Multi-harmonic Balance Method Research Articles

Nonlinear model updating of the rotor-bearing system by multi-harmonic balance method and analytical sensitivity derivation

Responses of the bearing system are usually predicted or tested, to analyze its nonlinear dynamic characteristics. However, responses predicted from the bearing model constructed by Hertz contact theory usually show differences with their tested counterparts, denoting the improper setting of the nonlinear contact parameters. It is compulsory to reduce such differences and obtain a precise bearing model using nonlinear model updating techniques. In this paper, a novel nonlinear model updating framework is proposed following the Multi-Harmonic Balance Method (MHBM) and the analytical sensitivity analysis, for updating of the rotor-bearing system using the nonlinear responses in the frequency domain. With the derivation of bearing response sensitivities to the nonlinear contact parameters, increments of the nonlinear parameters can be ascertained by the sensitivity inverse and updating of the nonlinear bearing model can be implemented in an iterative way, until a precise bearing model with the response differences reduced to minima is obtained. Firstly, a novel MHBM model is derived for the bearing system, which is fully described by matrix operations and used to further deduce the analytical sensitivity formulations of the bearing responses to the design parameters. Afterwards, the bearing responses and sensitivities in the frequency domain are predicted by the continuation strategy. Finally, model updating is conducted for the bearing system using the traditional inverse-sensitivity framework. Two cases are used to demonstrate the feasibility of the method. The first case is to update bearing parameters of contact stiffness, clearance and eccentricity of a 2-DOF bearing system in order to match the predicted bearing responses with the measured ones in strong nonlinearity with the multivalued phenomenon. The next is to update the Hertz contact parameters of the left and right bearings, together with the disc eccentricity, of a Finite Element (FE) rotor-bearing system. These results show the validity and superiority of the proposed method.

A semi-analytical multi-harmonic balance method on full-3D contact model for dynamic analysis of dry friction systems

Dry friction damping structures are widely-used in aero-engines to mitigate vibration. The nonlinear nature of friction and the two-dimensional in-plane motion on the contact interface bring challenges to accurately and efficiently predict the forced response of frictionally damped structures. The state-of-the-art Multi-Harmonic Balance Method (MHBM) on quasi-3D contact model in engineering cannot precisely capture the kinematics on the friction interface although the efficiency is high. The full-3D contact model can describe the constitutive relationship of the interface in a more accurate manner; however, the efficiency and convergence are not guaranteed for large-scale models. In this paper, a semi-analytical MHBM on full-3D contact model is proposed. The original Trajectory Tracking Method (TTM) for evaluating the contact force is reformulated to make the calculation more concise and the derivation of the Analytical Jacobian Matrix (AJM) feasible. Based on the chain rule of derivation, the AJM which is the core to upgrade the performance is deduced. Through a shrouded blade finite element model, the accuracy and efficiency of the proposed method are compared with both the MHBM on full-3D contact model with numerical Jacobian matrix and the MHBM on quasi-3D contact model with AJM. The results show that the AJM improves significantly the efficiency of the MHBM on full-3D contact model. The time cost of the proposed method is in the same order of magnitude as that of the MHBM on quasi-3D contact model. We also confirm that the full-3D contact model is necessary for the dynamic analyses of shrouded blades. If one uses the quasi-3D model, the estimation relative error of damping can even reach 31.8% in some cases. In addition, the AJM also brings benefits for stability analysis. It is highly recommended that engineers use the MHBM on full-3D contact model for the dynamic analysis and design of shrouded blades.

Experimental and Numerical Investigations on the Dynamic Response of Blades with Dual Friction Dampers

Friction dampers are widely employed to reduce blade resonance vibration amplitude in turbomachinery. In this paper, a study was performed on the forced response of two blades with dual friction dampers. Numerical simulation and experimental testing were conducted. Firstly, the dynamics of the blade and dual friction damper system assembly are modeled. A nonlinear code based on the multi-harmonic balance method was developed to calculate the resonance response. In this analysis, both the blade and the damper are modeled with the finite element and the matrices reduced with the component mode synthesis method, while the contact forces are modeled with a one-dimensional variable normal load array element. Secondly, a test rig made of two blades and dual friction dampers, the material of which was steel, was established to measure the nonlinear frequency response function curves of the blade system. The results indicate that when a dual friction damper is applied, superior vibration reduction characteristics are demonstrated, with the system exhibiting an average 21% reduction in the response amplitude levels and an increase of 3% in the frequency shifting range compared to a single damper. Dampers positioned at relatively higher locations contribute significantly to the vibration reduction process. In the end, the numerical predictions match very well with the experimental ones.

