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Non-harmonic resonance of viscoelastic structures subjected to time-dependent exponentially decreasing transverse distributed loads

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Abstract

In this paper, the non-harmonic resonance of Bernoulli viscoelastic beams, Kirchhoff viscoelastic plates, Timoshenko viscoelastic beams, and Mindlin viscoelastic plates subjected to time-dependent exponentially decreasing transverse distributed load is investigated for the first time. The constitutive equations are expressed utilizing Boltzmann integral law with a constant bulk modulus. The displacement vector is approximated by employing the separation of variables method. The Laplace transformation is used to transfer equations from the time domain to the Laplace domain and vice versa. The novel point of the proposed method is to express, prove and calculate the critical time in which the displacement will be several times the displacement at time zero. In addition, this new method calculates the maximum deflection at the critical time, explicitly and exactly, without any need to follow the time-displacement curve with a low computational cost. Additionally, the proposed method introduces the critical range of time so that the responses are greater than the responses at time zero.

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References

  • Aboudi J and Cederbaum G (1990), “Dynamic Stability of Viscoelastic Plates by Lyapunov Exponents,” Journal of Sound and Vibration, 139(3): 459–467.

    Article  Google Scholar 

  • Alibeigloo A (2015), “Effect of Viscoelastic Interface on Three-Dimensional Static and Vibration Behavior of Laminated Composite Plate,” Composites Part B, 75(15): 17–28.

    Article  Google Scholar 

  • Amabili M (2016), “Nonlinear Vibration of Viscoelastic Rectangular Plates,” Journal of Sound and Vibration, 362: 142–156.

    Article  Google Scholar 

  • Amabili M (2018a), “Nonlinear Damping in Nonlinear Vibrations of Rectangular Plates: Derivation from Viscoelasticity and Experimental Validation,” Journal of the Mechanics and Physics of Solids, 118: 275–292.

    Article  Google Scholar 

  • Amabili M (2018b), “Nonlinear Vibrations and Stability of Laminated Shells Using a Modified First-Order Shear Deformation Theory,” European Journal of Mechanics-A/Solids, 68: 75–87.

    Article  Google Scholar 

  • Amabili M, Balasubramanian P and Giovanni Ferrari (2021), “Nonlinear Vibrations and Damping of Fractional Linear Viscoelastic Rectangular Plates,” Nonlinear Dynamics, 103: 3581–3609.

    Article  Google Scholar 

  • Amoushahi H and Azhari M (2015), “Static and Instability Analysis of Moderately Thick Viscoelastic Plates Using a Fully Discretized Nonlinear Finite Strip Formulation,” Composite: Part B, 56: 223–231.

    Google Scholar 

  • Brilla J (1976), “Convolutional Variational Principles and Stability of Viscoelastic Plates,” Archive of Applied Mechanics, 45: 275–280.

    Google Scholar 

  • Donolato C (2002), “Analytical and Numerical Inversion of the Laplace-Carson Transform by a Differential Method,” Computer Physics Communications, 145: 298–309.

    Article  Google Scholar 

  • Eshmatov BK (2006), “Dynamic Stability of Viscoelastic Plates Under Increasing Compressing Loads,” Journal of Applied Mechanics and Technical Physics, 47(2): 289–297.

    Article  Google Scholar 

  • Eskandari M, Jafari N and Azhari M (2021), “Time-Dependent Three-Dimensional Quasi-Static Analysis of a Viscoelastic Solid by Defining a Time Function,” Mechanics of Time-Dependent Materials.https://doi.org/10.1007/s11043-021-09515-y

  • Feri M, Krommer M and Alibeigloo A (2021), “Three-Dimensional Static Analysis of a Viscoelastic Rectangular Functionally Graded Material Plate Embedded Between Piezoelectric Sensor and Actuator Layers,” Mechanics Based Design of Structures and Machines. https://doi.org/10.1080/15397734.2021.1943673

  • Gupta AK, Khanna A, Kumar S, Kumar M, Gupta DV and Sharma P (2010), “Vibration Analysis of Visco-Elastic Rectangular Plate with Thickness Varies Linearly in One and Parabolically in Other Direction,” Advanced Studies in Theoretical Physics, 4(13): 743–758.

