Cantilever Blade Research Articles

Reduced Order Modeling and Vibration Analysis of Mistuned Bladed Disk Assemblies With Shrouds

This paper presents important improvements and extensions to a computationally efficient reduced order modeling technique for the vibration analysis of mistuned bladed disks. In particular, this work shows how the existing modeling technique is readily extended to turbomachinery rotors with shrouded blades. The modeling technique employs a component mode synthesis approach to systematically generate a reduced order model (ROM) using component modes calculated from a finite element model (FEM) of the rotor. Based on the total number of degrees of freedom, the ROM is typically two or three orders of magnitude smaller than the FEM. This makes it feasible to predict the forced response statistics of mistuned bladed disks using Monte Carlo simulations. In this work, particular attention is devoted to the introduction of mistuning into the ROM of a shrouded assembly. Mistuning is modeled by projecting the mistuned natural frequencies of a single, cantilever blade with free shrouds onto the harmonic modes of the shrouded blade assembly. Thus, the necessary mistuning information may be measured by testing individual blades.

The creep problem in the VIRGO suspensions: a possible solution using Maraging steel

Each optical component of the interferometric gravitational wave detector VIRGO is suspended from a cascade of mechanical filters designed to suppress the transmission of seismic vibrations. Each mechanical filter supports the weight of the filters below it by means of a set of steel cantilever blade springs. The stress from the load acting on the blades was found to induce a drooping of the blade tips of several microns per day due to a series of microscopic yielding events (micro-creep). This process induces a mechanical displacement shot-noise on the optical component which can dominate the small displacements produced by gravitational waves. The use of a special precipitation hardened steel (Maraging C250), instead of common spring steel, allows the construction of blades that show an acceptable stability under stress.

Extending the VIRGO gravitational wave detection band down to a few Hz: metal blade springs and magnetic antisprings

The detection band of the interferometric gravitational wave detector VIRGO can be extended down to a few Hz by suspending each optical component of the interferometer from a chain of mechanical filters designed to suppress the transmission of seismic vibrations. Each mechanical filter supports the weight of the stages below it through a set of cantilevered blade springs. A system of permanent magnets, providing an “antispring” force, helps to reduce the highest vertical resonance of the chain from 7 Hz to about 2 Hz. This improvement allows VIRGO to reduce the frequency detection threshold from 10 Hz to about 4 Hz. A characterization of the mechanical filters is provided in this paper.

Flap-lag-torsion stability in hover and forward flight with three-dimensional wake

The contributions of full-wake dynamics in trim analysis are demonstrated for finding the control inputs and periodic responses simultaneously, as well as in Floquet eigenanalysis for finding the damping levels. The equations of flap bending, lag bending, and torsion are coupled with a three-dimensional, finite state wake, and low-frequency ( 1/rev) multiblade modes are considered. Full blade-wake dynamics is used in trim analysis and Floquet eigenanalysis. A uniform cantilever blade in trimmed flight is investigated over a range of thrust levels, advance ratios, number of blades, and blade torsional frequencies. The investigation includes the convergence characteristics of control inputs, periodic responses, and damping levels with respect to the number of spatial azimuthal harmonics and radial shape functions in the wake representation. It also includes correlation with the measured lag damping of a three-bladed untrimmed rotor. The parametric study shows the dominant influence of wake dynamics on control inputs, periodic responses, and damping levels, and wake theory generally improves the correlation.

Laser Vibrometry Measurements of Rotating Blade Vibrations

One of the most important design factors in modern turbomachinery is the vibration of turbomachinery blading. There is a need for developing an in-service, noncontacting, noninterfering method for the measurement and monitoring of gas turbine, jet engine, and steam turbine blade vibrations and stresses. Such a technique would also be useful for monitoring rotating helicopter blades. In the power generation industry, blade failures can result in millions of dollars of downtime. The measurement of blade vibrations and dynamic stresses is an important guide for preventive maintenance, which can be a major contributor to the availability of steam turbine, gas turbine, and helicopter operations. An experiment is designed to verify the feasibility of such a vibration monitoring system using the reference beam on-axis laser-Doppler technique. The experimental setup consists of two flat, cantilever blades mounted on a hub attached to the shaft of a dc motor. The motor rests on a linear bearing permitting motion only in the direction of the motor shaft. The motor and blade assembly is then excited via an electrodynamic shaker at the first natural frequency of the blades. The resulting blade vibration is then detected using a laser vibrometer. The vibration frequencies and amplitudes of the two rotating blades are successfully measured.

Vibration and damping analysis of layered isotropic twisted blades

A layered triangular plate element with six degrees of freedom per node is presented. This element has been used to analyse vibration and damping characteristics of pre-twisted cantilever blades. The damping layer is assumed to be made of a viscoelastic material, the elastic modulus of which is expressed in complex form. A parametric study is conducted to study the influence of twist angle, aspect ratio, layer thickness and skew angle on the natural frequencies and loss factors of the system. The variation in non-dimensional frequencies of a layered blade is similar to that of an unlayered blade for aspect ratios of 1 and 2. The loss factor decreases with an increase of the twist angle for the first flexural frequency.

