Arnaldo Mazzei

Effects of internal viscous damping on the stability of a rotating shaft driven through a universal joint

Abstract A rotating flexible shaft, with both external and internal viscous damping, driven through a universal joint is considered. The mathematical model consists of a set of coupled, linear partial differential equations with time-dependent coefficients. Use of Galerkins technique leads to a set of coupled linear differential equations with time-dependent coefficients. Using these differential equations some effects of internal viscous damping on parametric and flutter instability zones are investigated by the monodromy matrix technique. The flutter zones are also obtained on discarding the time-dependent coefficients in the differential equations which leads to an eigenvalue analysis. A one-term Galerkin approximation aided this analysis. Two different shafts (“automotive” and “lab”) were considered. Increasing internal damping is always stabilizing as regards to parametric instabilities. For flutter type instabilities it was found that increasing internal damping is always stabilizing for rotational speeds v below the first critical speed, v1. For v>v1, there is a value of the internal viscous damping coefficient, Civ, which depends on the rotational speed and torque, above which destabilization occurs. The value of Civ (“critical value”) at which the unstable zone first enters the practical range of operation was determined. The dependence of Civ critical on the external damping was investigated. It was found for the automotive case that a four-fold increase in external damping led to an increase of about 20% of the critical value. For the lab model an increase of two orders of magnitude of the external damping led to an increase of critical value of only 10%. For the automotive shaft it was found that this critical value also removed the parametric instabilities out of the practical range. For the lab model it is not always possible to completely stabilize the system by increasing the internal damping. For this model using Civ critical, parametric instabilities are still found in the practical range of operation.

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.

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.

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 Youngs 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 Youngs 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.

Vibrations of Discretely Layered Structures Using a Continuous Variation Model

Recently, there has been a large body of work directed towards the use of non-homogeneous materials in controlling waves and vibrations in elastic media. Two broad categories have been studied, namely, media with continuous variation of properties and those with discrete layers (cells). Structures with both a finite and infinite number of cells (periodic layout) have been examined. For the former, direct numerical simulation or transfer matrix methods have been used. The current work focuses on one-dimensional cases, in particular a two-layer cell. Transfer matrix methods require writing solutions for each layer of the basic cell and then matching them across the interface, a process that can be quite lengthy. Here an alternate strategy is explored in which the discrete cell properties are modeled by continuously varying functions (here logistic functions), which has the advantage of working with a single differential equation. Natural frequencies have been obtained using a forced motion method and are in excellent agreement with those found using a transfer matrix approach. Mode shapes for the continuous variation model have been obtained using a finite difference scheme and compare well with those obtained via the transfer matrix approach.

Harmonic Forcing of a Two-Segment Euler-Bernoulli Beam

This study is on the forced motions of non-homogeneous elastic beams. Euler-Bernoulli theory is employed and applied to a two-segment configuration subject to harmonic forcing. The objective is to determine the frequency response function for the system. Two different solution strategies are used. In the first, analytic solutions are derived for the differential equations for each segment. The constants involved are determined using boundary and interface continuity conditions. The response, at a given location, can be obtained as a function of forcing frequency (FRF). The procedure is unwieldy. Moreover, determining particular integrals can be difficult for arbitrary spatial variations. An alternative method is developed wherein material and geometric discontinuities are modeled by continuously varying functions (here logistic functions). This results in a single differential equation with variable coefficients, which is solved numerically, for specific parameter values, using MAPLE®. The numerical solutions are compared to the baseline analytical approach for constant spatial dependencies. For validation purposes an assumed-modes solution is also developed. For a free-fixed boundary conditions example good agreement between the numerical methods and the analytical approach is found, lending assurance to the continuous variation model. Fixed-fixed boundary conditions are also treated and again good agreement is found.

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.

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 %.

Extracting Natural Frequencies of Layered Beams Using a Continuous Variation Model and Modal Analysis

This study involves the determination of the bending natural frequencies of beams composed by stacked cells of different materials. The focus is on cases with two cells. The analytical model is based on Euler-Bernoulli theory with the variations from one cell to another modeled via logistic functions. This approach leads to a single differential equation with variable coefficients, which is solved numerically using MAPLE®‘s differential equation solvers. A forced motion method is used. Forcing frequencies are changed until large motions and sign changes are observed, leading to the resonant frequencies. Of interest is the validation of the analytically obtained frequencies via experimental results. Here, an experiment is set in which a simple two-cell beam is analyzed, via modal analysis, in order to verify the analytically calculated frequencies. The beam is excited using an impact hammer and the response is recorded using accelerometers. Mode shapes are also obtained via digital image correlation. The beam, including distinct materials, is composed of one cell made of steel and another made of aluminum. The joining method is discussed and results for fixed-free beams are obtained.

How to Join Fiber-Reinforced Composite Parts: An Experimental Investigation

A coupler has been developed to prevent windshield wiper systems from being damaged by excessive loads that can occur when the normal wiping pattern is restricted. Unlike the traditional steel coupler used in wiper systems, the composite coupler will buckle at a prescribed compressive load threshold and become extremely compliant. As a result, the peak loading of the coupler and the entire wiper system can be greatly reduced. The coupler is composed of a pultruded composite rod with injection-molded plastic spherical sockets attached at either end. The sockets are used to attach the coupler to the crank and rocker of the windshield wiper linkage. Because the loads exerted on a coupler vary in magnitude and direction during a wiping cycle, the joint between the sockets and the pultruded composite rod must be robust. The paradigm for attaching sockets to steel couplers (i.e. over-molding the sockets around holes stamped into the ends of traditional steel couplers) was tested and found to produce inadequate joint strength. This paper details the methodology that was employed to produce and optimize an acceptable means to join the injection-molded sockets to the fiber glass pultruded rods. Specifically, a designed experiment based on the Robust Design Strategy of Taguchi was used to identify the process, processing parameters, and materials that yield a sufficiently strong joint at a reasonable manufacturing cost without damaging the integrity of the underlying composite structure.