Shock and Vibration - Volume 18, issue 5

Authors: El-Saeidy, Fawzi M.A.

Article Type: Research Article

Abstract: A lagrangian formulation is presented for the total dynamic stiffness and damping matrices of a rigid rotor carrying noncentral rigid disk and supported on angular contact ball bearings (ACBBs). The bearing dynamic stiffness/damping marix is derived in terms of the bearing motions (displacements/rotations) and then the principal of virtual work is used to transfer it from the bearing location to the rotor mass center to obtain the total dynamic stiffness/damping matrix. The bearing analyses take into …account the bearing nonlinearities, cage rotation and bearing axial preload. The coefficients of these time-dependent matrices are presented analytically. The equations of motion of a rigid rotor-ACBBs assembly are derived using Lagrange's equation. The proposed analyses on deriving the bearing stiffness matrix are verified against existing bearing analyses of SKF researchers that, in turn, were verified using both SKF softwares/experiments and we obtained typical agreements. The presented total stiffness matrix is applied to a typical grinding machine spindle studied experimentally by other researchers and excellent agreements are obtained between our analytical eigenvalues and the experimental ones. The effect of using the total full stiffness matrix versus using the total diagonal stiffness matrix on the natural frequencies and dynamic response of the rigid rotor-bearings system is studied. It is found that using the diagonal matrix affects natural frequencies values (except the axial frequency) and response amplitudes and pattern and causes important vibration tones to be missig from the response spectrum. Therefore it is recommended to use the full total stiffness matrix and not the diagonal matrix in the design/vibration analysis of these rotating machines. For a machine spindle-ACBBs assembly under mass unbalnce and a horizontal force at the spindle cutting nose when the bearing time-varying stiffness matrix (bearing cage rotation is considered) is used, the peak-to-valley variation in time domain of the stiffness matrix elements becomes significant compared to its counterpart when the bearing standard stiffness matrix (bearing cage rotation is neglected) is used. The vibration spectrum of the time-varying matrix case is marked by tones at bearing outer ring ball passing frequency, rotating unbalnce frequency and combination compared to spectrum of the standard stiffness matrix case which is marked by only the rotating unbalnce frequency. Therfore, it is highly recomended to model bearing stiffness matrix to be a time-dependent. Show more

DOI: 10.3233/SAV-2010-0577

Citation: Shock and Vibration, vol. 18, no. 5, pp. 641-670, 2011

Authors: Kropáč, Oldřich | Múčka, Peter

Article Type: Research Article

Abstract: A voluminous set of longitudinal road profiles gathered from the Long Term Pavement Performance (LTPP) program was processed using median filtering to separate individual large obstacles from the basic quasi-homogeneous random road unevenness. The shapes of 16590 obstacles were analysed partitioned into four sub-groups according to the asphalt-concrete vs. cement-concrete road surfaces, and positive (bumps) vs. negative (potholes) vertical elevations. The empirical obstacle shapes were fitted using seven symmetric and two asymmetric …analytical shapes. Based on the root mean square error (RMSE) criterion and accounting for some practical aspects connected with the possible use for the simulation of road profiles with obstacles generalized parabolic shapes (power function) and half-sine shapes seem to be recommendable as adequate approximation functions. Asymmetric obstacle shapes of cosine and parabolic shapes were also briefly discussed and their impact on the vibration transient response of a quarter-car model outlined. Show more

Keywords: Road profile, vibration response, obstacle shape, bump, pothole, quarter-car model, long term pavement performance program

DOI: 10.3233/SAV-2010-0587

Citation: Shock and Vibration, vol. 18, no. 5, pp. 671-682, 2011

Authors: Shahba, Ahmad | Attarnejad, Reza | Hajilar, Shahin

Article Type: Research Article

Abstract: Structural analysis of axially functionally graded tapered Euler-Bernoulli beams is studied using finite element method. A beam element is proposed which takes advantage of the shape functions of homogeneous uniform beam elements. The effects of varying cross-sectional dimensions and mechanical properties of the functionally graded material are included in the evaluation of structural matrices. This method could be used for beam elements with any distributions of mass density and modulus of elasticity with arbitrarily …varying cross-sectional area. Assuming polynomial distributions of modulus of elasticity and mass density, the competency of the element is examined in stability analysis, free longitudinal vibration and free transverse vibration of double tapered beams with different boundary conditions and the convergence rate of the element is then investigated. Show more

Keywords: Axially functionally graded materia, tapered beam element, free vibration, stability

DOI: 10.3233/SAV-2010-0589

Citation: Shock and Vibration, vol. 18, no. 5, pp. 683-696, 2011

Authors: Saeedi, Khodabakhsh | Bhat, Rama B.

