# Hysteretic Loops in Correlation with the Maximum Dissipated Energy, for Linear Dynamic Systems

## Abstract

**:**

_{0}sinωt, where F

_{0}is the amplitude of the harmonic force and ω is the excitation angular frequency. The linear viscous-elastic rheological system (m, c, k) is characterized by the fact that the c linear viscous damping—and, consequently, the fraction of the critical damping ζ—may be changed so that the dissipated energy can reach maximum ${W}_{d}^{\mathrm{max}}$ values. The optimization condition between the ${W}_{d}^{\mathrm{max}}$ maximum dissipated energy and the amortization ${\zeta}^{0}=\pm \left(1-{\Omega}^{2}\right)/2\Omega $ modifies the structure of the relation F = F(x), which describes the elliptical hysteresis loop F–x in the sense that it has its large axis making an angle less than 90° with respect to the x-axis in $\Omega <1$ ante-resonance, and an angle greater than 90° in post-resonance for $\Omega >1$. The elliptical Q–x hysteretic loops are tilted with their large axis only at angles below 90°. It can be noticed that the equality between the arias of the hysteretic loop, in the two representations systems Q–x and F–x, is verified, both being equal with the maximum dissipated energy ${W}_{d}^{\mathrm{max}}$.

## 1. Introduction

## 2. Parameter Analysis

_{0}sinωt and with the reaction force Q(t) in the liniar viscoelastic system, c, k, is presented in Figure 1.

#### 2.1. Amplitude of Displacement

#### 2.2. De-Phasing between Force and Displacement

#### 2.3. Dissipated Energy

_{d}may be expressed according to $\Omega $ and $\zeta $ as follows:

_{d}, which is $\frac{d{W}_{d}}{d\zeta}=0$, it emerges that ${\zeta}^{0}=\pm \frac{1-{\Omega}^{2}}{2\Omega}$, so that the maximum dissipated energy ${W}_{d}^{\mathrm{max}}$ becomes

#### 2.4. Equation of the F–x Elliptic Hysteretic Loop

- (a)
- For the ante-resonance regime $\Omega <1$ at ${\zeta}_{a}^{0}=\frac{1-{\Omega}^{2}}{2\Omega}$, obtaining$$F\left(x,\Omega \right)=\left(1-{\Omega}^{2}\right)\hspace{0.17em}k\left[x\pm \sqrt{{A}^{2}\left(\Omega \right)-{x}^{2}}\right],$$$$F\left(x,\zeta ,\Omega \right)=2{\zeta}_{a}^{0}\Omega \hspace{0.17em}k\left[x\pm \sqrt{{A}^{2}\left(\Omega ,{\zeta}_{a}^{0}\right)-{x}^{2}}\right].$$
- (b)
- For the post-resonance regime $\Omega >1$ at ${\zeta}_{p}^{0}=-\frac{1-{\Omega}^{2}}{2\Omega}$ the following is obtained:$$F\left(x,\Omega \right)=\left(1-{\Omega}^{2}\right)\hspace{0.17em}k\left[x\pm \sqrt{{A}^{2}\left(\Omega \right)-{x}^{2}}\right],$$$$F\left(x,\zeta ,\mathsf{\Omega}\right)=-2{\zeta}_{p}^{0}\mathsf{\Omega}k\left[x\pm \sqrt{{A}^{2}\left(\mathsf{\Omega},{\zeta}_{p}^{0}\right)-{x}^{2}}\right].$$$$A\left(\Omega \right)=\pm \frac{{F}_{0}}{k}\frac{1}{\left(1-{\Omega}^{2}\right)\sqrt{2}},$$$$A\left(\Omega ,{\zeta}_{a,p}^{0}\right)=\pm \frac{{F}_{0}}{k}\frac{1}{2{\zeta}_{a,p}^{0}\Omega \sqrt{2}}.$$

