Research on vibration isolation optimization design technology for the articulated link mount of turbofan engine

As an important structure connecting the aircraft and the engine, the turbofan engine mount often adopts a hinged multi-link mechanism, and its dynamic performance directly affects the transmission of engine vibrations to the fuselage. To study and improve the vibration isolation performance of this type of mount system, this paper proposes an optimization strategy based on modal design ideas, which takes the minimum distance between the natural frequencies of the mount system and the excitation frequency as the optimization objective, seeks the parameter solution that maximizes the optimization objective, and uses genetic algorithms to carry out optimization analysis. The results show that through optimization analysis, the effective vibration isolation range of the system is increased by approximately 125% near the excitation frequency of the low-pressure rotor; at the excitation frequency of the low-pressure rotor, the system transfer rate is reduced by approximately 58.4%; after optimization, due to the overall frequency deviation of the installation section system from the excitation frequency, the impact of small perturbations of the main parameters on its transfer rate is generally lower, with a maximum decrease of approximately 17.3%, greatly improving the robustness of the system.


Introduction
The engine mount is an important mechanical system on an aircraft that connects the pylon and the engine.During flight conditions, it needs to withstand both static and dynamic loads, including thrust loads from the engine, gravity loads, complex aerodynamic loads, and unbalanced excitation loads caused by the engine rotor.Therefore, the design of the installation section not only needs to meet the basic requirement of transferring static loads from the engine, ensuring the transmission of loads such as engine thrust, but also must isolate unbalanced excitation loads caused by the eccentric mass of the engine rotor, minimizing vibrations transmitted to the wings.Internationally, early research in this field was conducted by E. S. Taylor [1], who addressed the issues of force transmission and isolation in early piston engine mount.He pointed out the fatigue issues and physiological hazards caused by vibrations and proposed flexible design solutions.H. Ashrafiuon [2] simplified the engine into a 6-DOF rigid body and established a spring-damping installation system model to study the response of vibration caused by the unbalance of the rotating parts of the turboprop engine.D. A. Swanson and H. T. Wut [3] optimized the optimal connection stiffness of turbo prop engines based on a rigid body engine model and spring-damping assumptions, achieving relatively optimal force transmission and isolation effects.Taehyun Shima and Donald Margolis [4] proposed a new optimization method for high bypass ratio engine installation system isolation that differs from traditional stiffness-damping isolation.They controlled the balanced position of the mount system by controlling the hot air flow exiting the bypass duct, achieving better isolation effects.Luigi Iuspa [5] and others used genetic algorithms to optimize the design of engine installation sections, optimizing the force transmission path of the installation section and finding the optimal design configuration under given requirements.X.N.Zhao [6] and others designed active isolation control systems based on rigid body spring models.In recent years, domestic research has begun to focus on the design and computational analysis of installation sections, mainly establishing finite element models for installation sections and performing stress analysis.For hang-type installation sections, Zheng Jibo [7] established a finite element model to calculate and analyze the strength of hang-type mount, propose optimization methods, and conduct modal analysis on hang-type installations, initially studying the relationship between engine excitation frequencies and hang-type installation natural frequencies.Li Shizhe et al. [8] established a finite element model for load analysis of installation section components, considering that local stress is relatively complex for local parts and each load needs to be analyzed.They calculated multiple loads acting on the engine and screened out severe loads, providing computational and testing methods for local stress research.Currently, domestic research has mainly focused on load transfer and fatigue analysis of hang-type installation sections [9][10][11], with research on vibration transmission in turbofan engines limited to guiding the design of engine isolators using concentrated mass models [12].To date, there has been no research on dynamic performance optimization analysis of turbofan engine installation section systems.Currently, the low frequency modal frequency of an mount system is close to the operating frequency of the engine's low-pressure rotor, leading to resonance and poor isolation effects due to unbalanced excitation from the low-pressure rotor.Therefore, this article mainly focuses on the turbofan engine mount system mentioned above.It uses a modal design optimization method based on genetic algorithms with the objective of maximizing the distance between the natural frequencies of the system and the excitation frequencies.It carries out optimization analysis on the isolation performance of the installation section system; assesses changes in the effective isolation range after optimization; and conducts robustness analysis on the optimized mount system.

