The paper considers an impact of viscous linear and cubic nonlinear damping of the elastic support on nonlinear vibrations of a vertical hard gyroscopic unbalanced rotor, taking into account nonlinear stiffness of the support material. Analyzing the research results shows that linear and cubic nonlinear damping can significantly suppress the resonance peak of the fundamental harmonic, eliminate the jumping phenomena of the nonlinear system. In non-resonance areas where the velocity is higher than the critical one, cubic nonlinear damping, unlike linear one, can slightly suppress amplitude of the rotor vibration. Therefore, in the high-velocity area, only nonlinear damping can maintain performance of a vibration isolator. In resonance area, an increase in linear or cubic nonlinear damping significantly suppresses the ability to absolute displacement. In non-resonance area, where the rotational velocity is lower than the critical one, they have almost no impact on ability to absolute displacement. In high velocity area, an increase in nonlinear damping may slightly increase the moment of force transmissibility, but linear damping has almost no impact on it. The obtained results can be successfully used to produce passive vibration isolators used for damping the vibrations of rotary machines, including gyroscopic ones.

As you know, the rotary machines are widely used in many industries and have been studied for a long time. Despite this, there are many problems unsolved, in particular those associated with an action of mass imbalance on vibrations and stability, and subsequently with the stabilization of resonance vibrations of the rotary machines.

A simplified model with concentrated rotor system parameters is typically used to study the dynamics of a single rotor shaft on the bearing supports. It is very important to use the properties and characteristics of the support material for vibration attenuation and damping to stabilize a movement of the unbalanced rotor and the vibration systems. Supports are the mean for connecting devices between the rotor and the supporting structure, which have various shapes and designs depending on the specific assumptions. A convenient way is to introduce attenuation for supporting the bearings in the rotor system on viscoelastic flexible rubber supports (Zakaria et al., 2015). In parallel with the development of viscoelastic material simulation (Gil-Negrete et al., 2009; Richards and Singh, 1999), which helps to describe a complexity of material properties, the use of viscoelastic components in rotor dynamics and vibration systems (Ravindra and Mallik, 1994; Peng et al., 2012; Ho et al., 2012) as a whole also increased, including with nonlinear elastic characteristics and damping. For example, Ravindra and Mallick (1994) researched the parametric impact of different types of attenuation on the performance of nonlinear vibration isolators in case of harmonic excitation. The paper (Peng et al., 2012) considers an efficiency of passive vibration isolators with linear damping and cubic nonlinear damping in resonance and non-resonance vibration areas of the system with linear stiffness. It also provides an excellent overview of researches of the linear and nonlinear vibration isolation systems. In the paper (Ho et al., 2012), an impact of cubic nonlinear rigidity of the material on the performance of the isolator is additionally taken into account in the studies. The papers (Iskakov, 2015, 2017a, b) researched impacts of quadratic nonlinear damping on the resonance vibrations and stability of the vertical gyroscopic unbalanced rotor with quadratic and cubic nonlinear stiffness of an elastic support. Magnetorheological materials can be used as damping elements. The model is based on presentation of magnetorheological oil by bilinear material (Zapomel and Ferfecki, 2015). In the paper (Fujiwara et al., 2015), an experimental single-disc rotor system is prepared that is supported by ball bearings at both ends, and vibration is compared with a flexible support containing springs or rubber blankets and with a rigid support base, through simulation and experiment.

This paper is aimed at researching an impact of the material of the elastic bearing support with linear and cubic nonlinear damping on the main resonance curve and the moment of force transmissibility of the gyroscopic vertical hard unbalanced rotor, taking into account the nonlinear stiffness of the support.

The rotor flowchart shown in Fig. 1 is considered. Shaft with the length

Rotor geometry.

