applied sciences Article
Failure Identification of Dump Truck Suspension Based on an Average Correlation Stochastic Subspace Identification Algorithm Bingwen Liu 1 , Zhiyong Ji 1 , Tie Wang 1,2, * , Zhenghao Tang 1 and Guoxing Li 1,2 1
2
*
Department of Vehicle Engineering, Taiyuan University of Technology, Taiyuan 030024, China;
[email protected] (B.L.);
[email protected] (Z.J.);
[email protected] (Z.T.);
[email protected] (G.L.) Centre for Efficiency and Performance Engineering, University of Huddersfield, Huddersfield HD1 3DH, West Yorkshire, UK Correspondence:
[email protected]; Tel.: +86-0351-6010580
Received: 15 August 2018; Accepted: 24 September 2018; Published: 1 October 2018
Abstract: This paper proposes a fault identification method based on an improved stochastic subspace modal identification algorithm to achieve high-performance fault identification of dump truck suspension. The sensitivity of modal parameters to suspension faults is evaluated, and a fault diagnosis method based on modal energy difference is established. The feasibility of the proposed method is validated by numerical simulation and full-scale vehicle tests. The result shows that the proposed average correlation signal based stochastic subspace identification (ACS-SSI) method can identify the fluctuation of vehicle modal parameters effectively with respect to different spring stiffness and damping ratio conditions, and then fault identification of the suspension system can be realized by the variation of the modal energy difference (MED). Keywords: modal parameter identification; dump truck suspension; stochastic subspace identification; fault identification
1. Introduction The suspension system is the component of a vehicle that connects the vehicle body and the wheel, which supports the car body and isolates the shock and vibration caused by road unevenness [1]. The suspension system is crucial for vehicle safety and riding comfort, and also plays an important role in handling and braking. Due to a poor working environment and complex operating conditions, failures often occur during in dump truck suspension systems in operation, which have a huge impact on the efficiency and reliability of fleet operations, and also endanger property and driver safety [2]. Therefore, methods for condition monitoring of the vehicle suspension system have attracted much attention [3–6]. There are two main types of fault diagnosis algorithms: data-driven methods and model-based methods. If the diagnosis process does not require a physical model and instead depends on measured data, it is referred to as a data-driven fault diagnosis method [7]; for example, independent component analysis (ICA), and principle component analysis (PCA). Furthermore, many methods are published to be applied in the fault diagnosis of suspension systems. He et al. combine k-means clustering with Fisher discriminant analysis (FDA) to conduct fault diagnoses of suspension systems [8]. Likewise, canonical vitiate analysis (CVA) and dynamical principle components analysis (DPCA) are applied to detect early faults [9]. Moreover, adaptive fuzzy c-means (FCM) clustering has been developed for condition monitoring, and possibilistic c-means (PCM) clustering with fault lines have been designed to isolate the faults [10,11].
Appl. Sci. 2018, 8, 1795; doi:10.3390/app8101795
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Data-driven fault diagnosis methods have the advantage of not requiring a numerical model. However, these models are susceptible to noise and varied operation conditions. Unfortunately, because of the poor working environment and complex operation conditions, the noise is strong and excitation from the road is intensive and varied, so data-driven methods have not been well suited to the failure identification of dump truck suspension to date. By contrast, in model-based fault diagnosis methods, a precise model is needed to be built [12–17]. Several model-based fault diagnosis methods are implemented in research; a Kalman-filter based fault diagnosis method with high efficiency has been developed [18], and a distributed-observer based fault detection and an interaction multi-model approach are proposed respectively for the damper and spring [19,20]. However, the difficulty makes it hard to obtain the parameters of a dynamic system for modeling. Identification of modal parameters is an approach to obtain the dynamic characteristic of a system [21,22], and it can acquire accurate dynamic parameters via test signals, such as subspace identification (SI) [23,24]. Moaveni et al. [25] used deterministic subspace identification (DSI) to identify the modal parameters of nonlinear systems, and proposed a method of determining the nonlinear dynamic characteristics based on the identification of deterministic subspace and time-varying modal parameters. Moreover, the output-only identification methods can recognize the mode using the response only [26]. Sarmadi et al. [27] proposed a recursive adaptive stochastic subspace method to monitor the modes of the dynamic system online using wide-area synchronous vector data. Chen et al. [28,29] used the average correlated stochastic subspace method (ASC-SSI) to identify the operating modal of the frame online, obtained the frame modal parameters under constraint conditions, and matched the relevant components according to the identification result. Dong et al. [30] used the stochastic subspace method to identify vehicle modal parameters and calculate the variation in vehicle inertia parameters. This study illustrates that the noise and high damping ratio (20–30%) greatly influences the identification results of the stochastic subspace method. This paper puts forward an online monitoring and diagnosis method for early faults of truck suspension systems. The average correlation signal based stochastic subspace identification (ACS-SSI) is adopted because of its advantages, including the low testing hardware requirements, high identification accuracy, and suitability for random vibration conditions. However, the challenge is dealing with the impact of noise and high damping ratios, so the vehicle vibration model with 11 degrees of freedom (DOFs) is established to verify the performance under different signal-to-noise ratios (SNR) and damper damping ratios, and then the modal parameters of the vehicle are identified by both numerical simulation and experiments. Moreover, the sensitivity of each modal parameter is evaluated, and a fault diagnosis method based on the modal energy difference method is established. Finally, the feasibility of the developed fault diagnosis method is validated by experimental tests carried out on a full-scale vehicle. The rest of this paper is organized as follows. Section 2 introduces the principles of ACS-SSI briefly, and verifies its robustness under noise and high damping ratios. Section 3 proposes the failure identification method and builds the 11 DOF model to simulate the diagnosis process. Section 4 conducts the real-scale experiment. Finally, the paper ends with conclusions in Section 5. 2. ACS-SSI Algorithm and Its Verification Considering the advantages and disadvantages of covariance-driven stochastic subspace identification (COV-SSI), this paper improves the covariance-driven subspace method using the noise suppression characteristics of the average correlation function. The use of multiple average correlation functions to replace the original vibration signal as the input can theoretically reduce the influence of noise signals on the identification accuracy. Therefore, the average correlation signal based stochastic subspace identification (ACS-SSI) is applied to the identification of modal characteristics of dynamic systems. The effectiveness of the improved method is initially verified based on a multi-DOF dynamic model.
