Eigenvalue and stability analysis for transverse vibrations of axially noving strings based on Hamiltonian dynamics

The Hamiltonian dynamics is adopted to solve the eigenvalue problem for transverse vibrations of axially moving strings. With the explicit Hamiltonian function the canonical equation of the free vibration is derived. Non-singular modal functions are obtained through a linear, symplectic eigenvalue analysis, and the symplectic-type orthogonality conditions of modes are derived. Stability of the transverse motion is examined by means of analyzing the eigenvalues and their bifurcation, especially for strings transporting with the critical speed. It is pointed out that the motion of the string does not possess divergence instability at the critical speed due to the weak interaction between eigenvalue pairs. The expansion theorem is applied with the non-singular modal functions to solve the displacement response to free and forced vibrations. It is demonstrated that the modal functions can be used as the base functions for solving linear and nonlinear vibration problems.

A Methodology for Assessing Axial Compressor Stability with Inlet Temperature Ramp Distortion

Based on a small perturbation stability model for periodic flow,the effects of inlet total temperature ramp distortion on the axial compressor are investigated and the compressor stability is quantitatively evaluated.In the beginning,a small perturbation stability model for the periodic flow in compressors is proposed,referring to the governing equations of the Harmonic Balance Method.This stability model is validated on a single-stage low-speed compressor TA36 with uniform inlet flow.Then,the unsteady flow of TA36 with different inlet total temperature ramps and constant back pressure is simulated based on the Harmonic Balance Method.Based on these simulations,the compressor stability is analyzed using the proposed small perturbation model.Further,the Dynamic Mode Decomposition method is employed to accurately extract pressure oscillations.The two parameters of the temperature ramp,ramp rate and Strouhal number,are discussed in this paper.The results indicate the occurrence and extension of hysteresis loops in the rows,and a decrease in compressor stability with increasing ramp rate.Compressor performance is divided into two phases,stable and limit,based on the ramp rate.Furthermore,the model predictions suggest that a decrease in period length and an increase in Strouhal number lead to improved compressor stability.The DMD results imply that for compressors with inlet temperature ramp distortion,the increase of high-order modes and oscillations at the rotor tip is always the signal of decreasing stability.

Effect of fore/aft-loaded rotor on compressor stability under inlet circumferential distortion

A modified small perturbation stability prediction model for axial compressors with circumferential inlet distortions is established and applied to investigate the effect of fore/aft-loaded rotor on compressor stability under circumferentially distorted inlet conditions.The inlet total pressure distribution downstream of the distortion screen is measured in experiments and employed for simulations which are implemented via time-space spectral method.The stall inception prediction results via the stability model indicate that the compressor with aft-loaded rotor not only performs better in terms of stability under uniform inlet,but also maintains a larger stability margin under circumferentially distorted inlet.The experiments for compressors with fore-loaded and aft-loaded rotor are respectively carried out.The results validate the reliability of numerical simulations and the predicted conclusion that the aft-loaded rotor is beneficial for compressor stability.Besides,the ability of the developed theoretical model for compressor stability prediction under circumferential distortions is confirmed.In addition,dynamic pressure signals at rotor tip measured in experiments illustrate that the circumferential distortion has little effect on the compressor stall pattern.

Experimental and numerical study of tip injection in a subsonic axial flow compressor

Parametric study of tip injection was implemented experimentally on a subsonic axial flow compressor to understand the underlying flow mechanisms of stability improvement of the compressor with discrete tip injection.Injector throat height varied from 2 to 6 times the height of rotor tip clearance,and circumferential coverage percentage ranged from 8.3% to 25% of the annulus.Static pressure fluctuations over the rotor tip were measured with fast-response pressure transducers.Whole-passage time-accurate simulations were also carried out to help us understand the flow details.The combinations of tip injection with traditional casing treatments were experimentally studied to generate an engineering-acceptable method of compressor stall control.The results indicate that the maximum stability improvement is achieved when injectors are choked despite their different sizes.The effect of circumferential coverage percentage on compressor stability depends on the value of injector throat height for un-choked injectors,and vice versa.Tip blockage in the blade passage is greatly reduced by the choked injectors,which is the primary reason for stability enhancement.The accomplishment of blockage diminishment is maintained in the circumferential direction with the unsteady effect of tip injection,which manifests as a hysteresis between the recovery of tip blockage and the recovery of tip leakage vortex.The unsteady effect is primarily responsible for the effectiveness of tip injection with a partial circumferential coverage.Tip injection cannot enhance the stability of the rotor with axial slots significantly,but it can improve the stability of the rotor with circumferential grooves further.The combined structure of tip injection with circumferential grooves is an alternative for engineering application.