Also assume that linear eccentricity

Given the above, the projections of the angular velocity on the coordinate
axes will be written as

Moments of external forces are as follows

Mallick et al. (1999) experimentally confirmed that the restoring and damping
forces in elastomeric isolators should be described, using a nonlinear
model. Richards and Singh (1999) found that rubber dampers have both
nonlinear damping and nonlinear stiffness. To achieve higher performance,
any nonlinearities in the design should be taken into account. Consequently,
the elastic support of the upper bearing of the gyroscopic rotor can be made
of non-linear materials, such as rubber, resin and other polymers widely
used as the damper of the arising vibrations. Given all this, we set the
dissipative forces in the elastic support as

The Lagrange's Equations of Second Kind (Yablonsky, 2007) for the rotor
system are written as

Substituting the expressions (2)–(6) in (7), obtain the equations of
motion of the rotor

By entering the following dimensionless parameters

Thus, it turns out that the steady-state motion of the considered rotor is described by the Duffing type differential equation system (13) and (14). To determine a periodic solution with a period equal to the period of external action, a method of decomposition of solutions (13) and (14) into Fourier series with uncertain coefficients is usually used. The coefficients can be found by the harmonic balance method (Hayashi, 1964; Szemplinska-Stupnicka, 1968; Kydyrbekuly, 2006), taking into account the finite and usually small number of terms.

Expanding the solutions of the equations of motion (13) and (14) into the
Fourier series with indefinite coefficients, can verify that the
approximation of the solutions with a simple harmonic and oscillation
frequency equal to the disturbing moment frequency is quite satisfactory in
the case of the main resonance. Taking into account the following notation
of the vibration parameters

The formula for the amplitude-frequency characteristic (17) taking into
account the expressions of dimensionless parameters (10), the critical
velocity formula (11), and the formula for the resulting moment amplitude
(12), may be needed when experimentally determining the values of the linear
damping coefficients

The results of numerical solutions of the equations of motion (13) and (14)
of a rotor system with nonlinear rigidity and the dependency graphs (15) and
(16) taking into account expressions (17) and (18) are presented in Figures
2 and 3 for the following parameters:

Oscillogram

Oscillogram

The graphs in Figs. 2 and 3 show a sufficient approximation of the results of numerical modeling and analytically obtained results.

In the absence of nonlinear terms in Eqs. (13) and (14), the results for the linear rotor model are obtained from expressions (17) and (18) (Iskakov and Kalybaeva, 2010).

With the introduction of additional value notations

In case when

If shaft speeds are close to zero, the action of the gravitation moment is
more noticeable. Then, if

On the other hand, take the projection of the moment of transmitted force
(29) in a form of the following harmonics

The numerical solution of the Eq. (20) and the calculation from the
expression (32) were performed using MathLab to demonstrate an impact of
linear and nonlinear viscous damping on vibration isolation of the Duffing
type rotor system. The results are shown in Figs. 4–13. The rotary system
with linear stiffness of the elastic support as described in researches
(Peng et al., 2012; Iskakov, 2018a, b) is presented in Figs. 4–7. In Figures 4 and 5, the elastic support of the rotor has linear
stiffness and linear damping with

Dependence of the vibration amplitude

Dependence of the moment of force transmissibility of the system

Figures 8–9 shows the effects of linear damping in case of nonlinear stiffness
of the elastic support. A contrast between the effects of linear and
nonlinear viscous damping can be observed by comparing Figs. 8–9 with
Figs. 10–13, where the cubic viscous damping coefficient

To ensure the stable movement, avoiding jumping should be an important
feature of the vibration isolator. Researches of Ravindra and Mallick (1994)
have shown that the jumping phenomena can be eliminated by linear damping.
With pure linear viscous damping, the calculation results in Figs. 8 and 9
show jumps occurring at

Dependence of the vibration amplitude

Dependence of the moment of force transmissibility of the system

Dependence of the vibration amplitude

Dependence of the moment of force transmissibility of the system

In non-resonance area, where

Dependence of the vibration amplitude A on the rotor velocity

Dependence of the moment of force transmissibility of the system

Dependence of the vibration amplitude

Dependence of the moment of force transmissibility of the system

Diagrams for study of the phase-frequency characteristic of the rotor are
shown in Figs. 14–17. When plotting the diagrams, the properties of the
arctangent function were used. These diagrams show that in case of linear
stiffness of the support, a rotation of the vibration phase occurs when
passing through the critical velocity, i.e. when

Dependence of the vibration phase

Dependence of the vibration phase

Dependence of the vibration phase

Dependence of the vibration phase

The results of numerical solutions of the equations of motio (13) and (14)
of a rotor system with nonlinear rigidity with the output of the