#### 2.5. Equation of the Hysteretic Loop Q–x

## 3. Elliptic Hysteretic Loops for the Ante-Resonance Regime ($\Omega <1$)

^{4}kg, k = 2⋅10

^{7}N/m at the relative pulses Ω

_{1}= 0.8 with ${\zeta}_{optim}={\zeta}_{1}^{0}=0.225$; Ω

_{2}= 0.6 with ${\zeta}_{optim}={\zeta}_{2}^{0}=0.53$; and Ω

_{3}= 0.5 with ${\zeta}_{optim}={\zeta}_{3}^{0}=0.75$, the representation of the elliptical hysteretic loops Q–x and F–x, according to the graph of function W

_{d}= W(ζ), as shown in Figure 2, Figure 3 and Figure 4. In each figure functions F(x) and Q(x) are represented based on Equations (16) and (24), respectively, corresponding to the representation of function W

_{d}

^{max}= W(ζ), given by Equation (8). Thus, it is found that for the maximum value of the dissipated energy W

_{d}

^{max}according to ${\Omega}_{i}<1$ and ${\zeta}_{optim}={\zeta}_{i}^{0}$, i = 1, 2, 3, the areas of the ellipses F–x and Q–x are maximum and equal [25,26,27,28,29]

_{d}

^{max}, may be defined as follows:

## 4. Elliptic Hysteretic Loops for the Post-Resonance Regime ($\Omega >1$)

_{4}= 1.31 with ${\zeta}_{4}^{0}=0.28$; Ω

_{5}= 1.66 with ${\zeta}_{5}^{0}=0.53$; Ω

_{6}= 2 with ${\zeta}_{6}^{0}=0.75$, and Ω

_{7}= 2.32 with ${\zeta}_{7}^{0}=0.95$, in the Figure 6, Figure 7, Figure 8, Figure 9 and Figure 10 there were represented the functions W

_{i}(ζ) and Q

_{i}(x) were i = 4,5,6,7.

## 5. Conclusions

- (a)
- For this, there are established the functional relations of the perturbatory force F = F(x) in relation with the instantaneous displacement x = x(t), of the viscoelastic force Q = Q(t) in relation with x = x(t), of the viscoelastic force Q = Q(x) in relation with x = x(t), and of the dissipated energy in relation with the variation of the viscous dumping $\zeta $, namely W
_{d}= W(ζ); - (b)
- In this context, based on the graphical representations, it was possible to emphasize the fact that the ellipses F–x are rotating in relation to the origin of the axis system, according to the dynamic regime $\Omega <1$ or $\Omega >1$. The axis of the ellipse F–x rotation is caused by the inertial effect that is produced by the presence of the factor $1-{\Omega}^{2}$ in the expression of the function F = F(x);
- (c)
- The representation of the function Q = Q(x) highlights the that it constantly maintains an angle of inclination of the ellipse Q–x, axis, with a positive slope;
- (d)
- It can be noticed that the areas of the hysteretic loops F–x and Q–x are equal between them and equal with the maximum value of the dissipated energy W
^{max}(${\zeta}^{0}$) for ${\zeta}^{0}=\frac{1-{\Omega}^{2}}{2\Omega}$. - (e)
- The effective critical dumping ${\zeta}_{ef}$ depends both on the value of ${\zeta}^{0}$ as well as on the relative pulsation $\Omega <1$, for ante-resonance or $\Omega >1$, for post-resonance. In the case of the resonance regime, for $\Omega =1$, then ${\zeta}_{ef}={\zeta}^{0}$ is a sole situation that enables the assessment of the fraction from the optimal critical amortization ${\zeta}^{0}$.

## Funding

## Conflicts of Interest

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**Figure 2.**Variation curves in ante-resonance for W

_{1}

^{max}at Ω

_{1}= 0.8 and ζ

_{1}= 0.225. (

**a**) Variation of W

_{1}according to ζ. (

**b**) Elliptic hysteretic loop F

_{1}–x. (

**c**) Elliptic hysteretic loop Q

_{1}–x. x = [−0.08701 ÷ +0.08701].

