Dynamics of a simplified nonlinear model offering insights into the hammering type brake squeal initiation process

2021 ◽  
Vol 69 (3) ◽  
pp. 243-261
Author(s):  
Osman Taha Sen ◽  
Rajendra Singh
Keyword(s):  
Nonlinear Model ◽  
Harmonic Balance Method ◽  
Nonlinear Frequency ◽  
Frictional Surface ◽  
Brake Squeal ◽  
Governing Equations ◽  
Coupled Modes ◽  
Initiation Process ◽  
Wide Range ◽  
Linearized System

This article develops a simplified mechanical system model that offers insights into the hammering type brake squeal initiation process while overcoming a void in the literature. The proposed formulation derives a nonlinear two-degree-of-freedom model where a mass is in contact with a rigid frictional surface that moves with constant velocity. The kinematic nonlinearities arise from an arrangement of springs that support the mass, as well as from contact loss between the mass and frictional surface. First, the nonlinear governing equations are numerically solved for several normal force vector arrangements, and a wide range of dynamic responses are observed. Results show that some assumptions made in prior articles are not valid. Second, the nonlinear governing equations are linearized, and the existence of quasistatic sliding motion is sought for selected inclined spring arrangements. Third, the dynamic stability of the linearized system is examined and compared with the results of a nonlinear model. The coupled modes are found even though some contradictions between the model assumptions and linearized system solutions are observed. Finally, the nonlinear frequency responses are calculated using the multi-term harmonic balance method although only the contact loss nonlinearity is retained. Shifts in the resonant frequencies during the motion of the pad are clearly observed. In conclusion, the contact loss nonlinearity is found to be crucial, and as such, it must not be ignored for the squeal source investigation. Finally, the new model offers insight into the squeal initiation process while revealing the limitations of linearized system analyses.

scholarly journals Vibration of Rotating and Revolving Planet Rings with Discrete and Partially Distributed Stiffnesses

2020 ◽  
Vol 11 (1) ◽  
pp. 127
Author(s):  
Fuchun Yang ◽  
Dianrui Wang
Keyword(s):  
High Speed ◽  
Differential Operators ◽  
Natural Frequencies ◽  
Rotation Speed ◽  
Distribution Area ◽  
Vibration Modes ◽  
Governing Equations ◽  
Vibration Properties ◽  
Wide Range ◽  
Forced Responses

Vibration properties of high-speed rotating and revolving planet rings with discrete and partially distributed stiffnesses were studied. The governing equations were obtained by Hamilton’s principle based on a rotating frame on the ring. The governing equations were cast in matrix differential operators and discretized, using Galerkin’s method. The eigenvalue problem was dealt with state space matrix, and the natural frequencies and vibration modes were computed in a wide range of rotation speed. The properties of natural frequencies and vibration modes with rotation speed were studied for free planet rings and planet rings with discrete and partially distributed stiffnesses. The influences of several parameters on the vibration properties of planet rings were also investigated. Finally, the forced responses of planet rings resulted from the excitation of rotating and revolving movement were studied. The results show that the revolving movement not only affects the free vibration of planet rings but results in excitation to the rings. Partially distributed stiffness changes the vibration modes heavily compared to the free planet ring. Each vibration mode comprises several nodal diameter components instead of a single component for a free planet ring. The distribution area and the number of partially distributed stiffnesses mainly affect the high-order frequencies. The forced responses caused by revolving movement are nonlinear and vary with a quasi-period of rotating speed, and the responses in the regions supported by partially distributed stiffnesses are suppressed.

scholarly journals Discovering governing equations from data by sparse identification of nonlinear dynamical systems

2016 ◽  
Vol 113 (15) ◽  
pp. 3932-3937 ◽  
Author(s):  
Steven L. Brunton ◽  
Joshua L. Proctor ◽  
J. Nathan Kutz
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Dynamical Systems ◽  
Measurement Data ◽  
Climate Science ◽  
Model Complexity ◽  
Governing Equations ◽  
Canonical Systems ◽  
Nonlinear Dynamical ◽  
Balance Accuracy ◽  
Wide Range

