Calculating Natural Frequencies With Extended Tuplin’s Method

In this paper an efficient numerical procedure is described which yields the eigenvalue of a lumped mass torsional vibration directly from the frequency equation of the system. Special characteristics of Tuplin’s frequency equation allow all eigenvalues to be easily located and accurately evaluated from the frequency polynomial. In contrast to the general belief that extracting roots of polynomials is less efficient than matrix reduction methods, this paper demonstrates that the direct solution competes favorably with the modern eigenvalue routines such as QR and tridiagonal methods [15] [16] [17] [21] in torsional vibration problems. A BASIC program FUNG has been written based on the numerical concepts of this paper. The current version is able to solve multiple branch systems of many degrees of freedom subject to the restriction that no branch shall exceed 4 rotors and 4 shafts. The program has been tested for various examples and the output compared with the known results. Within the above range of applicability, this method beats the modern tridiagonal eigenvalue subroutines [16] [21] by a comfortable margin which ranges from 15 times to 120 times faster. The comparisons were made on the basis of solving the same problems on the same computer. FUNG assumes a tight tolerance of convergence for iteration (correct to approximately 14 significant digits).

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Simulation of an incomplete algebraic problem of eigenvalues and vectors by the method of frequency-dynamic condensation based on FEM in the form of the classical mixed method

Structural Mechanics of Engineering Constructions and Buildings ◽

10.22363/1815-5235-2019-15-1-62-68 ◽

2019 ◽

Vol 15 (1) ◽

pp. 62-68

Author(s):

Alexander V Ignatyev ◽

Artem V Chumakov ◽

Vadim V Gilka

Keyword(s):

Dynamic Analysis ◽

Mixed Method ◽

Degrees Of Freedom ◽

Dynamic Properties ◽

Frequency Equation ◽

Practical Implementation ◽

Algebraic Problem ◽

Static Condensation ◽

Dynamic Condensation ◽

Reduction Methods

Relevance . Dynamic analysis of complex structures using numerical methods leads to the solution of the algebraic problem of eigenvalues and the corresponding eigenvectors of high orders. The solution of this problem for high order matrices is performed using reduction methods. One of the most effective methods is the method of sequential frequency-dynamic condensation, which allows partial consideration of the dynamic properties of the structure in the minor degrees of freedom. This allows for more accurate results compared to static condensation. Frequency-dynamic condensation is traditionally used to reduce frequency equations derived from the finite element method in the form of the displacement method or the force method. Methods. The authors have developed an algorithm for the frequency-dynamic condensation method for the frequency equation obtained on the basis of the FEM in the form of the classical mixed method. That allows to obtain not only the spectrum of the lower vibration frequencies, but also the corresponding vibration modes and the stress-strain state of the structure. Results . This article describes the algorithm and its practical implementation in the problem of dynamic analysis of a rectangular plate. The results of the numerical analysis of the problem are presented. An assessment of the accuracy of the method and recommendations for its use are given.

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Modeling of nonlinear torsional vibration of the automotive powertrain

Journal of Vibration and Control ◽

10.1177/1077546316668687 ◽

2016 ◽

Vol 24 (9) ◽

pp. 1774-1786 ◽

Cited By ~ 10

Author(s):

Sérgio J Idehara ◽

Fernando L Flach ◽

Douglas Lemes

Keyword(s):

Natural Frequency ◽

Torsional Vibration ◽

Degrees Of Freedom ◽

Front Wheel ◽

Torsional Stiffness ◽

Bench Test ◽

Passenger Car ◽

Engine Torque ◽

Friction Moment ◽

Wheel Drive

A vibration model of the powertrain can be used to predict its dynamic behavior when excited by fluctuations in the engine torque and speed. The torsional vibration resulting from torque and speed fluctuations increases the rattle noise in the gearbox and it should be controlled or minimized in order to gain acceptance by clients and manufactures. The fact that the proprieties of the torsional damper integrated into the clutch disc alter the dynamic characteristic of the system is important in the automotive industry for design purposes. In this study, bench test results for the characteristics of a torsional damper for a clutch system (torsional stiffness and friction moment) and powertrain torsional vibration measurements taken in a passenger car were used to verify and calibrate the model. The adjusted model estimates the driveline natural frequency and the time response vibration. The analysis uses order tracking signal processing to isolate the response from the engine excitation (second-order). It is shown that a decrease in the stiffness of the clutch disc torsional damper lowers the natural frequency and an increase in the friction moment reduces the peak amplitude of the gearbox torsional vibration. The formulation and model adjustment showed that a nonlinear model with three degrees of freedom can represent satisfactorily the powertrain dynamics of a front-wheel drive passenger car.