**Figure 3.**Variation curves in ante-resonance for W

_{2}

^{max}at Ω

_{2}= 0.6 and ζ

_{2}= 0.53. (

**a**) Variation of W

_{2}according to ζ. (

**b**) Elliptic hysteretic loop F

_{2}–x. (

**c**) Elliptic hysteretic loop Q

_{2}–x. x = [−0.05542 ÷ +0.05542].

**Figure 4.**Variation curves in ante-resonance for W

_{3}

^{max}at Ω

_{3}= 0.5 and ζ

_{3}= 0.75. (

**a**) Variation of W

_{3}according to ζ. (

**b**) Elliptic hysteretic loop F

_{3}–x. (

**c**) Elliptic hysteretic loop Q

_{3}–x. x = [−0.04715 ÷ +0.04715].

**Figure 5.**Families of curves in ante-resonance for W

_{1}

^{max}, W

_{2}

^{max}, and W

_{3}

^{max}. (

**a**) Families of curves W according to the current variable ζ and the discreet variable Ω. (

**b**) The family of elliptical hysteretic loops F–x according to the current variable x and the pair of discreet variables Ω, ζ. (

**c**) The family of elliptical hysteretic loops Q–x according to the current variable x and the pair of discreet variables Ω, ζ.

**Figure 6.**Variation curves in post-resonance for W

_{4}

^{max}at Ω

_{4}= 1.31 and ζ

_{4}= 0.28. (

**a**) Variation of W

_{4}according to ζ. (

**b**) Elliptic hysteretic loop F

_{4}–x. (

**c**) Elliptic hysteretic loop Q

_{4}–x. x = [−0.04878 ÷ +0.04878].

**Figure 7.**Variation curves in post-resonance for W

_{5}

^{max}at Ω

_{5}= 1.66 and ζ

_{5}= 0.53. (

**a**) Variation of W

_{5}according to ζ. (

**b**) Elliptic hysteretic loop F

_{5}–x. (

**c**) Elliptic hysteretic loop Q

_{5}–x. x = [−0.02012 ÷ +0.02012].

**Figure 8.**Variation curves in post-resonance for W

_{6}

^{max}at Ω

_{6}= 2 and ζ

_{6}= 0.75. (

**a**) Variation of W

_{6}according to ζ. (

**b**) Elliptic hysteretic loop F

_{6}–x. (

**c**) Elliptic hysteretic loop Q

_{6}–x. x = [−0.01179 ÷ +0.01179].

**Figure 9.**Variation curves in post-resonance for W

_{7}

^{max}at Ω

_{7}= 2.32 and ζ

_{7}= 0.95. (

**a**) Variation of W

_{7}according to ζ. (

**b**) Elliptic hysteretic loop F

_{7}–x. (

**c**) Elliptic hysteretic loop Q

_{7}–x. x = [−0.008045 ÷ +0.008045].

**Figure 10.**Families of curves in post-resonance for the maximum values of the dissipated energy W

_{4}, W

_{5}, W

_{6}, and W

_{7}at parameters (Ω

_{4}, ζ

_{4}), (Ω

_{5}, ζ

_{5}), (Ω

_{6}, ζ

_{6}), and (Ω

_{7}, ζ

_{7}). (

**a**) Families of curves W parametrized by the discreet variable Ω. (

**b**) The family of elliptical hysteretic loops F–x for the pair of variables Ω, ζ. (

**c**) The family of elliptical hysteretic loops Q–x for the pair of variables Ω, ζ.

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**MDPI and ACS Style**

Bratu, P.
Hysteretic Loops in Correlation with the Maximum Dissipated Energy, for Linear Dynamic Systems. *Symmetry* **2019**, *11*, 315.
https://doi.org/10.3390/sym11030315

**AMA Style**

Bratu P.
Hysteretic Loops in Correlation with the Maximum Dissipated Energy, for Linear Dynamic Systems. *Symmetry*. 2019; 11(3):315.
https://doi.org/10.3390/sym11030315

**Chicago/Turabian Style**

Bratu, Polidor.
2019. "Hysteretic Loops in Correlation with the Maximum Dissipated Energy, for Linear Dynamic Systems" *Symmetry* 11, no. 3: 315.
https://doi.org/10.3390/sym11030315