# Stability and Dynamics of Viscoelastic Moving Rayleigh Beams with an Asymmetrical Distribution of Material Parameters

^{1}

^{2}

^{3}

^{4}

^{5}

^{6}

^{7}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Mathematical Modeling

_{ρ}and α

_{E}represent the density and elastic modulus gradient parameters, respectively. They can be declared as follows:

_{0,}and ρ = ρ

_{0}. Furthermore, E = E

_{L}and ρ = ρ

_{L}at x = L. The potential and kinetic energies of the system can be expressed as [5,18,40]:

_{f}are the rotary inertia factor and the flexural stiffness of the considered system, respectively. It is worth noting that the governing equation of the system reduces to that of AFG moving EB beam by setting β = 0. Furthermore, if the viscoelastic coefficient of the system is supposed to be zero (μ = 0), the dynamic equation degenerates to that of a moving elastic AFG Rayleigh beam.

## 3. Discretization Technique

_{r}, n, and φ

_{r}are dimensionless generalized coordinate, the number of basic functions, and acceptable mode shape for the transverse displacement of the system, respectively. The normalized mode shapes of a simply supported beam are given by [47]:

**q**,

**M**,

**C**, and

**K**are the vector of generalized coordinates, mass, damping, and stiffness matrices, respectively, and are given by:

## 4. Stability Examination

**Z**(τ) =

**A**e

^{i}

^{ωτ}yields the following eigenvalue problem:

**I**indicates the unity matrix and

**D**= −

**B**

^{−1}

**E**. Moreover, ω is the complex-valued natural frequency of the AFG moving Rayleigh beams and can be determined in terms of different key factors such as rotary inertia factor, flexural stiffness, and density and elastic modulus gradient parameters. The stability of the AFG moving Rayleigh beams is profoundly affected by the sign of imaginary and real parts of the natural frequency. Imaginary and real parts of the natural frequency are related to damping and the frequency of oscillation of the system, respectively. When the real part of one of the frequency branches becomes zero while its imaginary part is negative (Re(ω) = 0, Im(ω) < 0), the instability via a pitch-fork bifurcation occurs in the system. The minimum velocity at which the divergence instability happens is recognized as the critical velocity. Furthermore, when the imaginary part of at least one of the frequency branches is negative while its real part is positive (Real(ω) > 0, Image(ω) < 0), the system experiences flutter instability via a Hopf bifurcation [9].

## 5. Results and Discussion

#### 5.1. Model Verification

_{f}, the fundamental frequency of the isotropic system rises.

#### 5.2. Effect of Elastic Modulus Variation

_{f}= 0.5. As it is obvious, the vibrational frequencies of the considered system are purely real when the axial velocity is zero. Afterward, by increasing the velocity, the real part of the vibrational frequencies of the system declines gradually, while their imaginary part is still equal to zero. At the critical axial velocity (v

_{d}), the real part of system frequencies vanished, and the system loses its stability and consequently undergoes the divergence phenomenon. The induced divergence instability in moving structures due to ascending the axial velocity is analogous to that of buckling in classical beams under the compression load [48]. By further increasing the axial velocity, the fundamental frequency of the considered structure becomes purely imaginary, while the second natural frequency declines monotonically. Due to gyroscopic effects in the system, at higher velocities, the beam regains its stability again. In other words, the initiation and termination of divergence instability are correlated to the vanishing of real and imaginary parts of the fundamental vibrational frequency, respectively. Eventually, real parts of the two vibrational frequencies merge into each other via a Paidoussis coupled-mode flutter bifurcation while their imaginary parts divide into two branches with negative and positive values. This phenomenon is related to the flutter instability in the system, and the corresponding velocity is known as the flutter velocity (v

_{f}). In fact, in addition to the velocities lower than critical divergence velocity (v < v

_{d}), a narrow range of axial velocity (between the termination point of the divergence instability and the initiation of flutter instability) exists in which the system is stable at this operational velocity range. It is worth mentioning that the system is no longer stable beyond the critical flutter velocity. As a result, the moving beam experiences a stability evolution of “stable – first mode divergence – stable – coupled-mode flutter”.

