[b]1 Introduction[/b] There are many types of electric motors with different uses. There are AC motors and DC motors. AC motors include AC generators and AC motors, and DC motors also include DC generators and DC motors. Various thermal and hydroelectric power generation units account for more than 90% of the total power energy of the whole society, occupying an important position in the national economy. A 600 MW large steam turbine generator unit has been put into operation in China, and a 1400 MW steam turbine generator unit has been put into operation abroad. my country is currently developing the world's largest 700 MW hydroelectric generator unit, which will be connected to the grid at the Three Gorges Hydropower Station on the Yangtze River in 2003. The electrical energy of the large power grid system formed by generator units can be used to drive a wide variety of electric motors to drive various working machines. These motors range from deciwatt-level electric clock pointers to several MW rolling mill motors. In control systems, AC magneto amplifiers used for power amplification can reach a power of 20 MW. In addition, there are generator sets for mining and shipbuilding, etc. The vibration problem of electromechanical coupling and magneto-solid coupling in electric motors is quite complex, involving the theoretical foundations of multiple disciplines, including mechanics (general mechanics, continuum mechanics, and vibration dynamics) and electricity (electromagnetic field theory, circuit theory, and motor theory) and the resulting interdisciplinary fields. The study of the interaction laws between electromechanical systems requires addressing two main issues: First, correctly establishing a mathematical model of the coupled electromechanical system. Electromechanical dynamics analysis is a very effective tool for studying electromechanical coupling problems. From an energy perspective, it serves as a unified method to establish a system of differential equations coupling general mechanics and circuit theory, and continuum mechanics and electromagnetic field theory, to study the interaction laws of electromechanical coupling. For further research, see the relevant literature, which has achieved systematic results. The second issue is that the mathematical equations describing all electromechanical coupling systems are nonlinear. Therefore, the progress in research on quantitative and qualitative methods for nonlinear equation systems is a particular focus for researchers of electromechanical coupling vibration problems. In the case of weak nonlinearity, there are already relatively effective and mature methods. The methods generally used include Poincare's small parameter method and KBM asymptotic method. This method is very effective for solving single-degree-of-freedom and multi-degree-of-freedom systems and greatly simplifies the study of the stability of periodic solutions. Nagfeh and Mook systematized various forms of multi-scale methods and effectively solved some nonlinear vibration problems of continuous media systems. Strong nonlinear vibration and nonlinear dynamics are currently research topics. Literature on this topic can be found in the paper. In the nonlinear vibration of electromechanical coupling systems, the paper [11] has achieved systematic results. 2 Research progress on electromechanical coupling and magneto-solid coupling vibration problems of motors 2.1 Electromagnetically excited nonlinear vibration 2.1.1 Overview The vibration and noise of motors have always been an important issue in the international motor industry. Vibration and noise excited by electromagnetic force is one of the research focuses, but the main results are limited to the research field of linear vibration and noise. The paper discovered various nonlinear electromagnetic forces from theory and experiment, and then organically combined nonlinear vibration with motor theory, opening up a new research field. 2.1.2 Parametric Resonance Excited by Electromagnetic Force The mathematical model established by the parametric resonance of the motor rotor excited by electromagnetic force is characterized by the fact that the nonlinear differential equations of the periodic coefficient of vibration and the algebraic equations of the steady-state circuit of the motor constitute a unified mathematical system. Since the coefficients of the vibration differential equations change with the operating conditions of the motor, it belongs to the problem of slowly variable coefficient differential equations. For different types of motors (such as synchronous generators, asynchronous motors, DC motors, etc.) under different operating conditions (such as three-phase symmetry, three-phase asymmetry, and various electrical faults, fault short circuits, etc.), the air gap magnetic field of the motor is different, its magnetic field energy function is different, and therefore the vibration equation and voltage equation are also different, and its vibration law is different. References [14-32] obtained a series of parametric vibration laws. This paper studies the influence of electromagnetic parameters on natural frequency and resonance characteristics. This paper studies the parametric resonance law of multi-pole low-speed motors. The transient process and start-up process of AC motor vibration characteristics are obtained. This paper studies the parametric resonance of a motor rotor excited by electromagnetic force during three-phase symmetrical and three-phase asymmetrical operation. It obtains the correlation characteristics of rotor vibration and drive, and a stability criterion with obvious geometric characteristics. The vibration law of the coupling of parametric resonance and forced resonance in a high-speed motor is also derived. Due to the combination of parametric resonance and forced resonance, the two resonance branch curves of pure parametric resonance each split into two, which, when connected, form three branch curves with the two pure parametric resonance curves and the zero line as asymptotes. All the above theoretical research results have been verified by experimental studies. The diversity of the obtained resonance laws is unprecedented in mechanical systems. For example, the width of the resonance region is variable, changing with the magnitude of voltage and current. Not only does the width of the resonance region change, but the resonance region also shifts, and its amplitude-frequency characteristic shape changes with the electromagnetic parameters. When one phase suddenly short-circuits (reaching maximum asymmetry), the critical speed can be significantly reduced, and the resonance region can be greatly widened. One consequence is that the operating speed can easily fall into the resonance region, thus suddenly exciting a large-amplitude parametric resonance. This paper studies the parametric resonance of the generator rotor and stator coupling, revealing that the resonance amplitude-frequency characteristics change in various forms with variations in the generator's active and reactive power. It also investigates the combined bending and torsional vibration of the generator rotor shaft system. Furthermore, it examines the influence of generator magnetic saturation nonlinearity on parametric resonance and combined bending and torsional resonance. New findings are made regarding electromagnetic damping theory and its impact on lateral vibration, revealing that electromagnetic damping is significantly greater than mechanical vibration damping, by several times or even tens of times. The paper discovers that electromagnetic fields have a substantial influence on low-order torsional natural frequencies and identifies a zero-order natural frequency induced by electromagnetic fields. Due to its low frequency, this zero-order natural frequency has significant engineering application value for low-speed large-scale hydro-generator units. All the above results are experimentally proven. 2.1.3 Electromagnetically Excited Multiple Resonances In nonlinear vibration systems, in addition to the resonance generated when the natural frequency K and the disturbance force frequency are equal in linear systems, the following types of resonances also occur: (1) K»w/n, subharmonic resonance, where n is a positive integer; (2) K»nw, superharmonic resonance; The resonance relationship of the resonance mode is a system of r simultaneous algebraic equations, and this type of resonance is called multiple resonance. Multiple resonance can simultaneously excite multiple modes to resonate. Its occurrence requires two conditions: on the one hand, the mathematical relationship of the equation needs to be satisfied, and on the other hand, the modes of the system need to have appropriate coupling relationships. Since the bending and torsional vibration system of the motor shaft system has multiple natural frequencies and multiple electromagnetic disturbance forces and torques, it is very likely that various multiple resonances will occur. The characteristic of multiple resonance is that multiple modes are excited at the same time, and energy exchange occurs between modes, resulting in amplitude modulation and phase modulation. Due to the many difficulties in solving the mathematical problem, multiple resonance is a type of vibration that has been studied less. This paper combines the averaging method from nonlinear vibration theory with analytical mechanics methods to derive an energy method for solving multiple resonances in nonlinear vibrations of elastic systems. Using this method, if the work done by the nonlinear disturbance force of the vibrating system can be found, the first-order approximate solution and the exact approximate solution for multiple resonances of the nonlinear system can be obtained without establishing the partial differential equations of the elastic body vibration. This method can also solve multiple resonances in multi-degree-of-freedom systems. The paper studies double and triple resonance problems with various resonance relationships. It investigates single-frequency double resonance with the natural frequency splitting into two. Due to the coupling of the two modes, the resonance region and amplitude are significantly wider and larger than in single resonance, and beat vibration also occurs. The paper studies double resonance with one internal