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

Abstract: This paper presents the outcomes of the theoretical and experimental research carried out on a real model at natural scale using Voigt–Kelvin linear viscoelastic type m, c, and k models excited by a harmonic force F(t) = F0 sinωt, where F0 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 Wmax d values. The optimization condition between the Wmax d maximum dissipated energy and the amortization ζ 0 = ± ( 1−Ω2 ) /2Ω 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 Ω < 1 ante-resonance, and an angle greater than 90◦ in post-resonance for Ω > 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 Wmax d .


Introduction
For the vibration-driven technological processes with excitation harmonic forces, Voigt-Kelvin linear rheological models are adopted with the discretionary variation of the amortization or of the fraction of the critical amortization ζ = c/2 √ k/m.The functional optimization of the vibration-driven processes consists of reaching a maximum W max d dissipated energy for ζ = ζ 0 in direct correlation with the area of the elliptical hysteretic loop.By locating suitable transducers on the physical system, the elliptical hysteretic loops can be determined in such a manner that allows the assessment of their areas, respectively the maximum dissipated energy for a dynamic system may be assessed with rigidity k and amplitude A of the instantaneous displacement x = A sin(ωt − ϕ), where ϕ is the phase angle.
Depending on the optimization requirement, in order to achieve a maximum dissipation, i.e., W max d , can be determined the pairs of values ζ optim = ζ 0 for which are established the calculus relations of the significant parameters.Thus, the functional relations were established for A = A(ω), amplitude, the W d = W d (ζ, Ω) dissipated energy, the viscoelastic force Q = c .
x + kx = Q(x), and the excitation force F = F(x).The correlations between the maximum dissipated energy and the areas of the elliptical hysteretic loops are reflected in the graphical representations for the case of operation in ante-resonance at Ω < 1 and respectively for the case of post-resonance at Ω > 1 [1][2][3].
In this paper are highlighted the hysteretic loops families of the elastomeric devices that were designed, manufactured and tested, for a viaduct on the "Transilvania" Motorway, in Romania.Thus, based on dynamic testing devices and realized within ICECON Bucharest (Romania), there were determined the characeristic shapes of the hysteretic loops for the three dynamic regimmes, namely: ante-resonance, resonance, and post-resonance.
The characteristic function for dissipated energy was also established, highlighting its modification in relation to the fraction of critical damping ζ as a system parameter.
The novelty of the approach consists of the fact that for the elastomeric isolator system/device are highlighted the dissipative characteristics, the hysteretic loop orientation, as well as the dissipated energy variation depending on the parameter ζ, which can take different values in relation to mass m, elasticity k, and linear dissipation c.The graphical representation of the hysteretic loop in accordance with the dissipated energy allowed a clearer, more operational and efficient highlighting of the dynamic behavior during the tests, with meaningful and eloquent conclusions.
In this paper are highlighted the hysteretic loops families of the elastomeric devices that were designed, manufactured and tested, for a viaduct on the "Transilvania" Motorway, in Romania.Thus, based on dynamic testing devices and realized within ICECON Bucharest (Romania), there were determined the characeristic shapes of the hysteretic loops for the three dynamic regimmes, namely: ante-resonance, resonance, and post-resonance.
The characteristic function for dissipated energy was also established, highlighting its modification in relation to the fraction of critical damping ζ as a system parameter.
The novelty of the approach consists of the fact that for the elastomeric isolator system/device are highlighted the dissipative characteristics, the hysteretic loop orientation, as well as the dissipated energy variation depending on the parameter ζ, which can take different values in relation to mass m, elasticity k, and linear dissipation c.The graphical representation of the hysteretic loop in accordance with the dissipated energy allowed a clearer, more operational and efficient highlighting of the dynamic behavior during the tests, with meaningful and eloquent conclusions.

Parameter Analysis
The dynamic linear models m, c, and k, produced by the harmonic force F(t) = F0 sinωt and with the reaction force Q(t) in the liniar viscoelastic system, c, k, is presented in Figure 1.The movement equation at the dynamic balance is  +  +  =  sin . ( The solution of Equation ( 1) is where A is the amplitude of the instantaneous displacement x = x(t),ϕ is the phase shift between the instantaneous displacement x(t) and the excitation force F(t).
It is introduced the notation: , where m , so that the variable Ω to be used as excitation relative pulsation.

