Abstract
With wheeled and tracked selfpropelled guns as research object, the study carries out the experimental modal analysis by using traditional method of hammering and operational modal analysis method, and obtains loworder natural frequency of the guns, and thus lays the foundation for further research on the vibration characteristics of wheeled and tracked selfpropelled guns. By contrasting the loworder natural characteristics of wheeled and tracked selfpropelled guns, conclusions can draw as the following: the modal shapes (from low to high) of wheeled and tracked selfpropelled guns are pitch, translation and roll; when the modal shapes are identical, the natural frequency of tracked selfpropelled guns is greater than that of wheeled selfpropelled guns, which accords with the test results of the gun’s suspension equivalent stiffness; for wheeled selfpropelled guns, an accurate measure of the gun’s natural characteristics is feasible by either the traditional or the operational modal analysis method. When it comes to tracked selfpropelled guns, the operational modal analysis method is more accurate.
1. Introduction
The natural characteristics of selfpropelled guns have significant impact on its occupant comfort, guns vibration, as well as its dynamic strength, which have equal importance on its firing accuracy [1]. Therefore, it is of significant importance to study the dynamic characteristics of selfpropelled guns, and to master the law of the dynamic characteristics and the impact on vibration. The loworder modes of selfpropelled guns are mainly manifested in the rigid body motion, and elastic elements like tires or tracked rubber suspension feet, while the rigid body motion consists mainly of loworder modes like the pitch, translation and roll of the guns. Each modal and its corresponding natural frequency cast different effects on the vibration of selfpropelled guns. There are relatively sharp contrast between wheeled selfpropelled guns and tracked selfpropelled guns due to their respective suspension systems and supporting means. Selecting two wheeled selfpropelled guns of different caliber (called wheeled selfpropelled gun A and wheeled selfpropelled gun B) and two tracked selfpropelled guns of the corresponding caliber (called tracked selfpropelled gun A and tracked selfpropelled gun B) as the research objects, the paper studies the dynamic characteristics of the guns using experimental modal analysis method. The study finds the loworder modes of the two types of guns and the differences between them.
2. The traditional experimental modal analysis method
The traditional experimental modal analysis method, gains modal parameters by testing excitation force and structural response, and by fitting the frequency response function or impulse response function [2].
Traditional experimental modal analysis system is comprised of the excitation system, response system and measurement system as well as analysis computing systems. During testing, the measured structure is given an excitation force $\left\{f\right(t\left)\right\}$ with a certain band width through the hammer or the exciter, and its vibration response $\left\{x\right(t\left)\right\}$ is also a signal consisting of system modal information. Through Fast Fourier Transform (FFT) on the force and response, Fourier spectrum $F\left(\omega \right)$ and $X\left(\omega \right)$ are obtained, and the frequency response function matrix is defined as:
Thus, the transfer function matrix of the structure can be obtained through the analytical processing of the excitation signal and response signal of each measuring point, and then modal parameters of the structure can be obtained.
3. The operational modal analysis method
Operational Modal Analysis Method [37] which identify the modal parameter through the response signal in working condition or environmental exciting Operating deflection shape (ODS) obtained through operational modal analysis method is the performance of a particular frequency in the working state or moment, the relative vibration between the two or more points on the structure, or the vibration state of a point with respect to all other points.
The basic idea of the operational modal analysis lies in the fact that there are similar expressions of the crosscorrelation function and the impulse response function between two response points, modal parameter identification can be done in the time domain after the crosscorrelation function between two response points has been obtained. When the excitation is unknown, operating deflection shape frequency response function (ODS FRF) is an ideal method for obtained the natural characteristics of the gun. ODS FRF is a frequency response function calculating response data in the working state, with response signal of some points as the reference signal, and other as flow response signal. By calculating the crossspectrum between the flow response signal and the reference response signal, the ODS FRF substitutes the square root of the spectral amplitude of the flow response signal with the crossspectral amplitude, and retains the phase information of response data between different measurement groups in cross spectrum. Therefore, ODS FRF is constituted by combining selfspectrum amplitude of the flow response signal and crossspectrum phase information of the flow response signal as well as reference response signal, and its expression is:
where: ${G}_{xy}\left(\omega \right)={F}_{x}\left(\omega \right){F}_{y}^{*}\left(\omega \right)$ represents the crossspectrum of the flow response signal and reference response signal; ${G}_{xx}\left(\omega \right)={F}_{x}\left(\omega \right){F}_{x}^{*}\left(\omega \right)$ represents the selfspectrum of the flow response signal.
If the excitation spectrum is relatively flat in the vicinity of natural modal, that is, it has characteristics similar to white noise excitation, the fitting method of the frequency response function can be used to fit ODS FRF curve, and thus modal parameters can be obtained. The frequency response function matrix contains the frequency response function curve, excitation Fourier spectrum and response Fourier spectrum. Frequency response function matrix elements can be expressed as:
where: ${F}_{x}\left(j\omega \right)$ represents Fourier spectrum of the response signal; ${F}_{f}\left(j\omega \right)$ represents Fourier spectrum of the excitation signal; ${H}_{x,f}\left(j\omega \right)$ represents transfer function of incentive and response signal.