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Stochastic subspace identification (SSI) can achieve modal identification only with output 2.1. Basicrelying Principle ACS-SSI signals onofthe system input data. The process can be summarized as follows. First, the control matrix and the observable matrix are obtained by the singular value decomposition of the covariance Stochastic subspace identification (SSI) can achieve modal identification only with output signals of the response data; then, the modal parameters of the state matrix can be derived [29]. The kinematic relying on the system input data. The process can be summarized as follows. First, the control matrix equations can be written as the discrete-state space stochastic equation and the observable matrix are obtained by the singular value decomposition of the covariance of the response data; then, the modal parameters 1) state Ax(k )matrix w(k ) can be derived [29]. The kinematic x(kofthe , (1) space stochastic equation equations can be written as the discrete-state y (k 1) Cx(k ) v(k ) ( where A is the state space matrix, Cxis w(k) is the representative random (kthe + 1output ) = Ax (space k) + wmatrix, (k) , (1) input signal which includes the process of noise and k is the discrete y(knoise, + 1) =v(k) Cxis (k)the + vmeasurement (k) time value. The state space matrix A contains the modal information of the system. whereThe A isstate the state space C is the by output space matrix, w(k) ismethod the representative random input matrix A ismatrix, reconstructed the stochastic subspace and the modal parameters signal includes process noise, v(k) is themeasurement measurement of noise and k is discrete is time can bewhich extracted fromthe it. The matrix of l-channel signals collected bythe l channels y(t), value. Thethe state space Asignal-to-noise contains the modal of the system. in which one withmatrix a better ratio information (SNR) is chosen and set as the k-th signal yk(t). The The statefunction matrix A by the stochastic subspace and the modal parameters correlation ofisallreconstructed the original signals for the k-th channelmethod is: can be extracted from it. The matrix of l-channel measurement signals collected by l channels is y(t), 1 in which the one with a better signal-to-noise and set as the k-th signal yk (t). rk ( ) ratio ) yis [ y(t(SNR) )] , (2) k (tchosen n The correlation function of all the original signals for the k-th channel is: where τ = 1, 2,…, N − n is the discrete time series of the correlation function, with N being the total 1 l * y(t + τ ) · ryk k((t))], R rk (τ ) = [function is an l-way signal related to a(2) set length of the collected signal. The correlation n of τ correlation functions. In order to improve the denoising effect, a multi-group experiment was where τ =out 1, 2, . . , N −the n is the discrete of the correlation function, with N being the total carried to.obtain n-group signaltime rk1, rseries k2, …, rkn; the n-group signal was averaged to obtain an length of the collected signal. The correlation function rk (τ ) ∈ Rl ∗τ is an l-way signal related to a set average correlation function rk ( ) used to replace the original y(t) as the input for modal of τ correlation functions. In order to improve the denoising effect, a multi-group experiment was identification. carried out to obtain the n-group signal rk1 , rk2 , . . . , rkn ; the n-group signal was averaged to obtain an The subsequent steps rare the same as the covariance-driven stochastic subspace method [26]: average correlation function k ( τ ) used to replace the original y(t) as the input for modal identification. T1 i ( ) the using covariancesteps of rkare to same form as thethe matrix , the singular value decomposition (SVD) Thethe subsequent covariance-driven stochastic subspace method [26]:is using the covariance of rk (τ ) to form the matrix T1/i , the singular value decomposition (SVD) is carried out to obtain the discrete state matrix A. After decomposing matrix A, the eigenvalue i and carried out to obtain the discrete state matrix A. After decomposing matrix A, the eigenvalue λi and Ψi eigenvectorΨi are are obtained and then converted into natural frequencyfnif,nimodal , modaldamping dampingratio ratioξξ eigenvector obtained and then converted into natural frequency i , i, andmodal modalvector vectorviv. i. and 2.2. 2.2.Numerical NumericalSimulation SimulationininMulti-DOF Multi-DOFSystem System 2.2.1. 2.2.1.Multi-DOF Multi-DOFNumerical NumericalSimulation Simulation AAthree-DOFs three-DOFsnumerical numericalsimulation simulationmodel modelisisestablished establishedininMatlab/Simulink Matlab/Simulinksoftware softwaretotoverify verify the robustness of the improved stochastic subspace algorithm and its reliability for high damping ratios the robustness of the improved stochastic subspace algorithm and its reliability for high damping (Figure 1). The 1). mass ,m are connected by the fixed stiffness k1 , k2 ,k1k,3kand damping c1 , 1 , m2 m ratios (Figure Theblocks mass m blocks 1,3m2 m3 are connected by the fixed stiffness 2, k3 and damping c2c,1,cc3 2, ,respectively. TheThe three natural frequencies of the system fn1 =fn12.5 Hz,Hz, fn2 f= 5.8 Hz, f fn3==9.9 c3, respectively. three natural frequencies of the system = 2.5 n2 = 5.8 Hz,n3 9.9Hz, Hz, are determined by both mass and stiffness. Three low-pass filtered white noise signals, w , w , and w 1 2 3, are determined by both mass and stiffness. Three low-pass filtered white noise signals, w1, w2, and are applied to the three mass units to simulate dynamic behaviors under random excitation. w3, are applied to the three mass units to simulate dynamic behaviors under random excitation.
Figure1.1.Model Modelofofthe thethree threedegrees degreesofoffreedom freedom(DOFs) (DOFs)system. system. Figure
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The sampling frequency of the simulation system is 200 Hz, the solution time is 20 s, and the The sampling frequency of the simulation is 200 the solution time isof20the s, and the average correlation calculation is carried out system 50 times. TheHz, dynamic responses vehicle average correlation calculation is carried out 50 times. The dynamic responses of the vehicle suspension suspension are concentrated primarily in the frequency range of 0–50 Hz [31]. Therefore, according are concentrated primarily in the frequency rangefrequency of 0–50 Hz Therefore, according to the Nyquist to the Nyquist sampling theorem, the sampling of[31]. the simulation system is selected as 200 sampling theorem, the sampling frequency of the simulation system is selected as 200 Hz. As seen Hz. As seen in Figure 2, the calculated acceleration of the three masses exhibits obvious random in Figure 2, the calculated acceleration of the three masses exhibits obvious random characteristics, characteristics, while correlation function signals are more regular. The average correlation while correlation function signals component are more regular. average correlationresponse calculation reduces calculation reduces the random in theThe original acceleration signal, so the the random component in the original acceleration response signal, so the correlation function obtained correlation function obtained after averaging has a higher periodicity than the original acceleration after averaging has a higher periodicity than the original acceleration signal. signal.
