Uncovering inner detached resonance curves in coupled oscillators with nonlinearity
Introduction
Detached resonance curves (DRCs) appear as isolated loops of solutions in the frequency response curves (FRCs) of oscillating systems with nonlinearity. Their detection may be undisclosed either when applying classical numerical techniques or when performing sine-sweep experimental tests. An analytical approach is then convenient because it would show the effects of the main system parameters on the DRCs appearance. Furthermore, it would be helpful to predict DRCs in advance, prior to conduct any numerical or experimental analysis.
DRCs manifest as a result of multivaluedness in the FRC [1]. This means that, in case of harmonic excitation and provided the system response is predominantly harmonic at the excitation frequency, multiple solutions may appear in the steady-state amplitude responses at a single frequency. Depending on the values of the system parameters, multivaluedness can lead to closed DRCs, which can either lay inside or outside the main continuous FRC.
Analyzing the performance of a vibration absorber with a nonlinear damping characteristics attached to a linear host structure, Starosvetsky and Gendelman [2] reported theoretical prediction and numerical confirmation of outer DRCs. The same authors predicted similar features when analyzing a three degree-of freedom (DOF) nonlinear oscillating system [3]. The values of the system parameters greatly affect the appearance of these features, so that, for instance, in the two DOF experimental setup tested in [4] DRCs did not manifest. Alexander and Schilder [5] found out a family of outer DRCs for vanishing linear spring stiffness term when analyzing the performance of a nonlinear tuned mass damper with linear plus cubic stiffness nonlinearity. Detroux et al. [6] identified DRCs numerically in the forced response of a satellite structure. Kuether et al. [7] studied the connection between nonlinear normal modes and isolated resonance curves appearing outside the main frequency response curve of a forced nonlinear system. A family of isolated sub-harmonic branches in the nonlinear frequency response of an oscillator with clearance was studied in [8], and outer detachments were also observed in case of impact phenomena [9]. By studying the response of a harmonically excited system consisting of coupled linear and nonlinear oscillators with hardening characteristics, Gatti and Brennan [10] predicted the appearance of either inner or outer DRCs and investigated the effect of parameters in the appearance of such features. They studied the system under the assumption of very small ratio between the mass of the attachment and that of the main structure. An experimental test was also conducted by the use of a very heavy electro-magnetic shaker and a relative light mass for the attachment [11], but the physical system parameters were not suitable to uncover a DRC experimentally. In that case, it was evident that inner DRC manifests for low values of the linear stiffness and damping in the attachment. Despite the theoretical formulation used in [10] and [11] captures physically the phenomena of DRCs, its practical application to real engineering design problems is limited by the former assumption about the mass ratio.
To the best of the author׳s knowledge it seems that there is no comprehensive work which clearly shows the effects of the main parameters of a nonlinear oscillating system on the appearance of such features, and the conditions and limitations for those features to emerge in the main continuous FRC. Thus, with the aim to provide a series of analytical usable expressions related to such interesting phenomenon, this paper presents a detailed theoretical analysis on the response of a harmonically excited two DOF oscillating system with coupling nonlinear stiffness. By removing previous limitations, as the assumption of small mass ratio [11] or the assumption of a light primary suspension [12], a novel and complete closed-form solution for the amplitude-frequency response equation and detachments is derived using the harmonic balance method. Both the cases of nonlinear hardening and softening spring are considered and it is shown how this characteristics affects the occurrence of a DRC and the corresponding ranges for the values of the system parameters.
Uncovering the main mechanism underneath the appearance of DRC will help engineers to better design nonlinear system (or system expected to operate in nonlinear conditions) and thus avoiding (or exploiting) a detachment to arise in the frequency response with a consequent dramatic amplitude shift.
The theoretical approach adopted in this work can be extended to different configurations of nonlinear coupled oscillators, so as to investigate their specific dynamic behavior when subject to harmonic excitation.
Section snippets
System description and equations of motion
The system of interest in this work is shown in Fig. 1. A primary oscillating mass ms is connected to ground through a linear spring ks and damper cs and it is excited by a force f. A secondary mass m is attached to the first mass through a linear damper c1 and a nonlinear spring, which consists of a linear, k1, plus cubic, k3, stiffness term.
The equations of motion of the system may be written in terms of the displacement of the primary mass, xs, and the relative displacement between the two
Amplitude–frequency equations
Predictions for the steady-state solutions of the equations of motion given by Eqs. (2a and b), in terms of amplitude-frequency expressions, are found in the assumption that the response of the system is predominantly harmonic at the frequency of excitation. For this purpose, the non-dimensional absolute displacement of the primary mass and the non-dimensional relative displacement between the two masses are assumed to be in the form of and ,
Bifurcation curves
In Section 3, the amplitude-frequency equations for the system non-dimensional displacements were derived. In particular, it may be noted that Eq. (3) is an implicit cubic equation in terms of W squared, Eq. (4) gives the amplitude Ys as a function of the amplitude W, and Eq. (6) gives the amplitude Y as a function of Ys and W. Since Eq. (3) is cubic, it may yield up to three solutions for W and thus for Ys and Y, the type of which depends on the discriminant Δ of the cubic polynomial on the
Stability of the steady-state solutions
The stability of the steady-state solutions of (3), (4), (6) is calculated by applying Floquet theory as described in [15]. To this aim, disturbances and are introduced to the harmonic solutions of the equations of motions given in Eqs. (2a and b), leading to.
Substituting Eqs. (10a and b) into Eqs. (2a and b), the following linearized variational equations are obtainedwhich admit solutions of
Frequency response curves
The FRCs of the amplitudes W, Ys and Y of the non-dimensional displacements as functions of Ω are calculated using (3), (4), (6), respectively, for particular values of the system parameters.
To emphasize the relationship between the bifurcation curves in the plane and the FRCs in the plane, a three-dimensional plot involving the three variables is adopted and shown in Fig. 6, Fig. 7. In both figures, the damping ratios are set to 0.01 and the mass ratio is set to 0.05. Fig. 6
Insight into the appearance of inner detachments
To get a closer insight into the appearance of inner DRCs in the FRC of the relative displacement amplitude W of the nonlinear system considered in this paper, it is first recalled, as observed in Section 6, that the condition for an inner DRC to manifest is associated to the presence of a relative maximum or minimum in the bifurcation curve related to the jump-up frequencies. This is because an inner DRC is formed when the projection of the FRC on the plane intersects twice the same branch
Conclusions
This paper presents an investigation on the appearance of isolated detached resonance curves inside the main continuous frequency response curve of nonlinear coupled oscillators. A general relation between the frequency response curve and the bifurcation curve is emphasized, as also discussed in [6], and that is used to study the formation of detachments. The specific case considered in the paper, i.e. that of a two degree-of-freedom nonlinear system with cubic stiffness nonlinearity, allowed
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