_{E}leads to ascending the critical divergence and flutter velocities of axially AFG moving EB beams. In other words, it is feasible to hinder the occurrence of undesirable divergence phenomenon by increasing the elastic modulus gradient in moving structures. Since α

_{E}has an increasing role in the stiffness matrix; hence, any increment in α

_{E}leads to a stiffer system and also wider stability regions. In other words, increasing α

_{E}induces the stiffness-hardening effect in the system. Another important feature in Figure 4a,b is that the velocity bandwidth corresponding to the divergence and flutter phenomena in the system (v

_{d}< v < v

_{f}) would be expanded slightly by increasing α

_{E}. Moreover, based on these figures, compared with the exponential variation of elastic modulus, the linear variation leads to a more stable system. As demonstrated in Figure 4b, the damping ratio of the system is higher for α

_{E}> 1 and linear variation of elastic modulus. Accordingly, it is possible to determine the instability thresholds and vibrational behavior of the system by fine-tuning of the elastic modulus gradient parameter.

_{f}= 0.5, β = 0, α

_{ρ}= 1, and α

_{E}= 2. Unit static displacement and zero initial velocity for the first mode (${q}_{1}\left(0\right)=0$ and ${\dot{q}}_{1}\left(0\right)=0$) are considered for the initial conditions of the system. According to Figure 4a, for v = 1, Real(ω) > 0 and Image(ω) = 0; hence, the system is dynamically stable and generates stable harmonic oscillations. By increasing v, the effective stiffness of the system declines due to the centrifugal force effects. For v = 2.5, the real part of the fundamental frequency of the considered system equals zero, and the beam undergoes instability. In this condition, the dynamic response of the system dramatically grows without oscillation, and the static instability occurs in the system. Increasing the velocity to v = 4 leads to vanishing the imaginary part of frequency, and the beam regains its stability. As the velocity increases, when v = 5, the real part of the frequency grows, while the imaginary part of the vibrational frequency becomes zero. As a result, the oscillation amplitude of the system amplified exponentially by the time. In this case, unlike the divergence instability, the AFG moving beam experiences the flutter instability with growing oscillation. Therefore, the flutter instability is more dangerous than the divergence instability for axially moving structures.

_{d}–k

_{f}and v

_{d}–α

_{E}planes are drawn in Figure 6a,b, respectively. The indicated curves in the stability maps separated the stable and unstable regions, in which, above of each curve, the structure is in the divergence condition. The critical divergence velocity of the system in Figure 4 is consistent with Figure 6. According to Figure 6a, the greater the flexural stiffness, the more stable the system becomes. Therefore, one can conclude that increasing the flexural stiffness parameter has a stabilizing effect on the axially moving systems. Additionally, by increasing α

_{E}, the stability regions of the system expanded. Based on Figure 6a, the linear variation of elastic modulus expands the stability regions of the system more than the exponential one. Moreover, the effect of the elastic modulus gradient is more prominent at higher values of k

_{f}, and the stability borders separate from each other.

_{E}= 1), the AFG system is more stable when α

_{E}> 1 and ascending α

_{E}promotes the stability of the system. It is evident that the critical divergence velocity of the system increases by k

_{f}, which can be attributed to the stabilizing effects of the flexural stiffness. By approaching the value of elastic modulus gradient parameter to one (isotropic condition), the stability boundaries of the system are close to each other for linear and exponential distributions, and these boundaries separate from each other by increasing or decreasing of α

_{E}. Additionally, except at α

_{E}= 1, the system is more stable for the linear distribution of elastic modulus in comparison with the exponential one.

#### 5.3. Effect of Density Variation

_{ρ}. The influence of α

_{ρ}is more tangible in the vibrational frequencies of higher modes. As displayed, the natural frequencies of the system have a descending trend by increasing the density gradient parameter. Hence, the influence of α

_{ρ}and α

_{E}variations on the vibrational behavior of the system are opposite to each other. Furthermore, the stability region of the system shrinks by ascending the density gradient parameter. Moreover, in contrast to the case of variable elastic modulus scrutinized in the former section, in the case of variable density, the critical divergence and flutter velocities of the exponential distribution is higher than that of the linear distribution.

_{ρ}and α

_{E}, as well as the type of their distributions.

_{d}–α

_{ρ}and v

_{d}–β planes are plotted in Figure 8a,b, respectively. As demonstrated in Figure 8a, the AFG moving beam is more stable for α

_{ρ}< 1 in comparison with the isotropic one. It is crucial to mention that the stability borders of exponential and linear distributions approach to each other for the density gradient parameters close to one (i.e., the isotropic case). Moreover, the critical velocity of the structure is higher for the exponential distribution of density in comparison with the linear one, especially for the higher and lower values of α

_{ρ}. In other words, the difference between the stability borders of linear and exponential distributions can be magnified by selecting a higher or lower density gradient.