Model updating of dynamic structures with strong nonlinearities using fixed frequency continuation tests

Model updating using multivalued responses is an important approach to construct strongly nonlinear models. During updating, swept frequency tests are usually used to measure the multivalued responses, but the measured responses only contain the stable branches. Accordingly, updating mainly uses these stable responses for the nonlinear model construction. Unstable responses are rarely tested or participate in the updating process, which may lead to errors in the constructed model with strong nonlinearities. Recently, fixed frequency tests using a continuation scheme have been used to measure fully multivalued responses, including both the stable and unstable branches, for structures with the strong nonlinearity. However, the measured responses were then transformed to equivalent swept frequency tests for the existing updating schemes, where this transformation may induce errors in the updating process. In this paper, a novel method for strong nonlinear model updating using the results of the fixed frequency test is proposed. Such updating is based on the directly measured data of the fixed frequency tests and can handle both the stable and unstable responses. Fixed frequency continuation using the Multi-Harmonic Balance Method (MHBM) is developed to predict the multivalued responses and their sensitivities. The multivalued responses of the prediction and measurement are directly matched in sensitivity-based updating. The proposed method is numerically verified on a 2 DOF nonlinear model. Using the proposed method, the nonlinear model is updated from linear to strongly nonlinear and the updated parameters all converge to the simulated values after 7 iterations. The proposed method is then experimentally validated on a 3 DOF nonlinear system with a non-smooth nonlinearity. A nonlinear model with a 7th order polynomial stiffness represents the equivalent system and is directly updated based on the fixed frequency test results. The updated responses are generally a good fit to the tested responses. Compared with updating using only the stable responses, the proposed updating method achieves higher precision and its updated responses are closer to the tested responses. These results show the validity and superiority of the proposed method.

Array buoys with nonlinear stiffness enhance low-frequency wave attenuation and energy capture

Extraction of energy and elimination of ocean waves at low frequencies are challenges facing current wave energy devices. A recent idea based on reducing the equivalent stiffness has been applied to such devices for low-frequency wave attenuation and energy capture. This study investigates a model of an array of buoys with an additional nonlinear stiffness mechanism to this end. The problems of hydrodynamic interaction between multiple floating bodies and interactions among nonlinear wave structures are solved by a semi-analytical method that combines the eigenfunction matching expansion method with the multi-harmonic balance method. The physical mechanism of the proposed nonlinear system of multiple buoys was explored, and it was found to deliver good performance in terms of power capture and wave elimination due to its “phase control” feature. Bragg resonance occurred in the arrayed buoys, which was not conducive to hydrodynamic efficiency. The properties of the multi-buoy system were evaluated, and it was found to be superior to a single buoy of equal volume. The results of this study indicate that an attached mechanism with nonlinear stiffness can be beneficial both for exploiting wave energy and reducing transmitted waves.

A comparative study on multi- and variable-coefficient harmonic balance methods for quasi-periodic solutions

Quasi-periodic solutions widely exist in nonlinear dynamical systems, which are the one of the most important dynamical behaviors but much more difficult to obtain full- or semi-analytically than equilibria and periodic solutions. Recently, multi-harmonic balance method (MHBM) and variable-coefficient harmonic balance method (VCHBM) have been proposed to determine the quasi-periodic solutions semi-analytically. In this paper, after a brief review on the two methods, that is, MHBM uses a multi-dimensional Fourier series and VCHBM a variable-coefficient Fourier series, the relationship between them is unveiled particularly for the case of quasi-periodic solutions consisting of two frequency components. The transform matrices between the algebraic equations of the Fourier coefficients in the two methods are derived. Furthermore, a new formulation for alternating Frequency-Time method (AFT) and a phase condition for arc-length continuation method (ALC) in VCHBM are proposed to make the method more efficient and robust. Through the application of two examples, it is found that when the two methods choose equal harmonic order and thus have the same number of unknowns, MHBM demonstrates higher computational efficiency but VCHBM shows more robust and accurate, especially for the quasi-periodic solutions in the nonlinear dynamical systems with a single-frequency excitation. This paper provides insight views into the features of MHBM and VCHBM..

