Exploratory Structural Modification and Nonlinear Based Analysis of Tuned Mass Dampers

This article presents a structural modification-based analysis of tuned mass dampers that motivates to investigate the performance of a cubic stiffness absorber. The rationale behind this investigation is that the analysis of vibration absorber using structural modifications could pave the way for the use of nonlinear mechanical elements in vibration absorption applications. To achieve this, the article reviews the fundamentals of structural modification-based analysis of absorbers. It then examines the effects of using the Duffing oscillator as an absorber. The Duffing oscillator is integrated into the primary system using the describing function and the amplitude-dependent nonlinear frequency response functions are computed. The results of this analysis are compared with those of a linear absorber in order to clarify the differences and point out possible directions for further development.


Introduction
Vibration absorbers are one-degree-of-freedom mass-spring-damper (MSD) systems fitted on structures experiencing excessive vibrations due to narrow band dynamic loads.The excessive vibrations are absorbed by setting the natural frequency of the vibration absorber equal the excitation frequency.The vibration theory underlying the behavior of structures that incorporate vibration absorbers is widely presented in vibration theory textbooks [1].
Over the years the MSD absorber theory has been developed to address the effect damping on the vibration absorber performance [2], [3] and to facilitate the MDD application to continuous structures, namely, beams [4], plates, shells [5] and cylinders [6].The development of this theory has demonstrated that a successful absorber design necessitates precise estimation of the modal mass corresponding to the mode intended to be absorbed [7], [8].
The theory of MSD absorbers has been expanded to consider: (a) the impact of damping in the primary structure on the design of MSD absorber [2], [3], and (b) the challenges of applying it to more intricate structures such as beams [4], plates, shells [5], and cylinders [6].An important outcome of extending the theory to complex structures is the recognition of the significance of modal mass in the design of vibration absorber [7], [8].However, the modal and effective mass of a complex structure can be acquired through experimental modal analysis [9]- [13], which can time consuming and costly [14].
Another direction in the advancement of vibration absorber applications is the use of nonlinear vibrating systems as absorbers.A thorough assessment of passive vibration isolation technologies employing nonlinear structural components developed prior to 2008 is discussed in [15].Efforts to generalize the theory and application of linear absorbers to the case of nonlinear absorbers are described in [16].This paper demonstrates that vibration absorbers must share the same functional form as the primary systems, and cases are shown where the proposed nonlinear vibration absorbers outperform the linear ones.The limitations associated with nonlinear vibration absorbers are also highlighted.
This article presents and discusses the preliminary results of an attempt to (a) apply structural modification theory to the optimal design of linear vibration absorbers [17] and (b) analyse and design nonlinear vibration absorbers using the concepts of describing function and nonlinear FRF.The use of structural modification in the design of linear vibration absorbers eliminates the need to estimate the modal mass.This is achievable as the absorber modifies locally the primary structure, and the characteristics of the modifying structured could be determined by data obtained from local FRF measurements.On the other hand, the analysis of nonlinear vibration absorbers is much more complicated and challenging due to the amplitude dependency and unstable dynamics that could compromise the optimality of the design.However, using the describing function method to extract the nonlinear FRFs provides an initial understanding of how the nonlinear absorber behaves under different loading levels and can thus be used to design an optimal absorber.

Vibration Absorber as Structural Modification
The rationale of this approach is based on the assumption of a single mode excessive vibration due to a band limited excitation force, with a small frequency range near the natural frequency of the excited mode.Hence the primary structure is considered an one degree of freedom system, shown in Figure 1(a).This system is then modified by incorporating a mass-spring-damper absorber, Figure 1  The primary system is structurally modified by the addition of the vibration absorber, mass   , stiffness   , and damper .The frequency response function (FRF) of the modified structure, denoted by ℎ  (), can be derived from the FRF of the primary structure, denoted as ℎ  (), using the following relationship [18].
where,   , is the vibration absorber mass,   , its stiffness and  its damping coefficient.The example used in [17] is also used here to illustrate the theory of structural modification using mass-stiffness-damper absorber.In this particular example, the mass, , and stiffness, , of the primary system were set to 30 Kg and 300 kN/m respectively.With these parameter values, the natural frequency of the primary system is calculated to be 100 rad/s.The response of the primary system is characterized by the FRF in Figure 2(a).The natural frequency of the vibration absorber must be tuned to 100 rad/s and this is attained by setting its mass,   , to 10 Kg and its stiffness   to 100 kN/m.The modified FRF for the undamped vibration absorber is obtained from Equation (1) by setting the damping value c to 0. The computed modified FRF of the undamped absorber is the curve plotted with a solid line in Figure 2(b).
The resonances that appear in the modified frequency response function can be calculated from the FRF, ℎ  (), of the primary system by first removing the damping from the system and then equating the denominator of Equation (1) to zero.This yields the following equations where ‖. ‖ denotes the absolute value for real arguments or magnitude for complex arguments.The dash and dashed-dot lines in Figure 2(a) show the positive and negative functions of Equation ( 2).Their intersections with the magnitude ‖ℎ  ()‖ of the FRF of the primary system give the new natural frequencies, namely,  1 =75.2 rad/s and  2 =135.46 rad/s.The frequency values of   and   are calculated by making the damping value of the modified FRF, Equation (1), to zero and infinity to get an equation for each damping value.Subsequently, their amplitudes are equated as follows, Through the algebraic manipulation stemming from this equality, it can be shown that   and   can be calculated by solving the following equation, whose graphical solution is depicted in Figure 3.The solutions that correspond to the points of intersection are 78.85 for   and 117.3 for   .