resonance and one single-frequency resonance, revealing a jump phenomenon in the system's response curve. When the internal resonance is undertuned, the system's frequency response curve exhibits soft characteristics; when the internal resonance is fully tuned, the frequency response curve is symmetrical and "M"-shaped; when the internal resonance is overtuned, the frequency response curve exhibits hard characteristics. The paper also studies double resonance problems in various cases. This paper studies triple resonance that simultaneously satisfies internal resonance, combined resonance, and single-frequency resonance. It finds that when two internal resonance relationships are satisfied, a double-saturation phenomenon exists, and the vibrational energy of higher-order modes can be observed. This energy is transferred to lower-order modes through the internal resonance relationship, causing the lower-order modes, which are not excited by external forces, to gradually generate large-amplitude vibrations. Rich vibrational characteristics of the amplitude and stability of the three modes in the resonance region are obtained. The characteristics of triple resonance under various conditions are also studied. Applying the theory of multiple resonance, the law of low-frequency vibration of 19 Hz in a large steam turbine generator set was obtained in the "8.5" National Major Research Project, achieving good diagnostic results. In the major funded project of the Three Gorges hydroelectric generator set, the zero-order natural frequency caused by electromagnetic parameters was discovered through theoretical and experimental research. This frequency easily causes resonant vibration in low-speed large hydroelectric generator sets, which has important engineering application value. The application of the multiple resonance theory method to study the vibration of the generator stator system has also achieved good results. 2.2 Nonlinear Dynamics of Electromechanical Coupling 2.2.1 Overview The characteristic of electromechanical coupling dynamics is that the vibration differential equations of the rotor shaft system and the differential equations of the motor current are mutually coupled, forming a unified system of differential equations. That is to say, the dynamics of the rotor system and the transient theory of the motor are closely coupled. How to correctly establish this mathematical system and how to obtain the mathematical solution are the two major difficulties. In the literature [47-56], the electromechanical coupling dynamics of AC motors and synchronous generators were studied respectively, and the nonlinear vibration theory was organically combined with the transient theory of motors, creating a new research field. 2.2.2 Nonlinear Theory of Electromechanical Coupling Vibration of AC Motors The literature conducted theoretical and experimental research on the electromechanical coupling dynamics of AC motors. The method of establishing the equation system is to first obtain the air gap magnetic field of the motor from electromagnetic theory and given boundary conditions, and then obtain the energy function of the air gap magnetic field. Then, together with the kinetic energy and potential energy of the vibration system, the electromechanical analysis dynamics method is applied to establish this electromechanical coupling equation system. By applying linear and nonlinear transformations, solutions to these strongly nonlinear equations were obtained. Theoretical and experimental studies were conducted on the radial electromagnetic force, electromagnetic torque, torsional vibration, and transverse vibration during the motor starting process, yielding various experimentally verified laws. It was found that the current and torsional vibration during motor starting are very large, exceeding the magnitude of resonance, thus identifying the cause of the strong noise during motor starting. It was discovered that the amplitude of the alternating electromagnetic torque that induces torsional vibration is 4-5 times larger than the motor's rated torque, with a frequency of 50Hz. If the natural frequency of the shaft torsional vibration is close to this value, the shaft is highly susceptible to breakage. It was found that the alternating electromagnetic force with a frequency of 100Hz increases significantly with increasing air gap eccentricity. When the eccentricity exceeds a certain limit, the rotor is prone to rubbing against the stator, causing an accident. This paper presents theoretical and necessary experimental studies on the electromechanical coupling dynamics of steam turbine generators and hydro turbine generators. The transient processes of electrical faults such as three-phase sudden short circuit, two-phase line-to-line short circuit, and one-phase-to-center short circuit were studied. The transient variation laws of current, electromagnetic torque, speed, torsional vibration, and torque between shaft segments were obtained, and the influence of excitation current and stator resistance on the transient process was investigated. Theoretical and experimental results show that for some electrical faults, the electromagnetic torque and the resulting shaft segment torsional vibration torque can reach several times the rated torque. The research results are of great value in estimating the damage degree of steam turbine generator sets and hydro turbine generator sets, and are of great significance in preventing major damage accidents to generator sets. All the above results have been verified by experiments. The generator rotor of the steam turbine generator set is a relatively long elastic shaft and cannot be regarded as a concentrated mass. It needs to be described by partial differential equations with distributed parameters. How to establish the electromechanical coupling relationship of torsional vibration is an unsolved problem. This paper finds a solution. Considering the electromagnetic nonlinearity caused by the magnetic field saturation of synchronous generators, a nonlinear mathematical model of motor transient theory is obtained, which can broaden the research scope of motor transient theory and electromechanical coupling dynamics. 2.2.3 Electromechanical Coupling Instability Self-Excited Vibration: Instability self-excited vibration of motors and power grids, besides shortening mechanical lifespan, can also cause current and voltage overloads, leading to damage to motors and electrical equipment. This paper studies the electromechanical coupling instability self-excited vibration of AC motors and synchronous generator sets. Using electromechanical analysis dynamics, a set of differential equations for the electromechanical coupling of the generator set is established. The influence of electromagnetic parameters on instability is studied. The theory and experiments of asynchronous operation of generator sets without excitation are investigated. The theory and experiments of Hopf bifurcation in unstable self-excited oscillations of generator sets are also studied. Recently, modern theories of dynamic bifurcation have been applied, yielding many characteristics and laws of unstable oscillations. 2.3 Magnetically-Solid Coupled Vibration Motors in Generator Stator Systems: The motor consists of two main parts: the rotor and the stator. Although the stator housing is fixed to the foundation, unlike the rotor which is prone to vibration due to rotation... However, for the stator of large generator sets, due to its complex structure and huge size, vibration and noise problems are prominent under the action of electromagnetic force. The vibration of the stator core exacerbates the wear and electro-corrosion of the stator winding insulation, often causing short circuit faults in the end windings, thus damaging electrical equipment. Short circuit faults can also cause torsional vibration damage to the generator shaft system. The generator stator end windings and their binding and fixing structures are very complex, and their vibration mechanical models are also complex. Accurate calculation of their natural frequencies and mode shapes is difficult. The leakage magnetic field at the end is affected by the surrounding metal components, and its distribution is also very complex. Therefore, it is difficult to accurately calculate the electromagnetic force borne by the end windings, and currently, it is not possible to accurately predict the vibration response during operation at the design and manufacturing stage. Furthermore, in foreign countries, fiber optic testing methods have been used to monitor the vibration of the end windings of large generator sets under operating conditions. However, existing vibration monitoring sensors and signal transmissions in China cannot function properly under strong magnetic fields. Therefore, the research task on the vibration characteristics of the stator end windings remains arduous. This paper [conducts research on the vibration of the stator system of large generators based on the theory of magneto-solid coupling]. A coupling mechanism for the interaction and mutual influence between the deformation field and electromagnetic field of the stator double cylindrical shell was established, and research on magneto-solid coupling vibration was carried out, achieving relatively systematic results. Recently, some progress has also been made in the research on magneto-solid coupling vibration of the generator end windings. [b]3 Engineering Significance and Prospect of the Research[/b] In recent years, in order to rationally utilize energy, improve economic efficiency, and protect the environment, domestic and international power systems have increasingly developed towards large-scale units, ultra-high voltage, and long-distance power transmission. The structure of the power grid has become more complex, and the operational stability of units and the power grid has become particularly important. Some large power grids in the world (such as Japan, France, Sweden, and the United States) have successively experienced power grid collapse accidents characterized by voltage collapse, leading to large-scale blackouts, causing huge economic losses and social disorder. Voltage stability issues have once again attracted widespread attention from countries around the world. The development of steam turbine generator sets and hydro turbine