Nonlinear wave energy dissipator with wave attenuation and energy harvesting at low frequencies

It is important to determine how to reduce the wave load and utilize the ocean energy through the hybrid wave attenuation and energy harvesting structure (WAAEHS) as it is the basic scientific problem in the field of ocean engineering. The existing linear hybrid devices have poor performance due to the mismatch between the natural frequency and the excitation frequency in the low frequency region. The method of increasing the mass is unreliable and low economic, and the introduction of nonlinear mechanism to reduce the stiffness has complex fluid-structure coupling problem between waves and structures. In this paper, a new type of negative stiffness mechanism with a simple structure and good stability has been proposed and applied to a hybrid WAAEHS, and a nonlinear frequency domain method combining eigenfunction expansion matching method(EEMM)and multi-harmonic balance method(MHBM) has been proposed to solve the nonlinear wave–structure interaction dynamic model. The influence of the key parameters of the mechanism on the wave attenuation and energy harvesting performance is examined, and the “phase control” mechanism of the negative stiffness mechanism is revealed to improve the low-frequency wave attenuation and energy harvesting performance.

A frequency-domain reduced order model for joints by hyper-reduction and model-driven sampling

The dynamic behavior of jointed assemblies exhibiting friction nonlinearities features amplitude-dependent dissipation and stiffness. To develop numerical simulations for predictive and design purposes, macro-scale High Fidelity Models (HFMs) of the contact interfaces are required. However, the high computational cost of such HFMs impedes the feasibility of the simulations. To this end, we propose a model-driven method for constructing hyper-reduced order models of such assemblies. Focusing on steady-state analysis, we use the Multi-Harmonic Balance Method (MHBM) to formulate the equations of motion in frequency domain. The reduction basis is constructed through solving a set of vibration problems corresponding to fictitious interface conditions. Subsequently, a Galerkin projection reduces the order of the model. Nonetheless, evaluating the nonlinear forces is prohibitively expensive given the necessary fineness of the discretization of the interfaces. For this reason, we implement an adapted Energy Conserving Weighing and Sampling (ECSW) technique for Hyper Reduction (HR), by which frictional contact nonlinearities are evaluated on a substantially smaller set of weighted elements, thereby allowing significant speedups for meshes of arbitrary fineness. This feature is particularly advantageous since analysts typically encounter a trade-off between accuracy and computational cost when deciding on the mesh size, whose estimation is particularly challenging for problems of this type. To assess the accuracy of our method without resorting to the HFM solution, we propose an error indicator with thresholds that have proven reliable in our analyses. Finally, the accuracy and efficiency of the method are demonstrated by two case studies.

Low-frequency energy capture and water wave attenuation of a hybrid WEC-breakwater with nonlinear stiffness

Wave energy is attractive for its generous, sustainable and clean, the combination of floating breakwater (FB) and wave energy converter (WEC) is an economical approach to capture wave energy and attenuate waves. The conventional hybrid WEC-FB system is ineffective for low-frequency waves. To overcome this shortcoming, this paper proposed a novel WEC-FB with nonlinear stiffness mechanism to improve its wave attenuation and power capture performance. A hybrid semi-analytical nonlinear frequency-domain approach including the eigenfunction expansion matching method (EEMM) and multi-harmonic balance method (MHBM) is proposed to solve the nonlinear waves-structures interaction problem. The performance of the nonlinear WEC-FB is demonstrated by comparing with conventional linear counterpart, and the underlying reason for the effect of nonlinear mechanism is explored. The phase control mechanism for improving low frequency performance of wave elimination and energy capture is revealed by using the analytical method. This study shows that the introduction of the nonlinear mechanism can both improve the wave attenuation and energy absorption performance, especially in the low-frequency wave region.