Figure Figure 2. 2. Input Input signal signal of of stochastic stochastic subspace subspace identification identification (SSI) (SSI) and and average average correlation correlation signal signal based based stochastic subspace identification (ACS-SSI). m = mass blocks. Acc: Acceleration Correlation. 1–3 = mass blocks. Acc: Acceleration Correlation. stochastic subspace identification (ACS-SSI). m1–3
2.2.2. Effect of Noise on Identification Results 2.2.2. Effect of Noise on Identification Results The signal-to-noise ratio (SNR) can substantially affect the identification accuracy. Due to the The signal-to-noise ratio (SNR) can substantially affect the identification accuracy. Due to the complexity of actual testing and monitoring conditions, noise generation is inevitable and it is difficult complexity of actual testing and monitoring conditions, noise generation is inevitable and it is to remove from effective signals, which requires the identification algorithm to have good robustness difficult to remove from effective signals, which requires the identification algorithm to have good to noise suppression. Therefore, it is necessary to verify the identification correctness of the ACS-SSI robustness to noise suppression. Therefore, it is necessary to verify the identification correctness of algorithm under different SNRs. the ACS-SSI algorithm under different SNRs. The robustness of the ACS-SSI algorithm is checked under six different SNRs, i.e., 9, 4, 2.3, 1.5, 1, The robustness of the ACS-SSI algorithm is checked under six different SNRs, i.e., 9, 4, 2.3, 1.5, and 0.67 respectively. Each case of the experiment was repeated 10 times, and noise occupying 10%, 1, and 0.67 respectively. Each case of the experiment was repeated 10 times, and noise occupying 20%, 30%, 40%, 50% and 60% of the total signal power was added to the system response. The average 10%, 20%, 30%, 40%, 50% and 60% of the total signal power was added to the system response. The correlation calculation is still carried out 50 times, and the sampling time of each segment is 20 s. average correlation calculation is still carried out 50 times, and the sampling time of each segment is The comparison among the SNRs shows that the third order mode presents obvious divergent trends, 20 s. The comparison among the SNRs shows that the third order mode presents obvious divergent so the third mode is selected for the statistical verification. trends, so the third mode is selected for the statistical verification. As Table 1 shows, although the identification of frequency and mode shapes is still valid when As Table 1 shows, although the identification of frequency and mode shapes is still valid when the SNR in the simulation reached 0.67, the damping ratio shows a divergent trend. It shows that the SNR in the simulation reached 0.67, the damping ratio shows a divergent trend. It shows that the the SNR has a great influence on the identification accuracy. To improve the identification accuracy, SNR has a great influence on the identification accuracy. To improve the identification accuracy, tolerance values of frequency, mode shape, and damping ratio must be properly adjusted to achieve tolerance values of frequency, mode shape, and damping ratio must be properly adjusted to achieve better results. better results. Table 1. Identification error of ACS-SSI under different signal-to-noise ratios (SNRs). Table 1. Identification error of ACS-SSI under different signal-to-noise ratios (SNRs). SNR 9 4 2.3 1.5 1 0.67
SNR
Frequency error (%) 0.14 Frequency error Damping ratio error (%) (%) 5.03 Damping ratio (%) error (%)0.34 Modal shape error
Modal shape error (%)
9 4 2.3 0.55 0.64 0.14 42.89 0.55 25.520.64 5.03 0.46 42.89 1.21 25.52 0.34 0.46 1.21
1.5
1
0.57 0.57 40.57 40.57 1.41
1.21 1.21 28.01 28.01 2.43
1.41
2.43
0.67 1.02 1.02 30.2 30.2 3.81 3.81
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Due to the negative influence of high damping ratio on the identification results, the stochastic subspace identification is rarely applied in the case of high damping ratios [30]. The damping ratio of a vehicle’s is higher than that of other mechanical systems, and the highest 2.2.3. Effect of suspension Damping onsystem Identification Results value is even up to 20–30%. The higher system damping ratio tends to deteriorate the effect of modal Due to theTherefore, negative influence of high on theof identification results, the correlated stochastic identification. it is necessary to damping verify theratio reliability the improved average subspace identification is rarely applied in the case of high damping ratios [30]. The damping ratio random subspace algorithm under high damping ratios before its formal application. of a vehicle’s is higher than that of other mechanical highest First, ninesuspension damping system ratios varying between 2.5% and 40% are set, systems, as shownand in the Table 2, to value is even up to 20–30%. The higher system damping ratio tends to deteriorate the effect of modal compare the changes in identification error of SSI and ACS-SSI with the changes of damping ratio. identification. Therefore, it is necessary to verify the reliability of the improved average correlated random subspace algorithm underTable high2.damping its formal application. Dampingratios ratios before ( ) of test. First, nine damping ratios varying between 2.5% and 40% are set, as shown in Table 2, to compare 1 2 error of3SSI and ACS-SSI 4 5 the changes 6 7 8 9 theNumber changes in identification with of damping ratio. 2.5% 5% 10% 15% 20% 25% 30% 35% 40% Table 2. Damping ratios (ξ ) of test.
The sampling frequency was set to 200 Hz, and the modal assurance criterion (MAC) was set to Number 1 2 3 4 5 6 7 8 9 30% to avoid false modes being identified. The input data is set for a period of 100 s and is averaged ξ 2.5% 5% 10% 15% 20% 25% 30% 35% 40% 50 times. Monte Carlo tests were performed 10 times for each damping ratio, and corresponding mean value of the identification error is shown in Figure 3. Thethe sampling was (40%), set to 200 and theerror modal assurance(3.9%) criterion (MAC) was of setthe to At highest frequency damping ratio theHz, frequency of ACS-SSI is less than half 30%(8.7%). to avoid false modes identified. Theerrors inputin data is set for a period of 100 s andisisstill averaged SSI Although bothbeing of them have large damping ratio, ACS-SSI (29.3%) lower 50 times. were performed 10 times damping ratio, mean than SSI Monte (42.4%).Carlo The tests trend of modal vector error for foreach ACS-SSI (2.1%) andand SSIcorresponding (6.3%) is basically value of the identification errorerror. is shown in Figure 3. consistent with the frequency
Figure 3. Identification error varying the damping ratio.
At the damping ratio (40%), the frequency of ACS-SSI (3.9%) issuperior less than of As can highest be seen from Figure 3, the identification resulterror of ACS-SSI is obviously to half that of the SSI (8.7%). Although both of them have large errors in damping ratio, ACS-SSI (29.3%) is still SSI. It can be seen from the identification stabilization diagram (Figure 4) that the ACS-SSI algorithm lower thanidentification SSI (42.4%). The trendand of modal error for (2.1%) and than SSI (6.3%) has better stability better vector convergence of ACS-SSI recognition results SSI. is basically consistent with the frequency error. As can be seen from Figure 3, the identification result of ACS-SSI is obviously superior to that of SSI. It can be seen from the identification stabilization diagram (Figure 4) that the ACS-SSI algorithm has better identification stability and better convergence of recognition results than SSI.
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Figure Figure4. 4.Identification Identificationstabilization stabilizationdiagrams diagramsof ofACS-SSI ACS-SSIand andSSI. SSI.PSD: PSD:Power PowerSpectral SpectralDensity Density Figure 4. Identification stabilization diagrams of ACS-SSI and SSI. PSD: Power Spectral Density
In In algorithm maintains In the the case case of of high high damping damping ratio, ratio, the the ACS-SSI ACS-SSI algorithm algorithm isis still still applicable applicable and and maintains maintains considerable considerable accuracy. ititwas was selected considerableaccuracy. accuracy.Since Sincemodal modalvectors vectorsare arenot noteasily easilydisturbed disturbedby byspeed speedand andload, load,it wasselected selected for the development of condition monitoring algorithms. for for the development of condition monitoring algorithms. Simulations Simulations based on the three-DOFs model show that the ACS-SSI algorithm can effectively Simulationsbased basedon onthe thethree-DOFs three-DOFsmodel modelshow showthat thatthe theACS-SSI ACS-SSIalgorithm algorithmcan caneffectively effectively identify the modal parameters of the system containing random white noise and non-stationary identify the modal parameters of system containing random white noise non-stationary identify the modal parameters of the system containing random white noise and non-stationary impact impact ACS-SSI effectively signal and maintain good impactcomponents. components.ACS-SSI ACS-SSIcan caneffectively effectivelysuppress suppressthe thenoise noisein inthe theraw rawsignal signaland andmaintain maintainaaagood good identification identification identificationaccuracy accuracyin inthe thecase caseof ofhigh highdamping dampingratio. ratio. 3. Modal Simulation Analysis 3. 3.Modal ModalSimulation SimulationAnalysis Analysis To algorithm to fault diagnosis of dump truck To To verify verify the thefeasibility feasibility of of applying applying the theACS-SSI ACS-SSIalgorithm algorithm to to fault fault diagnosis diagnosis of of dump dump truck truck suspension, an eleven-DOFs dynamic model of a dump truck was established. Secondly, the suspension, suspension,an aneleven-DOFs eleven-DOFsdynamic dynamicmodel modelof ofaadump dump truck wasestablished. established.Secondly, Secondly,the the road road roughness signal simulated Matlab/Simulink software, was input into the dynamic model to roughness roughnesssignal signalsimulated simulatedusing usingMatlab/Simulink Matlab/Simulinksoftware, software,was wasinput inputinto intothe thedynamic dynamicmodel modelto to obtain vibrational responses [31,32]. Finally, the proposed ACS-SSI algorithm is used to diagnose obtain vibrational responses [31,32]. Finally, the proposed ACS-SSI algorithm is used to diagnose obtain vibrational responses [31,32]. Finally, the proposed ACS-SSI algorithm is used to diagnose possible possiblefaults faultsof ofthe thevehicle vehiclesuspension. suspension.Figure Figure55shows showsthe theflowchart flowchartof ofsimulation simulationanalysis. analysis.
