are shown to verify the importance of connecting rod ratio on system performances

RESEARCH ON CUTTING FORCE MODELLING AND INFLUENCE FACTORS FOR END MILLS WITH VARIABLE-HELIX ANGLE

Variable helix cutter can reduce the location error of the finished surface,leading to a more accurate machining operation.Considering the geometries of variable pitch and helix angle,and the piecewise continuous regions of the cutting,an analytical model for predicting the cutting force for milling process with variable helix angle cutter is presented.The validation of the proposed model is conducted via some cutting tests.Based on proposed model,the influence of variable helix character on cutting force is studied.Analyses show that cutting force coefficients obtained from the traditional cutting tool could be used to accurately predict the cutting forces of the variable helix cutting tool.The bigger helix angle corresponds to the smaller cutting force and bigger width of phase.The character of variable-helix angle has more significant influence on cutting force when axial depth of cut is larger

Investigation on Modeling and Formation Mechanism of Dynamic Rotational Error for Spindle-Rolling Bearing System

In the field of precision machining, the spindle-rolling bearing (SRB) system is widely used on the machine tool as one of the most fundamental and important components. The rotational error motions of the SRB system have significant effects on the machining accuracy (contour accuracy and surface roughness). Over the past decades, much work has been focused on the measurement of spindle balancing and rotational error motions, the vibrations response induced by the nonlinear stiffness and surface waviness of the bearing. However, the formative mechanism of the rotational error motions for the SRB system is not well understood. In this paper, the dynamic model of the SRB system considering the bearing nonlinearity is established. Seeking to reveal the effects of surface waviness of the bearing raceway, unbalance mass and disturbance force on the dynamic rotational error, the modeling method and formative mechanism of the dynamic rotational error for the SRB system is explored both theoretically and experimentally. Then, numerical simulation is performed to analyze the influence of the bearing raceway waviness, unbalance mass and disturbance force on the dynamic rotational error. An experimental setup is established based on a typical SRB system and a series of experiments are carried out. The experimental results are in good agreement with the theoretical and simulation results, which can demonstrate the feasibility and validity of the modeling method. Furthermore, this method can be effectively applied to the design and development phases of an SRB system to improve dynamic rotational accuracy

Stability Analysis Method for Periodic Delay Differential Equations with Multiple Distributed and Time-Varying Delays

Dynamic stability problems leading to delay differential equations (DDEs) are found in many different fields of science and engineering. In this paper, a method for stability analysis of periodic DDEs with multiple distributed and time-varying delays is proposed, based on the well-known semidiscretization method. In order to verify the correctness of the proposed method, two typical application examples, i.e., milling process with a variable helix cutter and milling process with variable spindle speed, which can be, respectively, described by DDEs with the multidistributed and time-varying delays are considered. Then, comparisons with prior methods for stability prediction are made to verify the accuracy and efficiency of the proposed approach. As far as the milling process is concerned, the proposed method supplies a generalized algorithm to analyze the stability of the single milling systems associated with variable pith cutter, variable helix cutter, or variable spindle speed; it also can be utilized to analyze the combined systems of the aforementioned cases