When an aircraft is turning,the pilot always undergoes a drastic vibration,which is induced by a maneuver load. Zhu and Chen^{[1, 2]} has modeled an aero-engine rotor system,considering the space maneuverable flight. It is shown that the additional inertial forces as well as the additional damping effect and the additional stiffness effect will be produced in the rotor system when the aircraft is maneuvering in the space. Especially,when the aircraft is hovering or doing a dive-hike flight at a fixed speed,the maneuver loads will be simplified as constants.

Wei and Fan^[3] set up the maneuver load formula for aircraft’s hovering and dive-hike flights for the first time. Thereafter,many researchers^{[4, 5, 6, 7]} used this formula to study the dynamic characteristics of the aero-engine’s rotor system under different operating conditions. However, none of these works use analytical methods. Moreover,in the numerical studies,only a fixed maneuvering flight condition was supposed with a fixed maneuver load,and the relationship between the change of the maneuver load and the periodic solution behavior of the rotor system has not been investigated.

Singularity theory^{[8, 9]} is an effective method for analyzing the bifurcation behaviors of the periodic solutions of a vibration system. Qin and Chen^[10] extended the singularity theory to two-variable systems,proposed formulae to calculate the transition sets,and studied the galloping of the transmission line successfully. Li and Chen^[11] derived the transition set for- mulae for the two-variable systems with constraints,and studied the bending torsion coupling vibration problems of a rotor system with these formulae. Zhang et al.^[12] successfully used the two-variable singularity method to investigate the synchronous full annular rubbing problem of a rotor system.

In this paper,the differential equations of a Jeffcott rotor system with nonlinear elastic supports are set up by use of the Lagrange equation. Under the condition of aircraft’s hovering, the maneuver loads are reduced to constants. The qualitative is between the system vibration characteristics and the parameters of the constant maneuver load G and the eccentricity E. In the EG-plane,two hysteresis point set curves and one double limit point set curve are determined,according to which the bifurcation diagrams are obtained. Finally,the system bifurcation mechanism is analyzed.

Figure 1 shows the model of the rotor system with a biased rotor and several nonlinear elastic supports.

Fig. 1 Rotor system model

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The motion equations of the system in the dimensionless form are presented as follows^[13]:

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In the above equations,the expressions of the parameters are given in Ref. [13].

It has been indicated that the impact of q3 and q₄ is not distinguished. Thus,q₃ and q₄ can be supposed to be the magnitude of ε,where ε is a small parameter. The damping,the nonlinearity,and the exciting forces can also be supposed to be the magnitude of ε in the case of studying the primary resonance of the system. When the plane is hovering at a fixed speed, c₁,c₂,c₃,c₄,c₅,and G1 are all zero in the differential equations of the system,and G₂ is a constant. Therefore,(1a) and (1b) can be simplified as follows:

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where the parameters are expressed as follows

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Take the following transformation for q₂:

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Let

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Then,(2a) and (2b) can be transformed into

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where

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Use the multiple-scale method to solve (6) at the primary resonance. Let

Suppose the tuning parameters σ₁ and σ₂ as follows:

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Suppose the solutions of (6) as follows:

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Substituting (8) and (9) into (6) and equating the coefficients of ε₀ and ε₁ to zero,we can obtain that when ε₀ = 0,

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and when ε₁ = 0,

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where

From (10),the equations can be supposed to be

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Substituting (12) into (11) and eliminating the secular terms,we can obtain

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where

Suppose that the motion track of the rotor is approximately square elliptic. Then,we have

Let the right-sides of (13) be zero,and eliminate φ₁ and φ₂. Then,the bifurcation equations can be obtained as follows:

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Take the left-sides of (14) as the engineering unfolding,we can get

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where X = A² and Y = B² are the two variables,λ = Ω ² is the bifurcation parameter,and β₁ = G₂ and β₂ = E₂ are the unfolding parameters.

