Design of dry friction dampers for thin-walled structures by an accelerated dynamic Lagrange method
Introduction
Thin-walled structures are predominantly used in aerospace/aeronautical engineering [1,2], automotive industries [3] and various other industries [4,5], due to their advantages of excellent mechanical performance, light weight and high manufacturability. Inevitably, the widely adoption of these components also causes the vibration of the structure more serious, leading to the attenuation of the structure performance and even failure [6,7]. Therefore, an effective technique to reduce the vibration has become more and more important for the reliable application of the thin-walled structures.
Dry friction dampers are recognized as one of the most effective way for vibration reduction due to their high damping performance, reliability and insensitivity to the temperature [[8], [9], [10]]. Vibration energy is dissipated due to the relative displacement at contact interfaces, which are generally held in contact under a normal preload. Different kinds of dry friction dampers are designed for diverse structures. Applications for bladed disks assemblies include underplatform dampers [[11], [12], [13], [14]], shrouds [[15], [16], [17], [18]], blade roots [[19], [20], [21]], ring dampers [7,22], etc. For trusses, frames and thin-walled structures, friction from installation surfaces is supposed to supply over 90% of the passive damping capacity [23]. However, it is important to keep the structure strong enough by increasing the clamping force, which will make the internal damping of the structure very low instead [[24], [25], [26]]. To take this issue, damping enhancing devices are proposed, such as Friction Damped Braced Frame (FDBF) device for trusses and frames [27] and friction sleeves or rings for cylindrical shells [28]. The damping generated by the frictional damping devices are , and with the additional mass of , and respectively [28]. It is necessary to balance the damping effect and the weight. These damping enhancing devices are only suitable for specific conditions, and the application of dry friction dampers for thin-walled structures were seldom studied.
There are several design parameters for dry friction dampers. The normal preload applied to the contact interfaces is an essential design parameter [29,30]. Under a specific excitation, the optimal normal preload exists to minimize the resonant amplitude. In addition, some studies [[31], [32], [33], [34], [35]] also show that the damping performance can be further improved by providing time-varying preload, which is called semi-active dry friction damper [36]. However, the control law requires several auxiliary equipment, which makes the semi-active dry friction damper not profitable for thin-walled structures. Another way is to modify the design parameters (including geometry, material, location, etc.) to improve the passive damping. Epurean et al. [37] applied sensitivity analyses to design V-shaped ring dampers, and the damping performance was enhanced significantly by geometric optimization. In the work of Petrov et al. [38], comparison of split dampers and cottage-roof dampers was carried out through steady-state response of a bladed disk. Gastaldi et al. [39] showed that geometric parameters had a great influence on the performance of friction dampers, and a pre-optimization work-flow was summarized for under-platform dampers to reduce the design space.
In the design phase, it is important to develop a nonlinear dynamic model with contact interfaces and relevant solving algorithm in order to evaluate the effectiveness of dry friction dampers. Although the vibration response of the structure can be obtained through a time integration method [40,41], performing optimization design through shape, material and location is still a great challenge. Especially when running a large scale finite element (FE) models with relatively small damping, this method always leads to a prohibitive computational cost. Therefore, various methods have been developed to obtain the steady-state response of structures with dry friction dampers attached. The harmonic balance method (HBM) and its variation, such as the multi-harmonic balance method (MHBM) have been employed extensively. These methods in the frequency domain can reduce the computational cost significantly [[42], [43], [44]]. An alternating frequency-time (AFT) domain method is introduced by Griffin [45], which overcomes the drawback of the MHBM (hard to express the contact states analytically with over 3 harmonics retained) by obtaining the nonlinear contact force in the time domain. Based on this method, significant advances in response predictions have been accomplished over the last years. Poudou et al. [40] applied the AFT-MHBM to the study on the vibration of a mistuned bladed disk, where the hysteresis spring contact model is used to describe the contact states in the time domain. In the work of Chen et al. [46], the rationality of the proposed three-dimensional contact model is validated by the AFT-MHBM. Nacivet et al. [47] presented the Dynamic Lagrange Frequency/Time (DLFT) method to calculate the steady-state response of a blisk with ring dampers. The advantage is that hysteresis springs become unnecessary for contact elements. The simulation results of these two methods were verified by experiments to show their accuracy [12,48]. In addition, Joannin et al. [49] developed a nonlinear model reduction technique to calculate the response of bladed disk in a wide frequency range. However, all these methods may suffer convergence problems when an iteration process (like Newton-Raphson algorithm) is used to solve the strong nonlinear algebraic equations, especially for large-scale systems. One reason is that the error of the Jacobian matrix is always obtained by the cumbersome finite difference process, leading to poor convergence performance of the Newton-Raphson algorithm [50,51].