Forced Response Vibration Analysis of the Turbine Blade with Coupling between the Normal and Tangential Direction

This paper is about the nonlinear dynamic of turbine blades due to the friction between blades. This paper investigates the coupling effect in an experimental turbine blade model. In this contact model, coupling between friction in tangential and normal directions and energy dissipation in the lateral direction are considered simultaneously, which has not been considered elsewhere in the contact model of the turbine blade. The mathematical model of the blade under-platform damper model is derived, and to solve nonlinear equations, the multi-harmonic balance method is used. The contact force is calculated by the alternative frequency time-domain method in this solution method. To solve nonlinear derived algebraic equations, a continuation algorithm is applied. It is shown that considering the coupling phenomenon causes the results to be different from the situation in which this physical phenomenon is ignored. To validate the algorithm of the solution, numerical results of previous references in the turbine-blade field are regenerated. Experimental analysis on the under-platform damper and turbine blade model is done to investigate the coupling effect. It is shown that the contact model with consideration with coupling effect between tangential and normal direction can predict experimental results (amplitude and frequency of resonance) most of the other contact models used in the turbine field. To accurately determine the amplitude and frequency of resonance, it is necessary to consider the coupling effect.

Analytical investigation on wave attenuation performance of a floating breakwater with nonlinear stiffness

Floating breakwaters play an important role in protecting harborage and marine structures from wave impact. Traditional floating breakwaters (FBs) are not applicative for their ineffectiveness in low frequency waves attenuation. Based on the idea of reducing equivalent stiffness, this paper proposes a FB with nonlinear stiffness to improve the wave attenuation performance. The eigenfunction expansion matching method (EEMM) and multi-harmonic balance method (MHBM) are corporately applied to solve the new model involving nonlinear coupling problem of water waves and floating structures. A pile restrained FB with quasi-zero-stiffness (QZS) is taken as an example to verify the feasibility of the hybrid analytical model and to reveal the wave attenuation mechanism of the nonlinear breakwater. The comparison between nonlinear FB and linear FB shows that nonlinear FB can effectively widen the attenuation frequency bandwidth. The conception of nonlinear floating breakwater proposed in this study provides a new technical idea for attenuating low frequency waves and the hybrid solution method adopted in this paper may be extensible to other wave-structure nonlinear coupling problems.

On the Performance of Wavy Dry Friction and Piezoelectric Hybrid Flexible Dampers

Abstract This paper proposes a flexible dry friction plate to mitigate the vibration of thin-walled structures for one resonance crossing. Based on a cantilever beam–friction damper finite element model, the geometry and material parameters of the friction plate are optimized numerically through steady-state response analyses by the widely used multiharmonic balance method (MHBM). In order to further improve the damping effect, piezoelectric material is distributed to the flexible damper, and two types of dry friction and piezoelectric hybrid dampers are explored, namely, semi-active and passive, respectively. For semi-active hybrid dampers, piezoelectric material is used as an actuator to adjust the normal load applied to the friction interface in real-time, so that the friction damping is improved. For passive ones, piezoelectric material is used as a transducer, which dissipates the strain energy stored in the wavy plate by the shunting circuit, additional shunted piezoelectric damping contributes to the total output damping accordingly. Better damping effect compared with the friction baseline is realized for the two types ideally. This damping module has a simple structure and avoids the problem of installation and maintenance of piezoelectric material which is generally bonded to the host structure. Technical challenges are the semi-active type requires excessive voltage applied to the piezoelectric actuator, while the passive one needs to connect a programable synthetic circuit.

Nonlinear vibration analysis of forced response for rubbing problems using the automatic differential frame

The multi-harmonic balance method is widely applied to obtain the forced responses of nonlinear systems undergoing rubbing problems. Despite large-scale time savings compared with the time marching method, it suffers from the complicated derivations of the Jacobian matrix. To solve this problem, this paper focuses on applying the automatic differentiation frame to the multi-harmonic balance method to implement the nonlinear vibration analysis of systems subjected to the rub phenomena. By establishing computational graph and utilizing the automatic differentiation process, tedious works such as the derivations of the complicated analytical expressions of the Jacobian matrix are avoided, which guarantees the efficiency and applicability of the presented method. A single-degree-of-freedom system with nonlinear force in the form of cubic is used to verify the accuracy of the method, and numerical analysis results reveal that the method is accurate enough compared with the time marching method. Furthermore, for the purpose of application, the responses of two common friction models, which are of great concern in some practical engineering fields, including a two-degree-of-freedom system containing a friction damper and a rotor disk system with circumferential rubbing, are obtained utilizing the presented approach. The influences of several model parameters on their responses are investigated as well. Numerical investigations demonstrate that the automatic differential solution framework developed in this research for solving nonlinear vibration equations has high accuracy and eliminates the need for a complicated partial derivative analytical formula derivation.