Establishment of the turbofan engine installation section system model
The articulated multi-link mounting system [8] used in a turbofan engine consists of a front mounting joint, a rear mounting joint, and a thrust rod, as shown in Figure 1.

Establishment of multi-body dynamics equations
The mounting joint system shown in Figure 1 is mainly composed of rod members, hanger joints, engine bodies, and connection ball joints between these parts.In multibody dynamics modeling, the rod members are modeled using MNF flexible bodies, and the generalized coordinates of each MNF flexible body are The number of degrees of freedom is  the number of deformation degrees of freedom for each flexible body.The engine body is modeled as a rigid body, and its generalized coordinates are: The number of degrees of freedom is 6.Additionally, the hanger joints are directly connected to the ground and are not modeled separately.The connection ball joints are used to model the elastic force between the components without generating additional degrees of freedom.Therefore, the total number of degrees of freedom for the system is: The generalized coordinates of the system are: According to the Lagrange equation, the dynamic equation of the engine mount system can be obtained as: Where F is the external excitation force.The optimization design goal of the mounting joint system is to improve the vibration isolation effect of the mounting joint.Therefore, the natural frequencies of each order of the actual system are far away from the excitation frequencies of the high-and lowpressure rotors, so the effect of damping on the system vibration can be neglected, resulting in a simplified dynamic equation for the mounting joint system: Calculate the natural frequencies and modal shapes of the system through the generalized eigenvalue problem: Furthermore, we obtain the modal matrix  , perform normalization with respect to the mass matrix, then apply the transformation  ξ Φq .By left-multiplying equation ( 6) with T Φ , we arrive at the differential equation for decoupling modal coordinates.
According to the amplitude-frequency response characteristics of a single degree of freedom, and ignoring the effect of damping on phase lag, we can obtain: Therefore, the steady-state response of the engine mount system with external excitation is The magnitude of the transfer rate is positively correlated with the vibration amplitude ξ of the system, and the norm  of the modal shape does not change significantly during the optimization process.Therefore, for a given external excitation, the most effective means to reduce the transfer rate of the system is to change the magnitude of the natural frequencies of each order of the system, moving them away from the excitation frequency of the low-pressure rotor, thereby reducing Hi(ω) to reduce the vibration of the system and improve the vibration isolation efficiency.

Establishment of analysis model
When establishing the simulation model, this article only considered the flexible effects of the eight rods in the front and rear mounting sections, and other components were treated as rigid bodies.The engine dummy was modeled as a rigid body, with only the mass and moment of inertia of the engine assigned.The front and rear mounting sections were subjected to fixed boundary conditions, and the simulation model is shown in Figure 2.

Optimization analysis of vibration isolation performance of mount system
The dynamic optimization design idea for the installation section of a turbofan engine is to adjust the geometric parameters (angle and length of the flexible rod) and mechanical parameters (mass, stiffness, and damping coefficient of the flexible rod, as well as the clearance and contact stiffness of the spherical joint) of the installation section system reasonably, while ensuring that the static load on each rod meets the strength requirements, so that the natural frequency of the system is far away from the excitation frequency, and the transfer rate curve meets the given vibration isolation efficiency requirements.

