Rotordynamics

The equation of motion, in generalized matrix form, for an axially symmetric rotor rotating at a constant spin speed Ω is $${\displaystyle \mathbf {M} {\ddot {\mathbf {q} }}(t)+(\mathbf {C} +\mathbf {G} ){\dot {\mathbf {q} }}(t)+(\mathbf {K} +\mathbf {N} )\mathbf {q} (t)=\mathbf {f} (t)}$$ where:

M is the symmetric mass matrix;

C is the symmetric damping matrix;

G is the skew-symmetric gyroscopic matrix:

K is the symmetric bearing or seal stiffness matrix;

N is the gyroscopic matrix of deflection for inclusion of e.g., centrifugal elements;

q(t) is the generalized coordinates of the rotor in inertial coordinates;

f(t) is a forcing function, usually including the unbalance.

The gyroscopic matrix G is proportional to spin speed Ω. The general solution to the above equation involves complex eigenvectors which are spin speed dependent. Engineering specialists in this field rely on the Campbell Diagram to explore these solutions.

An interesting feature of the rotordynamic system of equations are the off-diagonal terms of stiffness, damping, and mass. These terms are called cross-coupled stiffness, cross-coupled damping, and cross-coupled mass. When there is a positive cross-coupled stiffness, a deflection will cause a reaction force opposite the direction of deflection to react the load, and also a reaction force in the direction of positive whirl. If this force is large enough compared with the available direct damping and stiffness, the rotor will be unstable. When a rotor is unstable, it will typically require immediate shutdown of the machine to avoid catastrophic failure.

The Jeffcott rotor (named after Henry Homan Jeffcott), also known as the de Laval rotor in Europe, is a simplified lumped parameter model used to solve these equations. A Jeffcott rotor consists of a flexible, massless, uniform shaft mounted on two flexible bearings equidistant from a massive disk rigidly attached to the shaft. The simplest form of the rotor constrains the disk to a plane orthogonal to the axis of rotation. This limits the rotor's response to lateral vibration only. If the disk is perfectly balanced (i.e., its geometric center and center of mass are coincident), then the rotor is analogous to a single-degree-of-freedom undamped oscillator under free vibration. If there is some radial distance between the geometric center and center of mass, then the rotor is unbalanced, which produced a force proportional to the disk's mass, m, the distance between the two centers (eccentricity, ε) and the disk's spin speed, Ω. After calculating the equivalent stiffness, k, of the system, we can create the following second-order linear ordinary differential equation that describes the radial deflection of the disk from the rotor centerline.^[2]

$${\displaystyle m\mathbb {\ddot {r}} +k\mathbb {r} =m\varepsilon \Omega ^{2}\sin(\Omega t)}$$

If we were to graph the radial response, we would see a sine wave with angular frequency ${\displaystyle \Omega /2\pi }$. This lateral oscillation is called 'whirl', and in this case, is highly dependent upon spin speed. Not only does the spin speed influence the amplitude of the forcing function, it can also produce dynamic amplification near the system's natural frequency.

While the Jeffcott rotor is a useful tool for introducing rotordynamic concepts, it is important to note that it is a mathematical idealization that only loosely approximates the behavior of real-world rotors.^[2]

The Campbell diagram, also known as "Whirl Speed Map" or a "Frequency Interference Diagram", of a simple rotor system is shown on the right. The pink and blue curves show the backward whirl (BW) and forward whirl (FW) modes, respectively, which diverge as the spin speed increases. When the BW frequency or the FW frequency equal the spin speed Ω, indicated by the intersections A and B with the synchronous spin speed line, the response of the rotor may show a peak. This is called a critical speed.

The history of rotordynamics is replete with the interplay of theory and practice. W. J. M. Rankine first performed an analysis of a spinning shaft in 1869, but his model was not adequate and he predicted that supercritical speeds could not be attained. In 1895, Dunkerley published an experimental paper describing supercritical speeds. Gustaf de Laval, a Swedish engineer, ran a steam turbine to supercritical speeds in 1889, and Kerr published a paper showing experimental evidence of a second critical speed in 1916.

Henry Jeffcott was commissioned by the Royal Society of London to resolve the conflict between theory and practice. He published a paper now considered classic in the Philosophical Magazine in 1919 in which he confirmed the existence of stable supercritical speeds. August Föppl published much the same conclusions in 1895, but history largely ignored his work.

Between the work of Jeffcott and the start of World War II there was much work in the area of instabilities and modeling techniques culminating in the work of N. O. Myklestad^[3] and M. A. Prohl^[4] which led to the transfer matrix method (TMM) for analyzing rotors. The most prevalent method used today for rotordynamics analysis is the finite element method.

Modern computer models have been commented on in a quote attributed to Dara Childs, "the quality of predictions from a computer code has more to do with the soundness of the basic model and the physical insight of the analyst. ... Superior algorithms or computer codes will not cure bad models or a lack of engineering judgment."

Prof. F. Nelson has written extensively on the history of rotordynamics and most of this section is based on his work.

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There are many software packages that are capable of solving the rotor dynamic system of equations. Rotor dynamic specific codes are more versatile for design purposes. These codes make it easy to add bearing coefficients, side loads, and many other items only a rotordynamicist would need. The non-rotor dynamic specific codes are full featured FEA solvers, and have many years of development in their solving techniques. The non-rotor dynamic specific codes can also be used to calibrate a code designed for rotor dynamics.

• Axle
• Balancing machine
• Bearing (mechanical)
• Driveshaft
• Exoskeletal engine
• Magnetic bearing
• Turbine
• Wind Turbine