Amplitude response of oscillations at

Amplitude response of oscillations at

Consider motion stability using an approximate theory (Iskakov, 2018b;
Iskakov, 2019; Van Dooren, 1971). The geometric location of the points at
which the amplitude curves for the oscillations of the principal resonance
have vertical tangents is determined by the equation

The solutions of Eq. (43) have the form

Figure 21 shows the boundaries of the instability region of an oscillatory
system with non-linear stiffness

The boundaries of the instability region for a system with a coefficient of non-linear stiffness of the support

The boundaries of the instability region for a system with a coefficient of non-linear stiffness of the support

Experimental studies of an effect of the support material on dynamics of the centrifugal gyroscopic rotor were carried out on a test unit which is generally shown in Fig. 22, and a flow chart with all measuring instruments is shown in Fig. 23.

Test Unit Overview: 1 – centrifuge with rotor, 2 –

Experimental installation flowchart.

The unit consists of an electrically powered rotor, a channel for measuring and recording the precession motion performance, a channel for measuring and recording the rotor shaft speed, a channel for measuring and recording current in driven electric motor.

The rotor to be studied is made in form of a cylindrical cup made of
duralumin D 16 T with a top transparent PMMA cover. Distance between
supports

The unit uses an electromagnetic speed sensor, voltage pulses from which are
fed to an input of frequency meter

Strain-gauge and variable inductance transducers of displacement were used
to record precession motion performance. Transducers are coupled by a
half-bridge circuit and are connected to an input of Topaz-3 (4) amplifier.
Offset voltage of the bridge between a movable rotor and variable inductance
transducer of displacement is fed to the input of UT-4 (5) amplifier and
from its output to

To measure current intensity in circuit of the driving motor, compensation method was used since the capacity variations to be determined are much less than the primary engine capacity.

Dynamic performance of the rotor motion and parameters of interaction with the driving motor were read according to the following procedure.

According to the results of measuring the amplitude of signals received in

Sensitivity and stability of amplifier channels was monitored and precession performance was recorded before and after each experiment by displacing the rotor shaft from an equilibrium position towards the sensor using a pusher with a distance meter. Value of voltage applied to the motor from LATR-1M laboratory autotransformer was recorded at all shaft velocities.

Along with determination of the rotor oscillation amplitude and the shaft
velocity, the current in the circuit of the motor was recorded by

The above operations were repeated at different damping of upper elastic support and values of radial clearance of bearing. Changing in damping was achieved by installing the PTFE support material in one case, and the soft rubber support material in another one. Changing in radial clearance was carried out by changing the bearings. Dynamic values of the rotor motion were compared for two limiting options of these parameters – a fluoroplastic support and bearings with 0.1 mm clearance in one case and a soft rubber elastic support and bearings with 0.01 mm clearance in another one.

As can be seen from the above, the measured values were amplitude and frequency of the rotor precession, shaft speed, voltage and current in the driving motor circuit.

Amplitude-frequency characteristic of the centrifugal rotor in different versions of the support material.

Amplitude-frequency characteristics of the rotor measured for cases of
fluoroplastic support, bearings with a clearance of 0.1 mm and a rubber
support, bearings with a clearance of 0.01 mm are shown in Fig. 24. Similar
curves were previously
obtained respectively in the papers by Kelzon and Pryadilov (1965), Tuleshov (1987). Comparison of the amplitude-frequency characteristic curves shows
that the replacement of the fluoroplastic material of the support with
rubber damps a resonance curve of the main harmonic in the area of the
critical velocity

Zones of higher vibration amplitudes are not related to a coincidence of the
shaft velocity with one of the rotor eigenfrequencies, but are related to
the ripple frequency of power in the electric motor because of the limited
energy source power. The presence of modes of coupled vibrations of power
and precession amplitude (Felix et al., 2015; Tuleshov, 1987), at
frequencies that are multiples of the frequency of the alternating component
of the actuating moment (628 s

Figure 25 presents a dependence of the angular shaft velocity on the tension
applied during acceleration and braking of the engine. The obtained
dependence indicates the quadratic character of an increase in losses in the
engine with an increase in its power, which is consistent with (Vishnevsky,
1955). Significant deviation from this law when accelerating the rotor in
the range of angular velocities 550 s

Swinging of the centrifugal rotor by torsional vibrations occurs due to the
cyclicity of the instantaneous power in the engine when powered from the AC
network, i.e., there is a phenomenon of capture on a ripple frequency of the
engine power. When replacing the support material with rubber, the movement
of an area of increased amplitudes to the area of the shaft velocity value

Dependence of the shaft velocity on the tension applied to the engine in case of the fluoroplastic material of the rotor support.