Extracting governing equations from data is a central challenge in many diverse areas of science and engineering. Data are abundant whereas models often remain elusive, as in climate science, neuroscience, ecology, finance, and epidemiology, to name only a few examples. In this work, we combine sparsity-promoting techniques and machine learning with nonlinear dynamical systems to discover governing equations from noisy measurement data. The only assumption about the structure of the model is that there are only a few important terms that govern the dynamics, so that the equations are sparse in the space of possible functions; this assumption holds for many physical systems in an appropriate basis. In particular, we use sparse regression to determine the fewest terms in the dynamic governing equations required to accurately represent the data. This results in parsimonious models that balance accuracy with model complexity to avoid overfitting. We demonstrate the algorithm on a wide range of problems, from simple canonical systems, including linear and nonlinear oscillators and the chaotic Lorenz system, to the fluid vortex shedding behind an obstacle. The fluid example illustrates the ability of this method to discover the underlying dynamics of a system that took experts in the community nearly 30 years to resolve. We also show that this method generalizes to parameterized systems and systems that are time-varying or have external forcing.

What Kind of Friction Model Is Needed to Predict Brake Squeal?

2012 ◽  
Author(s):  
Tore Butlin ◽  
Jim Woodhouse
Keyword(s):  
Large Scale ◽  
Dynamic Properties ◽  
Theoretical Work ◽  
Friction Model ◽  
Coupled Systems ◽  
Brake Squeal ◽  
Model Framework ◽  
Wide Range ◽  
Pin On Disc ◽  
Almost All

Predictive models of friction-induced vibration have proved elusive despite decades of research. There are many mechanisms that can cause brake squeal; friction coupled systems can be highly sensitive to small perturbations; and the dynamic properties of friction at the contact zone seem to be poorly understood. This paper describes experimental and theoretical work aimed at identifying the key ingredients of a predictive model. A large-scale experiment was carried out to identify squeal initiations using a pin-on-disc test rig: approximately 30,000 squeal initiations were recorded, covering a very wide range of frequencies. The theoretical model allows for completely general linear systems coupled at a single sliding point by friction: squeal is predicted using a linearised stability analysis. Results will be presented that show that almost all observed squeal events can be predicted within this model framework, but that some subsets require innovative friction modelling: predictions are highly dependent on the particular choice of friction model and its associated parameters.

On the Application of Frequency-Domain Methods to Multistage Turbomachinery

2015 ◽  
Author(s):  
Laura Junge ◽  
Graham Ashcroft ◽  
Peter Jeschke ◽  
Christian Frey
Keyword(s):  
Frequency Domain ◽  
Time Domain ◽  
Harmonic Balance ◽  
Axial Compressor ◽  
Harmonic Balance Method ◽  
Nonlinear Frequency ◽  
Solution Quality ◽  
Multi Stage ◽  
Frequency Domain Methods ◽  
Blade Row

Due to the relative motion between adjacent blade rows the aerodynamic flow fields within turbomachinery are normally dominated by deterministic, periodic phenomena. In the numerical simulation of such unsteady flows (nonlinear) frequency-domain methods are therefore attractive as they are capable of fully exploiting the given spatial and temporal periodicity, as well as capturing or modelling flow nonlinearity. Central to the efficiency and accuracy of such frequency-domain methods is the selection of the frequencies and the circumferential modes to be resolved in simulations. Whilst trivial in the context of the simulation of a single compressor- or turbine-stage, the choice of solution modes becomes substantially more involved in multi-stage configurations. In this work the importance of mode scattering, in the context of the unsteady aerodynamic field, is investigated and quantified. It is shown that scattered modes can substantially impact the unsteady flow field and are essential for the accurate modelling of wake propagation within multistage configurations. Furthermore, an iterative approach is outlined, based on the spectral analysis of the circumferential modes at the interfaces between blade rows, to identify the dominant solution modes that should be resolved in the adjacent blade row. To demonstrate the importance of mode scattering and validate the approach for their identification the unsteady blade row interaction within a 4.5 stage axial compressor is computed using both the harmonic balance method and, based on a full annulus midspan simulation, a time-domain method. Through the inclusion of scattered modes it is shown that the solution quality of the harmonic balance results is comparable to that of the nonlinear time-domain simulation.

