Transient Response Analysis of Damped Rotor Systems by the Normal Mode Method

1975 ◽  
Author(s):  
A. J. Dennis ◽  
R. H. Eriksson ◽  
L. H. Seitelman
Keyword(s):  
Steady State ◽  
Normal Mode ◽  
Transient Response ◽  
Transient Analysis ◽  
Equations Of Motion ◽  
Forced Response ◽  
Mode Shapes ◽  
Response Analysis ◽  
Rotor Systems ◽  
Incremental Formulation

A method to determine the transient response of damped single or multi-shaft rotor systems is presented. The rotor systems are idealized as rotating concentrated masses connected by massless beams, discrete springs, and dampers. The springs may have piecewise constant springs rates to simulate the stiffening effect of parts coming in contact after displacement through an initial offset. Arbitrary forcing functions are allowed. The method employs an incremental formulation in which damping gyroscopic and nonlinear terms are treated as external loads which are lagged in time. The equations of motion are uncoupled by performing a normal mode expansion of the response solution in terms of the non-rotating, undamped eigenvectors and their associated eigenvalues; modes and natural frequencies are obtained from a standard Prohl analysis. An analytical solution is used for each step of the incremental analysis. This technique has been used to study the response of a number of rotor systems to the sudden application of a rotating imbalance load. The systems studied include a dual shaft model of a rig, a single-shaft case from the written literature and a large multi-line (multi-shaft) system. The transient analysis was run out to steady-state and close agreement obtained with results from an independent steady-state forced response analysis. Orthogonality relations between the mode shapes were observed to be critical to the quality of the results. It was observed that transient analysis of multi-line systems can be accurately predicted only if the higher frequency modes which are participating in the response are included in the normal mode solution.

A Case Study of the Unsteady Response of a Hingeless Helicopter Rotor Blade

2019 ◽  
Author(s):  
Pratik Sarker ◽  
Uttam K. Chakravarty
Keyword(s):  
Mathematical Model ◽  
Steady State ◽  
Rotor Blade ◽  
Degrees Of Freedom ◽  
Forced Response ◽  
Helicopter Rotor ◽  
Mode Shapes ◽  
Response Analysis ◽  
Steady State Condition ◽  
Helicopter Rotor Blade

Abstract The helicopter is an essential means of transport for numerous tasks including carrying passengers and equipment, providing air medical services, firefighting, and other military and civil tasks. While in operation, the nature of the unsteady aerodynamic environment surrounding the rotor blades gives rise to a significant amount of vibration to the helicopter. In this study, the unsteady forced response of the Bo 105 hingeless helicopter rotor blade is investigated at the forward flight in terms of the coupled flapping, lead-lag, and torsional deformations. The mathematical model for the steady-state response of the rotor blade is modified to include the unsteady airfoil behavior by using the Theodorsen’s lift deficiency function for three degrees of freedom of motion. The nonlinear mathematical model is solved by the generalized method of lines in terms of the time-varying deflections of the rotor blade. The unsteady airloads are found to create larger deformations compared to that of the steady-state condition for a given advance ratio. The azimuth locations of the peak loadings also vary with different degrees of freedom. The first three natural frequencies and mode shapes of the rotor blade are presented. The model for the forced response analysis is validated by finite element results.

Accurate Interpolation of the Dependency of Modal Properties on the Rotation Speed for the Transient Response Analysis of Bladed Disks

2021 ◽  
Author(s):  
Jing Tong ◽  
Chaoping Zang ◽  
E. P. Petrov
Keyword(s):  
Transient Response ◽  
Rotation Speed ◽  
Spectral Composition ◽  
Forced Response ◽  
Reference Points ◽  
Mode Shapes ◽  
Response Analysis ◽  
Transient Vibration ◽  
Bladed Disks ◽  
Bladed Disk

Abstract During fast gas-turbine engine acceleration and deceleration the transient vibration effects in bladed disk vibration become significant and the transient response has to be calculated. In this paper an effective method is developed for efficient calculations of the transient vibration response for mistuned bladed disks under varying rotation speeds. The method uses the large-scale finite element modelling of the bladed disks allowing the accurate description of the dynamic properties of the mistuned bladed disks. The effects of the varying rotation speed on the natural frequencies and mode shapes of a mistuned bladed disk and its effects on the amplitude and the spectral composition of the loading are considered. The dependency of the modal characteristics on the rotation speed are based on the evaluation of these characteristics at reference points followed by the interpolation to obtain values at any rotation speed from the operating range. A new method has been developed for the interpolation of mode shapes while preserving the orthogonality and mass-normalization of the mode shapes. The method of mode shape interpolation is elaborated for tuned and mistuned bladed disks. The accuracy and efficiency of the method is demonstrated on test examples and on analysis of transient forced response of realistic bladed discs.

