*4.3. 1D Harmonic Shaking Motions*

To validate the assumption proposed above, the circular whirl around the static position with eccentricity ratio 50% is divided into two separate 1D shaking motions, as shown in Figure 22. One is the shaking motion in X direction; the other is in Y direction. The Y direction is also the tangential direction of the concentric whirl. Namely, the Y-directional shaking is somewhat similar to the circular whirl around seal center. The expressions of two shaking motions are, respectively, presented in Equations (17) and (18). The amplitude A is 10% *Cr* and the harmonic frequency Ω corresponds to the speed 10,200 r/min.

$$\begin{cases} \ x = s\varepsilon + A\cos(\Omega t) \\ y = 0 \end{cases} \tag{17}$$

$$\begin{cases} \quad \mathfrak{x} = \mathfrak{se} \\ \quad y = A \sin(\Omega t) \end{cases} \tag{18}$$

**Figure 22.** Two harmonic shaking motions.

According to motion equations of two harmonic shakings, fluid-induced forces can be obtained by nonlinear dynamic model and transient CFD simulations. They are shown in Figures 23 and 24. The comparison with transient CFD simulations shows that the reliability of nonlinear dynamic model is low in predicting fluid forces induced by X-directional shaking. Maximum differences of two approaches are 119 N for *Fx* and 30.8 N for *Fy*. However, as to the Y-directional shaking, the reliability of the nonlinear model is obviously improved, as shown in Figure 24. This can be attributed to the slight likeness of Y-directional shaking with the circular whirl around seal center. In Figure 20, maximum differences of two approaches are 18.5 N for *Fx* and 17.7 N for *Fy*, and they are much smaller than those in Figure 23. Namely, the nonlinear dynamic model is more applicable to rotor disturbances similar to the circular whirl around concentric position. The assumption proposed in Section 4.2 is confirmed and can explain the low reliability of nonlinear dynamic model. Under large eccentricity, the dynamic

characteristics of seals are varied with the motion ways of the rotor, and the nonlinear dynamic model based on a specific motion way is incompetent in dealing with all kinds of rotor disturbances.

**Figure 23.** Fluid forces induced by X-directional shaking.

**Figure 24.** Fluid forces induced by Y-directional shaking.