Thus, selecting the values of the stiffness coefficient

The invention is illustrated by drawings in Fig. 26. In Fig. 26, the driving electric motor (4) with the power take-off shaft (5) is strengthened on the platform (3) by means of the cylindrical ring (1) and the brackets (2). The electric motor shaft is connected to the power take-off shaft by a transient coupling (6). The driving electric motor with a power take-off shaft is attached to the frame by a lower hinge (7) and an upper spring (8) and rubber (9) elastic supports. A cylindrical vessel (10) is mounted on the upper conical part of the shaft (5) and is held on the shaft (5) by a lock nut (11). The cylindrical vessel has an internal lining (12) and is closed on top with a transparent cover (13). To measure characteristics of the precessional movement, the displacement sensors (14) are installed on the bracket (2), connected to the control device by wires (15), (16) and (17). To measure the shaft velocity (5), a magnetoelectric velocity sensor (18) is used, connected by wires (19) and (20) to the recording device in the control apparatus. The rotor velocity in the operating modes is set by the power source voltage regulator.

Gyroscopic rotor based centrifuge.

The electric motor is powered by wires (21) and (22). The shaft precession amplitude for the operating velocity and at full fill with liquid is regulated by tightening the spring of the upper elastic support (8) with the help of a nut (23), depending on the production process and product quality requirements. In this case, a rubber nozzle of the lower elastic support (9) is selected in such a way that the value of the vibration isolation coefficient (moment of force transmissibility of the system) of a special resin or rubber material is acceptable for safe passage through the critical velocity in case of determining the operating velocity beyond the critical velocity.

The following conclusions can be drawn based on the analysis and discussion of the research results:

The impact of an elastic bearing material with linear viscous damping and with cubic non-linear viscous damping on non-linear vibrations of a vertical hard gyroscopic rotor was studied, taking into account the non-linear stiffness of the elastic support.

The equations of motion of the rotor were composed based on the Lagrange's Equations of Second Kind, which were reduced to a dimensionless form.

The equations of motion of the rotor were solved by the harmonic balance method, and the expressions of amplitude-frequency and phase-frequency characteristics of the fundamental harmonic and the expression for determining the moment of force transmissibility of the system were obtained.

Researches of the amplitude-frequency characteristic depending on the coefficient of linear viscous damping and nonlinear viscous cubic damping of the elastic support showed that both linear and nonlinear cubic damping significantly suppresses the resonance amplitude of the fundamental harmonic, eliminate jumping effect of the nonlinear system.

In non-resonance areas, an impact of linear damping on the vibration amplitude is very weak and negligible, and nonlinear cubic damping in an area where the velocity is many times greater than its critical value can slightly suppress the amplitude of the rotor vibrations and, therefore, only nonlinear damping can maintain performance of the vibration isolator throughout the entire range of the rotor velocity.

The research results were used when preparing an application for a patent for invention of a gyroscopic rotor based centrifuge to develop a description, abstract and claims. The invented centrifuge can be used in pharmaceutical and food industries to effectively intensify the mixing of suspensions of medicinal herbs, dairy and cultured milk products, other liquid food products throughout the volume of a container due to controlled precession.

Our main research results prior to this publication are
stored in the following public data repositories:

ZI set a study objectives, formed differential equations of motion, worked on solution of differential equations of motion, prepared an analysis of the research results and a manuscript for the presentation. KB worked on numerical solution of differential equation of motion and participated in the analysis of the study results.

The authors declare that they have no conflict of interest.

This research has been supported by the The research work was funded by Ministry of Education and Science Republic of Kazakhstan on the basis of the Grant Financing of Scientific Research (grant no. 2263/GF4).

This paper was edited by Jahangir Rastegar and reviewed by Madan Lal Chandravanshi and one anonymous referee.