Oscillating system with a damper containing controlled damping fluid with hereditary electrorheological characteristics was investigated. Numerical calculation of the system in quasistationary approximation was carried out to prove the accuracy of applied assumptions

A. Bubulis

E. V. Korobko

G. K. Voronovich

G. N. Reizina

English

Vibroengineering PROCEDIA

 262 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. OCTOBER 2014. VOLUME 3. ISSN 2345-0533  
Dynamics of vibroprotecting system when using 
controlled damping fluids with hereditary factor 
G. N. Reizina1, A. Bubulis2, E. V. Korobko3, G. K. Voronovich4 
1, 3, 4A.V. Luikov Heat and Mass Transfer Institute of NAS Belarus, Minsk, Belarus 
2Kaunas University of Technology, Kaunas, Lithuania 
3Corresponding author 
E-mail: 1greizina@gmail.com, 2algimantas.bubulis@ktu.lt, 3evkorobko@gmail.com 
(Accepted 15 September 2014) 
Abstract. Oscillating system with a damper containing controlled damping fluid with hereditary 
electrorheological characteristics was investigated. Numerical calculation of the system in 
quasistationary approximation was carried out to prove the accuracy of applied assumptions. 
Keywords: vibration, damper, electrorheological fluids, viscosity, flow. 
1. Introduction 
One of the promising methods for targeted reduction of vibration exposure on the object is its 
vibroisolation, in particular, with the use of viscous damping. This method finds new control 
features through the use of modern intellectual damping materials with electric and 
magnetosensitive nanofillers [1, 2]. In the electrorheological materials (ERF) containing 
non-colloidal particles as the dispersed phase, under the influence of external field a strong 
structuring of particles is observed. Thanks to it, the material exhibits elastic and viscoplastic 
properties according to the level of external influence. This allows one to control its damping 
characteristics specifically and reversibly (response time less than 10-3 s). It is known, that ERFs 
are used as fluids for flow hydraulic systems and fluid dampers, squeeze dampers [3-5], including 
ground transportation systems and precision instruments. Analysis of rheological behavior of 
non-Newtonian ERFs shows that the hereditary factor becomes important [6], when the duration 
of the relaxation processes occurring in an environment commensurates with the time scale of 
hydrodynamic phenomena. For the considered electrosensitive flowing dispersions relaxation 
times are in the range from 10-2 to 10 s and heredity is manifested in conventional hydrodynamic 
regimes such as characteristic for the flow of the damping fluid in the channel of an electric shock 
absorber in the suspension system of the vehicle. The present paper is devoted to the modeling of 
oscillating system with ERF-damping in the tracking mode with the hereditary factor.  
2. Main part 
Consider the effect of the rheological features of hereditary media on the dynamics of 
vibroprotecting system of the damping device, shown in Fig. 1. 
 
Fig. 1. Scheme of damping device 
For example, in the damping device, the gap width ℎ is much less than the channel width and 
radius of curvature of the wall, ܴ݁ ≪ 1, we neglect the change in the shear stress across the gap 
can be neglected. Then the equation of motion of a body with mass ݉, connected to a damper, can 
be written as follows: 
݉ ሷܺ + ܵ߬(ߛሶ) + ߦܺ = 0, ߛሶ = ௑ି௎(௧)௛ , ܺ(0) = ሶܺ (0) = 0, (1)
DYNAMICS OF VIBROPROTECTING SYSTEM WHEN USING CONTROLLED DAMPING FLUIDS WITH HEREDITARY FACTOR.  
G. N. REIZINA, A. BUBULIS, E. V. KOROBKO, G. K. VORONOVICH 
 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. OCTOBER 2014. VOLUME 3. ISSN 2345-0533 263 
where ܺ – the displacement of the body connected to the spring; ܵ – the surface wetted by the 
fluid; ߛሶ  – the shear rate; ߬ – the shear stress; ߦ – the spring rigidity.  
Dependence of the shear stress versus the shear rate for nonlinear elastic electrorheological 
fluid is defined as: 
߬ = ෍ ߬௜(ݐ)
ஶ
௜ୀଵ
,   ݀߬௞݀ݐ +
ݍ௞
ߣ௞
߬௞ = ௞ܲߛሶ ,
݀ ௞ܲ
݀ݐ +
ݍ௞
ߣ௞ ௞ܲ
= ߟ௞
ߣ௞ଶ
௞݂, (2) 
where functions ݍ௞  and ௞݂  depend on the instantaneous value of the shear rate ߛሶ(ݐ); ߟ௞ , ߣ௞  – 
characteristics of the spectrum of relaxation times in the linear viscoelastic region. Values ݍ௞(ߛሶ) 
and ௞݂(ߛሶ ) incorporate the change of the shear modulus and relaxation time of the fluid with the 
increase of shear rate. 
Transition to dimensionless quantities in Eqs. (1), (2) yields: 
ܺ = ܷ଴߱଴
തܺ,   ݐ = ߱଴ݐ̅, ߛሶ =
ܷ଴
ℎ ̅ߛ
ሶ , ߬ = ߤ ܷ଴ℎ ߬̅, ௞ܲ =
ߟ
ߣ
തܲ௞,  
where ߱଴ = ඥݔ/݉ – the natural frequency of the body; ܷ଴ – velocity scale. The dynamics of the 
oscillating system and established characteristics of fluid flow are defined by the following 
dimensionless complex: ߝ = ܵఎ ݉߱଴ℎ⁄  – damping factor. 
It characterizes the ratio of viscous and inertial forces in the oscillating system and depends on 
[7]:  
• Modified Deborah number, which takes into account the relaxation processes in the fluid 
ܦ݁ = ߣ߱଴, which is the ratio of the maximum relaxation time of the fluid to the period of natural 
oscillations of the system;  
• Weissenberg number, showing the effect of shear rate on the rheological properties of the 
fluid ܹ݁ = ߣ ⋅ ߛሶ ;  
• As well as rheological parameters of damping fluid ߙ and ߙ′/ߣ.  
The problem task in this case is as follows: 
ܺ = −ܧ ቎෍ തܶ௞ + ൭ ෍
1
ߦ(ߙ) ⋅
1
ܭఈ ⋅
݂(ܹ݁, ߣ௞, ߛሶ )
ݍଶ ܹ݁, ̅ߣ௞, ̅ߛሶ
ேே
௄ୀ௄∗ାଵ
൱ ̅ߛሶ
௄∗
௄ୀଵ
቏ − തܺ,
തܶሶ௞ = −
1
ܦ݁଴
ݍ൫ܹ݁, ߣ௞, ̅ߛሶ ൯
ߣ௞
⋅ തܶ௞ +
1
ܦ݁଴
തܲ௞ ̅ߛሶ , 
തܲሶ௞ = −
1
ܦ݁଴
ݍ൫ܹ݁, ߣ௞, ̅ߛሶ ൯
ߣ௞
⋅ തܲ௞ +
1
ܦ݁଴ߦ(ߙ)
 ̅ߟ௞
̅ߣ௞
ଶ  ݂൫ܹ݁, ߣ௞, ̅ߛሶ ൯, 
̅ߛሶ = തܺሶ − ഥܷ(ݐ̅),    ܭ∗ = 6,   ݐ̅ = 0,    തܺ = തܺሶ = 0,   തܶ௞ = 0,    തܲ௞ =
1
ߦ(ߙ), 
̅ߣ௞ = ̅ߟ௞ =
1
ܭఈ ,    ߦ(ߙ) = ෍
1
߭ఈ
ஶ
జୀଵ
.
(3) 
Calculations are carried out for rheological equation of state (RES) of Macdonald-Berd-Carro: 
௞݂ =
1 − ߣ௞ᇱ |ߛሶ |
1 + ߣ௞|ߛሶ |
 ,    ݍ௞ =
(1 − ߣ௞ᇱ |ߛሶ |)ଷ/ଶ
(1 + ߣ௞|ߛሶ |)ଵ/ଶ
,
ߣ௞ =
ߣ
݇ఈ  ,    ߣ௞
ᇱ = ߣ′ܭఈ , ߟ௞ =
ߟ
ܧ(ߙ)ܭఈ ,
ߣ′
ߣ = 0, 2,
(4) 
DYNAMICS OF VIBROPROTECTING SYSTEM WHEN USING CONTROLLED DAMPING FLUIDS WITH HEREDITARY FACTOR.  
G. N. REIZINA, A. BUBULIS, E. V. KOROBKO, G. K. VORONOVICH 
264 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. OCTOBER 2014. VOLUME 3. ISSN 2345-0533  
where ߙ – the parameter of distribution of relaxation time spectrum in the linear area; ߟ − initial 
Newtonian viscosity; ߦ(ߙ) – Riemann zeta-function. The ratio ߣ′/ߣ characterizes the influence of 
shear rate on shear modulus. When ߣ′ = ߣ, ௞݂ = 1 the stationary shear viscosity in the model 
Eq. (4) is defined by the dependence: 
ߟ(ߛሶ) = ߟߦ(α) ෍
ܭఈ
(ܭఈ + ߣ′|ߛሶ |)ଶ
ஶ
௄ୀଵ
. (5)
In the numerical calculation of the row of Eq. (3), the first ܭ∗ members are determined from 
differential equations, and the rest are considered already relaxed. This simplification applies 
when the time scale of hydrodynamic phenomena is substantially bigger then ߣ/(ܭ∗ + 1)ఈ. 
When ܦ݁଴ ≪ 1, the elasticity of the fluid does not appear, the parameter ߙ determines only 
dependence of viscosity on the shear rate, i.e. the degree of “non-linearity” of the oscillating 
system. In this case, it has the same effect as the parameter ݌ in simplified nonlinear viscous model.  
When ܦ݁଴ ≫ 1, the value of ߙ determines the fraction of the reversible component of the total 
amount of deformation of the fluid. 
We carry out calculation of the dynamics of the oscillating system with ERF, rheological 
parameters of which depend on the strength of the external electric field ܧ with correction factor 
ܭ(ܧ): 
ߟ(ܧ) = ߟ
(ܧ)
ሾ1 + ߣ଴ଶ(ܧ)ߛሶ ଶሿ௣
= ߟ ⋅ ܭଵ
(ܧ)
ሾ1 + (ߣ଴ܭଶ(ܧ))ଶሿ௣
, ߟ = ߟ ⋅ ܭଵ(ܧ), ߣ = ߣ଴ ⋅ ܭଶ(ܧ). (6)
Previous experimental studies [8] showed that the greatest effect of damping is achieved when 
the viscous component of ERF has the largest value. For a given fluid, this is achieved at ܧ = 0.3. 
Increased viscosity results in an increase of the damping characteristics of the oscillating 
system. The growth of the electric field intensity also leads to an increase in the elastic component 
of ERF (Fig. 2), and hence to a decrease in the characteristic relaxation time ߣ. 
 