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Interface Reduction in Craig-Bampton Component Mode Synthesis by Orthogonal Polynomial Series

Volume 7B: Structures and Dynamics ◽

10.1115/gt2017-64583 ◽

2017 ◽

Author(s):

Luigi Carassale ◽

Mirko Maurici

Keyword(s):

Degrees Of Freedom ◽

Axial Compressor ◽

Component Mode Synthesis ◽

Reduction Techniques ◽

Interface Reduction ◽

Reduction Methods ◽

Interface Displacement ◽

Polynomial Series ◽

Efficient Representation

The component mode synthesis based on the Craig-Bampton method has two strong limitations that appear when the number of the interface degrees of freedom is large. First, the reduced-order model obtained is overweighed by many unnecessary degrees of freedom. Second, the reduction step may become extremely time consuming. Several interface reduction techniques addressed successfully the former problem, while the latter remains open. In this paper we tackle this latter problem through a simple interface-reduction technique based on an a-priory choice of the interface modes. An efficient representation of the interface displacement field is achieved adopting a set of orthogonal basis functions determined by the interface geometry. The proposed method is compared with other existing interface reduction methods on a case study regarding a rotor blade of an axial compressor.

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Adaptive Control of a Sliding Seat Using Magnetorheological Energy Absorbers

ASME 2010 Conference on Smart Materials, Adaptive Structures and Intelligent Systems, Volume 2 ◽

10.1115/smasis2010-3896 ◽

2010 ◽

Author(s):

Min Mao ◽

Norman M. Wereley ◽

Alan L. Browne

Keyword(s):

Adaptive Control ◽

Degrees Of Freedom ◽

Adaptive Controller ◽

Degree Of Freedom ◽

Lumped Mass ◽

Adaptive Controllers ◽

Bingham Number ◽

Control Objective ◽

Payload Mass ◽

Energy Absorbers

Feasibility of a sliding seat utilizing adaptive control of a magnetorheological (MR) energy absorber (MREA) to minimize loads imparted to a payload mass in a ground vehicle for frontal impact speeds as high as 7 m/s (15.7 mph) is investigated. The crash pulse for a given impact speed was assumed to be a rectangular deceleration pulse having a prescribed magnitude and duration. The adaptive control objective is to bring the payload (occupant plus seat) mass to a stop using the available stroke, while simultaneously accommodating changes in impact velocity and occupant mass ranging from a 5th percentile female to a 95th percentile male. The payload is first treated as a single-degree-of-freedom (SDOF) rigid lumped mass, and two adaptive control algorithms are developed: (1) constant Bingham number control, and (2) constant force control. To explore the effects of occupant compliance on adaptive controller performance, a multi-degree-of-freedom (MDOF) lumped mass biodynamic occupant model was integrated with the seat mass. The same controllers were used for both the SDOF and MDOF cases based on SDOF controller analysis because the biodynamic degrees of freedom are neither controllable nor observable. The designed adaptive controllers successfully controlled load-stroke profiles to bring payload mass to rest in the available stroke and reduced payload decelerations. Analysis showed extensive coupling between the seat structures and occupant biodynamic response, although minor adjustments to the control gains enabled full use of the available stroke.

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Application of Laplace Transform to the Free Vibration of Continuous Beams

Archives of Civil Engineering ◽

10.1515/ace-2017-0011 ◽

2017 ◽

Vol 63 (1) ◽

pp. 163-180 ◽

Cited By ~ 1

Author(s):

H.B. Wen ◽

T. Zeng ◽

G.Z. Hu

Keyword(s):

Free Vibration ◽

Laplace Transform ◽

Reaction Force ◽

Bending Moment ◽

Frequency Equation ◽

Simplified Model ◽

Continuous Beam ◽

Transform Method ◽

Continuous Beams ◽

Vibration Problems

AbstractLaplace Transform is often used in solving the free vibration problems of structural beams. In existing research, there are two types of simplified models of continuous beam placement. The first is to regard the continuous beam as a single-span beam, the middle bearing of which is replaced by the bearing reaction force; the second is to divide the continuous beam into several simply supported beams, with the bending moment of the continuous beam at the middle bearing considered as the external force. Research shows that the second simplified model is incorrect, and the frequency equation derived from the first simplified model contains multiple expressions which might not be equivalent to each other. This paper specifies the application method of Laplace Transform in solving the free vibration problems of continuous beams, having great significance in the proper use of the transform method.