Optimization of variable selection
Optimization variables should meet the following conditions: they can significantly change the vibration isolation efficiency, meet the engineering improvement requirements, and cannot have a significant impact on the static strength of the installation section.Since the mechanical parameters of the system (including the mass, stiffness, and damping coefficients of the flexible rod, as well as the clearance and contact stiffness of the spherical joint) are largely dependent on material properties, optimizing them does not have much practical significance; while changing the length of the MNF in the geometric parameters of the flexible rod requires manual adjustment of the mesh division, which is not convenient for large-scale calculations.Therefore, this article mainly focuses on optimizing the angles of the connecting rods in the front and rear installation sections.As shown in Figure 3, the left and right connecting rods of the front and rear mounting sections are rotated counterclockwise relative to the vertical configuration for positive inclination, and the middle connecting rod of the rear mounting section has only positive inclination.Since the mounting section system mainly transmits the thrust and gravity loads of the engine during operation, the inclination of the left and right connecting rods of the front and rear mounting sections should not be too large.The range of inclination of the front and rear mounting section connecting rods studied in this article is defined as shown in Table 1.Table 1.Value range of each variable.
Parameters existing configuration(°) lower limit(°) upper limit(°) The angle of the left and right connecting rods of the front mounting section α1 30 -60 60 The angle of the left and right connecting rods of the rear mounting section α2 -21 -60 60 The angle of the intermediate connecting rod of the rear mounting section α3 65 0 80

Optimization analysis
This article is mainly based on the modal design idea, taking the minimum distance between the natural frequencies ( ω1 ， ω2 ， … ， ωN ) of each order in the actual modal frequency and the excitation frequency ωEx of the low-pressure rotor as the optimization objective.For the parameter space formed by the left and right link inclination angles of the front mounting section and the left, middle, and right link inclination angles of the rear mounting section, we seek the parameter solution that maximizes the optimization objective within the parameter space, that is, This optimization analysis is implemented using genetic algorithms.During optimization, MATLAB is used to initialize the generation of parent populations, and then the solver code is modified and parallel operations are performed based on the generated populations.After the calculation is completed, the solver calculation results are automatically read and the frequency is calculated as the fitness of each individual.Then, the selection, crossover, and mutation processes are completed within MATLAB, thus iteratively improving the population.The flowchart of genetic algorithm is shown in Figure 4.The binary genetic algorithm is used for optimization, and the optimization objective is the population fitness.The population size is 20, the chromosome length is 7, the crossover probability is 0.8, and the mutation probability is 0.05.The average fitness curve of the population after 500 generations of genetic is shown in Figure 5.It can be seen that the population has been close to stable after about 100 generations, and the optimal angles of each connecting rod are [-60°, 9.9213°, 20.1575°].The absolute value of the difference between the corresponding natural frequency and the excitation frequency of the low-pressure rotor is 28Hz.The excitation frequency ωEx corresponding to a turbofan engine low-pressure rotor is 59Hz, which means that the optimization limit is to shift the modal frequencies of the mounting joint system to 31Hz and 87Hz respectively.

Optimization results
Based on the analysis results in 3.2, the optimized link angles were substituted into the analysis model for sweep frequency analysis.The transmission rate curves of the mounting system before and after optimization are shown in Figure 6.The transmission rate and effective vibration isolation range of the system at the low-pressure rotor frequency before and after optimization are shown in Table 2.    6 and Table 2, it can be seen that after optimization, the system transmission rate of the mounting joint system significantly decreased at the low-pressure rotor excitation frequency of 59 Hz, by approximately 58.4%.At the same time, the effective vibration isolation range of the system increased significantly near the low-pressure rotor excitation frequency, by approximately 125%.Therefore, when the low-pressure rotor excitation frequency shifted slightly, the mounting joint system could still effectively isolate vibration.

Discussion on system robustness
Based on the above research, in order to ensure that the installation system still has high vibration isolation efficiency when the main parameters of the system undergo small perturbations, it is necessary to conduct sensitivity verification on each parameter.The contact stiffness, contact damping, and clearance of the connecting ball joint are selected for verification, allowing them to fluctuate up and down by 10% around the original parameters, and the change in transmission rate is calculated.The results are shown in Table 3.As shown in Table 3, after optimization, due to the overall frequency deviation of the mount system from the excitation frequency, the influence of small perturbations on its transfer rate is generally low.

Figure 1 .
Figure 1.The overall layout of the engine mount system.

Figure 2 .
Figure 2. Dynamics model of the engine mounting system.

Figure 5 .
Figure 5.The average fitness curve of the population in the past.

Figure 6 .
Figure 6.Vibration transmission rate curve of the mount system before and after optimization.

Table 2 .
Comparison of main parameters of the mount system before and after optimization.