Efficient Energy Balancing Across Multiple Harmonics of Nonlinear Normal Modes

Predicting the forced responses of nonlinear systems is a topic that attracts extensive studies. The energy balancing method considers the net energy transfer in and out of the system over one period, and establishes connections between forced responses and nonlinear normal modes (NNMs). In this paper, we consider the energy balancing across multiple harmonics of NNMs for predicting forced resonances. This technique is constructed by combining the energy balancing mechanism with restrictions (established via excitation scenarios) on external forcing and harmonic phase-shifts; a semi-analytical framework is derived to achieve both accurate/robust results and efficient computations. With known inputs from NNM solutions, the required forcing amplitudes to reach NNMs at resonances, along with their discrepancy, i.e. the harmonic phase-shifts, are computed via a one-step scheme. Several examples are presented for different excitation scenarios to demonstrate the applicability of this method, and to show its capability in accurately predicting the existence of an isola where multiple harmonics play a significant part in the response.

Data-Driven Approach for Identifying Mistuning in As-Manufactured Blisks

Sector-to-sector geometry or material property variations in as-manufactured bladed disks, or blisks, can result in significantly greater vibration responses during operation compared to nominally cyclic symmetric designs. The dynamics of blisks are sensitive to these unavoidable deviations, known as mistuning, making the identification of mistuning in as-manufactured blisks necessary for accurately predicting their vibration. Previous approaches to identify such mistuning parameters often require the identification of modal information or blade-isolation techniques such as blade detuning using masses or adding damping pads. However, modal information can be difficult to obtain accurately even in optimal bench conditions. Additionally, in practice it can be difficult to isolate individual blades by restricting blade motion or detuning individual blades through added masses due to geometric constraints. In this paper, we present a method for mistuning identification using a data-driven approach based on a neural network. Here, mistuning in all sectors of blisks with the same nominal geometry can be identified by using a small number of forced responses and the forcing phase information from traveling-wave excitation. In this approach, no system or sector-level modal response information, restrictive blade isolation, or mass detuning are required. Validation of this approach is presented using a finite element blisk model containing stiffness mistuning within the blades to create computationally generated surrogate data. It is shown that mistuning can be predicted accurately using forced responses containing a significant amount of absolute and relative measurement noise, mimicking responses collected from experimental measurements.

Existence of quasi-periodic responses in quasi-periodically forced nonlinear mechanical systems

Forced responses of mechanical systems are crucial design and performance criteria. Hence, their robust and reliable calculation is of vital importance. While numerical computation of periodic responses benefits from an extensive mathematical basis, the literature for quasi-periodically forced systems is limited. More specifically, the absence of applicable and general existence criterion for quasi-periodic orbits of nonlinear mechanical systems impedes definitive conclusions from numerical methods such as harmonic balance. In this work, we establish a general existence criterion for quasi-periodically forced vibratory systems with nonlinear stiffness terms. Our criterion does not rely on any small parameters and hence is applicable for large response and forcing amplitudes. On explicit numerical examples, we demonstrate how our existence criterion can be utilized to justify subsequent numerical computations of forced responses.

Dynamic analysis of balance rope under multiple constraints with friction

Dynamic model of balance rope under multiple constraints with friction is established via using non-equal-length element division method (NEL-EDM) based on absolute nodal coordinate formulation. Then, the natural frequency of balance rope under multiple constraints is derived by the proposed method. The KDP generalized-alpha scheme is expanded to differential algebraic equations (DAEs) with friction constraint equations and used to solve the DAEs proposed by this paper. Compared with the frequencies, lateral vibration displacements at four observation points, the analysis of the NEL-EDM is carried out by MATLAB, ANSYS, and RECURDYN software, and the feasibility of NEL-EDM is verified. The frequencies of balance rope with installed bushing constraints will occur frequency veering phenomenon when the balance rope moves up and down with the conveyance. Last, free responses of the balance rope under multiple constraints due to the effects of conveyance vertical motion, and those of in-plane excitation on forced responses of balance rope under multiple constraints with friction are investigated.

