As frequency-varying sine excitations in rotating machines are always emerging during run-ups and shutdowns, the multi-input-multi-output (MIMO) swept-sine test is of utter significance in product validation. At present, swept-sine vibration tests are mostly conducted with frequency-domain methods, where drive spectra are generated and updated by frequency response function (FRF), and drive signals are then generated with sinusoid oscillators. In this paper, a time-domain approach using an inverse system method based on a multistep prediction model is developed to realize the MIMO continuous swept-sine vibration test. First, the multistep prediction model of the original system is estimated in the time domain. Then, the inverse multistep prediction model is derived. After that, this model is truncated to guarantee the robustness of the inverse system and the smoothness of the generated drive signals. At last, the proposed method is validated by a simulation example with a cantilever beam and an actual test by using a three-axis shaker. The results show that the MIMO continuous swept-sine vibration test can be operated effectively by the proposed method.
Rotating machines employed in helicopters, propeller-driven airplanes, washing machines, and so on constantly produce swept-sine excitations to the whole structures during their run-ups and shutdowns. The vibrations induced by swept-sine excitations must be taken into consideration for the safety and the durability of the affected structures [
The MIMO swept-sine test applies frequency-varying sinusoid excitations upon a test article and controls the response spectra at certain points to be within the acceptable limits. As a simplified substitution for the MIMO swept-sine test, the MIMO stepped-sine test is sometimes used in literatures [
It is noted that the FRF is needed in these frequency-domain methods to connect the input and the output. However, theoretically this relationship is based on the assumption that both the input and output are of the same fixed frequency and in steady state. As this assumption is no longer valid in the swept-sine test, the relationship between the input and output should not be constructed by FRF in the frequency domain.
Hence, it seems more reasonable to control the swept-sine test by the time-domain method, even though the frequency-domain methods had been successfully used for years in engineering. There seems to be no examples of the MIMO swept-sine test controlled by the time-domain method in the existing literature, to the best of the authors’ knowledge.
Among time-domain approaches, the inverse system method is suitable to solve the problem at hand, as it directly takes the targeted system responses as its input to generate drive signals. This type of method is initially developed and applied in the force estimation area. As a typical inverse system method, the inverse structure filter (ISF) which utilized the inverse of the original system’s state-space model was introduced by Kammer and Steltzner [
Naturally, using the inverse system for drive signals generation in the vibration test is different from force estimation, where forces are estimated in a relatively short period of time as in most of the studies mentioned previously. Several recent studies on the MIMO random test conducted by Chen et al. [
The main contents of the paper are arranged as follows. In Section
The MIMO swept-sine vibration test imparts frequency-varying sinusoid excitations upon a test article and controls the response spectra at certain points. At these control points, response signals should match the predefined reference swept-sine signals with some tolerable errors within the given margins.
These reference swept-sine signals are given in the form of envelope
The change rate of phase
For the MIMO swept-sine vibration test, the amplitude
As mentioned in the introduction, stepped sine is often used as a simplified replacement of continuous swept sine, where a number of frequencies are defined prior to the test according to each time step
Therefore the stepped-sine test is flawed despite its simplicity. The key problem is that the stepped-sine test cannot achieve continuous sweep frequency and hence cannot reflect the actual continuous change process of rotating machinery. This paper focuses on solving this problem.
The finite difference model is a representation of the vibration system which describes the relationship between the current responses and previous drives and responses. If we let
From the finite difference model, the multistep prediction model can be derived. Shifting equation (
Substituting equation (
Keep shifting and substituting equation (
The coefficient matrices can be computed by
Equation (
The initial coefficients
T matrix in equation (
For actual vibration systems, some of the singular values might be very close to zero, and the direct inverse of the T matrix might induce numerical instability, so the truncated SVD or TSVD method [
Then,
When using this method, the parameter
From the multistep prediction model of the test system, the inverse multistep prediction model can be derived and used to generate the drive signals. Combining the inverse multistep prediction model with the control strategy based on matrix power, a new time-domain MIMO sweep vibration test method can be constructed.
If the T matrix in equation (
It can be seen from equation (
Needless to say, equation (
In the real world, test articles start vibrating from rest conditions which cannot achieve stable response immediately; a segment of transient response will always exist. The reference signals for the swept-sine test actually specify the stable response of the system, so if the reference signals are used as the input for the inverse multistep prediction model directly, the drive signals generated will be unrealistic. Hence, a transition segment should be added to the forepart of the original reference signals in order to generate reasonable drive signals as illustrated in Figure
Modification of the original reference signal.
To compose this transition segment, a segment of fixed frequency sine signal which matches the initial amplitude and phase of the original reference signal is generated, and a 1/4 sine window is applied to ensure the amplitude of this segment varies smoothly from zero to the same initial amplitude of the original reference signal. This transition segment is inserted before the original reference signal to modify the reference signal. Obviously, the response of this transition segment is redundant and should be ignored.
Since equation (
Drive signal is generated by the inverse system method and the reference signal is used in an overlapped manner.
