Showing papers on "Critical speed published in 1996"

Dynamic Response of Planetary Trains to Mesh Parametric Excitations

Abstract: An extended three-dimensional model is used for calculating dynamic tooth loads on a planetary gear set. Time dependent mesh stiffnesses are determined and an original Ritz method aimed at solving large parametrically excited differential systems is proposed. Results from the Ritz method compare favorably with those given by direct integrations for highly reduced computation times. The difference between local critical speeds (for one individual mesh) and global critical speeds (for sun or ring gear-planet meshes) on a sequential spur gear train is pointed out. Finally, it is shown that, for linear behaviors, mesh stiffnesses are largely controlling dynamic tooth loads while the influence of a floating sun or ring gear is less important.

Analytical and Experimental Analysis of a Self-Compensating Dynamic Balancer in a Rotating Mechanism

Abstract: A theoretical and experimental approach was used to investigate the motion and effectiveness of a Self-Compensating Dynamic Balancer (SCDB). This is a device intended to minimize the effects of rotor imbalance and vibratory forces on a rotating system during normal operation. The basic concept of an automatic dynamic balancer has been described in many U.S. patents. The SCDB is composed of a circular disk with a groove containing massive balls and a low viscosity damping fluid. The objective of this research is to determine the motion of the balls and how this ball motion is related to the vibration of the rotating system using both theoretical and experimental methods. The equations of motion the balls were derived by the Lagrangian method. Static and dynamic solutions were derived from the analytic model. To consider dynamic stability of the motion, perturbation equations were investigated by two different methods: Floquet theory and direct computer simulation. On the basis of the results of the stability investigation, ball positions which result in a balance system are stable above the critical speed and unstable at critical speed and below critical speed. To determine the actual critical speed of the rotating system used in the experimental work, a modal analysis was conducted. Experimental results confirm the predicted ball positions. Based on the theoretical and experimental results, when the system operates below and near the first critical speed, the balls do not balance the system. However, when the system operates above the first critical speed the balls can balance the system.

Effects of slots on the lateral vibration of a circular saw blade

Abstract: In the majority of cases, wood is cut at a high feed rate and high rotational speed by circular saw blades with tungsten carbide tips brazed on the periphery of metal circular plates. Slots are made at several places from the periphery towards the center of a saw blade to prevent lateral vibration, both in idling and cutting, and to prevent a drop in critical rotational speed caused by heat expansion on the blade periphery during the cutting operation. In this paper, variations of lateral vibration mode, natural frequency and critical rotational speed of a slotted circular saw blade are investigated experimentally and by numerical calculations. The lateral deflection of a circular saw blade at the critical rotational speed and at the rotational speed near it is examined by a timber-cutting experiment.

Coupled Shaft-Torsion and Blade-Bending Vibrations of a Rotating Shaft–Disk–Blade Unit

Abstract: A new approach to analyzing the dynamic coupling between shaft torsion and blade bending of a rotating shaft-disk-blade unit is introduced. The approach allows the shaft to vibrate freely around its rotation axis instead of assuming a periodic perturbation of the shaft speed that may accommodate the shaft flexibility only to a limited extent. A weighted residual method is applied, and the receptances at the connections of blades and shaft-disk are formulated. Numerical examples are given for cases with between two and six symmetrically arranged blades. The results show not only coupling between the shaft, disk, and blades, but also coupling between individual blades where the shaft acts as a rigid support and experiences no torsional vibration. The blade-coupling modes occurred only in repeated frequencies. Finally, the effect of shaft speed on the modal frequencies was investigated. Plots illustrating the occurrence of critical speeds and flutter instabilities are presented.

Test Results of a New Damper Seal for Vibration Reduction in Turbomachinery

Abstract: A new type of labyrinth gas seal for damping vibration and whirl, called the TAMSEAL, has been evaluated in both non- rotating and rotating tests at Texas A&M University. Test results of the prototype, along with comparison tests of a conventional labyrinth seal, show up to one hundred times more direct damping than the conventional bladed seal. The new design also has a feature that blocks swirl of the working fluid, which is known to be rotordynamically destabilizing in machines with conventional seals. Coastdown tests of the new seal were conducted at various pressures on a rotordynamic test apparatus with a critical speed at 4000 rpm and compared with identical testing of a conventional labyrinth seal. Rap tests of both seals were also conducted to measure the logarithmic decrement of free vibration, and the leak- age of both seals was measured. Test results show large reductions in peak vibration at the critical speed in all cases, with the critical speed being completely eliminated by the TAMSEAL at some pressure drop conditions. The leakage rate of the tested TAM- SEAL is higher than the conventional seal at the same clearance, but the large reductions in vibration and whirl amplitudes suggest that the TAMSEAL could be operated with smaller clearances than conventional labyrinth seals.

Method for determining rotational speed from machine vibration data

Abstract: A method of determining the rotational speed of a rotating shaft from machine vibration data. The vibration produced by the shaft is sensed to produce a test vibration signal at an unknown rotational speed of the shaft, which test vibrational signal is converted to a test frequency spectrum. A reference frequency spectrum, corresponding to a known rotational speed of the shaft, is provided, and a stretch factor is determined. The stretch factor provides optimum correlation between the test frequency spectrum and the reference frequency spectrum. The unknown speed of the rotating shaft is calculated using the known speed of the reference frequency spectrum and the stretch factor.

Draft-speed relationship of simple tillage tools at high operating speeds

Abstract: Field experiments were conducted with five different blades operated vertically in sandy loam and clay soil types up to a speed of 18 m s −1 to determine the draft-speed relationships at high speeds. The soils were Calcic Chernozem and Haplic Kastanozem according to FAO classification. Mechanical properties of the soils were determined to correlate the results with model predictions. The results showed that the draft increased less above a critical speed range of 3 to 5 m s −1 . This critical speed range was compared with: the propagation speed of a longitudinal pressure wave in the soil, the speed of the soil particles caused by this pressure wave in the soil, and the speed of disruption of the soil. It was shown that these speeds are related to the observed critical speed range. As previously proposed, the amount of soil deformation due to tillage would decrease above the critical speed range. It is, therefore, important to examine the dynamic effect of tillage at higher operating speeds.

Torsional Instabilities in a System Incorporating a Hooke’s Joint

Abstract: Dynamic stability analysis of a driveline which incorporates a Hooke ’s joint is presented in this paper. In particular, torsional instabilities due to fluctuating angular velocity ratio across the joint are examined. The method of averaging is used to establish the critical speed ranges by considering the linearized equations which govern the torsional motion of the system. Within these ranges, parametric instability characterized by exponential build up of response amplitudes of the torsional modes occur. Closedform conditions for onset of sub-harmonic as well as sum-type combination resonance have been established. The significance of the sum-type combination resonance in a typical diesel hydraulic locomotive driveline has been demonstrated. Difference-type combination resonance has been shown to be absent. The instability conditions indicate the range of driveshaft speeds to be avoided during the design of a driveline which employs a Hooke’s joint.

Forced oscillations of a vertical continuous rotor with geometric nonlinearity

Abstract: Nonlinear forced oscillations of a vertical continuous rotor with distributed mass are discussed The restoring force of the rotor has geometric stiffening nonlinearity due to the extension of the rotor center line The possibility of the occurrence of nonlinear forced oscillations at various subcritical speeds and the shapes of resonance curves at the major critical speeds and at some subcritical speeds are investigated theoretically Consequently, the following is clarified: (a) the shape of resonance curves at the major critical speed becomes a hard spring type, and (b) among various kinds of nonlinear forced oscillations, only some special kinds of combination resonances have possibility of occurrence

Modeling and Analysis of the Intermediate Shaft Between Two Universal Joints

Abstract: Unlike a single universal joint, in which the output shaft rotates with a fluctuating speed depending on the misalignment angle, a double universal joint is used to connect two offset shafts. If a double universal joint is correctly arranged, with equal joint angles in perfect alignment, the output shaft rotates in a synchronous motion with uniform speed. However, in practice, the intermediate shaft of a double universal joint system suffers from bending moments induced by joint friction and velocity fluctuation induced by the joint angles. These factors distort the linear relationship between the input and output shafts. In this paper, the effects of joint angles and joint friction on the steady-state responses of a double universal joint system are examined. The mass matrix, damping matrix, stiffness matrix, and nonconservative force of a universal joint are derived analytically for finite element modeling. The formulation introduced generates an elementary matrix suitable for the analysis of complex rotor-bearing systems that include double universal joints.

Poloidal rotation of tokamak plasmas at super poloidal sonic speeds

Abstract: A tokamak plasma rotating poloidally at speeds in excess of the poloidal sound speed is studied. The study is motivated by the propositions that shear in the rotation could suppress turbulence and improve confinement and that the poloidal sound speed is the critical speed to exceed for flow profiles distributed over the entire minor radius. For such rapidly rotating plasma regimes, the Grad-Shafranov equation is examined, the damping rate of the rotation is calculated, particle orbits are investigated, and a heuristic MHD stability study is done. It is found that for rotation speeds exceeding the poloidal sound speed but less than the poloidal Alfven speed no deleterious effects can be expected from the rapid rotation as regards equilibrium or MHD stability. The damping rate of poloidal rotation is shown to fall off over the peak rate, in all collisionality regimes, as up-2. This feature makes more efficacious the external driving of such poloidal flow and allows favourable extrapolation to reactor relevant regimes. Finally, it is shown that the fraction of trapped ions is significantly depleted in this regime, suggesting a reduction in the neoclassical heat transport as well

The Role of Non-Linearities in the Dynamics of a Single Railway Wheelset.

Abstract: This paper presents the results of the analytical and semi-analytical investigation of a single railway wheelset. The non-linearities appear in the expression of the creep force law and in the flange contact model. Smoothing process is used to obtain a sufficiently smooth (C 4 ) creep force law. Above a certain critical speed, the stationary motion of the wheelset loses its stability. The methods of Hopf bifurcation theory provide the parameter domains where unstable periodic motions appear around the stable stationary motion, and also those domains where stable periodic motions exist above the critical speed. The amplitudes of these periodic motions are calculated in closed form and are checked by numerical simulation. In case of great amplitude increments for small speed variation, the non-linearity related to the flange contact becomes much more important than the non-linearity of the creep force law. By means of a linear creep force law and a flange contact modelled by a spring with dead band, almost one dimensional Poincare maps are constructed and presented on a realistic parameter domain. These maps describe and explain the often chaotic wheelset behaviour in a small range of the speed just above its critical value. Computer algebra is involved because of the complexity of the equations of motion.

Effect of Intermittent Contact on the Stability of Thermoelastic Sliding Contact

Abstract: In many sliding systems, the sliding surfaces are not coextensive, so that points on one surface experience alternating periods of contact and separation. This intermittent process can be expected to influence the sliding speed at which the system is susceptible to frictionally-induced thermoelastic instability (TEI). This question is explored in the context of a simple system consisting ofa rotating thin-walled cylinder whose end face slides against a rigid surface. The results show that at low Fourier number-i.e., when the frequency of the process is high compared with the thermal transient of the system-only the time-averaged frictional heat input is important and the critical speed is an inverse linear junction of the proportion of time in sliding contact. At higher Fourier number, lower critical speeds are obtained, but the dependence on Fourier number is relatively weak.

Using Finite Elements to model circular saw roll tensioning

Abstract: The objective of this study is to demonstrate that the finite element method can be used to model tensioning in circular sawblades. The finite element analysis is used to determine the residual stresses in the blade; the introduction of these stresses into a modal analysis allows us to predict the dynamic behavior (natural frequencies and critical speed) of the sawblades.

Determination of Blade Stresses Under Constant Speed and Transient Conditions With Nonlinear Damping

Abstract: Determination of resonant stresses is an important step in the life estimation of turbomachine blades. Resonance may occur either at a steady operating speed or under transient conditions of operation when the blade passes through a critical speed. Damping plays a significant role in limiting the amplitudes of vibration and stress values. The blade damping mechanism is very complex in nature, because of interfacial slip, material hysteresis, and gas dynamic damping occurring simultaneously. In this paper, a numerical technique to compute the stress response of a turbine blade with nonlinear damping characteristics, during steady and transient operations of the rotor, is presented. Damping is defined as a function of vibratory mode, rotor speed, and strain amplitude. The technique is illustrated by computing the stress levels at resonant rotor speeds for typical operation of a turbomachine.

High-speed circular sawing using temporary tensioning

Abstract: Workpiece feed speed can be increased while maintaining product dimension accuracy by proportionately increasing saw rotation speed. However, the required saw speed increase is acceptable only if the speed does not approach the saw critical speed. Tensioning can be used to increase critical speed, but the amount of tensioning possible is limited by saw “dishing.” This dishing makes saw leveling very difficult. This study demonstrates the concept of using temporary tensioning to overcome the dishing limitation. Normally, the saw is only moderately tensioned. However, after installation in the sawing machine, the tensioning can be increased temporarily, here by thermal means. In this way, the saw is highly tensioned while in operation, but only lightly tensioned during maintenance. The preliminary experiments presented in this study demonstrate this concept, and confirm that a workpiece feed speed increase of more than 50% is possible by using temporary tensioning.

A study on free vibration of a spinning disk

Abstract: Without a logical jump, we have derived the governing equation for free vibration of a spinning circular disk by using the variational formulation based upon the Kirchhoff plate theory and von Karman strain one. It has been found during the derivation that the governing equation is theoretically valld under the assumption that in plane deflections are steady and axisymmetric, and that internal forces are linearized while the strains remain nonlinear. The natural frequencies and the critical speeds of a freely spinning disk are obtained approximately and their dependencies on the spinning speed, mode number, and natural freqeuncy of the stationary disk are analyzed.

Thermal Vibration Balancing Method for Turbine Generator Rotor. Rotor Vibration Analysis Method due to Thermal Unbalance.

Abstract: A middle-load thermal power turbine generator frequently exceeds its critical speed, and it must be balanced at the critical speed carefully. Also, it is desirable to thermally balance the generator rotor to reduce the shaft vibration stroke caused by a load change at rated speed. In particular, when the generator capacity exceeds 500MVA, the generator rotor shifts from the secondary mode to the tertiary mode, and the in-phase unbalanced sensitivity increases due to the thermal bending of the shaft and other causes. As a result, thermal balancing is required in addition to conventional mechanical balance. This paper deals with 3 items. The first is the three-dimensional rotor vibration analysis method considered as asymmetry bearing supporting condition and thermal and magnetic unbalanced force. The second is to clarify bearing support condition using the actual rotor vibration response. The last is to explain an agreement between theory and experiment for rotor vibration phenomenon due to an artificial field coil layer short.

Self-diagnostic method for angular speed sensor

Abstract: PROBLEM TO BE SOLVED: To self-diagnose the vibration system of an angular speed sensor by driving a vibratory mass for a predetermined time and measuring the damping vibration of vibratory mass after stopping the driving. SOLUTION: In an angular speed sensor for detecting Coriolis force being generated in the direction of second axis of an orthogonal coordinate system when a vibratory mass is driven in the first axial direction and an angular speed is applied about the third axis, the vibratory mass is driven for a predetermined time and the damping vibration thereof is measured after stopping the driving thus self-diagnosing the vibration system of an angular speed sensor. Consequently, the amplitude and frequency of vibration system can be calculated. For example, the vibration frequency can be determined by measuring the generation time interval of zero-displacement point of vibration mass and the zero-displacement point can be determined based on the output from a C-V converter when the vibratory mass is stopped. Relation between the damping constant and natural angular speed at the time of no damping can be determined from the measurements of vibration frequency. COPYRIGHT: (C)1997,JPO

Stability Analysis of Symmetrical Rotor-Bearing Systems With Internal Damping Using Finite Element Method

Abstract: This paper deals with the stability analysis of self-excited bending vibrations of linear symmetrical rotor-bearing systems caused by internal damping using the finite element method. The rotor system consists of uniform circular Rayleigh shafts with internal viscous damping, symmetrical rigid disks, and discrete undatnped isotropic bearings. By combining the sensitivity method and the matrix representation of the rotor dynamic equations in complex form to assess stability, it is proved theoretically that the stability threshold speed and the corresponding whirling speed coincide with the first forward critical speed regardless of the magnitude of the internal damping.

Rotordynamic experiments on composite shafts

Abstract: Rotordynamic experiments are performed on two hollow tubular Carbon/Epoxy filament wound composite shafts. The experiments comprise of determination of critical speed and unbalance response as well as orbital and spectrum analysis. Step-wise progressive balancing near the critical speed was needed to operate the shaft in the super-critical region. Critical speeds determined from experiments match extremely well with theoretical values obtained from the equivalent modulus beam theory.

Composite Driveshafts: Technology and Experience

Abstract: Trucks, buses, automobiles and industrial equipment are negatively affected by the intrinsic weight, vibrational characteristics and critical speed of metal driveshafts. It has been proven that composite driveshafts are effective in overcoming these limitations. Indeed, the very nature of the composite materials (fiber and resinous binder) allows driveshafts to be designed to meet specific critical operational characteristics, and thus tailored to match the requirements of individual applications. 1.0 SUMMARY AND BACKGROUND Weight, vibrational, fatigue, and critical speed limitations have been recognized as serious problems in automotive and industrial drivetrains for many years. The associated effects and possible solutions have been subjected to detailed analysis. Numerous solutions such as flywheels, harmonic dampers, multiple shafts with additional bearings, and heavy rubber shock (vibration) absorbers have shown limited success in overcoming the problems, but always at the expense of increased weight, rotational inertia, and resistance in the drivetrain. Composite tubing has long been recognized to offer the potential of lighter weight driveshafts. Aerospace development efforts also demonstrated that correctly designed composite components have inherently superior fatigue and vibration damping characteristics to metals. Finally, the advent of higher modulus graphite fibers combined with these lighter weight and vibration damping characteristics allowed the design of driveshafts with much higher critical speed capabilities. These improvements have been realized and the reliability of composite driveshafts has been proven in heavy trucks, on race tracks, in automobiles and light trucks, and in industrial applications. ACPT, Inc. has been designing and producing carbon fiber composite driveshafts for these applications since 1982. 2.0 DRIVETRAIN VIBRATION PROBLEMS Vibration in drivetrains has been recognized as a major problem and has for many years been the subject of much theoretical analysis and trial-and-error vibrational control/reduction experimentation. 2.1 TRUCKS Mazziotti 1 in 1960 published a review and analysis of torsional vibrations associated with drivelines. He delineated some of the sources of non-uniform motion, which result in vibrational excitation of the drivetrain and presented a detailed mathematical analysis relating those sources of excitation to the physical dimensions, mechanical properties, and rotational speeds of driveshafts. He reported data firmly establishing the relationships between non-uniform motion sources and the natural frequency of the driveline components. He concluded that vibrations can be amplified or subdued while being transmitted through the driveline and recommends that (with metal shafts) the driveline be operated at no less than 1.5 times the natural frequency (torsional) of the system. Herein it was assumed that rubber springs, flywheels, flexible couplings and other natural frequency reduction additions were the best way to modify the natural frequency. In 1960 the technology did not exist to design or produce carbon fiber composite driveshafts. Mazziotti also stated that, “a resonant condition can produce objectionable disturbances as follows: 1. The high oscillating torque value can result in failure in rotating members. 2. Variable reactions on supporting members can be a source of objectionable noise and vibration. 3. Damage to gears, bearings, and other components can occur because of non-uniform loading.” All of these predictions have been proven to be accurate and are still sources of aggravation for truck designers, builders and operators. In recognition of, and in order to assist in the design of better truck drive systems, SAE paper #942322, 2 published by Spicer, division of Dana, describes a detailed torsional analysis computer simulation of truck drivetrains. The paper supports Mazziotti’s work and concludes, “torsional vibrations cause comfort problems for occupants and produce component failures.” “Torsionals also introduce dynamic loads on top of the mean static torque transmitted through the power train.” “...could easily cause catastrophic dynamic fatigue failures.” “At least responsible for wear problems at springs, splines, gear teeth, etc, eventually leading to the failure of these components.” This paper also presents a fairly detailed list of references on the subject of vibrations and their effect on automotive drivelines. Higher specific modulus (modulus/density) gives carbon fiber shafts the ability to run longer one piece lengths than metal shafts. A composite shaft, of the same length as a metal shaft, will start to resonate laterally at a much higher speed and have correspondingly increased margin of safety at the higher RPMs. This allows one piece composite shafts to replace two piece steel shafts. The benefits of eliminating the two piece shafts are significant reductions in weight, noise, vibration and harshness. The composite shafts have also proven to dampen vibration and absorb shock, greatly reducing wear on other drivetrain components as well as increasing tire traction. In a test started in August of 1994, a one piece carbon fiber driveshaft was installed in a garbage truck operating in Texas, Figure 2-1 and 2-2. This shaft replaced a two piece steel shaft and a center bearing, Figure 2-3. The resultant weight savings was about 80 lbs. The shaft has now seen daily use for the last two years in what has been described as one of the most torturous commercial truck applications possible. Absolutely no problems have been recorded. The operator is maintaining records of repairs with which to compare histories with other driveline components of identical trucks in the fleet. It is still too early to draw any conclusions from these records. It can be said, however, that the carbon fiber shaft is performing flawlessly. Figure 2-1: Garbage Truck Operating with One Piece Carbon Fiber Driveshaft The garbage truck shaft replacement was a cooperative effort with Inland Empire Driveline Services, Ontario, CA. They have been instrumental in helping us to develop and supply carbon fiber driveshafts to the automotive community. Inland has successfully developed and employs aluminum welding techniques, which resulted in moving torque testing failures from the weld joint to the U-joint. They have provided constant support in obtaining information and hardware mating carbon fiber tubing to the correct driveshaft fittings. Inland Empire Driveline Services continues to be of great technical assistance and currently performs all of the welding and balancing required in the manufacture of our carbon fiber driveshafts. Other direct experience is being obtained from OEM’s. Three major OEMs have run composite driveshafts on test tracks and have performed extensive laboratory evaluations. One of these companies is now running “on the road” fleet evaluations. 2.2 AUTOMOTIVE AND LIGHT TRUCKS The vibration problems in automobiles and light trucks are not so much “catastrophic failures”, but rather more of weight, noise, harshness and passenger discomfort. Component failure can still be a problem at high speeds, as natural resonant frequencies of the driveshaft are approached. GKN 3 states that, “the Mark VIII is top speed limited by its long steel driveshaft to 128 mph. Above 128 mph the driveshaft gets into a bending and vibration frequency that would eventually tear it apart.” They continue that, “to eliminate this problem most high speed European cars usually have a two piece shaft connected through a center bearing.” Carbon fiber driveshafts can alleviate this problem. 2.3 INDUSTRY Heavy duty industrial drivetrains, such as pump shafts and cooling tower driveshafts, suffer from similar torsional vibrations, natural frequency, and critical speed problems as heavy duty trucks and buses. The industrial community has demonstrated that composite driveshafts will reliably solve these problems. Figure 2-2: One Piece Carbon Fiber Driveshaft, Garbage