Frequency-domain nonlinear model updating based on analytical sensitivity and the Multi-Harmonic balance method

The predicted dynamic response of the structure with nonlinearity can be in some difference from the experimental one due to the discrepancies of nonlinear parameters between the constructed model and the real structure. The gradient-based nonlinear model updating procedures is an effective tool to reduce the difference. However, the computational costs for the sensitivity matrix could be too expensive. In this paper, a novel approach of the frequency-domain nonlinear model updating using the analytical sensitivity and the Multi-Harmonic Balance Method (MHBM) is proposed. The analytical sensitivity, directly derived from the frequency domain algebraic function obtained by the MHBM and Gamma matrix-based DFT-AFT methods, can significantly reduce the iteration time in model updating process. The proposed nonlinear model updating procedure is easily carried out and also considered as the FRF-based model updating framework. To illustrate the method, a simulated 3DOF nonlinear structure is utilized and differences between the updated nonlinear parameters and the original ones are reduced below 0.4% after updating. Besides, the proposed updating procedure has the best performance compared with the numerical method and the Nelder-Mead optimization. The simulation shows that the proposed analytical sensitivity calculation is generally above 4 times faster than the numerical one. Finally, the updating process of a real 3DOF nonlinear lumped parameter structure is described in detail. After the updating, the overlay of the prediction responses and the experimental ones is in a good agreement. Results indicate the efficiency and superiority of the proposed method to update nonlinear structures.

Analysis of the Vibration Suppression of Double-Beam System via Nonlinear Switching Piezoelectric Network

In this paper, a method to suppress the vibration of a double-beam system with nonlinear synchronized switch damping on the inductor via a network (SSDI-net) is proposed. Unlike the classical linear piezoelectric shunt damping, SSDI-net is a nonlinear piezoelectric damping. A double-beam system with SSDI-net was simplified to a lumped parameter electromechanical coupling model and analyzed by using the multi-harmonic balance method, at first with alternating frequency–time techniques (MHBM/AFT). Then, a new lower-power autonomous switching control circuit board was designed, based on SSD technique, and vibration control experiments using a double-beam system with an SSDI network are conducted, to verify the validity of the proposed analysis method and its calculation results. The nonlinear switching piezoelectric network proposed in this article can increase the voltage inversion factor. Furthermore, future applications of this switching piezoelectric network technology in the vibration suppression of bladed-disk structures in aero engines can reduce the number of switches by at least half and obtain almost the same damping effect.

A new numerical approach to estimate stress intensity factors under very high cycle fatigue testing

Ultrasonic very high cycle fatigue testing finds many advantages in fatigue studies especially in very low fatigue crack growth rate investigations. In this application, the determination of the stress intensity factor has a key role and presenting a general approach to calculate stress intensity factor in reasonable time and ultimate accuracy is necessary. The present research proposes a new numerical approach (hybrid method) based on reduce order modeling (dynamic substructuring), multi-harmonic balance method, and M-integral to evaluate stress intensity factors under very high cycle fatigue testing in the most accurate and fast way possible. The verification of the hybrid method was done through a benchmark and implicit solver implemented in ABAQUS commercial software as the reference solution. Investigations proved that the presented hybrid method here could determine stress intensity factor in ultrasonic regime with the ultimate accuracy and save computational time at least 50%.

Topological Optimization of Under-Platform Dampers With Moving Morphable Components and Global Optimization Algorithm for Nonlinear Frequency Response

Abstract To limit the risk of high cycle fatigue, underplatform dampers (UDPs) are traditionally used in aircraft engines to control the level of vibration. Many studies demonstrate the impact of the geometry of the damper on its efficiency, thus the consideration of topological optimization (TO) to find the best layout of the damper seems natural. Because of the nonlinear behavior of the structure due to the friction contact interface, classical methods of TO are not usable. This study proposes to optimize the layout of an UDP to reduce the level of nonlinear vibrations computed with the multiharmonic balance method (MHBM). The approach of TO employed is based on the moving morphable components (MMC) framework together with the Kriging and the efficient global optimization algorithm to solve the optimization problem. The results show that the level of vibration of the structure can be reduced to 30% and allow for the identification of different efficient geometries.