Data-Driven Approach for Identifying Mistuning in As-Manufactured Blisks

Sector-to-sector geometry or material property variations in as-manufactured bladed disks, or blisks, can result in significantly greater vibration responses during operation compared to nominally cyclic symmetric designs. The dynamics of blisks are sensitive to these unavoidable deviations, known as mistuning, making the identification of mistuning in as-manufactured blisks necessary for accurately predicting their vibration. As in previous mistuning modeling and identification approaches, the mistuning of interest is small and is parameterized by using deviations in cantilever blade-alone frequencies. Such mistuning parameterization is popular because it can be applied through blade-to-blade stiffness deviations in computational reduced-order models used to predict blisk dynamics. Previous approaches to identify such mistuning parameters often require the identification of modal information or blade-isolation techniques such blade detuning using masses or adding damping pads. However, modal information can be difficult to obtain accurately even in optimal bench conditions. Additionally, in practice it can be difficult to isolate individual blades by restricting blade motion around the blisk or detuning individual blades through added masses due to geometric constraints. In this paper, we present a method for mistuning identification using a data-driven approach based on a neural network. The network is first trained using surrogate computational data. Thus, the data-driven portion of the approach is executed using surrogate computational methods. With the trained network, mistuning in all sectors of blisks with the same nominal geometry can be identified by using a small number of forced responses and the forcing phase information from traveling-wave excitation. In this approach, no system or sector-level modal response information, restrictive blade isolation, or mass detuning are required. We additionally present a method for forcing frequency selection and response conditioning to improve identification accuracy. Validation of this approach is presented using a finite element blisk model containing stiffness mistuning within the blades to create computationally generated surrogate data. It is shown that mistuning can be predicted accurately using forced responses containing a significant amount of absolute and relative measurement noise, mimicking responses collected from experimental measurements. In addition, it is shown that mistuning can be predicted independently and accurately using different engine orders of excitation in regions of high modal density.

Robustness of modelling the forced change of the ENSO-Indian monsoon teleconnection

<p>Beside means, the forced response of the internal variability of the climate system is also of considerable practical interest. Teleconnections are one aspect of internal variability, and they derive their importance partly from their role in seasonal predictability. We compare the forced response of the ENSO-Indian monsoon teleconnection — as a first step of investigating the robustness of its modelling — in two Earth System Models, making use of the Large Ensemble data sets of the MPI-GE and CESM1-LE. We find considerable similarities of climatologies and the forced responses with respect to spatial patterns, in terms of e.g. MCA (Maximum Covariance Analaysis) modes. However, because of the mismatch of these patterns, both in terms of their weight and shape, the teleconnection associated with restricted areas, such as the domain of the so-called All-India Summer Monsoon Rainfall (AISMR) differ very considerably in the two models. While most representations of the teleconnection involving the principal modes of variability show a strengthening in the MPI-GE, not much change is detectable in the CEMS1-LE. In fact, the second modes, EOF2 or MCA2, are associated with much more change in the CESM1.</p>

Experimental and Numerical Assessment of Mistuning Effects on Vibratory Response of a Bladed Disk with Underplatform Dampers

This paper presents experimental and numerical investigation of mistuned forced responses of an integrally bladed disk with full set of underplatform dampers (UPDs). This research aims at providing: 1. An experimental benchmark for nonlinear dynamics of a mistuned bladed disks with UPDs. 2. A numerical model that can account for features of a mistuned forced response level. Accordingly, a detailed experimental campaign is conducted on a static test rig called Octopus. This rig is specifically designed to investigate the dynamics of a full-scale integrally bladed disk (blisk) with UPDs in a noncontact manner so that the dynamic response of the system is not modified. The effect of mistuning on experimental forced response levels is assessed and a linearized model is proposed to predict the modulation of frequency response functions (FRFs) due to the frequency splitting. In the development of the model, the mistuning pattern identified from the linear blisk without UPDs is used and it is assumed that adding the dampers does not change the structural mistuning of the blisk. In this study, the fundamental mistuning model identification (FMM ID) was employed to identify the mistuning pattern of the blisk. It is shown that the proposed model successfully predicts the modulation of linear mistuned FRFs. The linearized model is also able to predict the modulation of nonlinear mistuned FRFs in stick condition (when nonlinear friction damping is negligible) with a good accuracy validating this assumption that adding the dampers does not change the mistuning pattern.