From the above discussions, the generation of drive signals for the MIMO swept-sine test can be concluded as follows (also illustrated in Figure Estimate the finite difference model of the testing system Obtain Calculate the pseudoinverse of T by the TSVD method to get Establish the inverse multistep prediction model as the inverse system using equation ( Modify the reference signals and use them as the inputs of the inverse system to produce drive signals
Generation of drive signals for the MIMO swept-sine test.
Due to the noises in the input and output signals and other errors in the modeling of the system, the responses of the first loop may not be satisfactory without control. An offline control strategy is put forward, in which the response spectra are obtained and compared to the reference counterparts, and thus the errors in frequency domain are obtained. Then, the reference spectra for next control loop are updated according to these errors, and the drive signals are updated afterwards. To avoid damaging the test article and equipment, it advisable to control the swept-sine responses within acceptable range at a lower level and then restore the original level test.
The reference spectra is updated as
The control scheme for the swept-sine test is shown in Figure
Offline swept-sine control scheme where A and
A two-input-two-output swept-sine vibration test is simulated while using an aluminum cantilever beam as the test object. The parameters of the beam are listed in Table
Physical properties of the beam.
Property | Value |
---|---|
Density (kg/m3) | 7850 |
Elastic modulus (MPa) | 720 |
Length (mm) | 1000 |
Width (mm) | 50 |
Height (mm) | 15 |
Cantilever beam used in simulation.
The finite element model (FEM) of the cantilever beam is used in the simulation. The beam is divided evenly into 10 elements and has 20 freedoms, and all modal damping ratios are set as 0.8%. Thus, the mass (M), stiffness (K), and damping (
In the example, the sampling frequency is 5120 Hz, the multistep prediction model of the system is estimated by a length of 1024 steps, the inverse multistep prediction model is truncated to a length of 512 steps, the sweep mode is logarithm, and the sweep speed is 2 octaves per minute (a sweep from 5 Hz to 250 Hz takes roughly 169 seconds). The reference spectra at two control points are defined in Tables
Reference spectrum at control point 1.
Freq. (Hz) | Acc. (g) | Disp. (mm) | Phase (°) | |
---|---|---|---|---|
1 | 5 | — | 0.75 | 20 |
2 | 25.73 | 1 | — | 20 |
3 | 150 | 1 | — | 20 |
4 | 250 | 0.5 | — | 20 |
Reference spectrum at control point 2.
Freq. (Hz) | Acc. (g) | Disp. (mm) | Phase (°) | |
---|---|---|---|---|
1 | 5 | — | 0.75 | 40 |
2 | 25.73 | 1 | — | 40 |
3 | 250 | 1 | — | 40 |
In practice, the vibration at low frequency is usually given in the form of the peak-peak displacement restrictions to avoid damaging test equipment. In addition, the crossover frequency (25.73 Hz in the example) is where both requirements on acceleration and displacement are met. However, since the vibration in the test is often measured by using an accelerometer, the displacement defined in the low frequency must be transformed into acceleration. The transformation formula is
The results of the two-input-two-output swept-sine vibration test are shown in Figure
Simulation results of the two-input-two-output swept-sine vibration test example.
From Figure
The key step in the frequency-domain methods is to get the impedance matrix by inversing the frequency response function matrix H(
The drive signals can be obtained by substituting the amplitude and the initial phase into equation (
The condition number of FRF.
By comparing the inverse multistep prediction method proposed in this paper with the general frequency-domain method (Figure
Comparison between the inverse multistep prediction method (IMSP) and frequency-domain method (FD).
A two-input-two-output swept-sine test is conducted with a three-axis shaker to validate the method proposed in the paper. The test system exhibited in Figure
Test system constitution.
Due to the limited number of the VXI channels, only the accelerations of
Block diagram for the test system.
The reference spectra given in the simulation example in Section
A MIMO vibration test generally starts from a level lower than the normal references and climbs up to the normal reference level after it has been controlled. This is due to the fact that in MIMO cases the test cannot be controlled instantly, so in order to avoid any possible damage to the equipment, a preliminary test at a lower level is necessary. In this test, the preliminary test is conducted at the –6 dB level of the references. After 4 iteration loops, the response spectra are controlled within acceptable tolerances, and then the normal level test can be performed. The response spectra before and after control in the lower level test are shown in Figures
Responses of the lower level (−6 dB) preliminary test without control.
Responses of the lower level (−6 dB) preliminary test after control (4 iterations).
Responses of the normal level test after control.
An inverse multistep prediction model to represent the inverse system for the generation of drive signals for the MIMO continuous swept-sine test is proposed in the paper. The
A numerical example of a MIMO continuous swept-sine test on a cantilever beam is put forward, and the results show that the proposed method can perform excellently if the inverse multistep prediction model can be obtained precisely. The method is also verified by a MIMO continuous swept-sine test on a 3-axis shaker. The test results show that the response spectra can be controlled within the acceptable limits.
The test data used to support the findings of this study are available from the corresponding author upon request.
This study was performed in the State Key Laboratory of Mechanics and Control of Mechanical Structures.
The authors declare that they have no conflicts of interest.
This project was funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions.