## 1 Introduction  
Tilting pad bearings have been widely applied in many kinds of rotating machines, such as gas turbines [1], compressors [2], and so on, due to their ability to improve stability [3]. Tilting pad bearings are also suitable for oil-free operations [4; 5; 6], so they have the potential for new fields such as electrical submersible pumps [7; 8].  
The dynamic stiffness and damping coefficients of bearings prominently affect the dynamic behavior of a rotor system [9], including critical speed and stability [10; 11]. The pad assembly method was firstly established by [PERSON] [12] for calculating the synchronous dynamic coefficients of tilting pad bearings. [PERSON]'s method requires that the excitation frequency of the pad and shaft is assumed synchronous with the shaft rotation frequency. However, the use of synchronous coefficients tends to overestimate stability, especially in the case of a high Sommerfeld number [13] so the sub-synchronous dynamic coefficients are quite important. [PERSON] studied the effects of thermal boundary condition on the stiffness and damping coefficients for tilting pad bearings [14]. Experimental studies of tilting pad bearings also indicated the frequency dependence of dynamic coefficients [15]. [PERSON] advanced the partial derivative method for the prediction of a complete set of dynamic coefficients of tilting pad gas bearings [16]. Though this method successfully solved the perturbation frequency effect of dynamic coefficients, it is not suitable for turbulent lubrication.  
The turbulence effect on static and dynamic properties of bearings cannot be neglected in heavy loads or high rotational speeds [17]. [PERSON] developed an approximate one-dimensional finite element method for Reynolds equation with turbulence correction factors based on Hirs bulk-flow theory. Then the dynamic characteristics under half speed whirl were investigated [18]. [PERSON] and [PERSON] presented a comprehensive thermodynamic model for double-film floating disk thrust bearings based on Reynolds and energy equations considering turbulence effect [19]. [PERSON] investigated heavy-duty bearings by the transient mixed-lubrication model considering both turbulence in the large film thickness area and contact of asperities near the minimum nominal film thickness [20]. [PERSON] developed the formula for the transition region of the turbulent lubrication model [21].  
However, the dynamical effect of turbulence on stiffness and damping coefficients of high-speed tilting pad bearings was not considered in the turbulent lubrication studies above. The dynamical variations of turbulence coefficients along with the shaft vibration at different frequencies were incorporated into the methods for dynamic coefficients of fix pad bearings lubricated by organic refrigerant [22] and supercritical CO\({}_{2}\)[23]. In this research, by introducing the pad-pivot structural perturbations to the partial derivative method in Ref. [23], it was furthered to calculate the frequency-dependent stiffness and damping coefficients of oil-lubricated tilting pad bearings with laminar and turbulent states. Both the perturbed viscosity and the perturbed turbulence coefficients were embraced in the dynamic model of tilting pad bearings and then the importance of each perturbed variable was studied.