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Calculation of Natural Frequencies on Transverse Vibration for drill strings in Non-Uniform Temperature Field

It is necessary to consider the effect of temperature which changes the mechanical properties of materials when drill strings work in a non-uniform temperature field. In this paper, the natural frequencies of transverse vibration are studied under axial compression in the nonuniform temperature field. Using an asymptotic solution method, first-order approximation formula of the nonlinear vibration equation is obtained. The effects of temperature changes and axial pressure on the natural frequencies are discussed by an example. The results show that the influence of temperature on the critical load is small; change rate of natural frequency increases with the increase of axial pressure; while the axial pressures become larger, the influence of temperature on the first-order frequency is greater than on the second-order’s.


Introduction
Drill string structures can be simplified into elastic straight rods or elastic beams in drilling engineering. Because of increasing temperature, structures are usually in a state of thermal expansion which changes natural frequencies of straight rods or beams [1][2][3]. Domestic research is mainly divided into two categories. Considering the change of temperature in constitutive relations or material elastic modulus, one type obtained the nonlinear motion differential equations of vibration, which were solved in different ways. Another type paid attentions to linear vibration of straight rods or beams and discussed the effect of elastic modulus and geometry change on solution. Li G. Q. et al. [4] introduced the influence of axial force on the beam-column unit displacement function, and derived the geometric nonlinear beam-column unit considering the temperature influence. Based on the Hamilton variational principle, Zhao Y. B. et al. [5] obtained the nonlinear differential equation of motion which introduced temperature variation to the stress-strain relationship of beams and investigated temperature effects on the vibration characteristics to three different boundary conditions. Zhou D. [6] studied the natural frequency of transverse, longitudinal and torsional free vibrations of elastic straight bars in nonuniform temperature field using the asymptotic solution method, but did not consider the axial compression.
Drill strings are large-volume and high-quality pipes in oil drilling. It is extremely important to study their dynamic characteristics [7][8][9]. As temperature increases with the depth of the well, temperature change has a critical impact on drill strings and downhole equipments [10]. Li J. Q. et al. [11] analyzed the influence of high temperature on the natural frequency of drill string transverse vibration, and got the calculation formula of the natural frequency under axial force at any temperature. Temperature distribution and influence on the vibration of drill strings were researched by Dong Z. et al. [12] when their temperature of gas drilling was higher than that of fluid drilling. In this paper, the lateral nonlinear vibration of drill strings subjected to axial compression in a non-uniform temperature AEMCME 2019 IOP Conf. Series: Materials Science and Engineering 563 (2019) 052049 IOP Publishing doi:10.1088/1757-899X/563/5/052049 2 field is studied and the first-order approximation formula of natural frequency is given. The influence of temperature on the natural frequency is discussed.

Basic equation of dynamics
The drill string is simplified into Eular beam model for the non-uniform temperature fields, so the basic dynamic equation of lateral vibration is Suppose there is a separate variables solution, let ( , ) U( ) i t u x t x e ω = , Dimensionless quantity: The vibration equation of beams can be written as For the general temperature distribution, it is expressed as: Experiments had shown that for most metallic materials, the relationship between elastic modulus and temperatures is expressed as: The change of elasticity modulus along length is: Using the asymptotic solution method, consider α as a small parameter.
If only considering the temperature linear distribution along the beam ( ) 1 m = , the first order approximation solution of the natural frequency is: The solution of Eq. (4) The general solution of Eq. (6) is: The special solution of Eq. (6) is: The solution of Eq. (6) is: 1 Therefore, the first order approximate solution of lateral vibration when the temperature is linearly distributed along the beam is  Fig. 1 shows the curve of first-order natural frequency with axial pressure. It can be seen that the natural frequency of the beam decreases with axial pressure when the temperature is constant, namely 1 25 T C =  . The buckling occurs and axial pressure is the critical load when the natural frequency is zero. When the temperature changes, namely temperature difference between the two ends of the beam is 200  , the natural frequency is slightly smaller than that of constant temperature with same axial pressure. It also decreases with axial pressure and the critical load is same with constant temperature. Fig. 2-3 show the change rate curves of first two natural frequencies with temperature. As axial pressure increases, the natural frequency change rate increases gradually. When the axial force is zero, the first two orders of natural frequency change rates are identical. When axial pressure becomes large, the influence of temperature on the first-order natural frequency of the lateral vibration is much greater than it on the second-order. Compared with the paper [11], the influence of temperature on natural frequency in the non-uniform temperature field is smaller than that in the uniform temperature field, which is more suitable for actual engineering conditions.

conclusion
For the problem of transverse nonlinear vibration of drill strings subjected to axial compression in non-uniform temperature field, the first-order approximation formula of natural frequency is given by the progressive solution method, and the influence of temperature on the natural frequency is discussed. From the calculation results, the following conclusions can be got: (1) The critical load of the drill string can not be affected with temperature.
(2) The change rate of natural frequency becomes greater with the axial pressure, but the change is less than it in the uniform temperature field.
(3) The effect of temperature on the first-order frequency is greater than it on the second-order frequency with same axial pressure.