Mathematical Problems in Engineering

Volume 2015, Article ID 839105, 8 pages

http://dx.doi.org/10.1155/2015/839105

## Controlled Passage through Resonance for Flexible Vibration Units

^{1}Institute of Problems in Mechanical Engineering, 61 Bolshoy Avenue V.O., Saint Petersburg 199178, Russia^{2}Saint Petersburg State University of Architecture and Civil Engineering (SPSUACE), 2nd Krasnoarmeiskaya Street 4, Saint Petersburg 190005, Russia^{3}University ITMO, St. Petersburg State University and Institute of Problems in Mechanical Engineering, 61 Bolshoy Avenue V.O., Saint Petersburg 199178, Russia

Received 28 December 2014; Revised 24 March 2015; Accepted 24 March 2015

Academic Editor: Tomasz Kapitaniak

Copyright © 2015 Dmitry A. Tomchin et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The problem of controlled passage through resonance zone for mechanical systems with several degrees of freedom is studied. Control algorithm design is based on speed-gradient method and estimate for the frequency of the slow motion near resonance (Blekhman frequency). The simulation results for two-rotor flexible vibration units illustrating efficiency of the proposed algorithms and fractal dependence of the passage time on the initial conditions are presented. The novelty of the results is in demonstration of good behavior of the closed loop system if flexibility is taken into account.

#### 1. Introduction

A typical problem for control of vibration units is passing through resonance at startup mode of vibroactuators, in the case of the operating modes belonging to a postresonance zone. Such a problem arises in the case when the power of a motor is not sufficient for passage through resonance zone due to Sommerfeld effect [1]. Dynamics of near resonance behavior are nonlinear and very complicated. Their analysis attracts attention of researchers for about 50 years [2–4].

Perhaps the first approach to the problem of controlled passage through resonance zone was the so-called “double start” method due to Gortinskii et al. [5]. The method is based on the insertion of time relay into motor control circuit for repeatedly switching on and off motor at precalculated time instants. Basically, this and other feedforward (nonfeedback) methods are characterized by difficulties in calculation of switching instants of a motor and sensitivity to inaccuracies of model and to interferences. A prospective approach to the problem is based on feedback control. Feedback control algorithms for passing through resonance zone of mechanical systems were considered in [6–8]. In [6] an optimal control algorithm for passage an unbalanced rotor through critical speed was proposed. For the same problem several control methods are evaluated, and the necessary number of dampers and their optimal location were determined in [7]. In [8] a nonlinear controller reducing resonance effects during despin of a dual-spin spacecraft was designed. A method of vibration suppression for rotating shafts passing through resonances by switching shafts stiffness was proposed in [9]. In [10] the dynamics of passage through resonance of a vibrating system with two degrees of freedom was examined. However, the early algorithms did not have enough robustness with respect to uncertainties and were hard to design.

For practical implementation of control system, it is important to develop reasonably simple passing through resonance zone control algorithms, which have such robustness property: keeping high quality of the controlled system (vibration unit) under variation of parameters and external conditions. Perhaps the first such a simple controller was proposed in [11] based on the speed-gradient method previously used for control of nonlinear oscillatory systems [12]. A number of speed-gradient algorithms for passage through resonance in 2-DOF systems were proposed in [13]. The approach of [11, 13] was applied to two-rotor vibration setup in [14].

This work is dedicated to further extension of the results of [14]. Problem statement for control of passage through resonance zone for mechanical systems with several degrees of freedom is adopted from [13]. The control algorithms based on the speed-gradient method for two-rotor vibration units with flexible cardan shafts are described. The simulation results illustrating efficiency and robustness of the proposed algorithms are presented. Previous results [14] are related to two-rotor vibration units with rigid cardan shafts.

