Closed-form solutions for the optimal design of inerter-based dynamic vibration absorbers

Barredo, Eduardo; Ortega, Andrés Blanco; Colín, J.; Penagos, Victor M.; Abúndez, Arturo; Vela, L.G.; Meza, Virgilio; Cruz, Roller H.; Mayén, Jan

doi:10.1016/j.ijmecsci.2018.05.025

Cited by 102 publications

(77 citation statements)

References 25 publications

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“…With the knowledge of α , β , ν , and h , the mechanical damping ratio can be evaluated by transforming Equation into the following form:

ξ_{3} = \sqrt{- \frac{A - h C}{B - h D}},

where the only unknown parameter is the frequency λ , which should be properly chosen. According to the extended fixed points technique, the optimal ξ 3 should be imposed as the RMS value of damping levels evaluated at three reference frequencies λ 1 , λ 2 , and λ 3 . According to Krenk and Hfigsberg, the reference frequencies correspond to the real eigenvalues of two particular dynamical systems,

{(|, G)}_{ξ_{3} \to \infty}

and

{(|, G)}_{ξ_{3} = 0}

.…”

Section: Optimization Of G‐sdtmdi Under Harmonic Vibrationmentioning

confidence: 99%

“…The characteristic equation relevant to

{(|, G)}_{ξ_{3} \to \infty}

is a quadratic polynomial in λ 2 , ie,

P_{1} false(λ^{2} false) = ((), μ + η + 1) {((), λ^{2})}^{2} - 2 ((), μ + η + 1) λ^{2} + 1 = 0

from which two reference frequencies can be obtained, as follows:

λ_{1}^{2} = 1 - \sqrt{\frac{μ + η}{μ + η + 1}}, λ_{2}^{2} = 1 + \sqrt{\frac{μ + η}{μ + η + 1}} .

And the eigenvalue of

{(|, G)}_{ξ_{3} = 0}

should satisfy the following expression of order 3 in λ 2 :

P_{2} false(λ^{2} false) = {((), γ + 1)}^{2} {((), λ^{2})}^{3} - ((), γ + 1) ((), 4 γ + 3) {((), λ^{2})}^{2} + ((), γ + 1) ((), 2 γ + 3) λ^{2} - 1 = 0,

where an intermediate variable γ = μ + η is introduced in order to facilitate the curve fitting process, which stands for the sum of the total mass ratio and the total inertance‐to‐mass ratio. Clearly, Equation has three possible roots; however, only the one between λ 1 and λ 2 is chosen as the third reference frequency λ 3 . Although the exact solutions to the roots of P 2 ( λ 2 ) could be analytically derived, their formulae are extremely cumbersome and irreducible.…”

Section: Optimization Of G‐sdtmdi Under Harmonic Vibrationmentioning

confidence: 99%

“…Apparently, the classic fixed points theory could not be directly employed, as it is dedicated to systems characterized by two fixed points. Nevertheless, the extended fixed points technique 34 can efficiently tackle the optimization problem of systems featured by four invariant points, which will be used to optimize both SDTMDIs hereafter. In practice, the total mass ratio and the total inertance-to-mass ratio are predefined at the initial phase of design.…”

Section: Figurementioning

confidence: 99%

“…Clearly, Equation (27) has three possible roots; however, only the one between 1 and 2 is chosen as the third reference frequency 3 . 34 Although the exact solutions to the roots of P 2 ( 2 ) could be analytically derived, their formulae are extremely cumbersome and irreducible. Therefore, only the curve-fitted solution is retained, which is in the form of linear polynomial in :…”

Section: Optimal Solution Tomentioning

confidence: 99%

“…By vanishing the inertance, ie, = 0, the expressions of h, , , and in Equations (34) to (37) are exactly the same as optimal parameters of SDTMD (given in Table 1). Again, the optimal mechanical damping ratio 3 is chosen as the RMS value of three mechanical damping ratios evaluated at reference frequencies, which are the real roots of characteristic equations of G | 3 →∞ and G| 3 =0 , respectively:…”

Section: Optimization Of I-sdtmdi Under Harmonic Vibrationmentioning

confidence: 99%

See 4 more Smart Citations

Influence of inerters on the vibration control effect of series double tuned mass dampers: Two layouts and analytical study

Zhou

Jean‐Mistral

Chesné

2019

Struct Control Health Monit

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Summary In this paper, two configurations of series double tuned mass damper with inerters (SDTMDI) are investigated: grounded type (G‐SDTMDI) and inserted type (I‐SDTMDI). For the former layout, two inerters relate each tuned mass to the ground, while they are inserted between any two adjacent masses in the latter layout. Implementing on a harmonically forced primary system of single degree of freedom, their optimal design is conducted by using an extended version of fixed points theory. The present study suggests that for a given total tuned mass, the vibration control effect of G‐SDTMDI is solely controlled by the total inertance, because of which one of the two grounded inerters can be removed, and the vibration control performance will not be influenced under the condition that the amount of inertance remains unchanged. Meanwhile, the control effect delivered by I‐SDTMDI is dictated by the inerter relating the primary system and its adjacent tuned mass. Finally, it is analytically and numerically proven that grounded inerters can render the vibration control effect more pronounced, featured by a decreased peak amplitude and an enlarged frequency bandwidth of vibration reduction. The only potential detrimental effect observed is by incorporating an inerter between the host structure and its adjacent tuned mass. In this case, the vibration amplitude of host structure is amplified, and the effective plateau of vibration reduction is narrowed.

