# Dynamic Response of Zener-Modelled Linearly Viscoelastic Systems under Harmonic Excitation

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

_{n}is the natural pulsation of the system and $\zeta $ is the fraction of critical damping, such that $c=2\zeta {\omega}_{n}m$.

## 2. Dynamic Response with Respect to Displacements

_{1}, k

_{2}, and by the dissipative behaviour expressed by the dissipation coefficient c [4,5,6].

_{1}= k rigidity of the Hooke elastic element, and k

_{2}= kN is the rigidity of the elastic element in the structure of the Maxwell model, where N is a multiplication coefficient for rigidity k

_{1}[7,8].

^{3}kg; m

_{0}r = 20 kgm; k = 10

^{8}N/m; N = 10; c = (7, 9, 11, 13) × 10

^{5}Ns/m; ζ = 0,15; 0,18; 0,22; 0,32; ω = 0...500 rad/s; Ω = 0...10 [1,7,8,9].

#### Variation of Displacement Amplitudes ${X}_{0}$ and ${Y}_{0}$

_{1}, k

_{2.}

## 3. Transmitted Dynamic Force

#### Variation of transmitted force amplitude ${Q}_{0}$

## 4. The Capacity of Dynamic Insulation

#### Variation of Force Transmissibility

## 5. Amplitude of Instantaneous Deformation for a Linearly Viscous Buffer.

## 6. Dissipated Energy

## 7. Conclusions

- (a)
- The analytical model and parametric curves lead to the following conclusions:
- (b)
- (c)

_{0}amplitude presents a stable layer at low variations of the excitation pulsation, suggesting that Y

_{0}continuously increased with ω.

- (d)
- The maximal dynamic transmitted force Q
_{0}in the post-resonance field for ω >> ω_{n}or Ω >> 1 shows the stable values set by the size of the viscous amortization. - (e)
- Transmissibility decreases at high values of the excitation pulsation once a discrete value is set for amortization.
- (f)

_{0}= 0,005 m, the force transmitted to the field Q

_{0}= (1,5 ÷ 2,5) × 10

^{3}kN, and the dissipated energy W

_{d}= (10 ÷ 25) kJ.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Plots of ${\text{}\mathrm{X}}_{0}\left(\mathrm{c},\mathsf{\omega}\right)$ for continuously varying ω and discrete values of c.

**Figure 3.**Plots of ${\mathrm{Y}}_{0}\left(\mathrm{c},\mathsf{\omega}\right)$ for continuously varying ω and discrete values of c.

**Figure 4.**Plots of ${\text{}\mathrm{X}}_{0}\left(\mathsf{\zeta},\mathsf{\Omega}\right)$ for continuously varying Ω and discrete values of ζ.

**Figure 5.**Plots of ${\text{}\mathrm{Y}}_{0}\left(\mathsf{\zeta},\mathsf{\Omega}\right)$ for continuously varying Ω and discrete values of ζ.

**Figure 6.**Plots of transmitted force amplitude ${\mathrm{Q}}_{0}\left(\mathrm{c},\mathsf{\omega}\right)$ for continuously varying $\mathsf{\omega}$ and discrete values of c. The excitation force amplitude F

_{0}is expressed as ${F}_{0}={m}_{0}r{\omega}^{2}$.

**Figure 7.**Plots of transmitted force amplitude ${\mathrm{Q}}_{0}\left(\mathsf{\zeta},\mathsf{\Omega}\right)$ for continuously varying Ω and discrete values of ζ. Excitation force amplitude F

_{0}is expressed as ${F}_{0}=\frac{{m}_{0}r}{m}k{\Omega}^{2}$.

**Figure 8.**Plots of force transmissibility $\mathrm{T}\left(\mathrm{c},\mathsf{\omega}\right)$ for continuously varying ω and discrete values of c. Excitation force amplitude Fo is expressed as ${F}_{0}={m}_{0}r{\omega}^{2}$.

**Figure 9.**Plots of the dissipated energy per unit cycle ${\mathrm{W}}_{\mathrm{d}}\left(\mathrm{c},\mathsf{\omega}\right)$ for continuously varying ω and discrete values of c. The excitation force amplitude F

_{0}is expressed as ${F}_{0}={m}_{0}r{\omega}^{2}$.

**Figure 10.**The plots of the dissipated energy per unit cycle ${W}_{d}\left(\zeta ,\Omega \right)$ for continuously varying Ω and discrete values of ζ. The excitation force amplitude F

_{0}is expressed as ${F}_{0}=\frac{{m}_{0}r}{m}k{\Omega}^{2}$.

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**MDPI and ACS Style**

Bratu, P.; Dobrescu, C.
Dynamic Response of Zener-Modelled Linearly Viscoelastic Systems under Harmonic Excitation. *Symmetry* **2019**, *11*, 1050.
https://doi.org/10.3390/sym11081050

**AMA Style**

Bratu P, Dobrescu C.
Dynamic Response of Zener-Modelled Linearly Viscoelastic Systems under Harmonic Excitation. *Symmetry*. 2019; 11(8):1050.
https://doi.org/10.3390/sym11081050

**Chicago/Turabian Style**

Bratu, Polidor, and Cornelia Dobrescu.
2019. "Dynamic Response of Zener-Modelled Linearly Viscoelastic Systems under Harmonic Excitation" *Symmetry* 11, no. 8: 1050.
https://doi.org/10.3390/sym11081050