#### 2. Problem Statement and Approach to Solution

To describe the dynamics of a mechanical system and to carry out the control algorithm synthesis, it is easier to use standard Euler-Lagrange form, leading to the following model of controlled system dynamics with n degrees of freedom:where is -dimensional input vector, representing elastic forces depending, in turn, on controlling forces; is -vectors of generalized coordinates; is – inertia matrix; is the -vector of Coriolis and centrifugal forces; is the -vector of gravity forces; is the control matrix. For synthesis of control algorithms, it is often convenient to use equations in Hamiltonian form:where is the generalized momenta vector and denotes the Hamiltonian function (total energy of the system):where is the potential energy. For design an idea of speed-gradient energy control is used and introduced in [12]. It is based on an auxiliary control goal: approach of free energy of the system to a surface of the given energy level:Introducing the objective functionthe goal (4) is reformulated asControl algorithm for passing through resonant frequencies for an unbalanced rotor is based on the speed-gradient method [10], which allows us to synthesize control algorithms for significantly nonlinear objects. At the same time objective functional is chosen on the basis of total energy of a mechanical system excluding friction losses, because in this case total energy is invariant for a mechanical system that is required in the speed-gradient method in the version of [12].

It is assumed that in dynamics model of the system two subsystems are allocated: carrier and rotating body, such that the total energy is represented in the following form:where is the energy of rotating subsystem, is the energy of a carrier subsystem, and is the energy of interaction.

The solution of [6, 8] adopted in this paper is based upon usage of the speed gradient algorithms [12] and a motion separation into fast and slow components, which occurs near resonance zone [15]. Quantitative analysis of slow “pendular-like” movements was performed by Blekhman et al. in [15] where the frequency of an “internal pendulum” was evaluated. It will be further called the “Blekhman frequency.” An approach of [15] is briefly described below for completeness.

In [15] the following system with inertial excitation of oscillations was considered:where is the rotation angle of a rotor, is a platform deflection, is the mass of a platform, is the rotor mass, is the inertia moment of a rotor, is the system mass, is the coefficient of axial stiffness, is the damping coefficient of the spring, is the eccentricity of rotors, is the rotation torque of a motor (static characteristics), and is the torque of resistance forces. Assuming that frequency of rotation varies slowly in terms ofand using the method of direct separation of motions [1], Blekhman et al. derived the equation of slow “pendular-like” oscillations in the following form [15]:where is slow variable: “addition” to rotation frequency, , is the “total damping coefficient,” obtained by linearization of expressions for and near value : is a resonance frequency, , and is the first approximation to amplitude of oscillations of a platform. If the relation holds in the preresonance zone then the valueappears to be a frequency of small free oscillations of an “internal pendulum” (excluding resistance force), which is called the “Blekhman frequency” thereafter.

As seen before, that frequency vanishes to zero when . The necessary condition of the validity of (10) is that the Blekhman frequency should be significantly less than frequency of rotations (usually, is typically sufficient).

The idea of control algorithms described below is to extract the slow motion and to swing it with the aim to increase the energy of the rotating subsystem. To isolate slow motions, low-pass filter is inserted into the energy control algorithms. Particularly, if slow component appears in oscillations of angular velocity of a rotor , then the control algorithm proposed in [11] is used:where denotes the Hamiltonian (total energy of the system), is the variable of a filter that is an estimate of the slow motions satisfying (10), and is the time constant of a filter. At low damping, slow motions also fade out slowly, which gives control algorithm an opportunity to create suitable conditions to pass through resonance zone. Thus, the effect of “feedback resonance” [16] is created. After passing the resonance zone it is suggested to turn off the “swinging” and then to switch off control and to apply the constant drive torque. For a proper work of a filter, it should suppress fast oscillations with frequency and pass slow oscillations with frequency, where is the Blekhman frequency. That is, time constant of a filter should be chosen from the inequalityAlgorithms of passing through resonance zone for the two-rotor vibration units are described in [14]. Below we analyze the algorithm of [14] taking into account elasticity of the cardan shafts.

#### 3. Passing through Resonance Control Algorithm of Two-Rotor Vibration Unit

Consider the two-rotor vibration unit in the startup/spin-up mode [17]. The unit consists of two rotors 4 installed on the vibrating platform 5. The rotors are elastically connected with fixed basis 2, and frame 3 by springs 8 (Figure 1).