The efficiency and accuracy of common time and frequency domain methods that are used to simulate the response of a rotor system with malfunctions are compared and analyzed. The Newmark method and the incremental harmonic balance method are selected as typical representatives of time and frequency domain methods, respectively. To improve the simulation efficiency, the fixed interface component mode synthesis approach is combined with the Newmark method and the receptance approach is combined with the incremental harmonic balance method. Numerical simulations are performed for rotor systems with single and double frequency excitations. The inherent characteristic that determines the efficiency of the two methods is analyzed. The results of the analysis indicated that frequency domain methods are suitable single and double frequency excitation rotor systems, whereas time domain methods are more suitable for multifrequency excitation rotor systems.
Numerical simulation plays an important role in the study of rotor systems with malfunctions. With complex structures of the modern rotor systems (e.g., multidisc or parallel shafts) and the inherent strong nonlinear characteristics of malfunctions [
The numerical simulation methods commonly used in rotor dynamics can be divided into two types: time domain methods and frequency domain methods. The time domain methods mainly include the RungeKutta method [
When analyzing largescale rotor systems with complex structures, both the time domain and the frequency domain methods will encounter the challenge of solving a large system of equations with numerous interdependent variables. Therefore, a dimension reduction approach must be adopted. There are two kinds of dimension reduction approaches that are widely used: the dynamical substructure approach and the receptancebased approach. The former is often combined with time domain methods and the latter is often combined with frequency domain methods. The fixed interface component mode synthesis approach [
Both the types of methods have advantages and disadvantages. Frequency domain methods are highly efficient because they can skip the transient response and obtain the steady state response directly. However, these methods can only seek periodic and quasiperiodic responses. They also generally need prior information on the behavior of the system, such as which harmonic terms in the responses are dominant [
Modern rotor systems have various forms and excitation conditions. Therefore, selecting an appropriate numerical method with good precision and high efficiency is very important when analyzing a rotor system with fault. Few studies have compared the efficiency and accuracy of the time and frequency domain methods or discussed the application scope of these methods when applied to malfunctioning rotor systems. In this study, the two types of methods are compared and analyzed. Typical time domain and frequency domain methods are applied and compared under different conditions of excitation. The inherent characteristic that determines the efficiency of the two methods is analyzed. The scopes of application of the two kinds of methods are discussed.
The Newmark
The equation of motion of a malfunctioning rotor system can be written as
Letting
Based on known values of
Let
Equation (
Assuming that
Expanding (
With
There are four DOFs in one node and if the rotor system is made up of
The DOFs of the rotor system are separated into two distinct groups. The linear group
The nonlinear DOFs are fixed and the modes of the system comprised of the remaining DOFs are calculated. The modes that contribute most to the responses of the former rotor system are chosen to construct the fixed interface normal mode set:
Assuming unit displacement in each nonlinear DOF and setting the movement of the linear DOFs to zero, the constrained mode of the nonlinear DOFs can be obtained:
The constrained mode set can then be constructed from all the constrained modes:
The modal matrix of the dimensionreduced system can then be obtained by combining the normal mode set and the constrained mode set:
Using the modal matrix, (
The response
In the derivation process above, the dimension reduction mainly depends on the reduction of the normal modes. However, this may result in a loss of accuracy.
In the three commonly used frequency domain simulation methods (HB, DF, and IHB), the IHB method is equivalent to the HB method plus the NewtonRaphson method [
The equation of motion of a multifrequency excited rotor system is expressed as
When the rotor system is in steady state, the response
Assuming
Let
Substitute (
Let
Equation (
The receptance data is used in the dimension reduction strategy in the IHB method. The DOFs of the rotor system are separated into two distinct groups. The nonlinear group
When the nonlinear system is in steady state, the response vector
Assume that
Rewrite (
Assume that
Substituting (
For the fundamental harmonic term
From (
From (
The dimensions of (
Equation (
Since the characteristics of local nonlinearity are utilized, the harmonic terms of all DOFs of the system can be represented by the responses of the DOFs with nonlinearities. Therefore, only the nonlinear DOFs are retained and the computation speed increases greatly. In addition, the accuracy of the results is maintained.
A doublespan rotor is comprised of two rotors that are connected by a membrane coupling as shown in Figure
Shaft segment dimension of the twospan rotor system.
Segment  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15 


80  80  290  100  200  240  150  180  160  180  130  100  80  75  75 

140  180  210  220  235  250  250  300  240  235  230  190  160  140  140 


Segment  16  17  18  19  20  21  22  23  24  25  26  27  28  29  



75  75  62  360  90  390  300  500  300  700  80  360  62  100  

225  225  270  300  360  420  450  460  450  440  380  310  270  225 
The model of the twospan rotor system.
Assuming that the elastic modulus of the steel material is
Assuming that the full angular rub occurs on node 24 and that the clearance is
Figure
The rotor system response under single frequency excitation.
Comparison of the time response
Amplitudefrequency response curve
Stable and unstable time responses
The frequency domain methods combined with the arclength algorithm [
A comparison of the efficiency of the two methods is listed in Table
CPU time for the two methods under single frequency excitation.
Method  Time (s) 