Figure 5.5.Schematic Schematic diagram of simulation analysis. Figure Figure5. Schematicdiagram diagramof ofsimulation simulationanalysis. analysis.
3.1. Eleven-DOFs Eleven-DOFs Dump Truck Truck Model 3.1. 3.1. Eleven-DOFsDump Dump TruckModel Model The dump dump truck is is simplified to to an 11-DOFs 11-DOFs model comprising comprising three DOFs DOFs associated with with the The The dumptruck truck issimplified simplified toan an 11-DOFsmodel model comprisingthree three DOFsassociated associated withthe the vehicle body (Z , θ, Φ), six DOFs associated with the bounce and rolling motions of three integral axles bb, θ, Φ), six DOFs associated with the bounce and rolling motions of three integral vehicle body (Z vehicle body (Zb, θ, Φ), six DOFs associated with the bounce and rolling motions of three integral (ZA , Z(Z , Z , ZZ,C,ZZ , ZF ),E, and two DOFs describe the pitch motions of the balanced suspension (θ C , θ D ), axles axles B(ZAA, ,CZZBB, ,D ZC, EZDD, ,Z ZE,ZZF), F),and andtwo twoDOFs DOFsdescribe describethe thepitch pitchmotions motionsof ofthe thebalanced balancedsuspension suspension as seen in Figure 6. (θ (θCC, ,θθDD),),as asseen seenin inFigure Figure6.6.
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Figure 6. Eleven-DOFs dynamic model of a dump truck. Figure 6. 6. Eleven-DOFs Eleven-DOFs dynamic dynamic model model of of aa dump dump truck. truck. Figure
To To simplify simplify the the modeling modeling of of the the balanced balanced suspension, suspension, the the leaf leaf spring spring is is divided divided into into two two separate separate To simplify the modeling of the balanced suspension, the leaf spring is divided into two separate stiffnesses (k E and kF) and damping (cE and cF). Both ends of the leaf springs are separately linked with stiffnesses (k (kEE and and kkFF))and anddamping damping (c (cEEand andccFF).).Both Bothends endsofofthe theleaf leafsprings springsare areseparately separatelylinked linked with with stiffnesses the front and rear axle through the rigid stabilizer rod. The front and rear axle vibrations are coupled the front and rear axle through the rigid stabilizer rod. The front and rear axle vibrations are coupled the front and rear axle through the rigid stabilizer rod. The front and rear axle vibrations are coupled together 7. together with with the the stabilizer stabilizer rod rod [33]. [33]. The The equivalent equivalent balance balance suspension suspension model model is is shown shown in in Figure Figure 7. together with the stabilizer rod [33]. The equivalent balance suspension model is shown in Figure 7. ZC ZC
m m θC θC
kE kE
d1 d1
d2 d2
cE cE
kF kF mF mF
mE mE kEt kEt
cF cF
cEt cEt
kFt kFt
cFt cFt
Figure Figure 7. 7. Dynamic Dynamic model model of of balanced balanced suspension. suspension. Figure 7. Dynamic model of balanced suspension.
The dynamicmodel modelof of dump considering the integral and thesuspension, balanced The dynamic the the dump truck,truck, considering the integral axle and axle the balanced The dynamic model of the dump truck, considering the integral axle and the balanced suspension, caninbematrix written in matrix form as: can be written form as: suspension, can be written in matrix form as: .. . . , Z CZ KZ KZ K tQ Q C (3) gg,, MZM CtttQ Q (3) M+ZCZC+ Z KZ=K Kt t Qggg + C Q (3) g where M is the matrix representing vehicle body mass, Z is the vector of vehicle dynamic responses, matrix representing representing vehicle vehicle body body mass, mass, Z Z is is the the vector vector of of vehicle vehicle dynamic dynamicresponses, responses, where M is the matrix C is the matrix of vehicle damping, K is the stiffness matrix of the vehicle system, Ct is the matrix of C is is the matrix matrix of vehicle damping, damping, K K is the stiffness matrix of the vehicle system, Ctt is is the the matrix matrix of of C tire damping, Kt is the stiffness matrix of the tire and Qg is the vector of displacement inputs acting tire damping, K is the stiffness matrix of the tire and Q is the vector of displacement inputs acting on tire Ktt the stiffness matrix of the tire and Q g g is the vector of displacement inputs acting on the tires. thethe tires. on tires. The differential equations of Equation (3) can be transformed into the state space equation The differential equations of Equation (3) can be transformed into the state space equation (Equation (4)) to facilitate the simulation, reducing the need for solution time and computing (Equation (4)) to facilitate the simulation, reducing the need for solution time and computing resources under the premise of ensuring system integrity and accuracy. resources under the premise of ensuring system integrity and accuracy.
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The differential equations of Equation (3) can be transformed into the state space equation (Equation (4)) to facilitate the simulation, reducing the need for solution time and computing resources under the premise of ensuring system integrity and accuracy. (
.
x = Ax + Bu , Y = Cx + Du
(4)
The conventional method for equation transformation is achieved by selecting an appropriate intermediate variables x, regardless of the derivative term in the input variable u. The road roughness . input Qg was directly converted into the force input acting on the vehicle Kt Q g + Ct Q g = f (t) through numerical differential operation. Then: .
"
x=
0 − M−1 K
I − M−1 C
#
( x+
b 0 M−1
) f ( t ),
(5)
where I and 0 represent identity and null matrices with similar dimensions as the property matrices, respectively; b 0 is a null column vector whose length is the same as that of forcing vector f, and x is a vector containing the states (displacement and velocity vectors). The state variables x were selected as Equation (6): x
.