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Practical Solution of Torsional Vibration Problems. W. Ker Wilson, Ph.D., M.Sc, M.I.Mech.E. Two volumes. Chapman and Hall. Vol.I, 1940, 42/-; Vol.2, 1941, 42/-.

Journal of the Royal Aeronautical Society ◽

10.1017/s0368393100101610 ◽

1941 ◽

Vol 45 (366) ◽

pp. 231-232

Keyword(s):

Torsional Vibration ◽

Practical Solution ◽

Vibration Problems

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Coupled Torsional Vibration and Fatigue Damage of Turbine Generator Due to Grid Disturbance

Journal of Engineering for Gas Turbines and Power ◽

10.1115/1.4026214 ◽

2014 ◽

Vol 136 (6) ◽

Cited By ~ 7

Author(s):

Chao Liu ◽

Dongxiang Jiang ◽

Jingming Chen

Keyword(s):

Fatigue Damage ◽

Torsional Vibration ◽

Coupled System ◽

Lumped Mass ◽

Mass Model ◽

Turbine Generator ◽

Failure Prevention ◽

Lumped Mass Model ◽

Continuous Mass ◽

Fatigue Evaluation

Crack failures continually occur in shafts of turbine generator, where grid disturbance is an important cause. To estimate influences of grid disturbance, coupled torsional vibration and fatigue damage of turbine generator shafts are analyzed in this work, with a case study in a 600MW steam unit in China. The analysis is the following: (i) coupled system is established with generator model and finite element method (FEM)-based shafts model, where the grid disturbance is signified by fluctuation of generator outputs and the shafts model is formed with lumped mass model (LMM) and continuous mass model (CMM), respectively; (ii) fatigue damage is evaluated in the weak location of the shafts through local torque response computation, stress calculation, and fatigue accumulation; and (iii) failure-prevention approach is formed by solving the inverse problem in fatigue evaluation. The results indicate that the proposed scheme with continuous mass model can acquire more detailed and accurate local responses throughout the shafts compared with the scheme without coupled effects or the scheme using lumped mass model. Using the coupled torsional vibration scheme, fatigue damage caused by grid disturbance is evaluated and failure prevention rule is formed.

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Identification of firefly algorithm-based fluctuation coefficient of the exciting torque for vehicle driveline

Proceedings of the Institution of Mechanical Engineers Part K Journal of Multi-body Dynamics ◽

10.1177/1464419318808172 ◽

2018 ◽

Vol 233 (2) ◽

pp. 317-326

Author(s):

Qiaobin Liu ◽

Wenku Shi ◽

Zhiyong Chen

Keyword(s):

Torsional Vibration ◽

Firefly Algorithm ◽

Vibration Response ◽

Theoretical Modelling ◽

Lumped Mass ◽

Mass Model ◽

Engine Torque ◽

Lumped Mass Model ◽

Vibration Performance ◽

Engine Excitation

The unbalanced excitation force and torque generated by an engine that resonate with the natural frequency of drivetrain often causes vibration and noise problems in vehicles. This study aims to comprehensively employ theoretical modelling and experimental identification methods to obtain the fluctuation coefficients of engine excitation torque when a car is in different gear positions. The inherent characteristics of the system are studied on the basis of the four-degree-of-freedom driveline lumped mass model and the longitudinal dynamics model of vehicle. The correctness of the model is verified by torsional vibration test. The second order's engine torque fluctuation coefficients are identified by firefly algorithm according to the curves of flywheel speed in different gears under the acceleration condition of the whole open throttle. The torque obtained by parameter identification is applied to the model, and the torsional vibration response of the system is analysed. The influence of the key parameters on the torsional vibration response of the system is investigated. The study concludes that proper reduction of clutch stiffness can increase clutch damping and half-axle rigidity, which can help improve the torsional vibration performance of the system. This study can provide reference for vehicle drivetrain modelling and torsional vibration control.

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