If excitation in the vicinity of the natural modal meets the white noise excitation conditions, the amplitude of the Fourier spectrum can be substituted by a constant:
By Eq. (3) the relationship between the response and the frequency response function can be:
By the above analysis, it is evident that the information of the structure’s modal parameters is contained in the ODS FRF which has similar expression with frequency response function. Therefore, the frequency response function fitting method can be used in fitting the ODS FRF, and modal parameters can be obtained.
4. The experimental modal analysis of the wheeled and tracked selfpropelled guns
With regard to the differences of natural characteristics of wheeled and tracked selfpropelled guns, two differentcaliber wheeled selfpropelled guns and two tracked selfpropelled guns with corresponding caliber are selected for testing the natural characteristics by using experimental modal analysis method, and studying the law of the loworder natural characteristics of those guns, and then analyze the differences of the loworder natural characteristics of those guns.
During the test, selfpropelled guns are parked on a flat floor, analyzing the modes of the guns by using the traditional modal analysis method and operational modal analysis method. With the traditional modal analysis method, the topdeck corner of the chassis is to be selected as an excitation point for its large stiffness. With operational modal analysis method, engine of selfpropelled guns is to be started to excite the guns, while the acceleration response of all the measuring points is to be recorded. For those guns, the acceleration responses in the same position are picked up, and been recorded and processed by the same dynamic signal analyzer.
In order to verify the feasibility of obtaining the natural characteristics of selfpropelled guns through the operational modal analysis method, with wheeled selfpropelled gun A as the test object, the natural characteristics of guns within 50 Hz frequency are tested by a contrastive test of traditional experimental modal analysis method and operational modal analysis method respectively.
Fig. 1 shows the measuring points of Gun A. Fig. 2 to Fig. 4 display the modal shapes of Gun A. Table 1 manifests loworder natural frequencies of the guns obtained by the two methods. The test results show that the results obtained by operational modal analysis method are consistent with those of traditional modal analysis method. For the first three vibration modes of the gun, the first mode is the pitching modes; the second mode is the upanddown translation modes; the third mode is the roll mode.
Fig. 1Layout of testing points for wheeled selfpropelled gun A
Fig. 2Firstorder mode of selfpropelled gun A
Fig. 3Secondorder mode of selfpropelled gun A
Fig. 4Thirdorder mode of wheeled selfpropelled gun A
Similarly, through the operational modal analysis method, the modal parameters of selfpropelled gun A and selfpropelled gun B are obtained. Likewise, by using traditional modal analysis method, the modal parameters of tracked selfpropelled gun A and wheeled selfpropelled gun B have also been obtained, and the test results are shown in Table 2. Clearly, their loworder modal shapes are similar; when the modal shape are identical, the natural frequency of tracked selfpropelled guns is greater than wheeled selfpropelled guns.
Table 1Analysis results of operational modal and traditional modal methods
Modes order  Frequency / Hz  Frequency errors / %  Vibration modes  
Operational modal method  Traditional modal method  
1  2.90  2.86  1.4  Pitch mode 
2  3.89  3.87  0.5  Upanddown mode 
3  5.2  5.2  0.0  Roll mode 
The experiment has tested the suspension equivalent stiffness of tracked selfpropelled guns and wheeled selfpropelled guns with same caliber, and the results show that the value of the chassis suspension equivalent stiffness of the tracked selfpropelled guns is 1.7 to 2.1 times of that of the wheeled selfpropelled guns. The test results are as Fig. 5 shows.
Table 2Comparison between natural frequencies of wheeled and tracked selfpropelled guns
Order  Natural frequencies / Hz  Description  
Wheeled A  Tracked A  Wheeled B  Tracked B  
1  2.90  5.32  2.50  3.79  Pitch 
2  3.89  8.45  4.12  8.76  Upanddown 
3  5.20  12.91  5.64  13.66  Roll 
Fig. 5The chassis suspension equivalent stiffness test curve of two types of guns
The frequency response curves of wheeled and tracked selfpropelled guns are shown in Fig. 6 and Fig. 7 by using traditional modal analysis method. It is clear that when tracked selfpropelled guns is tested by traditional modal analysis method, there will be low signaltonoise ratio as well as a larger margin of modal parameter errors.
Fig. 6The frequency response curve of wheeled selfpropelled guns using the hammering method
5. Test results analysis and conclusions
Traditional modal analysis method and operational modal analysis method are used to test the natural characteristics of wheeled and tracked selfpropelled guns. Their modal parameters are obtained, and the following conclusions can be drawn:
1) Of the four selected selfpropelled guns, their loworder modal shapes are reflected in such an order (in terms of frequency from low to high) as: pitch, up and down translation and roll.
Fig. 7The frequency response curve of tracked selfpropelled guns using the hammering method
2) Of the four selected selfpropelled guns, the corresponding natural frequencies of the gun’s pitch, upand –down translation and roll modes in tracked selfpropelled guns are greater than those of wheeled selfpropelled guns respectively. The natural frequency of the wheeled guns is 1.7 to 2.1 times of the tracked guns. The chassis suspension equivalent stiffness of tracked selfpropelled guns is greater than that of wheeled selfpropelled guns, which accounts mainly for the higher loworder natural frequency of tracked selfpropelled guns.
3) With regard to wheeled selfpropelled guns, an accurate measure of the gun’s modal parameters is feasible by either the traditional experimental modal analysis method or the operational modal analysis method. When it comes to tracked selfpropelled guns, the operational modal analysis method is more accurate.
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