# The Characteristics and Mechanisms of High-Intensity Sound in a High-Speed Multistage Compressor
## 2 Numerical Calculation Theory
### Rotating Sound Source Theory  
The instability pressure wave \(p(\varphi^{F},t)\) shows the difference at different reference coordinates. The pressure wave form which is locked in the fixed coordinate frame can be expressed as [33]:  
\[p(\varphi^{F},t)=\sum_{n=1}^{\infty}\sum_{\alpha=-\infty}^{+\infty}A_{n\alpha }^{F}\cos(\alpha\varphi^{F}-\omega_{n}^{F}t-\phi_{n\alpha}^{F}) \tag{9}\]  
the direction of rotation during the process of blade vibration, making the time to reach the sensor change, so the vibration parameter information such as blade amplitude, phase, frequency, and so on, are obtained. The schematic diagrams of different circumferential positions of the vibration measuring points of the first-stage rotor blades are shown in Figure 2.  
The adjustment of the deflection angle of the zero-stage stator vanes is associated with the air flow state in the flow channel of the high-pressure compressor in the experiment. The noise signals in the flow channel significantly vary with different deflection angles, and abnormal vibration occurs on R1 at the same time. At the condition of a certain deflection angle of IGV of the compressor and a certain rotational speed, the noise signal with characteristic frequency structure in the flow channel is detected, and high-amplitude vibration occurs on the R1. The time series of the character frequency noise signal monitored by the microphones at the four measuring points is shown in Figure 3. The noise signal spectrum with the characteristic frequency monitored in the experiment is shown in Figure 4, and is characterized with typical broadband. 1 BPF is the blade passing frequency of the first stage of rotor blades, AR is a special characteristic frequency with the highest peak value. The values measured at the four measuring points all present the characteristic frequencies with multiple high-peak discrete pure tone components. These characteristic frequencies are 1402 Hz for the highest peak value of the pure tone component, 6285 Hz for the high-pressure R1 passing frequency (1 BPF) and 12,595 Hz for the multiplier of the high-pressure R1 passing frequency. Moreover, 4883 Hz had a certain combined relationship with the pass frequencies of the high-pressure R1. Figure 1: Measuring points for sound pressure in compressor casing. Figure 2: Measuring points for rotor blade vibration. The internal noise and vibration of R1 in the high-pressure compressor were simultaneously measured under different rotational speed conditions in the experiment. The relationship between the vibration displacement of R1 and the sound pressure level (SPL) of the internal characteristic frequency with the rotational speed was obtained through the arranged vibration and noise measurement points, as shown in Figure 5. It is obvious that the vibration amplitude of the high-pressure R1 and the sound pressure level of the noise signal at the characteristic frequency measured directly above the rotor blade increased and decreased synchronously against the rotational speed, and the evolution trends were consistent. When the blade vibration amplitude reached the maximum value, the SPL of the noise signal at the characteristic frequency was also the highest. Therefore, there may be a relation between this characteristic frequency and the vibration of R1. Figure 6 shows the noise spectrum in the compressor with different rotor speeds. The evolution rhythms of the characteristic frequencies and SPLs corresponding with the increasing rotor speeds are derived in Figure 7. It was found that this characteristic frequency increased with the rotor speed. The SPL corresponding to characteristic increased with increasing rotor speed; it reached the maximum value of 153 dB at the rotor speed of 10,020 r/min. Then, SPL suddenly decreased with the rotor speed. Additionally, there was a special phenomenon in a certain rotor speed range which the characteristic frequency did not change with the rotation speed. Figure 4: Sound pressure spectrum derived above R1.