Modeling Complex Contact Conditions and Their Effect on Blade Dynamics

Abstract Dry friction devices such as underplatform dampers are commonly included in turbine bladed disks designs to mitigate structural vibrations and avoid high cycle fatigue failures. The design of frictionally damped bladed disks requires adequate models to represent the friction contact. A widely used approach connects contact node pairs with normal and tangential springs and a Coulomb friction law. This simple model architecture is effective in capturing the softening behavior typically observed on frictionally damped structures subjected to increasing forcing levels. An unexpected hardening behavior was observed on the frequency response functions (FRFs) of a two-blades-plus-damper system tested by the authors in a controlled laboratory environment. The reason behind this unexpected behavior will be carefully analyzed and linked to the damper kinematics and to the dependence of contact elasticity on the contact pressure. The inadequacy of contact models with constant spring values will be discussed and alternatives will be proposed. The importance of being able to represent complex contact conditions in order to effectively predict the system dynamics is shown here using a laboratory demonstrator; however, its implications are relevant to any other case where large contact pressure variations are to be expected. The nonlinear steady-state simulations of the blades-plus-damper system will be carried out using an in-house code exploiting the multiharmonic balance method in combination with the alternating frequency time method.

Design of wave-like dry friction and piezoelectric hybrid dampers for thin-walled structures

This paper proposes a wave-like hybrid damper containing both dry friction and piezoelectric damping mechanisms for thin-walled structures. The idea is to distribute piezoelectric material on the wave-like friction plate, so that the elastic deformation of the plate can be further utilized to generate additional shunted piezoelectric damping. The proposed damper has the following advantages: simple structure, easy installation and maintenance, convenient to adjust the normal preload, and feasible to mount piezoelectric materials. Combined with damped nonlinear normal modes (dNNMs), the nonlinear modal electromechanical coupling factor (nonlinear MEMCF) is proposed, and its relationship with piezoelectric damping is established for electromechanically coupled systems with contact nonlinearities. Based on the extended periodic motion concept (E-PMC) of dNNMs for non-conservative systems, the preliminary design guidelines of wave-like hybrid dampers are given through nonlinear modal analyses. The modal damping ratio and the nonlinear MEMCF are used to evaluate the damping generated by friction and piezoelectric mechanisms, respectively. The damping effect is also verified by steady-state response analyses through the Multi-Harmonic Balance Method (MHBM). By using a cantilever beam finite element (FE) model, the spatial distribution of piezoelectric material is optimized for a single wave-like hybrid damper. Four optimized hybrid dampers are then implemented to an industrial thin-walled structure. Taking the normal preload as an example, the influence of parameter distribution patterns at different interfaces on the damping effect is also discussed. Compared with the underlying friction damper, the wave-like hybrid damper not only provides greater damping, but is also less sensitive to the variation of excitation amplitude and the normal preload. Whether friction or piezoelectric damping is inactive, the proposed damper can still generate considerable damping.

Upgraded Reduction Technique for Dynamic Analysis of Structures with Friction Contact

A recent reduction technique for the nonlinear forced response analysis of structures with contact interfaces is upgraded. The evaluations of the existing method (Dual formulation), which is based on dual Craig–Bampton method, show its considerable off-line computational time saving because of no matrix manipulation, whereas the resulting reduced model is less accurate than Rubin’s in approximating the full finite element model. The accuracy of this formulation is significantly improved with a low additional computational cost in the proposed upgraded formulation. That is obtained by consistent projection of all structural matrices on a reduction basis made up of two sets of vectors: free-interface normal modes and residual flexibility attachment modes. Assembly approach can be considered as dual because the interface forces are kept as generalized coordinates in opposition to primal methods, where interface displacements are retained. To reduce the computational cost, the nonlinear governing equations are also solved in frequency domain using the multiharmonic balance method and alternating frequency time method. Contact elements are introduced between adjacent interface nodes to find the nonlinear contact forces. The performance of the new formulation is demonstrated through two numerical models with friction contact.