On the Effect of Functionally Graded Materials on Resonances of Rotating Beams

Radially rotating beams attached to a rigid stem occur in several important engineering applications. Some examples include helicopter blades, turbine blades and certain aerospace applications. In most studies the beams have been treated as homogeneous. Here, with a goal of system improvement, non-homogeneous beams made of functionally graded materials are explored. The effects on the natural frequencies of the system are investigated. Euler-Bernoulli theory, including an axial stiffening effect and variations of both Young\u27s modulus and density, is employed. An assumed mode approach is utilized, with the modes taken to be beam characteristic orthogonal polynomials. Results are obtained via Rayleigh-Ritz method and are compared for both the homogeneous and non-homogeneous cases. It was found, for example, that allowing Young\u27s modulus and density to vary by approximately 2.15 and 1.15 times, respectively, leads to an increase of 23% in the lowest bending rotating natural frequency of the bea

On the Effect of Functionally Graded Materials on Resonances of Rotating Beams

Radially rotating beams attached to a rigid stem occur in several important engineering applications. Some examples include helicopter blades, turbine blades and certain aerospace applications. In most studies the beams have been treated as homogeneous. Here, with a goal of system improvement, non-homogeneous beams made of functionally graded materials are explored. The effects on the natural frequencies of the system are investigated. Euler-Bernoulli theory, including an axial stiffening effect and variations of both Young's modulus and density, is employed. An assumed mode approach is utilized, with the modes taken to be beam characteristic orthogonal polynomials. Results are obtained via Rayleigh-Ritz method and are compared for both the homogeneous and non-homogeneous cases. It was found, for example, that allowing Young's modulus and density to vary by approximately 2.15 and 1.15 times, respectively, leads to an increase of 23% in the lowest bending rotating natural frequency of the beam

Passage through resonance in a universal joint driveline system

A driving shaft coupled to a driven shaft by a universal joint is considered. The shafts are taken to be rigid and motion is restricted to one plane. The non-homogeneous differential equation of motion has time-dependent coefficients and both parametric and forced resonances can occur. Here the question of whether one can ‘‘drive’’ through the resonances using a driving angular velocity linear variation is addressed. Also, how long can one ‘‘dwell’’ at a potential resonance before actually encountering it is investigated. Numerical studies led to the following conclusions. For linearly increasing speed profiles no practically feasible sweep rates to avoid resonance build up were found for the forced motion resonances. For certain torque and damping values, parametric resonances are seen for slow angular velocity variations but they are not observed for practically feasible fast variations, thus raising the possibility that one can accelerate through them. For a trapezoidal speed input the dwell time is key in building up instabilities. As the dwell time increases larger response amplification is observed. For the case studied, it was shown that it is possible to drive through the instability if the dwell time is equal to or less than forty times the period of the parametric excitation

Teaching and Learning Experiences of an Integrated Mechanism and Machine Design Course

The objective of this paper is to discuss some of the issues concerning the teaching and learning experiences of an integrated mechanical assemblies and mechanical engineering design course taught at Kettering University. The integration into one discipline of subjects, which are otherwise commonly taught as separate courses, is also discussed. A course entitled Analysis and Design of Machines and Mechanical Assemblies is used as an example. Course objectives and learning outcomes are included along with an example course outline. While the two-course integration into a single one can pose some challenging issues with respect to the pre-requisites needed by the students, it provides a great opportunity to bring out new teaching materials conducive to active learning. The course is designed in such a way that the students are required to complete regular homework, class work and carry out simulation exercises using CAE tools. An example student project will be presented and the learning outcomes discussed

Bending Frequency Alteration of Rotating Shafts

Rotating shafts are commonly employed in several automotive systems. An important issue when utilizing these shafts is their NVH (noise, vibration and harshness) characteristics. The resonant frequencies of these components, which can be excited externally (due, for example, to an imbalance force or vehicle external loads) or internally (due to parametric excitations, a possibility in drivelines) should be avoided for optimum NVH performance. Provided the exciting frequencies are known, one can attempt to design the shafts, or alter an existing design, to achieve this goal. In this work, which is exploratory in nature, two items are investigated. A simple shape optimization is conducted with the objective of increasing the shaft resonant frequency, which could drive it away from the range of excitations. Also a study involving an exploration of the use of functionally graded materials is presented. An existing model, which allows for changes in the material\u27s Young\u27s Modulus and density via three parameters, is utilized. The sensitivity of the lowest bending natural frequency to the parameters is analyzed using an assumed mode method. The study showed that the effect of the spin of the shaft was negligible. It was found that an increase of about 43% of the first bending frequency can be achieved by increasing Young\u27s modulus by a factor of 2 and density by a factor of 1.15. In the study of shape optimization it was found that increases of the order of 5% can be obtained with realistic changes of the shape of the shaft