Article Type: Research Article

Abstract: A study of the natural frequencies and mode shapes of a multi-span beam is carried out by introducing constrained beam characteristic functions. The conventional method used for the dynamic analysis of such a beam is to consider span-wise characteristic function solution and then to solve it by using compatibility conditions such as the continuity in the a slope and bending moment at the intermediate supports and boundary conditions at the ends. In the method proposed here, …the matrix size is reduced and, if the support conditions are symmetric about the midpoint, the symmetry and anti-symmetry conditions at the midpoint can be conveniently exploited for computational economy. The natural frequencies occur in clusters, each one containing the number of natural frequencies equal to the number of spans. The results are presented and discussed. Show more

Keywords: Multi-span beam, vibration, intermediate support, reaction force

DOI: 10.3233/SAV-2010-0592

Citation: Shock and Vibration, vol. 18, no. 5, pp. 697-707, 2011

Authors: Yesilce, Yusuf

Article Type: Research Article

Abstract: The structural elements supporting motors or engines are frequently seen in technological applications. The operation of machine may introduce additional dynamic stresses on the beam. It is important, then, to know the natural frequencies of the coupled beam-mass system, in order to obtain a proper design of the structural elements. The literature regarding the free vibration analysis of Bernoulli-Euler and Timoshenko single-span beams carrying a number of spring-mass system and multi-span beams carrying multiple …spring-mass systems are plenty, but the free vibration analysis of Reddy-Bickford multi-span beams carrying multiple spring-mass systems has not been investigated by any of the studies in open literature so far. This paper aims at determining the exact solutions for the natural frequencies and mode shapes of Reddy-Bickford beams. The model allows analyzing the influence of the shear effect and spring-mass systems on the dynamic behavior of the beams by using Reddy-Bickford Beam Theory (RBT). The effects of attached spring-mass systems on the free vibration characteristics of the 1–4 span beams are studied. The natural frequencies of Reddy-Bickford single-span and multi-span beams calculated by using the numerical assembly technique and the secant method are compared with the natural frequencies of single-span and multi-span beams calculated by using Timoshenko Beam Theory (TBT); the mode shapes are presented in graphs. Show more

Keywords: Eigenvalue problem, free vibration, numerical assembly technique, Reddy-Bickford multi-span beam, spring-mass system

DOI: 10.3233/SAV-2010-0593

Citation: Shock and Vibration, vol. 18, no. 5, pp. 709-726, 2011

Authors: Dalli, Uğur | Yüksel, Şefaatdin

Article Type: Research Article

Abstract: An active control method utilizing the multiple trailing edge flap configuration for rotorcraft vibration suppression and blade loads control is presented. A comprehensive model for rotor blade with active trailing edge flaps is used to calculate the vibration characteristics, natural frequencies and mode shapes of any complex composite helicopter rotor blade. A computer program is developed to calculate the system response, rotor blade root forces and moments under aerodynamic forcing conditions. Rotor blade system response …is calculated using the proposed solution method and the developed program depending on any structural and aerodynamic properties of rotor blades, structural properties of trailing edge flaps and properties of trailing edge flap actuator inputs. Rotor blade loads are determined first on a nominal rotor blade without multiple active trailing edge flaps and then the effects of the active flap motions on the existing rotor blade loads are investigated. Multiple active trailing edge flaps are controlled by using open loop controllers to identify the effects of the actuator signal output properties such as frequency, amplitude and phase on the system response. Effects of using multiple trailing edge flaps on controlling rotor blade vibrations are investigated and some design criteria are determined for the design of trailing edge flap controller that will provide actuator signal outputs to minimize the rotor blade root loads. It is calculated that using the developed active trailing edge rotor blade model, helicopter rotor blade vibrations can be reduced up to 36% of the nominal rotor blade vibrations. Show more

DOI: 10.3233/SAV-2010-0594

Citation: Shock and Vibration, vol. 18, no. 5, pp. 727-745, 2011

Authors: Palmer, Jeremy A. | Paez, Thomas L.

Article Type: Research Article

Abstract: Modern electro-optical instruments are typically designed with assemblies of optomechanical members that support optics such that alignment is maintained in service environments that include random vibration loads. This paper presents a nonlinear numerical analysis that calculates statistics for the peak lateral response of optics in an optomechanical sub-assembly subject to random excitation of the housing. The work is unique in that the prior art does not address peak response probability distribution for stationary random …vibration in the time domain for a common lens-retainer-housing system with Coulomb damping. Analytical results are validated by using displacement response data from random vibration testing of representative prototype sub-assemblies. A comparison of predictions to experimental results yields reasonable agreement. The Type I Asymptotic form provides the cumulative distribution function for peak response probabilities. Probabilities are calculated for actual lens centration tolerances. The probability that peak response will not exceed the centration tolerance is greater than 80% for prototype configurations where the tolerance is high (on the order of 30 micrometers). Conversely, the probability is low for those where the tolerance is less than 20 micrometers. The analysis suggests a design paradigm based on the influence of lateral stiffness on the magnitude of the response. Show more

Keywords: Coulomb damping, lens, optomechanical, random vibration, retainer

DOI: 10.3233/SAV-2010-0595

Citation: Shock and Vibration, vol. 18, no. 5, pp. 747-758, 2011