# Improved FULMS Algorithm for Multi-Modal Active Control of Compressor Vibration and Noise Reduction
## 2 Modal Analysis of Vibration and Sound Radiation  
Taking the undamped \(n\)-order degree of freedom system as an example, the differential equation of motion is as follows:  
\[[m]\big{\{}\ddot{x}(t)\big{\}}+[k]\big{\{}x(t)\big{\}}\ =\ \{Q(t)\} \tag{1}\]  
where \([m]\) and \([k]\) are the mass matrix and the stiffness matrix of each node respectively. \(\{Q(t)\}\) is the excitation matrix. When the external force received by the system is a simple harmonic force, after decoupling and transformation of the equation, the steady-state response of the system can be calculated as follows:  
\[\{x(t)\}\ =\ \sum_{r\ =\ 1}^{n}\{u^{r}\}\frac{N_{0r}/\omega_{r}^{2}}{\sqrt{1-( \omega/\omega_{r})^{2}}}sin(\omega t) \tag{2}\]  
among them, \(\omega\) is the excitation frequency. \(\omega_{r}\) is the \(r\)-th natural frequency, and \(\{u^{r}\}\) is the \(r\)-th modal vector. It can be seen that when the excitation frequency is close to the \(r\)-th natural frequency, resonance occurs, and a large displacement response is generated. The vibration mode shape and energy frequency response of each mode can be obtained through model calculation to evaluate the active position and frequency band of vibration mode, respectively. The acoustic radiation mode is generated during the derivation of acoustic radiation power. Taking the unit radiator method as an example, the following formula can be obtained:  
\[W\ =\ \big{(}Q^{T}v\big{)}^{H}\Lambda\Big{(}Q^{T}v\Big{)}\ =\ Y^{H}\Lambda Y\ =\ \sum_{i=1}\lambda_{i}|y_{i}|^{2}\ =\ \sum_{i=1}W_{i} \tag{3}\]  
where \(Q\) and \(\Lambda\) are the characteristic matrix after eigenvalue decomposition of acoustic radiation resistance matrix and the diagonal matrix composed of eigenvalues, respectively. \(v\) is the normal velocity vector of plane element. Formula (3) shows that the acoustic radiation on the structural surface is the superposition of each order of acoustic radiation modes [19], so the acoustic radiation of the structure can be reduced by suppressing the acoustic radiation modes. According to the modal superposition principle, the radiated sound power of the structure can also be expressed as  
\[W=\rho_{0}c_{0}5v^{2}\sigma \tag{4}\]  
where \(S\) is the area in the propagation direction, and \(v\) is the vibration velocity level of the structure, which can be calculated by the root mean square of the vibration velocity of the structure, and \(\sigma\) is the radiation efficiency. According to Equation (3) and the orthogonality of eigenvectors:  
\[\sigma_{i}\ =\ \frac{2\lambda_{i}}{(\rho_{0}c_{0})S} \tag{5}\]  
among them, \(\lambda_{i}\) is the eigenvalue of the acoustic radiation resistance matrix. From Formula (5), \(\sigma_{i}\) is only related to the eigenvalue and radiation area of the acoustic radiation resistance matrix R. According to the physical meaning and transformation process of R, the radiation efficiency of the acoustic radiation mode is only determined by the geometric characteristics of the radiator and the excitation frequency [20]. The sound power of each order of the radiation mode is the product of radiation efficiency and modal amplitude. According to the radiation efficiency calculation formula, the modal radiation efficiency has no strong positive correlation with the amplitude of vibration mode, but the amplitude of the radiation mode is related to the vibration velocity. Through the analysis of the vibration mode, it is easy to know that the vibration velocity is the superposition of each order of vibration modes, so the relationship between them can be studied. The acoustic radiation mode is controlled by suppressing the vibration mode.