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RESEARCH PAPERS

J. Vib., Acoust., Stress, and Reliab. 1986;108(2):117-124. doi:10.1115/1.3269311.

High-performance aircraft engine fan and compressor blades are vulnerable to aerodynamically forced vibrations generated by inlet flow distortions due to wakes from upstream blade and vane rows, atmospheric gusts, and maldistributions in inlet ducts. In this paper, an analysis is developed to predict the flow-induced forced response behavior of an aerodynamically detuned rotor operating in a supersonic flow with a subsonic axial component. The aerodynamic detuning is achieved by alternating the circumferential spacing of adjacent rotor blades. The total unsteady aerodynamic loading acting on the blading, due to the convection of the transverse gust past the airfoil cascade and the resulting motion of the cascade, is developed in terms of influence coefficients. This analysis is then utilized to investigate the effect of aerodynamic detuning on the forced response characteristics of a 12-bladed rotor, with Verdon’s Cascade B flow geometry as a uniformly spaced baseline configuration. The results of this study indicate that for forward traveling wave gust excitations, aerodynamic detuning is generally very beneficial, resulting in significantly decreased maximum amplitude blade responses for many interblade phase angles.

Commentary by Dr. Valentin Fuster
J. Vib., Acoust., Stress, and Reliab. 1986;108(2):125-131. doi:10.1115/1.3269312.

A comprehensive controlled study of the resonant vibratory structural response of a shrouded fan blade/disk due to known excitation was performed. A full circumferential definition of the inlet velocity field was obtained at five radial locations for three axial spacings and for four unique patterns of distortion and three mass flow rates. Harmonic analyses of the velocity patterns were used to establish a gust perturbation velocity normal to the blade chord. From these spanwise perturbation velocities, a normalized force parameter was established. In-vacuum, nonrotating testing of the fan assembly allowed identification of individual blade frequencies and system modes. This testing used strain gaging and holography to identify mistuning, damping and split factors for diametral patterns of the 3, 4, 5, and 6 diametral mode families. Dynamic strain signatures from rotating rig resonant responses were obtained for inlet flows having 3, 4, 5, and 6 distorted regions to simulate inlet struts. Groups of 1/4-in. rods were used to create these regions of distorted flow. System mode responses to these distortion patterns included occurrences of standing and traveling waves. Trends of gust perturbation force parameter and measured dynamic stress with varying axial gap, mass flow, and loading were established. The data collectively quantify the strong cause and effect relationship between force parameter and measured dynamic stress.

Commentary by Dr. Valentin Fuster
J. Vib., Acoust., Stress, and Reliab. 1986;108(2):132-139. doi:10.1115/1.3269313.

A model has been developed for studying the effect of mistuning on bladed disk vibration. Its unique feature is the extent of aerodynamic and structural interaction which it simulates can be readily varied from full coupling of all blades on the disk to coupling of each blade with only its nearest neighbors. Simulations utilizing the resulting algorithm shows that limited coupling models may be used to predict the statistical distribution of blade amplitudes that characterizes the mistuning effect, which in turn determines stage durability. This approach is used to study the effect of changing various system parameters on amplitude scatter. Gas density, the number of blades on the disk, disk stiffness, and the engine order of the excitation are considered. The results are used to draw some conclusions about how to improve laboratory tests and component design.

Commentary by Dr. Valentin Fuster
J. Vib., Acoust., Stress, and Reliab. 1986;108(2):140-149. doi:10.1115/1.3269314.

The effects of pretwist, precone, setting angle and Coriolis forces on the vibration and buckling behavior of rotating, torsionally rigid, cantilevered beams are studied in this investigation. The beam is considered to be clamped on the axis of rotation in one case, and off the axis of rotation in the other. Two methods are employed for the solution of the vibration problem: one based upon a finite-difference approach using second-order central differences for solution of the equations of motion, and the other based upon the minimum of the total potential energy functional with a Ritz type of solution procedure making use of complex forms of shape functions for the dependent variables. Numerical results obtained by using these methods are compared to those existing in the literature for specialized simple cases. Results indicating the individual and collective effects of pretwist, precone, setting angle, thickness ratio, and Coriolis forces on the natural frequencies and the buckling boundaries are presented and discussed. Furthermore, it is shown that the inclusion of Coriolis effects is necessary for blades of moderate-to-large thickness ratios while these effects are not so important for small thickness ratio blades. Finally, the results show the possibility of buckling due to centrifugal softening terms for large values of precone and rotation.

Commentary by Dr. Valentin Fuster
J. Vib., Acoust., Stress, and Reliab. 1986;108(2):150-154. doi:10.1115/1.3269315.

