Mathematic study of the rotor motion with a pendulum selfbalancing device

The rotary machines used in manufacturing may become unbalanced leading to vibration. In some cases, the problem may be solved by installing self-balancing devices (SBDs). Certain factors, however, exhibit a pronounced effect on the efficiency of these devices. The objective of the research comprised of establishing the most beneficial spatial position of pendulums to minimize the necessary time to repair the rotor unbalance. The mathematical research of the motion of a rotor with pendulum SBDs in the situation of their misalignment was undertaken. This objective was achieved by using the Lagrange equations of the second type. The analysis identified limiting cases of location of the rotor unbalance vector and the vector of housing's unbalance relative to each other, as well as the minimum capacity of the pendulum. When determining pendulums ’ parameters during the SBD design process, it is necessary to take into account the rotor unbalance and the unbalance of the machine body, which is caused by the misalignment of rotor axis and pendulum's axis of rotation.

When determining pendulums' parameters during the SBD design process, it is necessary to take into account rotor unbalance and unbalance of the machine body, which is caused by the misalignment of rotor axis and pendulum's axis of rotation.

Equations of plane motion 2.1. Equations of plane motion of a rotor with self-balancing device (SBD).
Let us consider a mechanical system consisting of a rotor and a housing. The rotor is mounted in bearings of the housing, which, in its turn, is elastically connected to a stationary support. Pendulums are mounted movably on the rotor.
To comprise the equations of motion, the Lagrange equations of the second type were used. The notations for the Figure 1 include: the rotor's pivot point О1, the center of rotor gravity С, and the pendulums' pivot point О2. The following coordinate systems are introduced:  is rigidly bound to the foundation; x1y1 performs translational pendulums, starting in the point О1; xRyR is rigidly bound to the rotor; xp1 yp1 and xp2 yp2 as a partial cases of xp yp in the Lagrange equations starting in the point О2; the axis xP is rigidly bound to the center of pendulum mass. The turning angle of the rotor is defined as angle , and that of the axis xP as angle k. Position of the point О2 relative to the point О1 is the angle  and the eccentricity  ( Figure 1). Assume that the housing moves only progressively. Masses of the housing, the rotor and the pendulums are denoted as МH, МR and m. Rigidities of the housing's connection to the stationary support are assumed to be С and С.
The inertia force is significantly larger, than the gravity force thus making the latter ignored.
In order to compose the Lagrange equations of the second type, the kinetic energy of the system T was determined: T are kinetic energy of the rotor, the housing, and the pendulum, respectively; n is the quantity of pendulums.
The rotor performing plane motion has the kinetic energy as follows: where I is the moment of the rotor's inertia relative to the axis passing through the mass center; CR V is the velocity of the rotor's center of mass.
The kinetic energy of the progressively moving housing is: The mass centers' coordinates were found as follows: where l is pendulum length, е is the displacement of the rotor's mass center relative to the axis of rotation.
After differentiating these expressions with respect to time with their subsequent substitution to the equations (1),,the squares of velocities were found as follows: h is the viscous drag force proportionality factor; motor M is motor torque; is the motor damping factor.
Further analysis was carried out for isotropic support stiffness (С =С).

Partial solution of a system of equations for plane motion of a rotor with self-balancing device
Let us address a case when the rotor acceleration has been completed, and the motion of the rotorpendulums system becomes stationary. This way, we assume that: Under these conditions, the system of equations becomes the following form: