Abstract
The synchronization and stability of two unbalanced rotors (URs) separately driven by induction motors rotating in opposite directions, in a new mechanism system with four rigid frames (RFs), are investigated in present work. Applying Lagrange’s equation, the differential equations of motion of the system are deduced. The criteria of synchronization and stability of the synchronous states are derived analytically by using the average method and Hamilton’s principle, respectively. Based on the theoretical results, the coupling dynamic characteristics of the system are given by numeric, it is shown that the abilities of synchronization and stability, are the best under the condition that the parameters of the system are completely symmetrical. The stable states and the corresponding motion types of the system in different resonant regions, are clarified in detail. A Runge-Kutta simulation routine is employed to verify the validity of the theoretical results, as well as the feasibility of the used theory method. The selecting principle of the stable region with zero phase difference for the two URs, can provide a novel reference for designing a new type of vibrating feeder with the anti-blocking function in engineering.
Highlights
In the field of the investigations on vibrations, which main aim generally includes two aspects, vibration suppression and vibration utilization. For the former, it refers to vibration reduction, such as dynamic vibration absorbers [1], [2]; while in light of the latter, many types of vibrating machines using in the industrial production process, can be designed and manufactured by utilizing the principle of vibrations
After the synchronous operating frequency ωm0 passes through the resonant point of ω0, i.e., in region II, phase lag angle γ4y is close to zero, and the other three phase lags are still at π, in this case, the phase relationship between the isolation rigid frames (RFs) (m4) and unbalanced rotors (URs) is the operation with the same phases in y-direction; that between the other three RFs (m1, m2 and m3) and URs is the operation with the opposite phases
Taking the curve with η = 1 for example, when the synchronous operating frequency ωm0 is in region I, the stable phase difference (SPD) is stabilized in the vicinity of π
Summary
In the field of the investigations on vibrations, which main aim generally includes two aspects, vibration suppression and vibration utilization. In (11), Tu = m0r2ωm2 0/2 denotes the kinetic energy of the standard UR, and on the left-hand side of the equal sign, TLi (i = 1, 2), represents the average load torque of the motor i. The synchronization criterion of implementing the synchronous operation of two URs is (16), which describes that the difference of the effect residual electromagnetic output torque between two motors, should be less than the maximum of the coupling torque between two URs. Summing the two formulae in (10), leads to the fact that the dimensionless average load torque of each motor, denoted by τa(α ), can be expressed as. The stability criterion of the synchronous states is (26), which describes the coefficient of stability ability should be greater than zero
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