Vibrating cantilever mass flow sensor

A key aspect of maintaining the correct ratio of air to fuel at the intake of fuel-injected internal combustion engines is measuring the mass flow (kg/s) of incoming air. In one method, the hotwire anemometer, utilizing the phenomenon of forced convection, has seen considerable development for this purpose. In this paper an alternate technique, using the effect of aerodynamic drag on a vibrating object, is explored. In particular, the drag damps the resonance oscillations of the vibrator in an amount proportional to the mass flow. The vibrators are small cantilever blades driven at resonance (1–5 kHz range) by piezoelectric actuators. Such factors as blade size, resonant frequency and gas density were experimentally explored. The technique is attractive because of its non-thermal nature and its potential for taking advantage of the economics of silicon batch fabrication methods.

Acoustically-driven cantilever for gas sensing

A thin cantilever blade attached to a ceramic piezoelectric bimorph is excited acoustically at resonance by a nearby piezoelectric driver. The amplitude of the induced vibration is electrically detected by the flexure induced in the bimorph. The amplitude falls monotonically toward zero as the pressure is reduced, allowing for pressure sensing over six orders of magnitude. Alternatively, at higher pressures, the acoustic pressure and corresponding induced amplitude depend strongly on the average molecular mass of the gas, allowing a measure of the latter parameter if the pressure is known. Applications are suggested, including the measurement of small concentrations of a heavy species in a lighter ambient ( e.g., gasoline vapor in air).

A vibrating cantilever magnetic-field sensor

A magnetic-field sensor is described in which a planar coil, photolithographically fabricated on a silicon cantilever blade, is caused to vibrate at a resonance frequency of the blade using a piezoelectric driver. In the presence of a magnetic field, an e.m.f. is induced in the coil proportional to the field strength. Operation is analogous to that of an a.c. generator. A reciprocal device, analogous to an a.c. motor, is also possible in which an a.c. current in the coil results in a vibration of the blade with amplitude proportional to the magnetic field and a corresponding output from the piezoelectric element. Applications, required electronics and advantages of this method are discussed.

Natural frequencies of cantilever blades with resilient roots

It is difficult to enforce infinite rigidity at the root of a cantilever during vibration tests. The “cantilever” frequencies measured from a single response curve are the resonant frequencies of the entire system, including the effects of indeterminate root conditions. In some structural systems, such as bladed disk assemblies, system resonances may occur in a cluster near a blade cantilever frequency, so that it is difficult to distinguish between assembly modes and the blade cantilever frequency. It is shown that in such cases, a more accurate determination of actual cantilever frequencies may be made by computing the ratio of tip and root displacements as a function of excitation frequency.

A Computer Analysis of Wind Turbine Blade Dynamic Loads

The flapping motion of a single wind turbine rotor blade has been analyzed and equations describing the flapping motion have been developed. The analysis was constrained to allow only flapping motions for a cantilevered blade, and the equations of motion are linearized. A computer code, called FLAP (Force and Loads Analysis Program), to solve the equations of motion and compute the blade loads, has been completed and compared to measured loads for a 3-bladed downwind turbine with stiff blades. The results of the program are presented in tabulated form for equidistant points along the blade and equal azimuth angles around the rotor disk. The blade deflection, slope and velocity, flapwise shear and moment, edgewise shear and moment, blade tension, and blade torsion are given. The deterministic excitations considered in the analysis include wind shear, tower shadow, gravity, and a prescribed yaw motion.

Impact and Bending of a Rigid-Plastic Fan Blade

A rigid-plastic fan blade is subjected to a transverse impact force at the tip in addition to radial centrifugal forces caused by fan rotation. Large and small deflection formulations for inplane bending of the cantilevered blade are calculated and compared. If the effects of centrifugal force are completely considered, the difference between these calculations is shown to be small within the beam. Most of the deflection originates from plastic deformation far from the tip so the large deflection effects near the tip have little influence. The largest curvature is inversely proportional to the centrifugal force, directly proportional to the third power of impact force, and always occurs near the point of impact.

Application of the Reissner method to derive the coupled bending-torsion equations of dynamic motion of rotating pretwisted cantilever blading with allowance for shear deflection, rotary inertia, warping and thermal effects

The dynamic Reissner functional in conjunction with variational calculus has been employed to derive the equations of motion of pretwisted cantilever blades of asymmetric aerofoil cross section. Shear deflection, rotary inertia, warping, thermal and rotational effects are considered, with the blade assumed to be mounted on a rotating disc at a stagger angle. It is observed that the thermal bending and twisting moments form an implicit set of additional terms in the respective conventional equations and further that these equations appear to be unwieldy when pretwist is present. A direct solution by using the dynamic Reissner functional appears to be far easier than the solution of the coupled equations in complex vibration problems.

