A mathematical model of calculating rotordynamic coefficients associated with leakage steam flow through labyrinth seals was presented. Particular attention was given to incorporating thermal properties of the steam fluid into prediction of leakage flow and subsequent derivation of rotordynamic coefficients, which quantitatively characterize influence of aerodynamic forcing of the leakage steam flow on the rotordynamics. By using perturbation analysis, we determined periodic and analytic solutions of the continuity and circumferential momentum equations for the time-dependent flow induced by non-axisymmetric rotation of the rotor encompassed by a labyrinth seal. Pressure distributions along labyrinth seal cavities and rotordynamic coefficients were compared at the same condition for air and steam flows. Influence of steam flow through the labyrinth seal cavities on rotordynamic coefficients was analyzed in terms of inlet pressure, inlet swirl velocity and rotor speed. List of symbolA unsteady cross-sectional area of the cavity [m 2 ] A 0 steady annular flow area [m 2 ] B tooth height [m] C 0 orifice contraction coefficient C 1 kinetic energy carry-over coefficient C r steady radial clearance [m] C x x , C yy damping coefficients [N s/m] C xy , C yx cross-couple damping coefficients [N s/m] Dh 0 , Dh steady and unsteady hydraulic diameter of the cross-sectional area of the cavity [m] F total air reaction force [N] F x , F y annular seal reaction force components [N] h 0 stagnant enthalpy [J/Kg] h iseal enthalpy at the ith orifice [J/Kg] H unsteady labyrinth seal radial clearance [m] H 1 (t, θ) perturbation clearance [m] K x x , K yy direct stiffness coefficients [N/m] K xy , K yx cross-couple stiffness coefficients [N/m] L cavity length [m] N tooth number n specific heat ratio P 0i , P i steady and unsteady pressure at the ith cavity [Pa] P 0iseal , P iseal steady and unsteady pressure at the ith orifice [Pa] P 1i (t, θ) perturbation pressure in the ith cavity [Pa] q 0i ,q i steady and unsteady leakage flow rate per unit length [kg/s m] q 1i (t, θ) perturbation leakage flow rate per unit length [kg/s m] R steam constant [J/kg K] R s rotor radius [m] T i temperature at the ith cavity [K] T iseal temperature at the ith orifice [K] U iseal velocity at the ith orifice [m/s] V 0i , V i steady and unsteady circumferential velocities at the ith cavity [m/s] V 1i (t, θ) perturbation pressure at the ith cavity [m/s] (x, y)x and y coordinates of rotor whirling motionGreek symbols ε eccentricity ratio η imaginary number ρ i , ρ 0i steady and unsteady gas density of the ith cavity [kg/m 3 ] ρ 1 (t, θ) perturbation gas density of the ith cavity [kg/m 3 ] ρ 0iseal , ρ iseal steady and unsteady gas density at the ith clearance [kg/m 3 ] ρ 1iseal (t, θ) perturbation gas density at the ith clearance [kg/m 3 ] τ r 0i , τ ri steady and unsteady shear stresses at the rotor wall [N/m 2 ] τ r 1i (t, θ) perturbation shear stresses at the rotor wall [N/m 2 ] τ s0i , τ si steady and unsteady shear stresses at the stator wall [N/m 2 ] ...

An extensive investigation of the influence of the leakage flow through a labyrinth seal at supply pressure of 12 bar on the rotordynamics was performed by using numerical calculations and experimental measurements. Toward this end, an experimental rotor setup was established in Shanghai Jiao Tong University. Two labyrinth seals were chosen for comparison, e.g., an interlocking seal and a stepped one. The numerical calculations based on the bulk-flow theory and the perturbation analysis were accomplished. Simultaneous acquisitions of the fluctuating static pressure at the stator wall and the displacement of the whirling rotor were made. The influence of the aerodynamic forcing on the rotor was analyzed in terms of the axial distribution of the mean static pressure, the circumferential distribution of the fluctuating pressure, the fist critical speed and the destabilization rotating speed of the rotor. The experimental results demonstrated that the sinusoidal distribution of the fluctuating static pressure on the stator wall was closely related to the whirling motion of the rotor. The first critical speed of the rotor was reduced by the aerodynamic forcing, resulting in intensified destabilization of the rotor system. Furthermore, the numerical analyses were in good agreement to the experimental measurements. List of symbols AUnsteady cross-sectional area of the cavity (m 2 ) A n Steady annular flow area (m 2 ) [C]Damping matrix C 0Orifice contraction coefficient C 1 Kinetic energy carry-over coefficient C r Steady radial seal clearance (m) C x x , C yy Damping coefficients (N-s/m) C xy , C yx Cross-couple damping coefficients (N-s/m)

This paper is concerned with the inverse problem for imaging multiple three-dimensional objects using the information of the far-field pattern of the scattered wave. A spatially dependent function, which has noticeably different values inside and outside the obstacle, is derived. A numerical method based on the characterization is developed to obtain a visualization of the obstacle. The most remarkable advantage of this method is that it does not need any prior knowledge about the geometry and physical properties of the scatterer, and requires only the information of the far-field measurements for a finite number of directions of incidence and observation distributed over a limited range. Furthermore, the scheme is very simple and fast since it avoids the use of the iterative procedure and requires only the solution of a linear system. Some numerical examples with synthetic far-field data are given showing the practicality and efficiency of this scheme.

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In this paper, the global behavior of relative equilibrium states of a three-body satellite with¯exible connection under the action of the gravitational torque is studied. With geometric method, the conditions of existence of nontrivial solutions to the relative equilibrium equations are determined. By using reduction method and singularity theory, the conditions of occurrence of bifurcation from trivial solutions are derived, which agree with the existence conditions of nontrivial solutions, and the bifurcation is proved to be pitchfork-bifurcation. The Liapunov stability of each equilibrium state is considered, and a stability diagram in terms of system parameters is presented. IntroductionThe attitude motion of a rigid body in the gravitational ®eld has been extensively studied over a long period. The study of the libration of the moon can be traced back to Lagrange. In [1], the problem of attitude stability of a rigid body in a circular orbit was analyzed and a stability diagram in a parameter plane was presented, which is famous in the ®eld of spacecraft attitude dynamics. As the construction of modern spacecraft becomes more complicated, multibody models have been developed to replace the single-body model. A linear formulation of the attitude motion of a two-body satellite with¯exible connection was proposed in [2], and some stability problems of the permanent rotation of a torque-free two-body satellite were discussed in [3±6]. In [7], the problem of the global motion behavior of relative equilibrium states of a two-body system with¯exible connection in the gravitational ®eld was investigated.The development of nonlinear theories, such as bifurcation and chaos, has enlarged the research range of spacecraft attitude dynamics. Usually, numerical simulations play an important role in nonlinear areas. On the other hand, analytical models can be of great help in obtaining a qualitative understanding of the complex dynamic behavior of a system. In this paper, analytical methods are used to study the planar attitude motion of a three-body satellite with¯exible connection subjected to the gravitational torque in a circular orbit. The trivial solutions to the equilibrium equations of the satellite relative to the orbital coordinate frame are determined. The conditions of the existence of nontrivial solutions and occurrence of bifurcation from trivial solutions in analytical form are derived. The type of bifurcation is judged and ®nally, Liapunov's direct method is used in the analysis of the stability of each relative equilibrium state. Thus, the global behavior of the motion of the system is described qualitatively. Equations of motionConsider a system composed of three bodies B i Y i 1Y 2Y 3, and two¯exible axes J i Y i 1Y 2, modeled as a spherical joint with spring. The principal axes of the bodies x 1 Y x 2 , (x 2 Y x 3 )