# Hysteresis Response Loops in Stationary Vibrator Regimes for Elastomeric Insulators

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## Abstract

**:**

## 1. Introduction

## 2. Test Stand Constructive Scheme

_{f}; 6—semi-circular clamping device; 7—elastomeric insulator; 8—displacement transducer T

_{x}; 9—force transducer T

_{q}; 10—fixed base (frame) [16,17,18,19,20].

_{f}measures the incident force on the elastomeric insulator, transducer T

_{q}measures the emergent force, or the force transmitted at the base and transducer T

_{x}measures the instantaneous displacement x = x(t) or the vertical deformation of the deformable assembly. Each column has its own system of transducers T

_{x}si T

_{f}.

_{f}—the incident force transducer which measures force F(t); T

_{q}—the displacement transducer that measures the instantaneous displacement x = x(t).

_{y}, and the moving axes O

_{c}, for compression, and O

_{f}, for shear, are connected to the elastomeric device. The three distinct positions are highlighted by the seating angle α. Thus, for compression α = 0, compression-shear α > 0 and α < 90, and for shear α = 90° [21,22].

## 3. Evaluation of Dissipated Energy

_{n}is its own pulsation, the dissipated energy W

_{d}may be expressed as:

_{r}—critical viscous damping). Usually, $\zeta $ is under the value of 0.7;

_{0}r—static moment of the exciter with eccentric bodies in rotation motion, kg·m;

#### 3.1. Dissipated Energy as a Function of Damping

- (a)
- ante-resonance with Ω
_{a}< 1 for which we have

- (b)
- Post-resonance with Ω
_{p}> 1 for which it is valid the relation

_{c}, compression—shear k

_{α}and shear k

_{f}are characteristic of the ante-resonance regime ${\mathsf{\Omega}}_{a}=0.8,{\zeta}_{a}^{0}=0.225$ and, respectively, of the post-resonance regime ${\mathsf{\Omega}}_{p}=1.5,{\zeta}_{p}^{0}=0.41$ [23,24].

#### 3.2. Dissipated Energy as a Function of Relative Pulsation

_{d}

^{ciclu}(Ω) depending on the relative excitation pulsation $\mathsf{\Omega}=\frac{\omega}{{\omega}_{n}}$ for the given values of stiffness k and the fraction of the critical damping $\zeta $ is given by relation, with the parameter order i, so

_{i}can also be written as

_{i}is the stiffness for i = c, α, f, for the three situations corresponding to angle α = 0°, α = 60°, α = 90°, respectively, compression, compression—shear and shear.

_{c}> k

_{α}> k

_{f}, it emerges that the relative pulsations are in order Ω

_{c}< Ω

_{α}< Ω

_{f}as shown in Figure 7.

## 4. Hysteretic Loops

- (a)
- Hysteretic loops in F-x coordinates

_{a}= 0.8 in the ante-resonance regime presents the family of elliptical loops for the three significant cases of the viscoelastic system ${\alpha}_{c}={0}^{\mathrm{o}}$, $\alpha ={60}^{\mathrm{o}}$ and ${\alpha}_{f}={90}^{\mathrm{o}}$ that is in situation ${k}_{c},{\zeta}_{c}$, ${k}_{\alpha},{\zeta}_{\alpha}$ and ${k}_{f},{\zeta}_{f}$. It is specified that all ellipses are inclined in quadrants I and III [23,24].

_{p}= 1.5 in the post-resonance regime presents the family of elliptical loops for the three significant cases, with the specification that all ellipses are inclined in quadrant II and IV as effect of the influence of the resonance regime [23,24,25].

_{c}, E

_{α}and E

_{f}with the values in Figure 10 coincides with the corresponding values in Figure 7.

- (b)
- Hysteretic loops in Q-x coordinates

_{a}< 1, Ω

_{p}> 1 and Ω = 1 [23,24,25].

_{a}= 0.8 in ante-resonance, for the three significant values and ${k}_{c},{\zeta}_{c}$, ${k}_{\alpha},{\zeta}_{\alpha}$ and ${k}_{f},{\zeta}_{f}$ it is presented as the family of elliptical loops, all inclined in quadrants I and III.

_{p}= 1.5 in post-resonance, for the three sets of significant values ${k}_{c},{\zeta}_{c}$, ${k}_{\alpha},{\zeta}_{\alpha}$ and ${k}_{f},{\zeta}_{f}$ there are presented the hysteretic loops in quadrants I and III.

## 5. Conclusions

- (a)
- The analytical expression of the dissipative energy offers the possibility of evaluation for two significant cases, namely:
- -
- The variation of the dissipated energy depending on the discrete change of the damping for the three dynamic regimes: ante-resonance, post-resonance and resonance;
- -
- The variation of the dissipated energy depending on the variation of Ω for discrete variable sets of values of k and $\zeta $;

- (b)
- The representation of the elliptical hysteretic loops in the F-x coordinate system for the three cases of the dynamic regimes, namely: ante-resonance, post-resonance and resonance. It was found that in post-resonance the inclination of the axes of the ellipses towards the ante-resonance regime changes due to the inertial effect of the mass, and in resonance, the ellipses are symmetrically centered in relation to the F-x axis system.
- (c)
- The elliptical hysteretic loops in the Q-x system are inclined only in quadrants I and III, regardless of the dynamic regime.
- (d)
- The areas of the ellipses represent the dissipated energy. The numerical results were verified by experimental lifting of hysteretic loops on the dynamic stand.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 3.**Schematization of the position of the force transducers T

_{f}, T

_{q}and of the displacement transducer T

_{x}.

**Figure 5.**Variation of dissipated energy per cycle, in ante-resonance regime, depending on the damping ratio $\zeta $ and stiffness k for Ω = 0.8.

**Figure 6.**Variation of dissipated energy per cycle, in post-resonance regime, depending on the damping ratio $\mathsf{\zeta}$ and stiffness k for Ω = 1.5.

**Figure 7.**Variation of dissipated energy per cycle, as a function of relative pulsation Ω and discrete variation of parameters k and ζ.

**Figure 10.**Centered elliptical hysteretic loops F-x in resonance regime for Ω = 1. α

_{c}= 0, α = 60° and α

_{f}= 90°.

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**MDPI and ACS Style**

Bratu, P.; Dobrescu, C.; Drăgan, N.
Hysteresis Response Loops in Stationary Vibrator Regimes for Elastomeric Insulators. *Symmetry* **2022**, *14*, 246.
https://doi.org/10.3390/sym14020246

**AMA Style**

Bratu P, Dobrescu C, Drăgan N.
Hysteresis Response Loops in Stationary Vibrator Regimes for Elastomeric Insulators. *Symmetry*. 2022; 14(2):246.
https://doi.org/10.3390/sym14020246

**Chicago/Turabian Style**

Bratu, Polidor, Cornelia Dobrescu, and Nicu Drăgan.
2022. "Hysteresis Response Loops in Stationary Vibrator Regimes for Elastomeric Insulators" *Symmetry* 14, no. 2: 246.
https://doi.org/10.3390/sym14020246