Application of the reissner method to derive the coupled bending-torsion equations of dynamic motion of rotating pretwisted cantilever blading with allowance for shear deflection, rotary inertia, warping and thermal effects

The dynamic Reissner functional in conjunction with variational calculus has been employed to derive the equations of motion of pretwisted cantilever blades of asymmetric aerofoil cross section. Shear deflection, rotary inertia, warping, thermal and rotational effects are considered, with the blade assumed to be mounted on a rotating disc at a stagger angle. It is observed that the thermal bending and twisting moments form an implicit set of additional terms in the respective conventional equations and further that these equations appear to be unwieldy when pretwist is present. A direct solution by using the dynamic Reissner functional appears to be far easier than the solution of the coupled equations in complex vibration problems.

Coupled bending-bending vibrations of pretwisted tapered cantilever beams treated by the Reissner method

Theoretical natural frequencies and mode shapes of the first four coupled modes of a uniform pretwisted cantilever blade and the first five coupled flexural frequencies of pretwisted tapered blading are determined by using the Reissner method. The shape functions for the bending moments and deflections are developed in series form and with these used in the dynamic Reissner functional, the frequency equation is obtained by minimizing it through the Ritz process. A convergence study made in the case of the pretwisted uniform blade indicates that there appears to be a quicker convergence of the natural frequencies and that a five-term solution yields a set of results that are in good agreement with the theoretical and experimental values of other authors, available in the literature. The mode shapes obtained from the present analysis are compared with those from an earlier investigation and the effect of ignoring the shear deflection and rotary inertia in the analysis is discussed. The effects of breadth taper and depth taper on the vibration characteristics of pretwisted cantilever blading are discussed from the results obtained in the present limited study and it is observed that an extensive investigation appears to be necessary to draw positive conclusions covering wide ranges of pretwisted blade parameters.

Analysis of lateral vibrations of rotating cantilever blades allowing for shear deflection and rotary inertia by reissner and potential energy methods

The lateral vibrations of a uniform rotating blade have been analysed applying the Reissner and the potential energy methods. Shear deflection and rotary inertia are taken into account. A convergence study of the two methods is made and the effects of shear deflection, rotary inertia, rotation and stagger angle on the blade vibration characteristics are discussed. Comparison of the results indicates a quicker convergence and better mode shapes by the Reissner method than the classical potential energy method.

Coupled bending-bending vibrations of pre-twisted cantilever blading allowing for shear deflection and rotary inertia by the Reissner method

The Reissner method and the total potential energy approach are applied to a pre-twisted cantilever blade executing coupled bending-bending vibrations. Shear deflection and rotary intertia are accounted for in the analysis and the natural frequencies and mode shapes of the first four coupled modes are determined. A comparison of the results obtained from this investigation with those available in the literature indicates that the Reissner method gives quicker convergence and better mode shapes than the potential energy method and it is further seen that the inclusion of shear deflection and rotary inertia into the analysis leads to a reduction in the frequency values, thus resulting in a closer agreement with experimental results.

A Fiber-Optic Laser-Doppler Probe for Vibration Analysis of Rotating Machines

The problem of obtaining vibratory information from remote, rotating, components within a machine has been overcome by means of a laser-Doppler fiber-optic probe. Laser light is transmitted down the fiber-optic and scattered light is returned for analysis in the same way. The Twyman-Green mode of operation of the laser-Doppler system has been shown to have serious shortcomings for this type of work and a new mode has been produced, and used successfully, in conjunction with Bragg cells for frequency shifting. The use of a single set of Bragg cells to compensate for the average velocity of a rotating system, limits the allowable speed of rotation to about 1000 rpm. However, other techniques for frequency shifting are available. An experimental investigation is reported in which a rotating system, consisting of a disk carrying flat cantilever blades, was rotated at about 1000 rpm while one of the blades was excited at 163 Hz. The measured frequency of the vibrating blade was 162.8 Hz and similar accuracy was obtained for the amplitude measurements.

Stability of Hingeless Rotor Blades in Hover with Pitch-Link Flexibility

A stability analysis of a single cantilevered helicopter rotor blade in hover is presented. The blade is represented by an elastic uniform beam, cantilevered in bending and having a torsional root spring to simulate pitch-link flexibility. Nonlinear equations are adapted for a linearized stability analysis about the blade equilibrium operating condition. Numerical results are obtained for hingeless rotor configurations having pitchlink flexibility, precone, droop, twist, and flap-lag structural coupling. The results indicate that hingeless rotor stability characteristics are sensitive to changes in most configuration parameters. For a given torsion frequency, the effect of pitch-link flexibility is generally found to be similar to the effect of blade torisional flexibility. Droop and precone, although physically similar, exhibit different effects on stability when pitch-link flexibility is present. Twist is shown to influence the stability by altering the flap-lag structural coupling.

Influence of Modeling and Blade Parameters on the Aeroelastic Stability of a Cantilevered Rotor

A set of equations describing the coupled flap-lag-torsional dynamics of a cantilevered rotor blade in hover is presented. This set of equations is used to evaluate the influence of structural damping, preconing, and offsets on the linearized aeroelastic stability of some representative blade configurations. The sensitivity of the stability boundaries to the assumptions of approximate linear vs exact nonlinear static blade equilibrium position is considered. Finally, results with the distributed torsional representation of blade properties are compared with those obtained when the root torsional model is used.