Amplitude of Displacement
An amplitude of the instantaneous displacement x = x(t) may be expressed by the A(ω) or A(Ω) relation, as follows: or The movement equation at the dynamic balance is m ..
x + c .
x + kx = F 0 sin ωt. ( The solution of Equation ( 1) is where A is the amplitude of the instantaneous displacement x = x(t), ϕ is the phase shift between the instantaneous displacement x(t) and the excitation force F(t).
It is introduced the notation: Ω = ω ω n , where ω n = k m , so that the variable Ω to be used as excitation relative pulsation.

Amplitude of Displacement
An amplitude of the instantaneous displacement x = x(t) may be expressed by the A(ω) or A(Ω) relation, as follows:

De-Phasing between Force and Displacement
The de-phasing ϕ between the excitation force F = F(t) and the instantaneous displacement x = x(t) may be calculated as follows: (6)

Dissipated Energy
The dissipated energy W d may be expressed according to Ω and ζ as follows: where It can be noticed that for certain values of Ω it correspond the values of ζ 0 for which the dissipated energy is at maximum.The representation of W d (ζ), so that the variation domain of ζ to be included ζ 0 , leads to a curve with a maximum point, having the coordinates ζ 0 at W max d .For the discrete values of Ω, a family of parametrical curves of W d (Ω, ζ) can be obtained.

Equation of the F-x Elliptic Hysteretic Loop
Phase ωt is eliminated between x = x(t) and F = F(t) as follows: where If we note the dimensionless sizes x A = X and F F 0 = Φ, then relation (10) becomes From of equation ( 11) Φ emerges as Symmetry 2019, 11, 315 where sin ϕ = cω By replacing x A = X, F F 0 = Φ, sin ϕ, and cos ϕ in relation (12), the expression of force F emerges according to pulsation ω, rigidity k, and amortization c as follows: or in terms of Ω and ζ measures we have For the condition that the dissipated energy to be maximum, (a) For the ante-resonance regime or the following is obtained: or where

Equation of the Hysteretic Loop Q-x
The instantaneous viscous-elastic force Q(t) = c .
x + kx may be expressed from the dynamic equilibrium equation.Thus, it can be expressed m ..
from where Q(t) emerges as x, or in relation with x = x(t), it is obtained In Equation (22) we insert the expression of force F(x) = F(x, ω) from ( 13) and have It is found that the family of ellipses according to Equation ( 24) is characterized by the fact that all ellipses have their major axis in the quadrant (0, π/2) with positive angular coefficient tgα = k > 0 for any value Ω > 0, where α is the angle between the ellipse axis and the Ox axis [24].16) and (24), respectively, corresponding to the representation of function

Elliptic Hysteretic
given by Equation ( 8).Thus, it is found that for the maximum value of the dissipated energy W d max according to Ω i < 1 and , the areas of the ellipses F-x and Q-x are maximum and equal [25][26][27][28][29] We specify the fact that the fraction of the critical amortization ζ e f , in this case for W d max , may be defined as follows: and taking into account Equations ( 8) and ( 20), we have It can be observed that only at a resonance, when In Figures 2-5, the functions and in relation with Ω Equation ( 23) becomes It is found that the family of ellipses according to Equation ( 24) is characterized by the fact that all ellipses have their major axis in the quadrant (0, π/2) with positive angular coefficient 0 > = k tgα for any value 0 > Ω , where α is the angle between the ellipse axis and the Ox axis [24].16) and (24), respectively, corresponding to the representation of function Wd max = W(ζ ), given by Equation ( 8).Thus, it is found that for the maximum value of the dissipated energy Wd max according to

Elliptic Hysteretic Loops for the
, the areas of the ellipses F-x and Q-x are maximum and equal [25][26][27][28][29] We specify the fact that the fraction of the critical amortization ef ζ , in this case for Wd max , may be defined as follows: ( ) and taking into account Equations ( 8) and ( 20), we have It can be observed that only at a resonance, when In Figures 2-5

Elliptic Hysteretic Loops for the Post-Resonance Regime ( 1 > Ω )
In the case of the above described system, for Ω 4 = 1,31 with

Elliptic Hysteretic Loops for the Post-Resonance Regime (
In the case of the above described system, for Ω 4 = 1,31 with

Figure 1 .
Figure 1.Linear dynamic model m, c, and k.

Figure 1 .
Figure 1.Linear dynamic model m, c, and k.

Figure 5 .
Figure 5. Families of curves in ante-resonance for W 1 max , W 2 max , and W 3 max .(a) Families of curves 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.
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.