_{ρ}. Generally, compared with the stability maps in v

_{d}–k

_{f}and v

_{d}–α

_{E}planes analyzed in the former section, it can be declared that the stability maps in v

_{d}–β and v

_{d}–α

_{ρ}are overall descending with increasing β and α

_{ρ}. This implies that increasing β and α

_{ρ}can destabilize the system and leads to the decrement of the critical divergence velocity of the structure. It should be mentioned that the indicated stability borders in Figure 7 and Figure 8 are in agreement. Based on Figure 7 and Figure 8, the critical velocity of the structure is substantially dependent on α

_{ρ}. This dependency is more pronounced for the higher and lower values of α

_{ρ}.

#### 5.4. Effect of Simultaneous Elastic Modulus and Density Variations

_{E}and α

_{ρ}, separately. Based on Figure 4, Figure 5, Figure 6, Figure 7 and Figure 8, variations of elastic modulus and density along the longitudinal direction of the beam have opposite effects on the stability of the system. As a result, these parameters can provide additional degrees of freedom to modify the dynamic characteristics of axially moving structures. In other words, it is possible to significantly improve the performance of axially moving systems by simultaneous fine-tuning α

_{E}and α

_{ρ}. Therefore, determining the role of simultaneous gradation of the material properties on the stability of the moving structures is of great significance. In this section, stability characteristics of the system are explored by considering the coupled variation of density and elastic modulus through the axial direction (simultaneous mass-addition and stiffness-hardening effects). Additionally, the exact analytical expression is presented for critical divergence velocity. Furthermore, a comparison is carried out between different solution procedures.

_{E}, α

_{ρ}, as well as k

_{f}of the system. To better inspect the stability of the two-dimensional contour plot for the critical divergence velocity in α

_{E}–α

_{ρ}and k

_{f}–β planes are drawn in Figure 9a,b, respectively. As shown in Figure 9a, the critical divergence velocity of the system enhances by increasing the α

_{E}and decreasing α

_{ρ}and vice versa. As illustrated in Figure 9b, the critical divergence velocity of the system increases by decreasing the rotatory inertia factor and increasing the dimensionless flexural stiffness and vice versa. Additionally, it can be deduced that the influence of the flexural stiffness and the rotary inertia factor on the stability of the system is opposite to each other. In other words, contrary to the effect of α

_{ρ}and β, increasing the elastic modulus gradient parameter and dimensionless flexural stiffness enhance the buckling strength of the system. Accordingly, simultaneously selecting larger values of α

_{E}and k

_{f}as well as choosing smaller values of β and α

_{ρ}leads to a more stable system and, consequently, improves the performance of the axially moving structures.

_{E}= α

_{ρ}= α) are demonstrated, and the validity of the analytical expressions are examined. According to these figures, the critical divergence velocities acquired by the numerical approaches are in a close agreement with those acquired analytically. As seen in Figure 10a, increasing the material gradient parameter, α, leads to a slight decrease in the critical divergence velocity of the structure. Therefore, compared with the stability maps in previous sections (Figure 6 and Figure 8), it can be inferred that the density gradation (mass-addition effect) shows a dominant role in the stability of the system, and elastic modulus gradation (stiffness-hardening effect) has less effect on the vibrational behavior of the structure. It should be mentioned that compared with previous sections, as density and elastic modulus vary simultaneously, stability borders are closer to each other and are less sensitive to the axial material gradation when the effect of flexural stiffness is highlighted. According to Figure 10b, the stability regions of the system shrink by increasing the gradient parameter, and the curves have a descending trend. Moreover, in the higher values of dimensionless flexural stiffness, variation in the gradient parameter has a minor influence on the instability threshold of AFG moving beam; this point can be helpful at the design stage. As depicted in Figure 10c,d, the stability regions shrink by increasing the gradient parameter and rotary inertia factor. Compared with the isotropic case, when α < 1, the axially moving beam is more stable. Furthermore, at sufficiently high gradient parameter, the divergence velocity of the structure experiences small changes.