Accurate Interpolation of the Dependency of Modal Properties On the Rotation Speed for the Transient Response Analysis of Bladed Disks

2021 ◽  
Author(s):  
Jing Tong ◽  
Chaoping Zang ◽  
Evgeny Petrov
Keyword(s):  
Transient Response ◽  
Rotation Speed ◽  
Spectral Composition ◽  
Forced Response ◽  
Reference Points ◽  
Mode Shapes ◽  
Response Analysis ◽  
Transient Vibration ◽  
Bladed Disks ◽  
Bladed Disk

Abstract During fast gas-turbine engine acceleration and deceleration the transient vibration effects in bladed disk vibration become significant and the transient response has to be calculated. In this paper an effective method is developed for efficient calculations of the transient vibration response for mistuned bladed disks under varying rotation speeds. The method uses the large-scale finite element modelling of the bladed disks allowing the accurate description of the dynamic properties of the mistuned bladed disks. The effects of the varying rotation speed on the natural frequencies and mode shapes of a mistuned bladed disk and its effects on the amplitude and the spectral composition of the loading are considered. The dependency of the modal characteristics on the rotation speed are based on the evaluation of these characteristics at reference points followed by the interpolation to obtain values at any rotation speed from the operating range. A new method has been developed for the interpolation of mode shapes while preserving the orthogonality and mass-normalization of the mode shapes. The method of mode shape interpolation is elaborated for tuned and mistuned bladed disks. The accuracy and efficiency of the method is demonstrated on test examples and on analysis of transient forced response of realistic bladed discs.

A Method for Use of Cyclic Symmetry Properties in Analysis of Nonlinear Multiharmonic Vibrations of Bladed Disks

2004 ◽  
Vol 126 (1) ◽  
pp. 175-183 ◽  
Author(s):  
E. P. Petrov
Keyword(s):  
High Efficiency ◽  
Equations Of Motion ◽  
Linear Part ◽  
Forced Response ◽  
Mode Shapes ◽  
Solution Technique ◽  
Disk Model ◽  
Sector Model ◽  
Bladed Disk ◽  
Symmetry Properties

An effective method for analysis of periodic forced response of nonlinear cyclically symmetric structures has been developed. The method allows multiharmonic forced response to be calculated for a whole bladed disk using a periodic sector model without any loss of accuracy in calculations and modeling. A rigorous proof of the validity of the reduction of the whole nonlinear structure to a sector is provided. Types of bladed disk forcing for which the method may be applied are formulated. A multiharmonic formulation and a solution technique for equations of motion have been derived for two cases of description for a linear part of the bladed disk model: (i) using sector finite element matrices and (ii) using sector mode shapes and frequencies. Calculations validating the developed method and a numerical investigation of a realistic high-pressure turbine bladed disk with shrouds have demonstrated the high efficiency of the method.

A Successive Merging and Condensation (SMAC) Method Applied to Transient Analysis of Multi-Shaft Rotor System

1993 ◽  
Author(s):  
Santosh Ratan ◽  
Jorge Rodriguez
Keyword(s):  
Transient Response ◽  
Transient Analysis ◽  
Rotor System ◽  
Time Step ◽  
Transient Dynamic Analysis ◽  
Numerical Examples ◽  
Linear Coupling ◽  
Rotor Systems ◽  
Coupling Constraints ◽  
Smac Method

Abstract A method for performing transient dynamic analysis of multi-shaft rotor system is proposed. The proposed methodology uses the reported Successive Merge and Condensation (SMAC) method [12] and a decoupling technique to decouple the shafts. Multi-shaft rotor systems are treated as systems of many independent single shaft rotor systems with external unknown coupling forces acting at the points of couplings. For each time step, first, the SMAC method is used to get the transient response in terms of the unknown coupling forces. This is followed by the application of the coupling constraints to calculate the coupling forces and, in turn, the response at the end of that time step. The proposed method preserves the efficiency advantages of the SMAC algorithm for single-shaft rotor system. Numerical examples to validate and illustrate the applicability of the method are given. The method is shown to be applicable to linear and non-linear coupling problems.

An Efficient Response Analysis Method for a Non-Linear/Parameter Changing System Using Sub-Structure Modes

1993 ◽  
Vol 207 (1) ◽  
pp. 41-52
Author(s):  
H K Kim ◽  
Y-S Park
Keyword(s):  
Equations Of Motion ◽  
Forced Response ◽  
Suggested Method ◽  
Response Analysis ◽  
Synthesis Methods ◽  
Lumped Mass ◽  
State Equations ◽  
Mass Model ◽  
Non Linear ◽  
Forced Responses

An efficient state-space method is presented to determine time domain forced responses of a structure using the Lagrange multiplier based sub-structure technique. Compared with the conventional mode synthesis methods, the suggested method can be particularly effective for the forced response analysis of a structure subjected to parameter changes with time, such as a missile launch system, and/or having localized non-linearities, because this method does not need to construct the governing equations of the combined whole structure. Both the loaded interface free-free modes and free interface modes can be employed as the modal bases of each sub-structure. The sub-structure equations of motion are derived using Lagrange multipliers and recurrence discrete-time state equations based upon the concept of the state transition matrix are formulated for transient response analysis. The suggested method is tested with two example structures, a simple lumped mass model with a non-linear joint and an abruptly parameter changing structure. The test results show that the suggested method is very accurate and efficient in calculating forced responses and in comparing it with the direct numerical integration method.