An expression to calculate the buffeting response of a multispan tube bundle with nonconstant linear mass density is derived by generalizing Powell’s joint acceptance concept. Application of the equation to lock-in vortex-induced vibration analysis is also discussed.

Commentary by Dr. Valentin Fuster
J. Vib., Acoust., Stress, and Reliab. 1986;108(2):155-164. doi:10.1115/1.3269316.

In this paper we primarily discuss a theory of power transmission and vibration energy distribution of dynamically loaded structures. The loads are random and the system comprises linked elements, which consist of machine-supported stiffened plates. Fundamentally, the theory is deterministic, but in addition it uses some features of the SEA. In fact, the analysis is intended to verify fundamental theorems of the Statistical Energy Analysis in the lower frequency range.

Commentary by Dr. Valentin Fuster
J. Vib., Acoust., Stress, and Reliab. 1986;108(2):165-170. doi:10.1115/1.3269317.

A digital signal processing technique is proposed for calculating the amplitude and phase modulation of the tooth meshing vibration of a gear from the signal average of the vibration with application to the early detection of local defects such as fatigue cracks. The importance of phase modulation in the detection of defects is confirmed.

Commentary by Dr. Valentin Fuster
J. Vib., Acoust., Stress, and Reliab. 1986;108(2):171-176. doi:10.1115/1.3269318.

The ability to analyze accurately the torsional-axial-lateral coupled response of geared systems is the key to the prediction of dynamic gear forces, shaft moments and torques, dynamic reaction forces, and moments at all bearing points. These predictions can, in turn, be used to estimate gear-tooth lives, shaft lives, housing vibrational response, and noise generation. Typical applications would be the design and analysis of gear drives in heavy-lift helicopters, industrial speed reducers, Naval propulsion systems, and heavy, off-road equipment. In this paper, the importance of certain linear dynamic coupling terms on the predicted response of geared rotor systems is addressed. The coupling terms investigated are associated with those components of a geared system that can be modeled as rigid disks. First, the coupled, nonlinear equations of motion for a disk attached to a rotating shaft are presented. The conventional argument for ignoring these dynamic coupling terms is presented and the error in this argument is revealed. It is shown that in a geared system containing gears with more than about 50 teeth, the magnitude of some of the dynamic-coupling terms is potentially as large as the magnitude of the linear terms that are included in most rotor analyses. In addition, it is shown that the dynamic coupling terms produce the multi-frequency responses seen in geared systems. To quantitatively determine the effects of the linear dynamic-coupling terms on the predicted response of geared rotor systems, a trial problem is formulated in which these effects are included. The results of this trial problem shows that the inclusion of the linear dynamic-coupling terms changed the predicted response up to eight orders of magnitude, depending on the response frequency. In addition, these terms are shown to produce sideband responses greater than the unbalanced response of the system.

Commentary by Dr. Valentin Fuster
J. Vib., Acoust., Stress, and Reliab. 1986;108(2):177-181. doi:10.1115/1.3269319.

A method is described for calculating critical speeds, unbalance response and damped natural frequencies of long rotors on a flexible foundation. The shaft and the foundation are calculated separately and coupled at the bearings through impedance matching. Included in the analysis is also a method for representing the shaft response by an expansion in its free-free modes.

Commentary by Dr. Valentin Fuster
J. Vib., Acoust., Stress, and Reliab. 1986;108(2):182-188. doi:10.1115/1.3269320.

A transfer matrix-direct integration combined method is proposed, which employs the transfer matrix method to derive the equations of motion of a “characteristic disk,” and uses the direct integration method to determine the critical speeds, modes and unbalance response of a rotor-bearing system, and to analyze its stability. Despite the complexity of the system, the number of governing equations is not greater than eight. For a single-spool rotating system, the number of equations is only four. A transfer matrix for a uniform shaft is derived to consider its distributed mass, moment of inertia and the effect of shearing force. An impedance matrix iteration method is proposed to consider the effect of a complicated bearing-supporting system on the rotor dynamics. Two examples are given, and the results agree satisfactorily with the experiments.

Commentary by Dr. Valentin Fuster
J. Vib., Acoust., Stress, and Reliab. 1986;108(2):189-196. doi:10.1115/1.3269321.

A theoretical analysis of the dynamics of a rotor-bearing system with a transversely cracked rotor is presented. The rotating assembly is modeled using finite rotating shaft elements and the presence of a crack is taken into account by a rotating stiffness variation. This stiffness variation is a function of the rotor’s bending curvature at the crack location and is represented by a Fourier series expansion. The resulting parametrically excited system is nonlinear and is analyzed using a perturbation method coupled with an iteration procedure. The system equations are written in terms of complex variables and an associated computer code has been developed for simulation studies. Results obtained by this analysis procedure are compared with previous analytical and experimental work presented by Grabowski.

Commentary by Dr. Valentin Fuster
J. Vib., Acoust., Stress, and Reliab. 1986;108(2):197-206. doi:10.1115/1.3269324.

When designing a rotor system it is frequently desirable to have at hand a set of design sensitivity coefficients which quantitatively predict a change in specific system characteristics to changes in design parameters. This paper presents eigenvalue sensitivity coefficients for the damped natural frequencies of whirl of general linear rotor system modelled by finite element discretization. In addition, a simple and direct method for calculation of the damped critical speeds is presented, which utilizes the eigenvalue sensitivity with respect to the spin speed. It is shown that the combination of design parameter and spin speed whirl frequency sensitivity coefficients may be used to also evaulate the damped critical speed sensitivity coefficients.

Commentary by Dr. Valentin Fuster
J. Vib., Acoust., Stress, and Reliab. 1986;108(2):207-212. doi:10.1115/1.3269325.

Axially moving materials (e.g., belts, chains, bandsaws, paper and magnetic tapes, etc.) arise in various mechanical systems, and their vibration and stability are often of engineering significance. Previous work describing the transverse vibration of axially moving materials is extended in this paper to include the elastic coupling between spans. The equations of motion are derived, an approximate solution method is developed and simulation results are presented. The simulation results predict the presence of a beating phenomenon at low transport velocities which is destroyed by higher velocities and/or tension differences in the two spans. The coupling mechanism described could potentially be exploited for vibration control.

Commentary by Dr. Valentin Fuster
J. Vib., Acoust., Stress, and Reliab. 1986;108(2):213-221. doi:10.1115/1.3269326.

The mechanism of tightening bolted joints and the stress distribution in the bolt are analyzed. Measurements were made of static and dynamic strength of joints. It is shown that all bolts behave elastically when external loads are applied to the joint even when the fastener was tightened to its torque-tension yield point. It is shown that joints generally fail when the external loads are sufficient to overcome the bolt preload. Fatigue reesults show that joint fatigue strength increases with preload, and that high fatigue bolts gave an improvement over standard fasteners at all preloads.

Commentary by Dr. Valentin Fuster
J. Vib., Acoust., Stress, and Reliab. 1986;108(2):222-229. doi:10.1115/1.3269328.

A common problem of fatigue failure of stamping dies was experienced during the stamping operation with socket-head screws. In order to establish a design standard for the stripper bolt, a methodology for determination of the loads and the fatigue strength of the stripper bolt was developed. Stresses due to an impulsive load and a rectangular pulse were calculated based on a simplified spring mass system and the appropriate corrections were made to elaborate the solution. This approximate solution was validated by a finite element analysis. The stripper bolt should have an infinite fatigue life to survive a half million stamping operations. The fatigue problem involves a stress concentration with combined mean and alternating stresses. The Gerber parabola and the residual stress method were employed to treat the combined loading and the stress concentration. In order to enhance the fatigue life of the stripper bolt, a cushion is introduced at the surface where an impact load is applied. The cushion is found very effective in improving the fatigue life of the stripper bolt. An interactive computer program was developed as a tool for designing stamping dies.

Commentary by Dr. Valentin Fuster

DISCUSSIONS

Commentary by Dr. Valentin Fuster

ERRATA

Commentary by Dr. Valentin Fuster

TECHNICAL BRIEFS

J. Vib., Acoust., Stress, and Reliab. 1986;108(2):230-231. doi:10.1115/1.3269329.

A method for minimizing forced harmonic vibration of a rotor-bearing system by the application of external control forces is presented. The frequency of the vibration is assumed known. In cases of mass unbalance or bend in the shaft this will be shaft rotation frequency and can usually be monitored without difficulty. The control forces could be provided by electromagnetic actuators. The control strategy presented does not require any knowledge of the system parameters and, provided the uncontrolled system is stable, cannot destablize the system. Results from a simulation are shown.

Commentary by Dr. Valentin Fuster

BOOK REVIEWS

J. Vib., Acoust., Stress, and Reliab. 1986;108(2):232-233. doi:10.1115/1.3269330.
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Commentary by Dr. Valentin Fuster
J. Vib., Acoust., Stress, and Reliab. 1986;108(2):233-234. doi:10.1115/1.3269331.
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Commentary by Dr. Valentin Fuster
J. Vib., Acoust., Stress, and Reliab. 1986;108(2):234-235. doi:10.1115/1.3269332.
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Abstract
Commentary by Dr. Valentin Fuster
J. Vib., Acoust., Stress, and Reliab. 1986;108(2):235-236. doi:10.1115/1.3269333.
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Commentary by Dr. Valentin Fuster

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