Harmonic Forcing of a Two-Segment Elastic Rod

This work is on the motions of non-homogeneous elastic rods. In a previous work the natural frequencies and associated mode shapes were determined for a two-segment rod, in which the geometric and material properties were constant in each segment. Here the steady state response due to harmonic forcing is investigated using two strategies. The first employs the exact displacement equations. For harmonic forcing in time, the response is periodic and general solutions to the resulting differential equations can, in principle, be found for each segment. The constants involved are found from boundary and interface conditions and then response, as a function of forcing frequency, can be obtained. The procedure is cumbersome and problematic if the forces vary spatially, due to difficulties in finding “particular integrals”. An alternative method is developed in which geometric and material discontinuities are modeled by continuously varying functions (here logistic functions). This leads to a single differential equation with variable coefficients, which is solved numerically using MAPLE®’s PDE solver. For free-fixed boundary conditions and spatially constant force good agreement is found between the two methods, lending confidence to the continuous varying approach, which is then used to obtain response for spatially varying forces

Natural Frequencies of Layered Beams Using a Continuous Variation Model

This work involves the determination of the bending natural frequencies of beams whose properties vary along the length. Of interest are beams with different materials and varying cross-sections, which are layered in cells. These can be uniform or not, leading to a configuration of stacked cells of distinct materials and size. Here the focus is on cases with two, or three cells, and shape variations that include smooth (tapering) and sudden (block type) change in cross-sectional area. Euler-Bernoulli theory is employed. The variations are modeled using approximations to unit step functions, here logistic functions. The approach leads to a single differential equation with variable coefficients. A forced motion strategy is employed in which resonances are monitored to determine the natural frequencies. Forcing frequencies are changed until large motions and sign changes are observed. Solutions are obtained using MAPLE®’s differential equation solvers. The overall strategy avoids the cumbersome and lengthy Transfer Matrix method. Pin-pin and clamp-clamp boundary conditions are treated. Accuracy is partially assessed using a Rayleigh-Ritz method and, for completeness, FEM. Results indicate that the forced motion approach works well for a two-cell beam, three-cell beam and a beam with a sinusoidal profile. For example, in the case of a uniform two-cell beam, with pin-pin boundary conditions, results differ less than 1 %

Resonances of Compact Tapered Inhomogeneous Axially Loaded Shafts

An important technical area is the bending of shafts subjected to an axial load. These shafts could be tapered and made of materials with spatially varying properties (Functionally Graded Material – FGM). Previously the transverse vibrations of such shafts were investigated by the authors assuming the shafts had large slenderness ratios so that Euler-Bernoulli theory could be employed. Here compact shafts are treated necessitating the use of Timoshenko beam theory. For constant axial load case analysis of the effects of both FGMs and tapering on frequencies, the value of the compressive load is chosen to be 80% of the smallest critical (buckling) value for the shafts considered. The equations of motion give rise to two coupled differential equations with variable coefficients. These equations in general do not have analytic solutions and numerical methods must be employed (here using MAPLE®) to find the natural frequencies. MAPLE®’s built-in solver for two-point boundary value problems does not directly provide the eigenvalues. The strategy used is to solve a harmonically forced motion problem. On varying the excitation frequency and observing the mid-span deflection the resonance frequency can be found noting where a change in sign occurs. For example, results for FGM cylindrical and tapered shafts show that for a compact cylindrical beam the resonant frequency obtained differs from the Euler-Bernoulli prediction by 11%, and for a tapered beam by 12%, indicating that the effects of compactness can be significant. Since Timoshenko theory requires a value for the shear coefficient, which is not readily available for FGM beams, a sensitivity study is conducted in order to access the effect of the value on the results. Some effects of axial load variations on frequencies are also presented

Harmonic Forcing of a Two-Segment Timoshenko Beam

This work treats the lateral harmonic forcing, with spatial dependencies, of a two-segment beam. The segments are compact so Timoshenko theory is employed. Initially the external transverse load is assumed to be spatially constant. The goal is the determination of frequency response functions. A novel approach is used, in which material and geometric discontinuities are modeled by continuously varying functions. Here logistic functions are used so potential problems with slope discontinuities are avoided. The approach results in a single set of ordinary differential equations with variable coefficients, which is solved numerically, for specific parameter values, using MAPLE®. Accuracy of the approach is assessed using analytic and assumed mode Rayleigh-Ritz type solutions. Free-fixed and fixed-fixed boundary conditions are treated and good agreement is found. Finally, a spatially varying load is examined. Analytic solutions may not be readily available for these cases thus the new method is used in the investigation

On the effects of non-homogeneous materials on the vibrations and static stability of tapered shafts

Shafts loaded by axial compressive forces constitute an area of considerable technical importance. The static stability and transverse vibrations of such shafts are the subjects of this work. Occasionally the shafts are tapered and of interest is the effect of employing functionally graded materials (FGMs), with properties varying in the axial direction, on the buckling load and lowest bending natural frequency. Here the shaft cross section is taken to be circular and three types of taper are treated: linear, sinusoidal and exponential. The shafts are assumed to have the same volume and length and to be subjected to a constant axial force. Euler–Bernoulli theory is used with the axial force handled by a buckling type approach. The problems that arise are computationally challenging but an efficient numerical strategy employing MAPLE®’s two-point boundary value solver has been developed. Typical results for a linear tapered pin–pin shaft where one end radius is twice the other, and the FGM model varies in a power law fashion with material properties increasing in the direction of increasing area, include doubling of the buckling load and first bending frequency increase of approximately 43%, when compared with a homogeneous tapered shaft