Thermal Balancing Method For Turbine Generator Rotor. Factor of Thermal Vibration Unbalance and Balancing Method.

Abstract: A middle-load thermal power turbine generator frequently exceeds its critical speed, and it must be balanced at the critical speed carefully. Also, it is desirable to thermally balance the generator rotor at a factory to reduce the shaft vibration stroke caused by a load change at rated speed. In particular, when the generator capacity exceeds 500MVA, the generator rotor shifts from the secondary mode to the tertiary mode, and the in-phase unbalanced sensitivity increases due to the thermal bending of the shaft and other causes. As a result, thermal balancing is required in addition to conventional mechanical balance. This study presents a method for adjustment in order to decrease the thermal vibration stroke. By analyzing the kind of thermal cause in vibration phenomena and diagnosing the rotor condition with a vibration measuring equipment, it was possible to predict the cause of the thermal vibration and decrease its stroke. By this method, a satisfactory vibration level was achieved for 600MW∼700MW class turbine generators.

The transfer matrix — component mode synthesis for rotordynamic analysis —

Abstract: The transfer matrix–component mode synthesis has been developed for the analysis of critical speed, response to imbalance and rotordynamic optimal design of multi–spool rotor system. This method adopted the advantages of the transfer matrix method for the train structure and the component mode synthesis for reducing degrees of freedom. In this method, the whole system is divided into several subsystems at the boundary coordinates. The constrained vibration modes and the static deflection curves of the constrained rotor subsystems are analysed by the improved transfer matrix method. The whole system is connected together by the component mode synthesis in accordance with the coordinate transformation. Numerical examples show that this method is superior to the traditional transfer matrix method and the component mode synthesis by FEM. This method has been successfully used for the rotordynamic analysis and optimal design of the compressors and the gas turbine aeroengines.Copyright © 1996 by ASME

Unbalance Response of a Multi-Mode Rotor Supported on Short Squeeze Film Dampers

Abstract: This paper describes a fast algorithm to obtain the steady state unbalance response of a multi-mode rotor supported on short squeeze film dampers (SFDs). The presented algorithm is developed based on planar modal theory. Undamped critical speed analysis is first performed to obtain the rotor critical speeds and their associated mode shapes. The modal analysis technique is then applied to the linear part of the rotor-SFD assembly to obtain the system differential equations. The rotor is assumed to execute circular centered orbits, hence all differential equations are reduced to algebraic ones. The resulting equations are manipulated algebraically to form a polynomial in rotor rotational speed. The roots of the polynomial are found and the full unbalance response is obtained. A conventional rotor is used to describe the developed algorithm numerically. Results show that the proposed algorithm gives accurate response in comparison to that obtained by integrating the system differential equations numerically. The great advantage of the proposed algorithm is the saving in the execution time which is extremely dramatic with respect to numerical integration, in addition to other advantages such as the possibility of obtaining all solutions occurring in regions of multiple steady state. Accuracy and speed of execution are quite advantageous regarding parametric studies on multi-mode rotors. These parametric studies can help in the optimization of SFDs design.© 1996 ASME