generator sets towards ultra-large sizes, especially with the further increase in high voltage and high current, will generate strong magnetic and electric fields. Under the influence of these strong magnetic and electric fields, new problems will arise in the electromechanical coupling vibration, stability, and dynamic strength of the rotor system. Due to the increased strength of the magnetic and electric fields, the dynamic problems of magneto-solid coupling vibration and stability of the generator stator and its end windings will become more prominent. The busbars and core laminations of transformers must also take magneto-solid coupling dynamics into account. Accidents of electromechanical coupling torsional vibration failure of generator sets caused by electrical faults and other reasons occur frequently both domestically and internationally. In 1970, a 300 MW steam turbine generator set at the Mohave Power Station in the United States experienced subsynchronous oscillations due to grid disturbances, resulting in two consecutive shaft breakage accidents. In 1976, a 960 MW steam turbine generator set suffered a crack fracture at the keyway of its shaft due to grid connection misoperation. Meanwhile, in 1970, the 600 MW unit at the Laspezia power station in Italy, in 1972, two 600 MW units at the Didcot power station in the UK, and in 1973, a 600 MW unit in West Germany all experienced torsional vibration failures of the aforementioned nature. In 1992, a 300 MW turbine generator unit at the Wujing Power Plant in Shanghai experienced two short-circuit accidents at the generator end, causing the exciter shaft to break. At the Fengzhen Power Plant in Inner Mongolia, a serious accident occurred where incorrect grid connection caused shaft torsion. In 1985, a major shaft breakage accident occurred at the Datong Power Plant, where the 200 MW turbine generator unit's shaft broke into five sections. According to the expert group's investigation and analysis, a key cause was that after the unit's load increased, the generator's excitation current was not adjusted accordingly, resulting in an underexcited operating state, causing the generator to lose synchronism and leading to torsional vibration and shaft breakage. However, it is worth pointing out that many destructive accidents have occurred domestically. Due to poor monitoring methods, there is a lack of recorded data and evidence of the damage. Experts' analyses of the causes of the damage are often limited to their respective areas of expertise, resulting in conflicting opinions and a lack of definitive conclusions. The following report from the 32nd International Conference on Large Electric Systems effectively illustrates that electromechanical coupling vibration of generator sets is a significant cause of generator set vibration damage. The Electric Power Research Institute (EPRI) in the United States installed nine torsional vibration monitoring systems (TVMS) on steam turbine generator sets in North America. Monitoring of 13 large steam turbine generator sets resulted in 108 torsional vibration accidents recorded over five years, as shown in the accident database summary table. This summary table illustrates the necessity of further emphasizing and strengthening research on grid-machine coupling torsional vibration of generators. Research on magnetic levitation bearings has made significant progress, with five or six related international academic conferences held. They have already been applied to large-scale high-speed power machinery abroad and are commercially available. Large magnetic levitation bearings developed in France have been tested on 900 MW steam turbine generator bearings, demonstrating numerous advantages in addressing vibration control and monitoring issues of the units. Therefore, research on the nonlinear dynamics of the coupling between the electromechanical dynamics and control system of magnetic levitation bearings has promising development prospects and application value. The achievements in high-temperature superconducting material research will bring hope to the production of superconducting motors; their large currents and strong magnetic fields will bring new research topics to the dynamic stability and vibration problems of motor structures. The application of high-temperature superconducting materials in long-distance high-voltage transmission lines will bring significant energy savings; the dynamic and static stability of superconducting structural materials will be a prominent problem that needs to be solved. To find more efficient methods for generating electrical energy, an attempt is being made to develop a magnetohydrodynamic generator that uses expanding hot gas as a moving conductor. Another method utilizes the interaction between charged particles and non-conductive flowing gas to generate electricity using electro-hydraulic power; this generator provides extremely high voltages of 20 MV for hospital treatment and physics research. Electromechanical propulsion schemes using magnetohydrodynamics or electro-hydraulic power to accelerate matter to obtain thrust, and their application in space, are also under research and development.