Nonlinear vibrations of a thin isotropic circular plate subjected to moving point loads

2020 ◽  
pp. 095440622097615
Author(s):  
Amit K Rai ◽  
Shakti S Gupta
Keyword(s):  
Boundary Conditions ◽  
Frequency Response ◽  
Circular Plate ◽  
Nonlinear Vibrations ◽  
Moving Load ◽  
Harmonic Balance Method ◽  
Angular Speed ◽  
Transverse Displacement ◽  
Governing Equations ◽  
Rotating Load

Here, we have studied the linear and nonlinear vibrations of a thin circular plate subjected to circularly, radially, and spirally moving transverse point loads. We follow Kirchoff’s theory and then incorporate von Kármán nonlinearity and employ Hamilton’s principle to obtain the governing equations and the associated boundary conditions. We solve the governing equations for the simply-supported and clamped boundary conditions using the mode summation method. Using the harmonic balance method for frequency response and Runge-Kutta method for time response, we solve the resulting coupled and cubic nonlinear ordinary differential equations. We show that the resonance instability due to a circularly moving load can be avoided by splitting it into multiple loads rotating at the same radius and angular speed. With the increasing magnitude of the rotating load, the frequency response of the transverse displacement shows jumps and modal interaction. The transverse response collected at the centre of the plate shows subharmonics of the axisymmetric frequencies only. The spectrum of the linear response due to spirally moving load contains axisymmetric frequencies, the angular speed of the load, their combination, and superharmonics of axisymmetric frequencies.

Bifurcation of Nonlinear Normal Modes by Means of Synge’s Stability

2011 ◽  
Author(s):  
Young S. Lee ◽  
Heng Chen
Keyword(s):  
Harmonic Balance ◽  
Coupled Oscillators ◽  
Normal Modes ◽  
Harmonic Balance Method ◽  
Orbital Stability ◽  
Nonlinear Normal Modes ◽  
Energy Transfers ◽  
Wide Range ◽  
Energy Domain ◽  
Nonlinear Normal

We study bifurcation of fundamental nonlinear normal modes (FNNMs) in 2-degree-of-freedom coupled oscillators by utilizing geometric mechanics approach based on Synges concept, which dictates orbital stability rather than Lyapunovs classical asymptotic stability. Use of harmonic balance method provides reasonably accurate approximation for NNMs over wide range of energy; and Floquet theory incorporated into Synges stability analysis predicts the respective bifurcation points as well as their types. Constructing NNMs in the frequency-energy domain, we seek applications to study of efficient targeted energy transfers.

On the Role of Enduring Contact in Powder Lubrication

2005 ◽  
Vol 128 (1) ◽  
pp. 168-175 ◽  
Author(s):  
J. Y. Jang ◽  
M. M. Khonsari
Keyword(s):  
Analytical Solution ◽  
Flow Velocity ◽  
Volume Fraction ◽  
Flow Characteristics ◽  
Kinetic Regime ◽  
Governing Equations ◽  
Wide Range ◽  
Fraction Distribution ◽  
Powder Lubrication

This paper is devoted to a study of the enduring contact between granules of powder lubricants in an effort to better understand the flow characteristics of powder lubricants. Appropriate formulation of the governing equations is reported that can be used for prediction of the flow velocity, pseudo temperature, and volume fraction distribution of powders for a wide range of operating speeds. A set of parametric simulations and a limiting analytical solution is presented for predicting the behavior of a powder lubricant under low operating speeds when the enduring contact tends to dominate the kinetic regime. The limiting solution shows that below a certain sliding speed the volume fraction remains unchanged due to the effect of the enduring contact. It is also shown that below this limiting speed the enduring contact plays a major role and should not be neglected.

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