Fig. 2. Dependence of rheological parameters of elastic modulus ܩ’ and loss modulus ܩ’’  
on the electric field intensity 
Assuming proportional dependence on the growth of the effective viscosity ߟ and decreasing 
of the characteristic relaxation time ߣ, we obtain a quantitative relationship (see Fig. 3). 
Voltage is applied in the tracking mode of the dynamic characteristics of the system, previously 
proposed in [9]. 
Test calculations were performed (Fig. 4) for oscillating system with the external action 
depending on the electric field varying in the form of a rectangular pulse. 
Accounting for the effects of rheological properties of the fluid dynamics of shear flow was 
carried out in a wide range of rheological parameters: ߝ = 0.314-3.14; ߣ′/ߣ =  0.2; ߙ = 2-3;  
௘ܹ = 0-100; ܦ݁଴ = 0.1-5.0 and ܧ = 0-0.3 kV/mm. Calculations were carried out at ܭ∗ = 6 to 
DYNAMICS OF VIBROPROTECTING SYSTEM WHEN USING CONTROLLED DAMPING FLUIDS WITH HEREDITARY FACTOR.  
G. N. REIZINA, A. BUBULIS, E. V. KOROBKO, G. K. VORONOVICH 
 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. OCTOBER 2014. VOLUME 3. ISSN 2345-0533 265 
increase the count rate, since the difference in the numerical values obtained for ܭ∗ = 23 and 
ܭ∗ = 6 is only 2 %. 
 
 
Fig. 3. Qualitative characteristics of elastic ܽ(ܧ) and viscous ܾ(ܧ) components of ERF, [ܧ] – kV/mm 
 
Fig. 4. Characteristics of established oscillations 
Thus the rheological model of MacDonald-Berd-Carro used in the calculation predicts the 
nonlinear behavior of the system. Indeed, the calculation results for the nonlinear viscous 
rheological equation of MacDonald-Berd-Carro are in good agreement with the results of 
calculations for a simple nonlinear viscous model: 
ߟ(ߛሶ) = ߟ(1 + ߣଶߛሶ ଶ)௣ , ݌ < 0,5.  
Thus, the accuracy of the data in the quasistationary approximation is shown, which can 
dramatically reduce the time of calculating the characteristics of the oscillating system. 
DYNAMICS OF VIBROPROTECTING SYSTEM WHEN USING CONTROLLED DAMPING FLUIDS WITH HEREDITARY FACTOR.  
G. N. REIZINA, A. BUBULIS, E. V. KOROBKO, G. K. VORONOVICH 
266 © JVE INTERNATIONAL LTD. VIBROENGINEERING PROCEDIA. OCTOBER 2014. VOLUME 3. ISSN 2345-0533  
3. Conclusion 
Studies performed have shown that the change in the controlling of the external impact by the 
electric field in the tracking mode leads to a change of the elastic and viscous components of the 
damping fluid, as well as the number of parameters of Deborah and Weissenberg numbers, and, 
consequently, to a change in the damping coefficient, which in turn affects the dynamics of the 
oscillating system. 
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Dynamics of vibroprotecting system when using controlled damping fluids with hereditary factor

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Dynamics of vibroprotecting system when using controlled damping fluids with hereditary factor

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