# Application of rotor dynamics in alloy centrifugal casting

The first paper about rotor dynamics in history is on centrifugal force of rotating shaft published by Rankine in 1869. Rankine thinks that the rotating shaft can operate stably only at the first critical speed. This conclusion limits the speed of rotor for a long time later. The first mock exam is the Jeffcott rotor model, which was proposed by FopplPl in 1895. This model consists of a rigid shaft with two ends and a disc rotor in the middle. After that, the dynamic characteristics of the rotor model are explained first, so the model becomes a Jeffcott rotor model, Je Ffcott pointed out that the rotor will produce automatic centering phenomenon in supercritical operation, so it can work stably. Under the guidance of this conclusion, the power and application range of rotating machinery have been greatly improved, which has played a great role in the industrial revolution. Later, it was found that when the rotor reached a certain speed in supercritical operation, it would have strong self-excited vibration and cause instability. Newkirk ＾ found that this instability was caused by oil film bearing, thus determined the important position of stability in rotor dynamics analysis. After the mid-20th century, the rapid development of electric power, aviation, machinery, chemical industry and other industries has greatly promoted the research of rotor dynamics. The rotor of rotating machinery is more and more flexible, the power is greater and greater, and the speed is higher and higher, which puts forward higher requirements for the research of rotor dynamics and greatly promotes the development of rotor dynamics.

The main contents of traditional rotor dynamics calculation and analysis are about the critical speed, unbalance response and stability of rotor bending vibration and the transient response of rotor under various excitations. Some rotor systems need to calculate the natural frequency and response of torsional vibration. From the point of view of mechanics, these calculation and analysis are to solve the eigenvalue and response problems of a mechanical system. In general, the differential equation of motion of mechanical system can be written as

Where m, C and K are the mass, damping and stiffness matrices of the system, Z is the generalized coordinate vector of the system, and F is the generalized external force acting on the system.

However, in rotor dynamics, sometimes the rotation effect caused by gyroscopic moment and the oil film force and damping force caused by bearing must be considered

Where C is the damping matrix, which is asymmetric, G is the gyroscopic matrix, which is antisymmetric, K is the symmetric part of the stiffness matrix, and S is the asymmetric part of the stiffness matrix. The above matrices are often functions of angular velocity.

The process of analyzing and solving the rotor system is the process of solving the above differential equations, but it is more difficult to solve the above equations, especially in the case of more degrees of freedom. In the development of rotor dynamics, there are many methods to solve the motion equation of rotor system

The first is analytical method, which uses structural mechanics and vibration analysis to establish physical equations, and solves these equations according to boundary conditions or initial conditions to obtain accurate analytical solutions. This method is only applicable to a few special simple cases;

The second is numerical method, including transfer matrix method and finite element method. It has become the most widely used rotor dynamics calculation method because of its good programmability and strong boundary adaptability;

The third kind is semi analytical method, which uses the continuous analytic function of analytical method and obtains approximate solution by energy principle, and the result is not very accurate.

At present, people mainly use transfer matrix method and finite element method to analyze rotor dynamics. 