The derivatives of (15) are as follows:

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According to the two-variable singularity theory^[10],the calculation formulae of the transition sets are as follows:

(i) The bifurcation set is

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(ii) The hysteresis set is

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(iii) The double limit set is

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According to (16) and (17),the relationship curves of β₁ and β₂,namely,G and E,can be derived (see Fig. 2,where a₄ = 0.037 6,and other system parameters are taken from Ref. [13]).

Fig. 2 Curves of transition sets (GE-plane)

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In Fig. 2,H₁,H₂,and DL are three lines,starting from the point P. That is to say,when a constant maneuver is loaded,one point from the hysteresis set line of the a₄E-plane will emit three lines (two hysteresis set lines and one double limit set line) onto the GE-plane. Moreover, there are two hysteresis set curves and one double limit set curve,dividing the GE-plane into four regions. Taking the parameter points from the four regions and the hysteresis set lines H₁ and H2 and the double limit set line D_L,respectively,we can derive the bifurcation diagrams (see Figs. 3−9).

Fig. 3 Bifurcation diagram of Region I (E = 0.06 and G = 2.5)

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Fig. 4 Bifurcation diagram of Region II (E = 0.13 and G = 3)

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Fig. 5 Bifurcation diagram of Region III (E = 0.25 and G = 2.1)

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Fig. 6 Bifurcation diagram of Region IV (E = 0.25 and G = 1.4)

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Fig. 7 Bifurcation diagram of Line H1 (E = 0.135 and G = 3)

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Fig. 8 Bifurcation diagram of Line H2 (E = 0.1 and G = 3)

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Fig. 9 Bifurcation diagram of Line DL (E = 0.25 and G = 1.98)

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Actually,the vibration characteristics of q₁ is identical to that of q₂ (q₃) in the rotor system without a maneuver load. However,when a constant maneuver load is induced into the rotor system,both the linear natural frequencies of q₁ and q₂ are expected to increase according to (7). In addition,the increase in the q₂ degree of freedom is three times as that in the q₁ degree of freedom. As a result,the jump zones on the q₁ degree of freedom and the q₂ degree of freedom will move to the right,and the movement on the q2 degree of freedom is faster than the movement on the q₁ degree of freedom. Therefore,the jump zones of both the degrees of freedom will misplace each other,and the secondary jump phenomenon will be produced due to the nonlinear coupling effects of the system (see Figs. 5-6). With the increase in the maneuver load,when the two jump zones separate from each other,the double limit set emerges (see Fig. 9). Moreover,the maneuver load will narrow the jump zones (see Figs. 3-4). With the disappear of the jump zones of q₂ and q₁ one after another,the hysteresis sets H₁ and H₂ appear correspondingly (see Figs. 7-8).

This paper focuses on the characteristics of an aero-engine rotor system under the condition that the aircraft is hovering at a fixed speed. The effect of the hovering flight on the rotor system is modeled to be a constant load. The effect of the constant maneuver load on the characteristics of the rotor system is studied by using the two variable singularity method. The conclusions are as follows:

Maneuver load will increase the linear natural frequencies of the system due to the nonlinear interaction. Thus,the jump zones of the system will move to the right. Therefore,the exciting force frequency will be contained within the jump zones,and cause to the multiplicity of the solutions of the system if the rotor works at a speed above the first critical speed of the system.

When the aircraft is hovering at a fixed speed,the movement of the jump zone on the q₂ degree of freedom is faster than that on the q₁ degree of freedom under the constant maneuver load. As a result,different bifurcation performances will be produced by the nonlinear inter- action of the system according to different overlapping modes of the two jump zones. Two hysteresis set lines and one double limit set line are derived in the GE-plane.

When the maneuver load is decreased to zero,the three lines of the transition sets degenerate to one point,which corresponds to the point P on the hysteresis set line in the a₄E-plane. That is to say,the new bifurcation modes are induced by the maneuver load. This result lays a theoretical foundation for controlling the stability of aero-engine’s rotor system under a maneuver load.