_{f}is plotted in Figure 11. It is detected that the instability threshold of moving beams is entirely dependent on the axial material gradation, dimensionless flexural stiffness, and rotary inertia factor. As it is obvious, the variation of the material gradient parameter and the rotary inertia factor have the same effects on the stability boundaries, and by increasing each of these parameters, the divergence strength of the system diminishes. While ascending the flexural stiffness enhances the stability of the structure. Consequently, in the simultaneous presence of higher values of dimensionless flexural stiffness, lower values of the material gradient, and rotary inertia factor, the performance of axially moving systems would be improved. Therefore, these parameters could be introduced as the key factors for the vibration control of moving continua. Generally, AFG moving beams are more flexible to adjust their vibrational behavior in comparison with isotropic ones. As a result, it can be claimed that, compared with isotropic materials, AFG ones have a better performance in axially moving structures.

#### 5.5. Effect of Viscoelastic Material

## 6. Conclusions

- Increasing the density/elastic modulus gradient parameter has a destabilizing/stabilizing effect on axially moving beams. Compared with isotropic axially moving beams, the system is more stable when density/elastic modulus decreases/increases along the axial direction.
- In the case of density/elastic modulus variation, exponential/linear distribution leads to a more stable system.
- In the case of simultaneous axial variation of elastic modulus and density, the effect of density gradation on the vibrational configuration of the system is dominant.
- The higher flexural stiffness, and the lower rotary inertia factor, the more stable the structure becomes. Moreover, the influence of axial material gradation on the stability boundaries of the system is more tangible at higher and lower values of flexural stiffness and rotary inertia factor.
- Compared with isotropic and moving axially graded beams, utilizing the viscoelastic material changes the stability evolution of the system.