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“…With the knowledge of α , β , ν , and h , the mechanical damping ratio can be evaluated by transforming Equation into the following form:

ξ_{3} = \sqrt{- \frac{A - h C}{B - h D}},

{(|, G)}_{ξ_{3} \to \infty}

and

{(|, G)}_{ξ_{3} = 0}

.…”

Section: Optimization Of G‐sdtmdi Under Harmonic Vibrationmentioning

confidence: 99%

“…The characteristic equation relevant to

{(|, G)}_{ξ_{3} \to \infty}

is a quadratic polynomial in λ 2 , ie,

P_{1} false(λ^{2} false) = ((), μ + η + 1) {((), λ^{2})}^{2} - 2 ((), μ + η + 1) λ^{2} + 1 = 0

from which two reference frequencies can be obtained, as follows:

λ_{1}^{2} = 1 - \sqrt{\frac{μ + η}{μ + η + 1}}, λ_{2}^{2} = 1 + \sqrt{\frac{μ + η}{μ + η + 1}} .

And the eigenvalue of

{(|, G)}_{ξ_{3} = 0}

should satisfy the following expression of order 3 in λ 2 :

P_{2} false(λ^{2} false) = {((), γ + 1)}^{2} {((), λ^{2})}^{3} - ((), γ + 1) ((), 4 γ + 3) {((), λ^{2})}^{2} + ((), γ + 1) ((), 2 γ + 3) λ^{2} - 1 = 0,

Section: Optimization Of G‐sdtmdi Under Harmonic Vibrationmentioning

confidence: 99%

Section: Figurementioning

confidence: 99%

Section: Optimal Solution Tomentioning

confidence: 99%

Section: Optimization Of I-sdtmdi Under Harmonic Vibrationmentioning

confidence: 99%

See 3 more Smart Citations

Influence of inerters on the vibration control effect of series double tuned mass dampers: Two layouts and analytical study

Zhou

Jean‐Mistral

Chesné

2019

Struct Control Health Monit

View full text Add to dashboard Cite

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Optimal design of a novel high‐performance passive non‐traditional inerter‐based dynamic vibration absorber for enhancement vibration absorption

Baduidana

Kendo‐Nouja

Kenfack-Jiotsa

et al. 2022

Asian Journal of Control

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This paper proposes a novel high‐performance passive non‐traditional inerter‐based dynamic vibration absorber (NIDVA), tuned by the H∞ optimization using the patternsearch solver in MATLAB with the starting points, the optimum parameters based on the extended fixed points theory (EFPT). Then, the control performance of the proposed NIDVA is compared with the classic DVA and the high‐performance passive non‐traditional inerter‐based dynamic vibration absorber (NIDVA‐C4). Based on the optimum parameters under harmonic force excitation, the relative dynamic performance index (%RDP) of the proposed NIDVA registered improvements of 31%–63% and 54%–72% with respect to the NIDVA‐C4 and classic DVA, respectively, considering the mass ratio range from 1% to 10%. Regarding the suppression bandwidth index (%SB) for the mass ratio of 5%, the NIDVA can enlarge the SB by 50% and 62% with respect to NIDVA‐C4 and classic DVA, respectively. For the white noise excitation, more than 38%–64% and 53%–69% improvement of random vibration reduction can be obtained from NIDVA with respect to NIDVA‐C4 and classic DVA, respectively. Accordingly, results of this study provide some technical information to design more effective vibration control device in engineering practices.

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Dynamic performance of structures with clutching inertia‐damping system: Theoretical study

Liang,

Bai,

et al. 2023

Earthq Engng Struct Dyn

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This paper addresses performance limitations in passive clutching inertia devices due to the absence of flywheel deceleration damping. A solution introducing viscous damping to serve as the flywheel deceleration mechanism is proposed, thus forming the “CIDS” (Clutching Inertia‐Damping System). The study establishes a multi‐body motion analysis model for the CIDS within a single‐degree‐of‐freedom structure. Simulations investigate the impact of control force model parameters on the computation. The impact of additional damping on structural dynamic responses and the system's relative motion is thoroughly elucidated and revealed through dynamic simulations and multi‐body motion decomposition. The “ELS” (Equivalent Linear System) is introduced for the simplified performance assessment of CIDS. Expressions for equivalent damping and inertance are derived based on the equivalence of vibration period and energy in sinusoidal steady‐state vibration. The feasibility and accuracy of ELS for assessing CIDS's performance are validated through response analyses under harmonic and seismic excitations. Notably, displacement indicators exhibit higher precision than acceleration. When subjected to seismic excitation, selecting a more suitable excitation frequency to determine the corresponding equivalent damping can enhance the accuracy of ELS. Considering the frequency dependence of equivalent damping, a frequency‐domain‐based random response evaluation method is proposed. The influence of additional rotational damping, flywheel inertia, and structural natural period on random response is evaluated and analyzed. In CIDS with similar rotational damping and inertia, its control effect is related to the excitation. In summary, this study offers a structured framework for comprehending and evaluating the performance of CIDS, offering valuable insights into its potential engineering design, applications, and analysis.

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Closed-form solutions for the optimal design of inerter-based dynamic vibration absorbers

Cited by 102 publications

References 25 publications

Influence of inerters on the vibration control effect of series double tuned mass dampers: Two layouts and analytical study

Influence of inerters on the vibration control effect of series double tuned mass dampers: Two layouts and analytical study

Optimal design of a novel high‐performance passive non‐traditional inerter‐based dynamic vibration absorber for enhancement vibration absorption

Dynamic performance of structures with clutching inertia‐damping system: Theoretical study

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