Theoretical and Numerical Analysis of Coupled Axial-Torsional Nonlinear Vibration of Drill Strings

Based on a lumped parameter model with two degrees of freedom, the periodic response of the coupled axial-torsional nonlinear vibration of drill strings is studied by HB-AFT (harmonic balance and alternating frequency/time domain) method and numerical simulations. The amplitude-frequency characteristic curves of axial relative displacement and torsional relative angular velocity are given to reveal the mechanism of bit bounce and stick-slip motion. The stability of periodic response is analyzed by Floquet theory, and the boundary conditions of bifurcation of periodic response are given when parameters such as nominal drilling pressure, angular velocity of turntable, and formation stiffness are varied. The results show that the amplitude of the periodic response of the system precipitates a spontaneous jump and Hopf bifurcation may occur when the angular velocity of the turntable is varied. The variation of parameters may lead to the complex dynamic behavior of the system, such as period-doubling motion, quasiperiodic motion, and chaos. Bit bounce and stick-slip phenomenon can be effectively suppressed by varying the angular velocity of turntable and nominal drilling pressure.

Periodic Temperature Responses in a Thermal System Under a Periodic Heating

In this paper, the periodic temperature responses of a thermal system under a periodic heating input are studied. Using the implicit mapping method, periodic temperature responses varying with excitation frequency are predicted for different input amplitudes. The corresponding stability of the periodic responses are discussed through eigenvalue analysis. The experimental and numerical results of the periodic response are presented for comparison to the analytical results.

Vibrational Amplitude Frequency Characteristics Analysis of a Controlled Nonlinear Meso-Scale Beam

Vibration response and amplitude frequency characteristics of a controlled nonlinear meso-scale beam under periodic loading are studied. A method including a general analytical expression for harmonic balance solution to periodic vibration and an updated cycle iteration algorithm for amplitude frequency relation of periodic response is developed. A vibration equation with the general expression of nonlinear terms for periodic response is derived and a general analytical expression for harmonic balance solution is obtained. An updated cycle iteration procedure is proposed to obtain amplitude frequency relation. Periodic vibration response with various frequencies can be calculated uniformly using the method. The method can take into account the effect of higher harmonic components on vibration response, and it is applicable to various periodic vibration analyses including principal resonance, super-harmonic resonance, and multiple stationary responses. Numerical results demonstrate that the developed method has good convergence and accuracy. The response amplitude should be determined by the periodic solution with multiple harmonic terms instead of only the first harmonic term. The damping effect on response illustrates that vibration responses of the nonlinear meso beam can be reduced by feedback control with certain damping gain. The amplitude frequency characteristics including anti-resonance and resonant response variation have potential application to the vibration control design of nonlinear meso-scale structure systems.

Forced-self-excited System of Iced Transmission Lines under Planar Harmonic Excitations

This paper is concerned with the analysis of the self-excited vibrations and forced vibrations of the iced transmission lines. By introducing the external excitation load, the effect of dynamic wind on the nonlinear vibration equations of motion is reflected by vertical aerodynamic force. The approximate analytical solution of the non-resonance, and the amplitude frequency response relation of the principal resonance of the forced self-excited system are obtained by using the multiple scale method. With the increase in excitation amplitude, the nonlinearity of the system is enhanced, and the forced-self-excited system experiences three vibration stages (self-excited vibration, the superposition form of self-excited vibration and forced vibration, forced vibration controlled by nonlinear damping). Among them, the accuracy of the approximate analytical solution decreases with the increase of the nonlinear strength. And the excitation amplitude is greater than the critical value, the quenching phenomenon appear in the forced-self-excited system, and the discriminant formula is derived in this paper. In addition, the frequency of excitation term determines the vibration form of the system. The principal resonance, super-harmonic resonance and sub-harmonic resonance of the forced-self-excited system are analyzed by using different excitation frequencies. Compared with the principal resonance and the harmonic resonance, the meaningful transition from periodic response to quasi periodic response is easy to appear with the condition of the 1/3-order sub-harmonic and the 3-order super-harmonic. The conclusions would be helpful to the practical engineering of the iced transmission lines. More important, as a combination of Duffing equation and Rayleigh equation, the forced-self-excited system also has high theoretical research value.

Bifurcation Analysis of a Micro-Machined Gyroscope with Nonlinear Stiffness and Electrostatic Forces

The bifurcation of the periodic response of a micro-machined gyroscope with cubic supporting stiffness and fractional electrostatic forces is investigated. The pull-in phenomenon is analyzed to show that the system can have a stable periodic response when the detecting voltage is kept within a certain range. The method of averaging and the residue theorem are employed to give the averaging equations for the case of primary resonance and 1:1 internal resonance. Transition sets on the driving/detecting voltage plane that divide the parameter plane into 12 persistent regions and the corresponding bifurcation diagrams are obtained via the singularity theory. The results show that multiple solutions of the resonance curves appear with a large driving voltage and a small detecting voltage, which may lead to an uncertain output of the gyroscope. The effects of driving and detecting voltages on mechanical sensitivity and nonlinearity are analyzed for three persistent regions considering the operation requirements of the micro-machined gyroscope. The results indicate that in the region with a small driving voltage, the mechanical sensitivity is much smaller. In the other two regions, the variations in the mechanical sensitivity and nonlinearity are analogous. It is possible that the system has a maximum mechanical sensitivity and minimum nonlinearity for an appropriate range of detecting voltages.

Periodic Response and Stability of a Maglev System with Delayed Feedback Control Under Aerodynamic Lift

In this research, the periodic response and stability of a nonlinear maglev system under the combined effects of steady and unsteady aerodynamic lifts is investigated, considering time delay in the feedback control loop. First, a nonlinear maglev system with a single levitation point that accounts for the nonlinearity of the electromagnetic force, time delay in the feedback control loop, and effect of aerodynamic lift is established. Then the periodic solutions of the maglev system with aerodynamic lift and time delays are obtained by an incremental harmonic balance analysis, in which the explicit time-delay action matrices used indicate that the effect of time delay on the response of the maglev system is periodic. The stability of the periodic solutions based on a finite difference continuous time approximation method and Floquet theory is studied, from which the critical time delay is obtained. Also, the relationship between the periodic vibration amplitude and the time delay is examined, along with the steady aerodynamic lift coefficient, and frequency of the unsteady aerodynamic lift, as well as the variation of critical delay with respect to the position feedback and velocity feedback with the control gain parameters. In addition, the stability boundary for the simultaneous time-delayed position and velocity feedback is obtained.