Newmark, CMS order 5  14.08 
Newmark, CMS order 10  16.11 
Newmark, CMS order 20  19.76 
IHB, 
0.29 
A model of the dual rotor system of an aeroengine is shown in Figure
The dimensions of the inner rotor.
Segment  1  2  3  4  5  6  7  

Node  1  2  2  3  3  4  4  5  5  6  6  7  7  8 


Length (mm)  150  100  250  1000  250  100  150  
Diameter (mm)  100  500  100  100  100  500  100 
The dimensions of the outer rotor.
Segment  1  2  3  4  5  

Node  9  10  10  11  11  12  12  13  13  14 


Length (mm)  200  100  400  100  200  
External diameter (mm)  200  400  200  400  200  
Internal diameter (mm)  150  150  150  150  150 
Model of the dualrotor system.
Bearings I, II, and III are fixed with supports. Bearing IV is an intershaft bearing that connects the outer and inner rotors. The stiffness of the bearings is
The IHB method is compared with the Newmark method to predict the steady state responses of the dual rotor system. When the IHB method is applied, the harmonic terms are chosen based on the method by Ushida and Chua [
In Figure
The responses under double frequency excitation.
Comparison of time response
Amplitudefrequency response curve
Threedimensional spectrum diagram
The computational time for each method is listed in Table
CPU time for the two methods under double frequency excitation.
Method  Time (s) 

Newmark, CMS order 5  6.49 
Newmark, CMS order 10  8.85 
Newmark, CMS order 20  10.48 
IHB, 
34.7 
The simulation results above show that the frequency domain methods are much more efficient than the time domain methods in single frequency excitation condition. This is because of the following.
The frequency domain methods can skip the transient responses. They can obtain the steady state responses directly, while the time domain methods must devote computational time and resources to calculate the transient responses, especially when the damping is small.
The size of the instantaneous stiffness matrix is very small. When the number of the nonlinear DOFs and harmonic terms is small, the size of the instantaneous stiffness matrix in frequency domain methods is also small. In contrast, the size of the instantaneous stiffness matrix in time domain methods is mainly determined by the number of normal modes.
The results also show that the efficiency of the IHB method is very high for single frequency excitation but descends quickly for double frequency excitation. This is because of the following.
Calculating the multiple integral for double frequency excitation is timeconsuming. The Galerkin integration procedure in the IHB method for the double frequency excitation involves a multiple integral process, which is very timeconsuming. For example, for single frequency excitation, one period is divided into 100 subintervals in both the Newmark and IHB method. Therefore, the two methods complete the integration over the same time frame. However, for double frequency excitation, 10,000 subintervals must be created for the multiple integral of the IHB method, which results in much higher time costs than in the Newmark method.
The dimensions of instantaneous stiffness matrix are very high in double frequency excitation when using the IHB method. The dimensions of the instantaneous stiffness matrix increase with the increase in the number of harmonic terms, which is much higher for double frequency excitation than for single frequency excitation. For example, when
If the number of DOFs of the rotor system is larger with fewer nonlinear DOFs, and the damping is small, the frequency domain methods can still have a high efficiency. Otherwise, time domain methods are more efficient.
There are some rotor systems in practice, such as the geared supercharger, that experience three or more than three frequencies excitations. If frequency domain methods are applied, the multiple integral and the large instantaneous stiffness matrix will require too much computational time to determine the response. Therefore, time domain methods are most appropriate in those scenarios.
Time domain and frequency domain methods are used to predict the responses of rotor system with malfunctions. The efficiency and accuracy of the two types of methods are compared based on the excitation type. The main conclusions are as follows.
The dimension reduction approach in time domain methods is a substructure approach. The high order modes are truncated, which may affect the accuracy of the methods. Frequency domain methods that use the receptance data for dimension reduction can retain accuracy.
The efficiency of the frequency domain methods is determined by the number of excitation frequencies and the number of harmonic terms. These two factors determine the complexity of the numerical integration process and the size of the instantaneous stiffness matrix. Frequency domain methods are highly efficient for single frequency excitations but suffer a decline in efficiency for multiple frequency excitations. The size of the instantaneous stiffness matrix remains the same in time domain methods. Therefore, the efficiency remains the same for both single and multifrequency excitation.
Frequency domain methods are recommended for single frequency excitation. Time domain methods are recommended in cases where there are three or more frequency excitations. For double frequency excitations, the choice of simulation method depends on the specific conditions.
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors would like to gratefully acknowledge the National Basic Research Program of China (2011CB706504), the Natural Science Foundation of China (51475085), and the Fundamental Research Funds for the Central Universities (N120403007) for the financial support for this study.