= [ Z, Z ] . . . . . . . . . . . , = [ Zb , θ, φ, Z f , θ f , Zm , θm , Zr , θr , θC , θ D , Z b , θ, φ, Z f , θ f , Z m , θ m , Zr , θ r , θ C , θ D ]
(6)
Although input order reduction has been realized in Equation (5) via the conversion from . Kt Q g + Ct Q g to f (t), the essence of this order reduction is an approximate computational process of a numerical differentiation algorithm based on the Runge-Kutta method. Order reduction itself could cause truncation error input and reduce the reliability and accuracy of the system output. Therefore, to oversimplify the input vector, it is essential to keep the input derivative for obtaining reliability and accuracy. At present, the transformation from a single-input single-output (SISO) differential equation containing an input derivative to state space representation is achieved mainly by selecting an appropriate intermediate variable. Based on the SISO transformation method, the input, output variable, and the status variable are expanded to the vector; the variable of each system is expanded to the coefficient matrix, and matrix operations are introduced into the conversion process. Then, . it became possible to eliminate the input derivatives of multivariable equations, Q g , and the state-space equation was achieved, as shown in Equation (7): .
x2 =
"
0 − M−1 K
I − M−1 C
#
( x2 +
M−1 Ct M−1 Kt − M−1 C × M−1 Ct
) Qg ,
(7)
where the state variable x2 was selected as Equation (8): .
x2 = [ Z, Z − b1 Q], "
0 − M−1 K
(8) I − M−1 C
#
After obtaining the state coefficient matrix A = and the input matrix, ( ) M−1 Ct B= , the appropriate output state matrix C and the output control M−1 Kt−M−1 C × M−1 Ct matrix D (general definition D = 0) were specified via the state space ss.m function in MATLAB, and an 11-DOFs state-space model of a dump truck was built on the basis of Equation (7).
modal parameters of the suspension system, i.e., natural frequency, damping ratio and mode shape, under different spring stiffnesses through simulation in MATLAB (R2017a, MathWorks, Natick, MA, USA). Firstly, the stiffness of the left rear spring is changed from 0% to 20% in the model to simulate Appl. Sci. 2018, 8, 1795 9 of 18 spring failure and to study its effect on modal parameters. Figure 8 shows the changes in natural frequency, damping ratio and mode shapes caused by the drop in spring stiffness when the left rear leaf spring fails; the blue line represents the bounce mode, the green line represents the pitch mode The state matrix A contains all the characteristic information of the model. Therefore, by decomposing and the red line represents the roll mode. A, the modal natural frequency fn, damping ratio ξ, and the modal vector v can be obtained. It can be seen from the first three modes that when the stiffness of the left rear spring is weakened, the natural all three decreases to varying degrees, among which the pitch mode 3.2. Influence frequencies of SuspensionofFaults on Modal Parameters changes the most. Because of the small amplitude of the natural frequency variation, it is difficult to During the measurements long-term operation of the applications. vehicle, loadThe conditions environmental obtain accurate in practical dampingand ratios of all three factors modes can cause fault phenomena such as cracks and breaks in leaf springs, resulting in changes in the increased significantly, but the above suggested that the identification error of the damping ratio is performance parameters of the suspension springs and result a decrease in spring stiffness. To realize larger than that of the other two. That is, the identified can hardly reflect the actual changethe in suspension faultsoidentification based on indicator modal parameter analysis, it is essential to first understand damping ratio, it is not suitable as an of fault diagnosis. From the change of the vibration the specific influence of the fault on the modal parameters. Therefore, it is necessary to calculate mode shapes in the three orders of modes in Figure 8, due to the mode shapes being consideredthe as modal parameters the suspension system, i.e., natural frequency, modefeature shape, can under eigenvectors, the of values are dimensionless, and the changes are damping obvious. ratio This and obvious be different springthe stiffnesses through in MATLAB Natick, MA, USA). used to judge fault, and from simulation the previous analysis in(R2017a, SectionMathWorks, 2.2, the identification error of the Firstly, the stiffness of the left rear spring is changed from 0% to 20% in the model to simulate modal mode is relatively small and thus suitable as a monitoring indicator. spring and to study effect in onthe modal Figure 8 shows the changes natural It failure can be seen from the its changes modeparameters. shape in Figure 8 that as a feature vector,inthe value frequency, damping ratio and mode shapes caused by the drop in spring stiffness when the left rear of the mode shape is dimensionless and its variation is more obvious. This distinctive feature can be leaf fails; the mode,indicator. the green line represents the pitch mode usedspring to diagnose theblue faultline andrepresents is suitablethe as abounce monitoring and the red line represents the roll mode.
Figure Changesininnatural natural frequency, damping ratio mode shape caused by front a leftspring front Figure 8. Changes frequency, damping ratio andand mode shape caused by a left spring fault. fault.
ItThen, can be seen from the first three modes thatdifference when the (MED) stiffnessisofproposed the left rear weakened, the variation called modal energy to spring reveal is the faults of the natural frequencies of all three decreases to varying degrees, among which the pitch mode suspension system more straightforwardly. In order to convey the imbalance of the modechanges shapes the most. the Because of the small amplitude of trends the natural frequency variation, it as is difficult to obtain between left and right sides, the rolling of mode shape are denoted the differential of accurate measurements in practical applications. The damping ratios of all three modes increased square of two mode shape, which is called modal energy difference (MED) in the following section. significantly, but thethe above suggested that the identification error of thebedamping ratio is larger than The MED between left and right suspensions in bounce mode can defined as Equation (9). that of the other two. That is, the identified result can hardly2 reflect the actual change in damping ratio, dEbf diagnosis. 2[(va2 vb2From ) / (va the vb2change )] so it is not suitable as an indicator of fault of the vibration mode shapes (9) 2 2 2 2 2[( v ) / ( v v )] in the three orders of modes in Figure dE 8, brdue tovthe mode shapes being considered as eigenvectors, c d c d , the values are dimensionless, and the changes are obvious. This obvious feature can be used to judge the fault, and from the previous analysis in Section 2.2, the identification error of the modal mode is relatively small and thus suitable as a monitoring indicator. It can be seen from the changes in the mode shape in Figure 8 that as a feature vector, the value of the mode shape is dimensionless and its variation is more obvious. This distinctive feature can be used to diagnose the fault and is suitable as a monitoring indicator. Then, the variation called modal energy difference (MED) is proposed to reveal the faults of suspension system more straightforwardly. In order to convey the imbalance of the mode shapes between the left and right sides, the rolling trends of mode shape are denoted as the differential of square of two mode shape, which is called modal energy difference (MED) in the following section. The MED between the left and right suspensions in bounce mode can be defined as Equation (9).
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where in, dE bf represents the difference between the vibration modes of left and right suspensions front of the vehicle in the bounce mode, and dE br represents the MED in rear suspensions. Appl. Sci. in 2018, 8, 1795
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In Figure 9, the solid line is the MED between the two front suspensions and the dashed line represents the corresponding MED at the rear of the vehicle. The left and right sides of each figure 2 + v2 )] dEenergy [(v2a − vin2b )the /(vfirst-order respectively represent the modal b f = 2changes a b ,mode (bounce mode) and the (9) 2 2 2 third-order mode (pitch mode). dEbr = 2[(vc − vd )/(vc + v2d )] As shown in Figure 9a, when the left front wheel leaf spring fails, the difference in modal energy in the mode increases by about 15%, andthe thevibration difference modes in modalofenergy in the pitch mode where in, dEbbounce the difference between left and right suspensions in f represents drops by about 10%. When the right front wheel leaf spring fails (Figure 9b), the modal energy of the front of the vehicle in the bounce mode, and dEbr represents the MED in rear suspensions. by about 10%, while the modal energy of the pitch mode increases from In bounce Figuremode 9, thedecreases solid line is the MED between the two front suspensions and the dashed line −5% to 3%, which is opposite to the trend presented in the two-order mode of the left front leaf spring. represents the corresponding MED at the rear of the vehicle. The left and right sides of each figure When the left rear leaf spring fails (Figure 9c), the modal energy of bounce mode increases by 3%, respectively modal energy in the first-order (bounce mode)ofand the and therepresent pitch modethe increases from −7% tochanges 3%. As shown in Figure 9d, themode decrease in the stiffness third-order mode mode). the right rear(pitch leaf spring causes a decrease in the modal energy of two modes, wherein the pitch mode decreases by 15%. Pitch mode
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Figure 9. Change of modal energydifference difference (MED) caused by spring faults atfaults different Figure 9. Change of modal energy (MED) caused by spring at positions. different(a): positions. MED changes caused by faults in left front wheel; (b): MED changes caused by faults in right front (a): MED changes caused by faults in left front wheel; (b): MED changes caused by faults in right front wheel; (c): MED changes caused by faults in left rear wheel; (d): MED changes caused by faults in wheel; (c): MED changes caused by faults in left rear wheel; (d): MED changes caused by faults in right right rear wheel. rear wheel.
It can be seen that the weakening of the leaf springs at different positions of the vehicle can cause
Asdifferent shown in Figure when the because left front springstiffness fails, the difference in modal changes in 9a, modal energy, thewheel changeleaf of spring at different positions will energy cause themode vehicleincreases to roll in different directions. It can concluded that the modal energy in the bounce by about 15%, and thebedifference in modal energy indifference the pitch mode is sensitive to leaf spring failure both the bounce pitchfails modes. Moreover, the position of the of the drops by about 10%. When the rightinfront wheel leafand spring (Figure 9b), the modal energy fault and the magnitude of the change in stiffness can be determined by the change in the modal bounce mode decreases by about 10%, while the modal energy of the pitch mode increases from −5% energy difference. to 3%, which is opposite to the trend presented in the two-order mode of the left front leaf spring. In order to simulate the influence of the damper fault on the modal parameters, the damping When the left leaf absorber spring fails (Figure 9c), the modal energy of to bounce value of rear the shock is gradually reduced from the original 100% 30%. mode increases by 3%, and the pitch increases from −7%damper to 3%. decreases, As shown Figure 9d, the inthe thefront stiffness of Asmode the damping of the left-front asin seen in Figure 10a, decrease the MED of rear in the bounce modeapresents a diametrically opposite In the pitch wherein mode, the the MED of mode the rightand rear leaf spring causes decrease in the modal energytrend. of two modes, pitch the front side gradually increases with the decrease of the damping value, and that of the rear side decreases by 15%. gradually decreases. If the right-front damper fails (Figure 10b), the MED in the bounce mode exhibits It can be seen that the weakening of the leaf springs at different positions of the vehicle can cause a tendency opposite to that of left-front side. In the pitch mode, unlike the left-front damper failure, different changes in modal energy, because the change of spring stiffness at different positions will the MED of the front and rear side increases slightly with the decrease of the damping. cause the vehicle to roll in different directions. It can be concluded that the modal energy difference is sensitive to leaf spring failure in both the bounce and pitch modes. Moreover, the position of the fault and the magnitude of the change in stiffness can be determined by the change in the modal energy difference. In order to simulate the influence of the damper fault on the modal parameters, the damping value of the shock absorber is gradually reduced from the original 100% to 30%. As the damping of the left-front damper decreases, as seen in Figure 10a, the MED of the front and rear in the bounce mode presents a diametrically opposite trend. In the pitch mode, the MED of the front side gradually increases with the decrease of the damping value, and that of the rear side gradually decreases. If the right-front damper fails (Figure 10b), the MED in the bounce mode exhibits a tendency opposite to that of left-front side. In the pitch mode, unlike the left-front damper failure, the MED of the front and rear side increases slightly with the decrease of the damping.
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Figure10. 10.Changes ChangesofofMED MEDin indifferent differentpositions positionsof of damper damper faults. faults. (a): MED changes Figure changes caused caused by byfaults faults in left-front damper; (b): MED changes caused by faults in right-front damper. in left-front damper; (b): MED changes caused by faults in right-front damper.
ItItcan seen that thethe modal energy difference can can clearly indicate the damping changes in a shock canbebe seen that modal energy difference clearly indicate the damping changes in a absorber and reveal thereveal fault behind it. Therefore, damper failure can befailure effectively and shock absorber and the fault behind it. the Therefore, the damper can monitored be effectively diagnosed thediagnosed method proposed in thisproposed paper. in this paper. monitoredby and by the method ToTosummarize thethe tendencies of stiffness and and damping ratioratio in theinTable 3, we can seecan the positive summarize tendencies of stiffness damping the Table 3, we see the and negative is distinctive in every situation. This shows proofthe that MED can indicate positive and direction negative direction is distinctive in every situation. Thisthe shows proof that MED can the faults.the faults. indicate Table 3. Tendencies of stiffness and damping ratio. Table 3. Tendencies of stiffness and damping ratio. Left
Left
Right
Right Pitch Mode Bounce Mode Pitch Mode Bounce Mode Pitch Mode Front MED Rear MED Front MED Rear MED Front MED Rear MED Front MED Rear MED Front Rear Front Rear Front Rear Front Rear Front spring + + + + Rear spring + MED + + + MED MED MED MED MED MED MED Damper + + + + + Front spring + + + + Rear spring + + + + ItDamper strongly suggests that the modal energy differences are susceptible to stiffness changes + + + + + and damping changes. Furthermore, the diverse trends of modal energy correspond to the occurrence of different faults, which help to achieve the purpose fault diagnosis positioning and condition monitoring. It strongly suggests that the modal energyofdifferences are susceptible to stiffness changes and Bounce Mode
Pitch Mode
Bounce Mode
damping changes. Furthermore, the diverse trends of modal energy correspond to the occurrence of 3.3. Verification of ACS-SSI Applied on MED different faults, which help to achieve the purpose of fault diagnosis positioning and condition monitoring. To further verify the correctness and accuracy of the proposed ACS-SSI algorithm, the failures of the leaf spring and the damper were simulated for error analysis. A noise signal with signal-to-noise 3.3. (SNR) Verification of ACS-SSI onaMED ratio equaling two is Applied added to select normal vibration response to simulate the fault vibration signal.To The sampling frequency of the simulation system is proposed 200 Hz, the sampling time is 20 and the further verify the correctness and accuracy of the ACS-SSI algorithm, thes,failures average correlation calculation is carried out 50 times. Then,analysis. the proposed ACS-SSI algorithm was of the leaf spring and the damper were simulated for error A noise signal with signal-toused to perform modal identification. The input data is set for a period of 100 s and averaged 50 times. noise ratio (SNR) equaling two is added to a select normal vibration response to simulate the fault Monte Carlo tests The weresampling performed 10 timesoffor simulation, and identification vibration signal. frequency theeach simulation system is its 200corresponding Hz, the sampling time is 20 error is shown in Tables 4 and 5. calculation is carried out 50 times. Then, the proposed ACS-SSI s, and the average correlation algorithm was used to perform modal identification. The input data is set for a period of 100 s and Table 4. Identification error of simulated suspension spring faults. averaged 50 times. Monte Carlo tests were performed 10 times for each simulation, and its corresponding identification in Tables Stiffness Changes error is shown 0% 5% 4 and 5.10% 15% 20% Natural Frequency Error 3.16% 2.28% 2.14% 2.07% Modal Shape Error 4.78% 3.67% 3.16% spring 2.96% Table 4. Identification error of simulated suspension faults.
3.25% 4.69%
Stiffness Changes 0% error5% 15% 20% Table 5. Identification of simulated10% absorber faults. Natural Frequency Error 3.16% 2.28% 2.14% 2.07% 3.25% Changes 4.78% 0% 3.67% 20% 3.16% 40% 2.96% 60% 4.69% ModalDamping Shape Error Natural Frequency Error 4.32% 3.34% 5.53% Modal Shape Error 3.46% 2.64% 4.28% Table 5. Identification error of simulated absorber faults.
3.45% 2.86%
Damping Changes 0% 20% It can be seen from Tables 4 and 5 that the identification error is40% always60% within a reasonable range, Natural Frequency Error 4.32% 3.34% 5.53% 3.45% and the recognition error does not increase due to the change of the suspension parameters. Modal Shape Error 3.46% 2.64% 4.28% 2.86%
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It can be seen from Tables 4 and 5 that the identification error is always within a reasonable range, and the recognition error does not increase due to the change of the suspension parameters. In order bebe clearly distinguished according to order to to investigate investigatewhether whetherthe thedifferent differentfault faultstates statescan can clearly distinguished according the modal energy difference in theinpresence of noise the Monte test was repeated to the modal energy difference the presence of interference, noise interference, theCarlo Monte Carlo test was 50 times for spring stiffness conditions. Theconditions. modal energy corresponding to repeated 50three timesdifferent for three different spring stiffness Thedifference modal energy difference the three springtostiffness states is shown in states Figureis11. corresponding the three spring stiffness shown in Figure 11.
Figure 11. Identification of rear modal energy difference.
4. Real-Scale Vehicle Vehicle Tests 4. Real-Scale Testsand andResult ResultAnalysis Analysis The fault identification and diagnosis of The feasibility feasibility of of the theapplication applicationofofthe theACS-SSI ACS-SSIalgorithm algorithminin fault identification and diagnosis the dump truck suspension systems However, of the dump truck suspension systemshas hasbeen beenproven provenby byaaseries seriesof ofdynamic dynamic simulations. simulations. However, in the actual operation of the dump truck, there are many non-stationary, nonlinear in the actual operation of the dump truck, there are many non-stationary, nonlinear dynamic dynamic characteristics and other other mechanical vibration events events aliased aliased in in the characteristics and mechanical vibration the vibration vibration response, response, which which will will affect the effectiveness of the diagnostic algorithm. To this end, it is necessary to further study affect the effectiveness of the diagnostic algorithm. To this end, it is necessary to further study the the feasibility application of the ACS-SSI ACS-SSI algorithm algorithm in the fault fault diagnosis diagnosis of of dump dump truck truck suspension suspension feasibility of of the the application of the in the systems systems through through real-scale real-scale vehicle vehicle tests. tests. 4.1. Sensor Arrangement 4.1. Sensor Arrangement In order to accurately obtain the dynamic response of the suspension system, multiple acceleration In order to accurately obtain the dynamic response of the suspension system, multiple sensors (Model No. X901, Sinocera Piezotronics, Inc., Yangzhou, China) needed to be placed at the acceleration sensors (Model No. X901, Sinocera Piezotronics, Inc., Yangzhou, China) needed to be axle, at the junction of the suspension and frame, and at the frame, as seen in Figure 12. During placed at the axle, at the junction of the suspension and frame, and at the frame, as seen in Figure 12. the operation of the dump truck, the suspension dynamic response is mainly caused by the vertical During the operation of the dump truck, the suspension dynamic response is mainly caused by the excitation of the road surface irregularity. Therefore, only unidirectional acceleration sensors were vertical excitation of the road surface irregularity. Therefore, only unidirectional acceleration sensors vertically arranged in all mounting points. Considering the structural characteristics of the dump truck were vertically arranged in all mounting points. Considering the structural characteristics of the with more degrees of freedom, 12 measuring points were symmetrically selected on both sides of the dump truck with more degrees of freedom, 12 measuring points were symmetrically selected on both vehicle, including: the front and rear lifting lugs of front suspension, front axle (unsprung), the fourth sides of the vehicle, including: the front and rear lifting lugs of front suspension, front axle beam of frame and the intermediate and rear axles (unsprung). The specific mounting position of (unsprung), the fourth beam of frame and the intermediate and rear axles (unsprung). The specific the acceleration sensors on the left side of the suspension is shown in Figure 12. All of the vibration mounting position of the acceleration sensors on the left side of the suspension is shown in Figure 12. signals were recorded simultaneously by a multiple channel acquisition system (Model No. YMC9232, All of the vibration signals were recorded simultaneously by a multiple channel acquisition system YMC Piezotronics, Inc., Yangzhou, China), and thereafter processed offline for making corresponding (Model No. YMC9232, YMC Piezotronics, Inc., Yangzhou, China), and thereafter processed offline comparisons against 11-DOFs dynamic model predictions. In keeping with the sampling rate of the for making corresponding comparisons against 11-DOFs dynamic model predictions. In keeping simulation system, the frequency of data acquisition is set to 200 Hz. with the sampling rate of the simulation system, the frequency of data acquisition is set to 200 Hz.
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Figure 12. 12. Position Position of of accelerometers. accelerometers. Figure
4.2. Testing Program 4.2. Testing Program In order to verify the identification accuracy and robustness of the ACS-SSI algorithm under In order to verify the identification accuracy and robustness of the ACS-SSI algorithm under complex operating conditions and a strong noise background, tests under various suspension faults complex operating conditions and a strong noise background, tests under various suspension faults were carried out. Suspension faults were identified by analyzing the vibration response caused by were carried out. Suspension faults were identified by analyzing the vibration response caused by random excitation of the road surface roughness. The real-scale vehicle testing programs under the random excitation of the road surface roughness. The real-scale vehicle testing programs under the condition of leaf spring failure (stiffness weakening) and shock absorber failure (damping reduction) condition of leaf spring failure (stiffness weakening) and shock absorber failure (damping reduction) are designed respectively. The two sets of testing cases are as follows: are designed respectively. The two sets of testing cases are as follows: In the first case, named Case 1, the left rear suspension spring was replaced by two springs with In the first case, named Case 1, the left rear suspension spring was replaced by two springs with 80% and 90% stiffness of the original spring to simulate a suspension failure. It should be noted that, 80% and 90% stiffness of the original spring to simulate a suspension failure. It should be noted that, corresponding to the simulation results, the sampling length of the test data for each case is 200 s/times corresponding to the simulation results, the sampling length of the test data for each case is 200 × 100 times = 20,000 s. s/times × 100 times = 20,000 s. In the second case (Case 2), the left and right dampers of the front suspension were replaced In the second case (Case 2), the left and right dampers of the front suspension were replaced by by adjustable dampers. The sampling length of each test is the same 20,000 s. Table 6 summarizes the adjustable dampers. The sampling length of each test is the same 20,000 s. Table 6 summarizes the changes of damping in the experiments. The damping of the adjustable damper can be set to three states, changes of damping in the experiments. The damping of the adjustable damper can be set to three and State 1 is the baseline state. The damping value of State 1 to State 3 decreases from 100% to 60%. states, and State 1 is the baseline state. The damping value of State 1 to State 3 decreases from 100% to 60%. Table 6. Changes of damping in the test. State 1 2 3
Table 6. ChangesChange of damping in the test.Damping Change of Left Front Damping of Right Front
State 1 2 3
0% Change of Left 20% Front 40%Damping 0% 20% 40%
0% Change of Right 0% Front Damping 0% 0% 0% 0%
Vehicle Speed 30 km/h
Vehicle Speed 30 km/h 30 km/h 30 km/h 30 km/h
30 km/h
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4.3. Test and and Result Result Analysis Analysis 4.3. 4.3. Test Test and Result Analysis 4.3.1. Leaf Leaf Spring Spring Failure Failure (Case (Case 1) 1) 4.3.1. Firstly,the themode modeshapes shapesand andmodal modalenergy energydifferences differencesof ofthe theleaf leafspring springwith withand andwithout withoutfailure failure Firstly, the differences Firstly, without failure were analyzed. The modal frequencies and mode shapes of the baseline suspension and Case 1 were were analyzed. The modal frequencies and mode shapes of the baseline suspension and Case 1 were compared, and and the the results resultsare areshown shownin inFigure Figure13. 13. compared, Figure 13.