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Chen, L.-Q. Analysis and control of transverse vibrations of axially moving strings. Appl. Mech. Rev.
**2005**, 58, 91–116. [Google Scholar] [CrossRef] - Marynowski, K.; Kapitaniak, T. Dynamics of axially moving continua. Int. J. Mech. Sci.
**2014**, 81, 26–41. [Google Scholar] [CrossRef] - Stylianou, M.; Tabarrok, B. Finite element analysis of an axially moving beam, part II: Stability analysis. J. Sound Vib.
**1994**, 178, 455–481. [Google Scholar] [CrossRef] - Sreeram, T.; Sivaneri, N. FE-analysis of a moving beam using Lagrangian multiplier method. Int. J. Solids Struct.
**1998**, 35, 3675–3694. [Google Scholar] [CrossRef] - Wickert, J. Non-linear vibration of a traveling tensioned beam. Int. J. Non-Linear Mech.
**1992**, 27, 503–517. [Google Scholar] [CrossRef] - Ghayesh, M.H.; Amabili, M. Post-buckling bifurcations and stability of high-speed axially moving beams. Int. J. Mech. Sci.
**2013**, 68, 76–91. [Google Scholar] [CrossRef] - Chen, L.-Q.; Yang, X.-D. Vibration and stability of an axially moving viscoelastic beam with hybrid supports. Eur. J. Mech. A Solids
**2006**, 25, 996–1008. [Google Scholar] [CrossRef] - Guo, X.-X.; Wang, Z.-M.; Wang, Y.; Zhou, Y.-F. Analysis of the coupled thermoelastic vibration for axially moving beam. J. Sound Vib.
**2009**, 325, 597–608. [Google Scholar] [CrossRef] - Öz, H.; Pakdemirli, M. Vibrations of an axially moving beam with time-dependent velocity. J. Sound Vib.
**1999**, 227, 239–257. [Google Scholar] [CrossRef] - Yang, T.; Fang, B.; Yang, X.-D.; Li, Y. Closed-form approximate solution for natural frequency of axially moving beams. Int. J. Mech. Sci.
**2013**, 74, 154–160. [Google Scholar] [CrossRef] - Chen, L.-Q.; Yang, X.-D.; Cheng, C.-J. Dynamic stability of an axially accelerating viscoelastic beam. Eur. J. Mech. A Solids
**2004**, 23, 659–666. [Google Scholar] [CrossRef] - Kiani, K. Divergence and flutter instabilities of nanobeams in moving state accounting for surface and shear effects. Comput. Math. Appl.
**2019**, 77, 2764–2785. [Google Scholar] [CrossRef] - Kiani, K. Longitudinal, transverse, and torsional vibrations and stabilities of axially moving single-walled carbon nanotubes. Curr. Appl. Phys.
**2013**, 13, 1651–1660. [Google Scholar] [CrossRef] - Chang, J.-R.; Lin, W.-J.; Huang, C.-J.; Choi, S.-T. Vibration and stability of an axially moving Rayleigh beam. Appl. Math. Model.
**2010**, 34, 1482–1497. [Google Scholar] [CrossRef] - Zinati, R.F.; Rezaee, M.; Lotfan, S. Nonlinear Vibration and Stability Analysis of Viscoelastic Rayleigh Beams Axially Moving on a Flexible Intermediate Support. Iran. J. Sci. Technol. Trans. Mech. Eng.
**2019**, 1–15. [Google Scholar] [CrossRef] - Tang, Y.-Q.; Chen, L.-Q.; Yang, X.-D. Natural frequencies, modes and critical speeds of axially moving Timoshenko beams with different boundary conditions. Int. J. Mech. Sci.
**2008**, 50, 1448–1458. [Google Scholar] [CrossRef] - An, C.; Su, J. Dynamic response of axially moving Timoshenko beams: Integral transform solution. Appl. Math. Mech.
**2014**, 35, 1421–1436. [Google Scholar] [CrossRef] - Zhu, K.; Chung, J. Vibration and stability analysis of a simply-supported Rayleigh beam with spinning and axial motions. Appl. Math. Model.
**2019**, 66, 362–382. [Google Scholar] [CrossRef] - Dehrouyeh-Semnani, A.M.; Dehrouyeh, M.; Zafari-Koloukhi, H.; Ghamami, M. Size-dependent frequency and stability characteristics of axially moving microbeams based on modified couple stress theory. Int. J. Eng. Sci.
**2015**, 97, 98–112. [Google Scholar] [CrossRef] - Fard, M.; Sagatun, S. Exponential stabilization of a transversely vibrating beam via boundary control. J. Sound Vib.
**2001**, 240, 613–622. [Google Scholar] [CrossRef] - Zhang, Y.-W.; Hou, S.; Xu, K.-F.; Yang, T.-Z.; Chen, L.-Q. Forced vibration control of an axially moving beam with an attached nonlinear energy sink. Acta Mech. Solida Sin.
**2017**, 30, 674–682. [Google Scholar] [CrossRef] - Li, T.-C.; Hou, Z.-C.; Li, J.-F. Stabilization analysis of a generalized nonlinear axially moving string by boundary velocity feedback. Automatica
**2008**, 44, 498–503. [Google Scholar] [CrossRef] - Zhang, Y.-W.; Zhang, Z.; Chen, L.-Q.; Yang, T.-Z.; Fang, B.; Zang, J. Impulse-induced vibration suppression of an axially moving beam with parallel nonlinear energy sinks. Nonlinear Dyn.
**2015**, 82, 61–71. [Google Scholar] [CrossRef] - Sarparast, H.; Ebrahimi-Mamaghani, A. Vibrations of laminated deep curved beams under moving loads. Compos. Struct.
**2019**, 226, 111262. [Google Scholar] [CrossRef] - Sedighi, H.M.; Daneshmand, F.; Abadyan, M. Dynamic instability analysis of electrostatic functionally graded doubly-clamped nano-actuators. Compos. Struct.
**2015**, 124, 55–64. [Google Scholar] [CrossRef] - Hosseini, R.; Hamedi, M.; Mamaghani, A.E.; Kim, H.C.; Kim, J.; Dayou, J. Parameter identification of partially covered piezoelectric cantilever power scavenger based on the coupled distributed parameter solution. Int. J. Smart Nano Mater.