Steady-State Characteristics of a Dual-Rotor System With Intershaft Bearing Subjected to Mass Unbalance and Base Motions

2018 ◽  
Author(s):  
Xi Chen ◽  
Mingfu Liao
Keyword(s):  
Steady State ◽  
High Efficiency ◽  
Equations Of Motion ◽  
Axial Rotation ◽  
Rotor System ◽  
Mode Shapes ◽  
Mass Unbalance ◽  
Moving Base ◽  
Lateral Rotation ◽  
Force Vectors

A dual-rotor system with an intershaft bearing subjected to mass unbalance and base motions is established. Using Lagrange’s principle, equations of motion for dual-rotor system relative to moving base are derived. Rotary inertia, gyroscopic inertia, transverse shear deformation, mass unbalance, and six components of deterministic base motions are taken into account. Using state-space vector, steady-state characteristics of dual-rotor system are analyzed through dual-rotor critical speed map, mode shapes, unbalance responses considering base rotations, frequency responses due to base motions, and shaft orbits. The results show that base translations just add external force vectors, while base rotations bring on parametric system matrices and additional force vectors. Base rotations not only change natural frequencies of dual-rotor system, but also break the symmetry of dynamic characteristics in the case of base lateral rotation. Excited by base harmonic translation, resonant frequencies correspond to whirl frequencies. The orbit remains circular under base axial rotation, while it becomes elliptical with a static offset under lateral rotation and then a complicated curve due to harmonic translation. When harmonic frequency of base translation gets close to dual-rotor excitation frequencies, obvious beat vibration appears. Overrall, this flexible approach can ensure calculation accuracy with high efficiency and good expandability.

scholarly journals Predicting the Dynamic Response of Dual-Rotor System Subject to Interval Parametric Uncertainties Based on the Non-Intrusive Metamodel

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 736 ◽  
Author(s):  
Chao Fu ◽  
Guojin Feng ◽  
Jiaojiao Ma ◽  
Kuan Lu ◽  
Yongfeng Yang ◽  
...  
Keyword(s):  
Steady State ◽  
Equations Of Motion ◽  
Computational Cost ◽  
Distribution Functions ◽  
Rotor System ◽  
Propagation Mechanism ◽  
Parametric Uncertainties ◽  
Scanning Method ◽  
Rotor Systems ◽  
Interval Uncertainties

In this paper, the non-probabilistic steady-state dynamics of a dual-rotor system with parametric uncertainties under two-frequency excitations are investigated using the non-intrusive simplex form mathematical metamodel. The Lagrangian formulation is employed to derive the equations of motion (EOM) of the system. The simplex form metamodel without the distribution functions of the interval uncertainties is formulated in a non-intrusive way. In the multi-uncertain cases, strategies aimed at reducing the computational cost are incorporated. In numerical simulations for different interval parametric uncertainties, the special propagation mechanism is observed, which cannot be found in single rotor systems. Validations of the metamodel in terms of efficiency and accuracy are also carried out by comparisons with the scanning method. The results will be helpful to understand the dynamic behaviors of dual-rotor systems subject to uncertainties and provide guidance for robust design and analysis.

Investigation of Damping Potential of Strip Damper on a Real Turbine Blade

2016 ◽  
Author(s):  
M. Afzal ◽  
I. Lopez Arteaga ◽  
L. Kari ◽  
V. Kharyton
Keyword(s):  
Boundary Conditions ◽  
Dynamic Properties ◽  
Equations Of Motion ◽  
Iterative Solution ◽  
Element Model ◽  
Forced Response ◽  
Contact Forces ◽  
Response Analysis ◽  
Bladed Disk ◽  
Different Types

This paper investigates the damping potential of strip dampers on a real turbine bladed disk. A 3D numerical friction contact model is used to compute the contact forces by means of the Alternate Frequency Time domain method. The Jacobian matrix required during the iterative solution is computed in parallel with the contact forces, by a quasi-analytical method. A finite element model of the strip dampers, that allows for an accurate description of their dynamic properties, is included in the steady-state forced response analysis of the bladed disk. Cyclic symmetry boundary conditions and the multiharmonic balance method are applied in the formulation of the equations of motion in the frequency domain. The nonlinear forced response analysis is performed with two different types of boundary conditions on the strip: (a) free-free and (b) elastic, and their influence is analyzed. The effect of the strip mass, thickness and the excitation levels on the forced response curve is investigated in detail.

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