CC BY
1
MSC 35Q74
DOI: 10.14529/ mmp240109
INVESTIGATION OF THE TRANSIENT RESPONSES OF A BEAM ON AN ELASTIC POLYMERIC FOUNDATION
A.D. Kashcheeva1, A.A. Zamyshlyaeva1
1 South Ural State University, Chelyabinsk, Russian Federation E-mail: kashcheevaad@susu.ru, zamyshliaevaaa@susu.ru
The negative impact of vibrations on various devices and mechanisms can be significant, so it is important to take this factor into account when designing, operating and maintaining various equipment and engineering systems. Various methods and technologies can be used to protect against the negative effects of vibrations. Special damping materials are often used. This research paper is devoted to the analysis of the effectiveness of vibration reduction taking into account the physical parameters of elastic polymeric materials. To conduct the study, a mathematical model describing motion of the beam resting on an elastic polymeric foundation is constructed. The model is based on a system of nonlinear differential equations. An algorithm was developed and applied for the numerical solution of this system of equations. Numerical experiments were carried out for the study of the system reaction to different cases of accelerations. As a result, the deflection structure for materials with different physical characteristics were obtained. These results can serve as a starting point for a deeper study of materials and creation of more complex structures.
Keywords: transient responses; elastic polymeric foundation; nonlinear differential equation; Galerkin method.
Introduction
Vibrations can have a negative impact on various devices and mechanisms. Numerous studies in various fields consider the methods of reducing the impact of vibrations. Vibration reduction is an important issue in the structures and buildings, the transport sector, as well as in the manufacture of precision equipment [1-4]. The classic way to reduce vibrations is to use elastic polymeric layers. Elastomers are commonly used for damping due to their high elasticity and damping properties. The damping efficiency depends on many factors [5]. An important factor is the initial compression stress and the area of the elastic polymeric element. However, the choice of damping material is initially important. Rubber used to be the most common material that can reduce vibration. But at the moment, many new materials have appeared, such as elastomers. In practice, dielectric elastomers (DE) [6,7], magnetorheological elastomers (MRE) [8] and thermoplastic elastomers (TPE) [9] are often used. However, researchers are now developing new materials for vibration isolation. This work is aimed at investigating the effectiveness of vibration isolation depending on the main characteristics of elastic polymeric materials - Young's modulus and material density.
1. Model Specification
Fig. 1 illustrates a structure that consists of an elastic beam and an elastic polymeric foundation. The elastic polymeric foundation is a damping layer. Fig. 1a provides a side view of the structure with dimensions: length L, width b, beam thickness hb and elastic
a) b)
Fig. 1. Side view of the structure a) before the force is applied, b) after the force is applied
polymeric foundation thickness hf, before it is subjected to external acceleration a0. Fig. 1b presents a side view after the force or external acceleration is applied.
This allows us to present the governing equation for the transverse motion of the beam as a nonlinear integro-differential equation:
tfm21
dx
+P„A + ao) + + Ff{Xi t) = o,
(1)
where Eb is the effective Young's modulus of the beam, Wb(x,t) is the deflection of the beam, defined to be positive in the direction to the right, A = bh is the cross-section area, I = j^dhl is the moment of inertia of the beam cross section, b is the beam width and hb is the beam thickness, a0 (t) is the external acceleration field, cb is equivalent damping coefficients of the beam in transversal vibration of the structure, pb is a density of the beam. Ff represents the force exerted by the foundation on the beam:
Ff (x,t) = -bEf
(0)
dW
f
+ -
1 dWf 2
f
dz 2 dz
z=0
Boundary conditions for equation (1) are as follows:
Wb(0, t) = 0,
dWb
Equation
dt
d2 Wf
= 0, Wb(L, t) = 0,
dWb
x=0
dt
= 0.
x=L
N2 (wf)=Pf~öpL + Cf
dWf dt
- Ef
( d2Wf d2Wf dWf \ dz2 dz2 dz
=0
(2)
(3)
describes transverse motion of the damping layer. Here Wf (x, z, t) and Ef are the deflection and effective Young's modulus of the elastic polymeric layer respectively, pf and Cf represent the density and equivalent damping coefficients of the elastic polymeric layer accordingly.
Boundary conditions for equation (3) are as follows:
Wf (x, 0, t) = Wb(x, t), Wf (x, hf, t) = 0.
(4)
2. Numerical Solution
The Galerkin method was used to study the behavior of the transverse motion of the beam associated with the compressive motion of the polymer layer of the structure under the influence of an external force. Partial nonlinear equations of mixed form were discretized using this method, turning the system into a discrete number of ordinary differential equations depending on the considered number of spatial basis functions. Since the accompanying boundary condition for the compressive motion of the elastic polymeric layer is mobile, we consider the following transformation for the compressive motion of the polymeric layer:
Wf(x, z, t) = U(x, z, t) + - -^j Wb(x, t). (5)
Introduce the following dimensionless quantities:
W U x z t
Wb = JTf>u = ve = Z'C= Vr = P (6)
where t* is a time scale, t* = ■sj'^^f, t = ¿r is the dimensionless time respectively.
Then the solutions of system (1), (3) can be expressed as follows in terms of the basis functions, satisfying the boundary conditions (2), (4):
= an(т)фи{0, (7)
n=1
) = E E bnm (т )фп (£ )Фш(С ), П =1, 2, 3,...,N,m = 1, 2, (8)
N M
) / / wnm\' JYn^JYmK n=1 m=1
The error functions are constructed by substituting solutions (6), (7) and (8) into the system of equations (1) and (3). As a result, we obtain nonlinear ordinary differential equations that describe the motion of the structure:
N N N M N
E Mm -àn+E cnk àn+^ Kn^anbnm+
n=l n=l n=l m=1 n=l
N N M N N M N M N
+ ^^ ^^ Znmkànam + ^^ ^^ ^^ Lnmpkbnmap + ^^ ^^ ^^ ^^ Fnmpskbnmbpq + (9)
m=ln=l m=1n=lp=l m=1n=lp=l s=l
N N N N
+ E E E N™pkànàmàp + £ D11}ao(T) = 0, m=1n=1p=1 k=1
M N N M N M N
r(2) b _L M(3) a _L C(2) b _L K(2)
n
Mmnkq bmn + ^^ Mnkq an + ^^ ^^ Cmnkqbmn + ^^ ^^ kq bmn +
m=1n=1 n=1 m=1n=1 m=1n=1
M N M N N
+ EEE E N^ÏU bmn bps + E EX a n+ (10)
m=1n=1p=1 s=1 n=1
M N N
+ S S S Gmnpkqbmn ap = m=1n=1p=1
where the coefficients M^k, C^k, K^ and so on are constructed by scalarly multiplying the error equations in L2 (G).
3. Numerical Experiments
In this section of the article, we interpret the findings of this study. We consider several different cases and explain the need to use elastic polymeric layers. The cases will be aimed at studying the relationship between deflection and material properties.
According to the model described previously, the elastic beam and the elastic polymeric layer have the properties described in Table 1 and Table 2. It is assumed that the elastic beam is made of steel, and the elastic polymeric layer is made of elastic material. It is worth noting that we do not use specific materials in the study. The study is based on changes in the Young's coefficient and the density of the material. These two parameters are among the main ones in the description of elastic polymeric foundation.
Table 1
Material and geometrical properties of elastic beam
Length L 100 /im
Thickness hb lfim
Width b 10 fim
Effective Young's modulus Eb 169 GPa
Density pb 2330kg/ms
Equivalent damping coefficients q, 0,052 - 0,054
Table 2
Geometrical properties of elastic polymeric layer
Length L 100 /im
Thickness hb 5 /im
Width b 10 fim
The basis functions are linearly independent and satisfy boundary conditions (2), (4) as well as the completeness condition. The following functions are selected as basis functions:
<pn(i) = 1 - cos(2nn£), ^n(Z) = sin(nnZ).
The applied force holds the structure in vertical position. The purpose of experiments is to study the deflection structure (wb and Wf) when subjected to different type of accelerations a0 over time t.
An algorithm for conducting computational experiments has been developed and implemented in Python. The program calculates the coefficients for equations (9), (10). The system of nonlinear differential equations is solved by the Runge-Kutta method of the 4-th order.
The following acceleration parameters were selected for the experiment:
mm -1
ao = 60--, a = 100--, ujn = 122sec .
sec 2 sec 2
Fig. 2 illustrates the effect of acceleration a0(r) on the deflection of the elastic beam wb(£,r) and elastic polymeric layer Wf (£,(,r) when effective Young's modulus Eb = 5kPa, 10kPa, 20kPa. The deflection value is determined precisely at the point of contact between the beam and the layer, where: £ = 0, 5,y = 0, ( = 0. As a result of the acceleration the beam experiences a strong initial force, resulting in a large displacement. However, as time progresses, the amplitude of these oscillations gradually diminishes. This gradual decrease in amplitude can be attributed to various factors. One key aspect is the inherent damping properties of the elastic polymeric layer involved. Graphs in Fig. 2 suggest that when the coefficient changes, the structure has different oscillation amplitudes. The higher is the coefficient, the greater is the amplitude of vibrations of the structure.
- Ff = 5 * 103 ---Ef= 104 ...... Ef=2* 104
f \Ч
• ,, ч.^' -— ^ ^ -V
V
a)
b)
- Ff = 5»103
1 •,» ..... Ef = 2* Ю'5
■J Шл if ! 11 i-. (MijJt^iHiiii шШШЩщ- gjs? ' ''ii-j —
c) d)
Fig. 2. The non-dimensional deflection of the structure under the influence of acceleration ao at the central point of connection of the elastic beam and the damping layer: a) ao(r) = aoH(r), where H(r) is the Heaviside function; b) a0(r) = a08(r), where 8(r) is the Dirac delta function; c) a0(r) = a0 sin(^rar); d) a0(r) = a0 + a sin(^rar)
The effects of changes in the density of an elastic polymeric material on the amount of bending of the structure were also examined. Fig. 3 represents the deflection of the elastic beam wb and elastic polymeric layer Wf under the influence of acceleration when
the material density p = 1000,1500 and 2000kg/m3. As shown by Fig. 3, a change in the density of the material has a negligible effect on the magnitude of the oscillation amplitude. This study represents the first step towards a deeper understanding of the properties of elastic polymeric materials. Further work is required to fully study the various materials used in the structure. In the future, additional research will be conducted, focusing on different materials, in order to determine their potential to be used in the described structure. This allows to determine more accurately the optimal materials to achieve the required characteristics and maximum efficiency in the structure.
7
c £ 6
0
1 5
5 2
-a
(u ? i
0
A — p= 1000
- p= 1500 ■ p= 2000
■ A A
W v ..... s— ~ —
V
1.0 1.5 2.0
Time, sec
2.5 3.0
a)
| 3 u
0j
I 2
T3 flj § 1
b)
— p~ 1000 1500 2000
— p.... p
M, m f, il ft
—
u
1
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 Time, sec
c) d)
Fig. 3. The non-dimensional deflection of the structure under the influence of acceleration a0 at the central point of connection of the elastic beam and the damping layer: a) a0(t) = a0H(t), where H(t) is the Heaviside function; b) ao(T) = a05(T), where 8(t) is the Dirac delta function; c) a0(T) = a0 sin(^rat); d) a0(T) = a0 + a sin(^raT)
Conclusion
Vibration is a common phenomenon that can have a negative impact on the operation of various systems and devices. It can cause wear and damage to components, as well as
lead to loss of energy and suboptimal operation of the equipment. Therefore, it is crucial to develop methods and materials that will reduce vibration and improve system performance. One of ways for improvement is to use elastic polymeric materials. Elastomers have the ability to absorb energy and reduce vibration transmission. Such materials are widely used in various fields, including the transportation, energy and building industries. In this article, a mathematical model that describes motion of the beam resting on an elastic polymeric layer is constructed. For this aim, nonlinear differential equations were used. They take into account the physical characteristics of elastic polymeric materials and their vibroisolating ability. The algorithm developed for the numerical solution of the system allows to calculate the deflection of the elastic beam and elastic polymeric layer under the influence of acceleration and take into account the features of elastomeric materials. Numerical experiments exploring the reaction of the system to external acceleration were carried out. The obtained results made it possible to determine the effectiveness of vibration isolation when using elastic polymeric materials with different physical characteristics. The values of the deflections of the structure for each of the materials are obtained. This gives information on which material can be most effective in a particular situation. These results represent a starting point for deeper research into elastic polymeric materials and the creation of more complex structures. Their analysis can help to improve system performance, reduce energy losses, and improve equipment reliability. The results of the study can also be used in engineering and in the design of various devices where vibration is a problem.
References
1. Savinov O.A. Sovremennye konstruktsii fundamentov pod mashiny i ikh raschet [Modern Designs of Foundations for Machines and Their Calculation]. Leningrad, Stroiizdat Publishing House, 1979. (in Russian)
2. Kudryavtsev I.A. Vliyanie vibratsiy na osnovaniya sooruzheniy [Impact of Vibration on Foundations Structures]. Gomel, BelGUT, 1999. (in Russian)
3. Borisov E.K., Alimov S.G., Usov A.G., Lysak L.G., Krylova T.V., Stepanova E.A. Eksperimental'naya dinamika sooruzheniy. Monitoring transportnoy vibratsii [Experimental Dynamics of Structures. Transport Vibration Monitoring]. Petropavlovsk-Kamchatsky: KamchatGTU, 2007. (in Russian)
4. Kaul S. Vibration Isolation - Background. Modeling and Analysis of Passive Vibration Isolation Systems, 2021, pp. 1-26. DOI: 10.1016/B978-0-12-819420-1.00007-8
5. Valizadeh S., Fathalilou M., Rezazadeh G. Material Dielectricity Effects on the Performance of Capacitive Micro-Devices; A Nonlinear Study. International Journal of Mechanics and Materials in Design, 2023, vol. 19, no. 3, pp. 537-552. DOI: 10.1007/s10999-023-09649-6
6. Yakovlev S.N., Mazurin V.L. Vibroisolating Properties of Polyurethane Elastomeric Materials, used in Construction. Magazine of Civil Engineering, 2017, vol. 6, no. 74, pp. 53-60. DOI: 10.18720/MCE.74.5
7. Shian S., Bertoldi K., Clarke D.R. Dielectric Elastomer Based Grippers for Soft Robotics. Advanced Materials, 2015, vol. 27, no. 43, pp. 6814-6819. DOI: 10.1002/adma.201503078
8. Jyh-Horng Wu, Chia-Hao Li, Hsien-Tang Chiu, Zhi-Jian Shong, Peir-An Tsai. AntiVibration and Vibration Isolator Performance of Poly (Styrene-Butadiene-Styrene)/Ester-Type Polyurethane Thermoplastic Elastomers. Polymers Advanced Technologies, 2010, vol. 21, no. 3, pp. 164-169. DOI: 10.1002/pat.1411
9. Zhiqiang Xu, Qiuliang Wang, Kejun Zhu, et al. Preparation and Characterization of Magnetorheological Elastic Polishing Composites. Journal of Intelligent Material Systems and Structures, 2019, vol. 30, no. 10, pp. 1481-1492. DOI: 10.1177/1045389X19835960
Received January 8, 2024
УДК 517.9 DOI: 10.14529/mmp240109
ИССЛЕДОВАНИЕ ПЕРЕХОДНЫХ ХАРАКТЕРИСТИК БАЛКИ НА ЭЛАСТИЧНОМ ПОЛИМЕРНОМ ОСНОВАНИИ
А.Д. Кащеева1, А.А. Замышляева1
1 Южно-Уральский государственный университет, г. Челябинск,
Российская Федерация
Негативное воздействие вибраций на различные устройства и механизмы может быть значительным, поэтому важно учитывать этот фактор при проектировании, эксплуатации и техническом обслуживании различного оборудования и инженерных систем. Для защиты от негативного воздействия вибраций могут использоваться различные методы и технологии. Часто используются специальные демпфирующие материалы. Данная исследовательская работа посвящена анализу эффективности снижения вибрации с учетом физических параметров эластомерных материалов. Для проведения исследования построена математическая модель, описывающая движение балки, опирающейся на эластомерное основание. Модель основана на системе нелинейных дифференциальных уравнений. В ходе работы был разработан и применен алгоритм численного решения этой системы уравнений. Были проведены численные эксперименты для изучения реакции системы на различные случаи ускорений. В результате были получены величины прогиба для материалов с различными физическими характеристиками. Эти результаты могут послужить отправной точкой для более глубокого изучения материалов и создания более сложных конструкций.
Ключевые слова: переходные процессы; эластичное полимерное основание; нелинейное дифференциальное уравнение; метод Галеркина.
Литература
1. Савинов, О.А. Современные конструкции фундаментов под машины и их расчет / О.А. Савинов. - Л.: Стройиздат, 1979.
2. Кудрявцев, И.А. Влияние вибраций на основания сооружений / И.А. Кудрявцев. - Гомель: БелГУТ, 1999.
3. Борисов, Е.К. Экспериментальная динамика сооружений. Мониторинг транспортной вибрации / Е.К. Борисов, С.Г. Алимов, А.Г. Усов, Л.Г. Лысак, Т.В. Крылова, Е.А. Степанова. - Петропавловск-Камчатский: КамчатГТУ, 2007.
4. Kaul, S. Vibration Isolation - Background / S. Kaul // Modeling and Analysis of Passive Vibration Isolation Systems. - 2021. - P. 1-26.
5. Valizadeh, S. Material Dielectricity Effects on the Performance of Capacitive Micro-Devices; A Nonlinear Study / S. Valizadeh, M. Fathalilou, G. Rezazadeh // International Journal of Mechanics and Materials in Design. - 2023. - V. 19, № 3. - P. 537-552.
6. Yakovlev, S.N. Vibroisolating Properties of Polyurethane Elastomeric Materials, used in Construction / S.N. Yakovlev, V.L. Mazurin // Magazine of Civil Engineering. - 2017. -V. 6, № 74. - P. 53-60.
7. Shian, S. Dielectric Elastomer Based Grippers for Soft Robotics / S. Shian, K. Bertoldi, D.R. Clarke // Advanced Materials. - 2015. - V. 27, № 43. - P. 6814-6819.
8. Jyh-Horng Wu. Anti-Vibration and Vibration Isolator Performance of Poly (Styrene-Butadiene-Styrene)/Ester-Type Polyurethane Thermoplastic Elastomers / Jyh-Horng Wu, Chia-Hao Li, Hsien-Tang Chiu, Zhi-Jian Shong, Peir-An Tsai // Polymers Advanced Technologies. - 2010. - V. 21, № 3. - P. 164-169.
9. Zhiqiang Xu. Preparation and Characterization of Magnetorheological Elastic Polishing Composites / Zhiqiang Xu, Qiuliang Wang, Kejun Zhu, et al. // Journal of Intelligent Material Systems and Structures. - 2019. - V. 30, № 10. - P. 1481-1492.
Анна Дмитриевна Кащеева, аспирант, кафедра «Прикладная математика и программирование:», Южно-Уральский государственный университет (г. Челябинск, Российская Федерация), kashcheevaad@susu.ru.
Алена Александровна Замышляева, доктор физико-математических наук, профессор, кафедра «Прикладная математика и программирование», ЮжноУральский государственный университет (г. Челябинск, Российская Федерация), zamyshliaevaaa@susu.ru.
Поступила в редакцию 8 января 2024 г-