**2017**, 8, 110–124. [Google Scholar] [CrossRef] [Green Version] - Li, Y.; Dong, Y.; Qin, Y.; Lv, H. Nonlinear forced vibration and stability of an axially moving viscoelastic sandwich beam. Int. J. Mech. Sci.
**2018**, 138, 131–145. [Google Scholar] [CrossRef] - Ghayesh, M.H. On the natural frequencies, complex mode functions, and critical speeds of axially traveling laminated beams: Parametric study. Acta Mech. Solida Sin.
**2011**, 24, 373–382. [Google Scholar] [CrossRef] - Lv, H.; Li, Y.; Li, L.; Liu, Q. Transverse vibration of viscoelastic sandwich beam with time-dependent axial tension and axially varying moving velocity. Appl. Math. Model.
**2014**, 38, 2558–2585. [Google Scholar] [CrossRef] - Esfahani, S.; Khadem, S.E.; Mamaghani, A.E. Size-dependent nonlinear vibration of an electrostatic nanobeam actuator considering surface effects and inter-molecular interactions. Int. J. Mech. Mater. Des.
**2019**, 15, 489–505. [Google Scholar] [CrossRef] - Mirtalebi, S.H.; Ebrahimi-Mamaghani, A.; Ahmadian, M.T. Vibration Control and Manufacturing of Intelligibly Designed Axially Functionally Graded Cantilevered Macro/Micro-tubes. IFAC Pap.
**2019**, 52, 382–387. [Google Scholar] [CrossRef] - Ebrahimi-Mamaghani, A.; Mirtalebi, S.H.; Ahmadian, M.-T. Magneto-mechanical stability of axially functionally graded supported nanotubes. Mater. Res. Express
**2020**, 6, 1250–1255. [Google Scholar] [CrossRef] - Ebrahimi-Mamaghani, A.; Sotudeh-Gharebagh, R.; Zarghami, R.; Mostoufi, N. Thermo-mechanical stability of axially graded Rayleigh pipes. Mech. Based Des. Struct. Mach.
**2020**, 1–30. [Google Scholar] [CrossRef] - Safarpour, M.; Rahimi, A.; Alibeigloo, A.; Bisheh, H.; Forooghi, A. Parametric study of three-dimensional bending and frequency of FG-GPLRC porous circular and annular plates on different boundary conditions. Mech. Based Des. Struct. Mach.
**2019**, 1–31. [Google Scholar] [CrossRef] - Safarpour, M.; Rahimi, A.; Alibeigloo, A. Static and free vibration analysis of graphene platelets reinforced composite truncated conical shell, cylindrical shell, and annular plate using theory of elasticity and DQM. Mech. Based Des. Struct. Mach.
**2019**, 1–29. [Google Scholar] [CrossRef] - Piovan, M.T.; Sampaio, R. Vibrations of axially moving flexible beams made of functionally graded materials. Thin Walled Struct.
**2008**, 46, 112–121. [Google Scholar] [CrossRef] - Sui, S.; Chen, L.; Li, C.; Liu, X. Transverse vibration of axially moving functionally graded materials based on Timoshenko beam theory. Math. Probl. Eng.
**2015**, 2015, 391452. [Google Scholar] [CrossRef] [Green Version] - Kiani, K. Longitudinal and transverse instabilities of moving nanoscale beam-like structures made of functionally graded materials. Compos. Struct.
**2014**, 107, 610–619. [Google Scholar] [CrossRef] - Yan, T.; Yang, T.; Chen, L. Direct Multiscale Analysis of Stability of an Axially Moving Functionally Graded Beam with Time-Dependent Velocity. Acta Mech. Solida Sin.
**2020**, 33, 150–163. [Google Scholar] [CrossRef] - Rezaee, M.; Lotfan, S. Non-linear nonlocal vibration and stability analysis of axially moving nanoscale beams with time-dependent velocity. Int. J. Mech. Sci.
**2015**, 96, 36–46. [Google Scholar] [CrossRef] - Ghayesh, M.H. Nonlinear forced dynamics of an axially moving viscoelastic beam with an internal resonance. Int. J. Mech. Sci.
**2011**, 53, 1022–1037. [Google Scholar] [CrossRef] - Mamaghani, A.E.; Zohoor, H.; Firoozbakhsh, K.; Hosseini, R. Dynamics of a Running Below-Knee Prosthesis Compared to Those of a Normal Subject. J. Solid Mech. Vol.
**2013**, 5, 152–160. [Google Scholar] - Mirtalebi, S.H.; Ahmadian, M.T.; Ebrahimi-Mamaghani, A. On the dynamics of micro-tubes conveying fluid on various foundations. SN Appl. Sci.
**2019**, 1, 547. [Google Scholar] [CrossRef] [Green Version] - Mamaghani, A.E.; Khadem, S.; Bab, S. Vibration control of a pipe conveying fluid under external periodic excitation using a nonlinear energy sink. Nonlinear Dyn.
**2016**, 86, 1761–1795. [Google Scholar] [CrossRef] - Mamaghani, A.E.; Khadem, S.E.; Bab, S.; Pourkiaee, S.M. Irreversible passive energy transfer of an immersed beam subjected to a sinusoidal flow via local nonlinear attachment. Int. J. Mech. Sci.
**2018**, 138, 427–447. [Google Scholar] [CrossRef] - Ebrahimi-Mamaghani, A.; Sotudeh-Gharebagh, R.; Zarghami, R.; Mostoufi, N. Dynamics of two-phase flow in vertical pipes. J. Fluids Struct.
**2019**, 87, 150–173. [Google Scholar] [CrossRef] - Esfahani, S.; Khadem, S.E.; Mamaghani, A.E. Nonlinear vibration analysis of an electrostatic functionally graded nano-resonator with surface effects based on nonlocal strain gradient theory. Int. J. Mech. Sci.
**2019**, 151, 508–522. [Google Scholar] [CrossRef] - Wickert, J.; Mote, C., Jr. Classical vibration analysis of axially moving continua. J. Appl. Mech.
**1990**, 57, 738–744. [Google Scholar] [CrossRef]

**Figure 2.**Fundamental frequency of an isotropic moving EB simply supported beam against dimensionless axial velocity, μ = 0.

**Figure 3.**Natural frequencies of an isotropic moving Rayleigh beam against dimensionless axial velocity for β = 0.001, k

_{f}= 0.8, μ = 0.

**Figure 4.**(

**a**) Real and (

**b**) imaginary parts of two vibrational frequency of the system against the axial velocity for β = 0, α

_{ρ}= 1, k

_{f}= 0.5, μ = 0.

**Figure 5.**Dynamic response of an AFG moving EB beam for β = 0, α

_{E}= 2, α

_{ρ}= 1, k

_{f}= 0.5, μ = 0.

**Figure 6.**Critical divergence velocity of an AFG EB beam against (

**a**) dimensionless flexural stiffness and (

**b**) elastic modulus gradient parameter for α

_{ρ}= 1, μ = 0.

**Figure 7.**(

**a**) Real and (

**b**) imaginary parts of two vibrational frequencies of the system against the axial velocity for β = 0, α

_{E}= 1, k

_{f}= 0.5, μ = 0.

**Figure 8.**Critical divergence velocity of an AFG Rayleigh beam against (

**a**) density gradation parameter and (

**b**) rotary inertia factor for k

_{f}= 0.5 and α

_{E}= 1, μ = 0.

**Figure 9.**Effect of (

**a**) elastic modulus and density gradations and (

**b**) dimensionless flexural stiffness and rotary inertia factor on the dimensionless stability of the structure, μ = 0.

**Figure 10.**Critical divergence velocity of an AFG moving Rayleigh beam against (

**a**) dimensionless flexural stiffness (

**b**) gradient parameter (

**c**) rotary inertia factor (

**d**) gradient parameter, μ = 0.

**Figure 11.**Effect of dimensionless flexural stiffness, rotary inertia factor, and material gradation parameter on the critical divergence velocity of the AFG moving Rayleigh beams, μ = 0.

**Figure 12.**(

**a**) Real and (

**b**) imaginary parts of two vibrational frequencies of a viscoelastic moving beam for k

_{f}= 0.5, α

_{E}= α

_{ρ}= 1, β = 0, μ = 0.001.

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Shariati, A.; Jung, D.w.; Mohammad-Sedighi, H.; Żur, K.K.; Habibi, M.; Safa, M.
Stability and Dynamics of Viscoelastic Moving Rayleigh Beams with an Asymmetrical Distribution of Material Parameters. *Symmetry* **2020**, *12*, 586.
https://doi.org/10.3390/sym12040586

**AMA Style**

Shariati A, Jung Dw, Mohammad-Sedighi H, Żur KK, Habibi M, Safa M.
Stability and Dynamics of Viscoelastic Moving Rayleigh Beams with an Asymmetrical Distribution of Material Parameters. *Symmetry*. 2020; 12(4):586.
https://doi.org/10.3390/sym12040586

**Chicago/Turabian Style**

Shariati, Ali, Dong won Jung, Hamid Mohammad-Sedighi, Krzysztof Kamil Żur, Mostafa Habibi, and Maryam Safa.
2020. "Stability and Dynamics of Viscoelastic Moving Rayleigh Beams with an Asymmetrical Distribution of Material Parameters" *Symmetry* 12, no. 4: 586.
https://doi.org/10.3390/sym12040586