    Google Scholar 

  • Heidari E and Ariaei A (2022), “A New Approach for Free Vibration Analysis of a System of Elastically Interconnected Similar Rectangular Plates,” Earthquake Engineering and Engineering Vibration, 21(4): 947–967. https://doi.org/10.1007/s11803-022-2129-9

    Article  Google Scholar 

  • Ilyasov MH and Akoz AY (2000), “The Vibration and Dynamic Stability of Viscoelastic Plates,” International Journal of Engineering Science, 38: 695–714.

    Article  Google Scholar 

  • Jafari N and Azhari M (2017), “Stability Analysis of Arbitrarily Shaped Moderately Thick Viscoelastic Plates Using Laplace-Carson Transformation and a Simple HP Cloud Method,” Mechanics of Time-Dependent Materials, 21(3): 365–381.

    Article  Google Scholar 

  • Jafari N and Azhari M (2019), “Time-Dependent Static Analysis of Viscoelastic Mindlin Plates by Defining a Time Function,” Mechanics of Time-Dependent Materials, 25: 231–248.

    Article  Google Scholar 

  • Jafari N and Azhari M (2021), “Free Vibration Analysis of Viscoelastic Plates with Simultaneous Calculation of Natural Frequency and Viscous Damping,” Mathematics and Computers in Simulation, 185: 646–659.

    Article  Google Scholar 

  • Li B, Cheng Y, Zhu Z and Zhang F (2019), “A Closed-Form Solution for a Double Infinite Euler-Bernoulli Beam on a Viscoelastic Foundation Subjected to Harmonic Line Load,” Earthquake Engineering and Engineering Vibration, 18(1): 129–140.

    Article  Google Scholar 

  • Mohajel Sadeghi S and Alibeiglo A (2019), “Parametric Study of Three-Dimensional Vibration of Viscoelastic Cylindrical Shells on Different Boundary Conditions,” Journal of Vibration and Control, 25(19–20): 2567–2579.

    Article  Google Scholar 

  • Nguyen SN, Lee J and Cho M (2015), “Efficient Higher-Order Zig-Zag Theory for Viscoelastic Laminated Composite Plates,” International Journal of Solids and Structures, 62: 174–185.

    Article  Google Scholar 

  • Norouzi H, Alibeigloo A (2017), “Three Dimensional Static Analysis of Viscoelastic FGM Cylindrical Panel Using State Space Differential Quadrature Method,” European Journal of MechanicsA/Solids, 61(c): 254–266.

    Article  Google Scholar 

  • Norouzi H and Alibeigloo A (2018), “Three-dimensional Thermoviscoelastic Analysis of a FGM Cylindrical Panel Using State Space Differential Quadrature Method,” Journal of Thermal Stresses, 41(3): 1–16.

    Article  Google Scholar 

  • Rashvand K, Alibeigloo A and Safarpour M (2020), “Free Vibration and Instability Analysis of a Viscoelastic Micro-Shell Conveying Viscous Fluid Based on Modified Couple Stress Theory in Thermal Environment,” Mechanics Based Design of Structures and Machines. https://doi.org/10.1080/15397734.2020.1745079

  • Shariyat M (2011), “A Double-Superposition Global-Local Theory for Vibration and Dynamic Buckling Analyses of Viscoelastic Composite/Sandwich Plates: A Complex Modulus Approach,” Archive of Applied Mechanics, 81: 1253–1268.

    Article  Google Scholar 

  • Sladek J, Sladek V and Zhang C (2008), “Local Integral Equation Method for Viscoelastic Reissner-Mindlin plate,” Computational Mechanics, 41: 759–768.

    Article  Google Scholar 

  • Szyszkowski W and Glockner PG (1985), “The Stability of Viscoelastic Perfect Column: A Dynamic Approach,” International Journal of Solids and Structures, 21(6): 545–559.