Duffing Oscillator as a Vibration Absorber: Formulation and Exploratory Results
The linear absorber is replaced with a Duffing oscillator to function as a nonlinear absorber.A cubic stiffness of 10 9 N/m 3 is selected for the Duffing oscillator to match the natural frequency of the linear system.Following this, the nonlinear frequency response functions are calculated using the describing function method [19], [20].
The nonlinear FRFs of a nonlinear multi-degree of freedom system can be expressed by, where [] is the mass matrix, [] the damping matrix and [] the stiffness matrix pertaining to the linear part of the system.The nonlinearity in the system is characterised by the matrix [Δ], which is constructed using the describing function corresponding to the specific type of nonlinearity.The describing function matrix for the system that includes the Duffing oscillator as the absorber is given by, where Xp is the amplitude of the mass of the primary system and   is the amplitude of the mass of the absorber.
It is evident that the nonlinear FRFs depend on the response amplitude, indicating a dependency on the excitation force.The nonlinear FRF is computed through iterative solutions of Equation ( 5) for various force excitation amplitudes.The results for two different damping values, namely, 250 Nm/s and 500 Nm/s are shown in Figures 4(a Both figures clearly show that the nonlinear FRFs vary with amplitude, as expected.It is noticeable that the nonlinear effects become more pronounced with increasing force amplitude.The amplitude level decreases as the amplitude of the force excitation increases.However, as the force excitation increases, unstable bifurcations occur, leading to responses that depend on the initial conditions and unpredictable results.One promising observation is the elimination of resonances introduced into the system by the linear absorber.This characteristic could prove valuable for applications where the excitation frequency may be uncertain.

Conclusion
This article focuses on the analysis of linear absorbers using principles of structural modification and examines the effects of a Duffing oscillator as a damped nonlinear absorber.The study effectively demonstrates that structural modification theory is applicable to the analysis and design of an optimised damped linear oscillator.By using the structural modification approach, the need for computational or experimental estimation of the modal mass is eliminated.Furthermore, the use of a Duffing oscillator effectively reduces the vibration amplitudes around the resonance.However, with increasing force excitation, nonlinear effects, especially unstable branches, become more pronounced. (b).
a) Single degree of freedom of mass m and stiffness k (b) MSD absorber: mass   , spring   and damper c

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International Conference on Noise, Vibration and Comfort (NVC 2023) Journal of Physics: Conference Series 2721 (2024) 012016 IOP Publishing doi:10.1088/1742-6596/2721/1/0120163 a) -FRF of the original system.---positive part Equation (2).-.-negative part of Equation (2) (b) modified FRF for damping values c: -0, ---250, …., 500 and -.-750 Nm/s The influence of damping on the modified FRF is shown in Figure 2(b).This figure illustrates the FRF amplitudes of the FRFs corresponding to varying damping coefficients c, namely, 250, 500 and 750 Nm/s.As expected, the amplitude of vibration decreases at the resonance frequencies and at the same time the amplitudes of the FRFs increase in the frequency range between the two new resonances.Furthermore, an important observation with regard to the definition of the optimal damping is that all curves of the modified FRFs intersect at two distinct points corresponding to specific frequency values, denoted by   and   .

Figure 4
FRFs: primary system, linear absorber and nonlinear absorber for 50, 200 and 600 N amplitude of force excitation (a) damping c, 250 Nm/s (b) damping c, 500Nm/s