In this study, we propose a new dry friction damper which can be used for general thin-walled structures. The friction patch is designed to reduce the vibration of the target mode. It operates as a damper rather than a stiffener since it works in the sliding state most of the time under the optimal normal preload. The damping performance of the damper is evaluated by the steady-state forced response of the structure under the optimal normal preload, which is calculated by a velocity-based DLFT method [52]. This method has the following advantages: the results are insensitive to penalty coefficients, and no iterative evaluation of the contact law is required in the time domain [53]. Inspired by the analytical Jacobian matrix applied to displacement-based AFT-MHBM [54,55], we propose another form of the matrix which significantly improves the convergence of the velocity-based DLFT method. The global parameter studies become feasible thanks to its computational efficiency. The design guidelines from numerical studies are verified experimentally. Finally, the well-designed dry friction dampers are implemented to a finite element model of an industrial thin-walled structure to illustrate its damping performance.
This paper is arranged as follows. Section 2 dedicates to the numerical tools in terms of forced response simulation for dry friction systems. In Section 3, two damping structures are compared. Parameter studies and experimental verification are also carried out in this section. Section 4 performs an industrial application.
Section snippets
Numerical method: response prediction for dry-friction-damped structures
The forced response of a system with frictional contact interfaces under periodic excitation is investigated in this section. The complex motion at the contact interfaces is considered, including variable normal force and possible separation. The velocity-based dynamic Lagrange frequency-time (DLFT) method proposed by Laxalde [52] is adopted. Compared with other numerical methods in the frequency domain, the results obtained by the DLFT are insensitive to penalty coefficients, and no iterative
Model description
Two different types of friction patches are shown in Fig. 5. For type 1 shown in Fig. 5a, the mounting surface of the friction damper is located at the center of the patch, and two contact interfaces are symmetrically distributed on both sides. For type 2, the friction patch is installed asymmetrically with only one contact interface. For both cases, the friction patch is fixed with the host structure tightly by the bolt and nuts to avoid relative displacement on the non-target contact surface
Application
In this section, the friction patches are applied to a more complex thin-walled structure which is widely used in engineering applications to further study the effectiveness of the damper.
Conclusion
In this paper, a dry friction patch for complex thin-walled structures is proposed. In order to carry out the steady-state response analysis of the friction-damped system more efficiently, the analytical Jacobian matrix of the velocity-based DLFT is derived. The simulation time of the advanced method has been significantly reduced compared to the DLFT with the Jacobian matrix estimated by the finite difference method. It is also more efficient than the EFT method which is widely-used.
Based on
CRediT authorship contribution statement
H.Y. Ma: Methodology, Software, Investigation, Writing - original draft. L. Li: Supervision, Project administration, Funding acquisition. Y.G. Wu: Methodology, Software, Investigation, Writing - original draft, Writing - review & editing, Visualization. Y. Fan: Conceptualization, Supervision, Resources, Writing - review & editing, Project administration. Q. Gao: Visualization, Validation.
Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgment
This work is funded by National Science Foundation of China (grant Nos. 51675122 and 11702011).
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