    Article  Google Scholar 

  • Tchamo JM and Zhou Y (2018), “An Alternative Practical Design Method for Structures with Viscoelastic Dampers,” Earthquake Engineering and Engineering Vibration, 17(3): 459–473.

    Article  Google Scholar 

  • Temel B and Sahan MF (2013), “Transient Analysis of Orthotropic, Viscoelastic Thick Plates in the Laplace Domain,” European Journal of MechanicsA/Solids, 37: 96–105.

    Article  Google Scholar 

  • Touati D and Cederbaum G (1994), “Dynamic Stability of Nonlinear Viscoelastic Plates,” International Journal of Solids and Structures, 31(17): 2367–2376.

    Article  Google Scholar 

  • Wang SJ, Chiu IC, Yu CH and Chang KC (2018), “Experimental and Analytical Study on Design Performance of Full-Scale Viscoelastic Dampers,” Earthquake Engineering and Engineering Vibration, 17(4): 693–706.

    Article  Google Scholar 

  • Xiao JR and McCarchy MA (2003), “Meshless Analysis of Timoshenko Beams Based on a Locking-Free Formulation and Variational Approach,” Computer Methods in Applied Mechanics and Engineering, 192: 4403–4424.

    Article  Google Scholar 

  • Yang Z and Lam ESS (2014), “Dynamic Responses of Two Buildings Connected by Viscoelastic Dampers Under Bidirectional Earthquake Excitations,” Earthquake Engineering and Engineering Vibration, 13(1): 137–150.

    Article  Google Scholar 

  • Ye K, Li L and Tang JX (2003), “Stochastic Seismic Response of Structures with Added Viscoelastic Dampers Modeled by Fractional Derivative,” Earthquake Engineering and Engineering Vibration, 2(1): 133–139.

    Article  Google Scholar 

  • Zamani HA, Aghdam MM and Sadeghi M (2017), “Free Damped Vibration Analysis of Mindlin Plates with Hybrid Material Foundation Viscoelasticity,” Int J Mech Sci, 121: 33–43.

    Article  Google Scholar 

  • Zenkour AM (2004), “Buckling of Fiber-Reinforced Viscoelastic Composite Plates Using Various Plate Theories”, J Eng Math, 50: 75–93.

    Article  Google Scholar 

  • Zenkour AM, Allam MNM and Sobhy M (2011), “Bending Response of a Fiber-Reinforced Viscoelastic Composite Plate Resting on Elastic Foundation,” Arch Appl Mech, 81: 77–96.

    Article  Google Scholar 

  • Zhang J, Ge R and Zhang L (2019), “Transverse Free Vibration Analysis of a Tapered Timoshenko Beam on Visco-Pasternak Foundations Using the Interpolating Matrix Method,” Earthquake Engineering and Engineering Vibration, 18(3): 567–578.

    Article  Google Scholar 

  • Zhang NH and Cheng CJ (1998), “Nonlinear Mathematical Model of Viscoelastic Thin Plates with Its Applications,” Comput. Methods Appl Mech. Eng, 16(5): 307–319.

    Google Scholar 

  • Zhou YF and Wang ZM (2014), “Exact Solutions for the Stability of Viscoelastic Rectangular Plate Subjected to Tangential Follower Force,” Arch Appl Mech, 84: 1081–1089.

    Article  Google Scholar 

Download references

Acknowledgment

The author would like to thank Professor Bijan Boroomand for his contributions in studying the manuscript and providing useful suggestions.

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Authors and Affiliations

  1. Department of Civil Engineering, Isfahan University of Technology, Isfahan, 84156-83111, Iran

    Nasrin Jafari

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  1. Nasrin Jafari
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Correspondence to Nasrin Jafari.

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Jafari, N. Non-harmonic resonance of viscoelastic structures subjected to time-dependent exponentially decreasing transverse distributed loads. Earthq. Eng. Eng. Vib. 22, 825–840 (2023). https://doi.org/10.1007/s11803-023-2200-1

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  • DOI: https://doi.org/10.1007/s11803-023-2200-1

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