Vibrations of a Rotating Beam

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Abstract
The object of this paper is to analyze the coupled flatwise (out of plane) edgewise (in plane) vibrations of a beam rotating about one if its ends. The hub is assumed motionless and the vibration is considered to occur about a large deflected position. That is, coning and lagging angles are allowed to be large. The equations of motion are derived by a combination of techniques. The kinetic energy of the system is expressed in terms of coordinates lying in the beam. This is done by making use of four coordinate transformations that relate the beam coordinate system to a fixed coordinate system. The inertial load distribution is obtained by application of the first two terms of Lagrange's equation. These loads are used to compute the bending moment distribution which is substituted into Euler's beam bending equation. The equations of motion are solved by assuming a solution in the form of a linear combination of orthogonal modes. These equations are multiplied by the jth mode shape and integrated over the beam length. There are four terms resulting; the mass and elastic stiffness terms form a diagonal array and the Coriolis' and centrifugal spring terms form a full array. These equations may be solved by easily available matrix techniques. The modes chosen for the solution are the normal modes of the nonrotating beam. The advantage of this choice is that each of the modes already satisfies the problem boundary conditions. Since the non‐rotating modes are a good approximation to the rotating modes the series converges rapidly and can be cut off after a few terms. Several sample problems are worked out. First, the beam is assumed rigid and free to flap. The classical formula for flapping frequency is verified with the addition that the terms due to large cone and lag angles are included. Second, the same problem is done except that instead of the flapping degree of freedom the lagging degree of freedom is analyzed. The classical formula for lagging is also verified for the zero cone angle. When the cone angle is large this degree of freedom becomes statically unstable. Third, the above problem is redone for the coupled lagging — flapping degrees of freedom. Fourth, a flexible beam is assumed with zero cone and lag angles. Mode shapes and frequencies are computed as a function of rotor speed. It is shown that as rotor speed increases the beam mode shapes and frequencies approach those of a chain. That is, the elastic stiffness becomes negligible relative to the centrifugal stiffness. The advantages of the formulation developed in this paper (in addition to allowing consideration of large coning and lagging angles) are: 1) that the terms that involve rotor speed (the centrifugal spring and the Coriolis coupling) have that parameter as a factor multiplying the whole matrix so that if frequencies and modes are required over a range of rotor speeds the centrifugal and Coriolis' terms need only be calculated once; 2) at large rotor speeds the Myklestad analysis has difficulty converging but in this procedure, because the non‐rotating modes already satisfy the boundary conditions, there is no difficulty in convergence.

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