Figure13. 13.Differences Differencesofof ofmode mode shape between baseline and spring fault. RF: Right-Front; Right-Front; LF: LeftLeftshape between baseline andand spring fault. RF: Right-Front; LF: Left-Front; Figure 13. Differences mode shape between baseline spring fault. RF: LF: Front; LR: Left-Rear. LR: Left-Rear. Front; LR: Left-Rear.
Figure 13a,b shows the bounce and pitch mode vibration shapes at baseline; Figure 13c,d show Figure 13a,b 13a,b shows shows the the bounce bounce and and pitch pitch mode mode vibration vibration shapes shapes at at baseline; baseline; Figure Figure 13c,d 13c,d show show Figure the bounce and pitch mode shapes of the spring with 80% stiffness. It is apparent that the amplitudes thebounce bounceand andpitch pitchmode modeshapes shapesof of the thespring springwith with 80% 80%stiffness. stiffness.ItItis isapparent apparent that that the theamplitudes amplitudes the of pitch and roll motions in Figure 13c,d are greater than that in Figure 13a,b, which verifies the results of pitch and roll motions in Figure 13c,d are greater than that in Figure 13a,b, which verifies the of pitch and roll motions in Figure 13c,d are greater than that in Figure 13a,b, which verifies the obtained by the simulation. results obtained by the simulation. results obtained by the simulation. Then, difference in bounce andand pitchpitch modemode was obtained to highlight the change Then,the themodal modalenergy energy difference in bounce bounce and pitch mode was obtained obtained to highlight highlight the Then, the modal energy difference in was to the of stiffness. It can be seen from Figure 14 that the MED differences change with the stiffness of the change of stiffness. It can be seen from Figure 14 that the MED differences change with the stiffness change of stiffness. It can be seen from Figure 14 that the MED differences change with the stiffness faulty suspension spring.spring. It can be that based the ACS-SSI modal identification method, of the the faulty faulty suspension suspension spring. can be be concluded concluded thaton based on the the ACS-SSI ACS-SSI modal identification identification of ItIt concluded can that based on modal the spring stiffness can fault be diagnosed by the change ofchange the modal energy difference. In practical method, the springfault stiffness fault can be be diagnosed diagnosed by the the change of the the modal energy difference. difference. In method, the spring stiffness can by of modal energy In applications, appropriate test calibration for different vehicle types and fault forms can make the practical applications, appropriate test calibration for different vehicle types and fault forms can practical applications, appropriate test calibration for different vehicle types and fault forms can algorithm more widely used and more suitable for online for monitoring. make the the algorithm algorithm more widely used and and more more suitable for online online monitoring. monitoring. make more widely used suitable
Figure 14. 14.Difference Difference of ofrear rearMED MED between betweenbaseline baselineand andspring spring fault. fault. Difference Figure
4.3.2. Damper Failure (Case 2) According to the testing program, the damping of the left front suspension damper was adjusted to three different states. It can be seen from the comparison of the modal energy differences in Figure 15a that the MED of the front bounce mode exhibits a significant rise, while the MED of the rear side Appl. Sci. 2018, 8, 1795 15 of 18 shows the opposite downward trend. In the pitch mode, the MED of front pitch mode gradually increases as the damping decreases, while the MED of rear pitch mode is slightly reduced. with the simulation predictions in Section 3.3, the MED of rear suspension failure in 4.3.2.Consistent Damper Failure (Case 2) the bounce mode exhibits a tendency opposite to that of front suspension failure. In the pitch mode, According thefront testing program, damping of the with left front suspension damper was adjusted the MED of bothtothe and rear sidethe increases slightly the decrease of the damping, which is to three different states. It can be seen from the comparison of the modal energy differences in also consistent with the simulation prediction. This shows that when the rear side damper fails, the Figure 15a that the MED of theexacerbate front bounce exhibits a significant rise, while the MEDThe of the rear reduction of damping will themode rolling motion of the front suspension. results side shows the opposite downward trend. In the pitch mode, the MED of front pitch mode gradually identified from the test data are basically consistent with the simulation, indicating that the MEDincreases as the damping decreases, whilefor themonitoring MED of rear pitch mode is reduced. based monitoring technology is effective and diagnosis ofslightly the damper faults.
(a) Left-front damper fails
(b) Right-front damper fails Figure 15. 15. Changes of MED caused by different different damper damper faults. faults. Figure
Consistent with the simulation predictions in Section 3.3, the MED of rear suspension failure 5. Conclusions in the bounce mode exhibits a tendency opposite to that of front suspension failure. In the pitch This paper proposes a fault identification method based on an average correlation signal based mode, the MED of both the front and rear side increases slightly with the decrease of the damping, stochastic subspace identification (ACS-SSI) algorithm and verifies its application feasibility in fault which is also consistent with the simulation prediction. This shows that when the rear side damper diagnosis of dump truck suspension through a series of simulations and an experiment. The main fails, the reduction of damping will exacerbate the rolling motion of the front suspension. The results conclusions can be made as follows: identified from the test data are basically consistent with the simulation, indicating that the MED-based 1. It can be concludedisthat the modal energy difference (MED)of is the sensitive to faults. the leaf spring failure monitoring technology effective for monitoring and diagnosis damper in both the bounce and pitch modes. Moreover, the position of the fault and the variation 5. Conclusions amplitude of stiffness can be determined by the combination of MED in bounce and pitch modes. The paper plus or minus asign the MEDs can indicate location of faults, and the This proposes faultofidentification method basedthe onspecific an average correlation signal based magnitude of change in MED can characterize the severity of the fault. stochastic subspace identification (ACS-SSI) algorithm and verifies its application feasibility in fault 2. The change in MED by damping is of different from that by stiffness diagnosis of dump truck caused suspension throughfailure a series simulations and caused an experiment. Thefailure, main but its location and fault degree can still be effectively identified by the combined analysis of conclusions can be made as follows: MED. 1. It can be concluded that the modal energy difference (MED) is sensitive to the leaf spring failure in In conclusion, the front and rear MEDs in bounce and pitch modes can serve as a vector to both the bounce and pitch modes. Moreover, the position of the fault and the variation amplitude indicate the location and severity of the fault that occurred in the dump truck suspension. It shows of stiffness can be determined by the combination of MED in bounce and pitch modes. The plus that the proposed average correlation signal based stochastic subspace identification (ACS-SSI) or minus sign of the MEDs can indicate the specific location of faults, and the magnitude of method can identify the changes in suspension modal parameters effectively with respect to different change in MED can characterize the severity of the fault. spring stiffness and damping ratio conditions, which paves a promising way for the application of 2. The change in MED caused by damping failure is different from that caused by stiffness failure, but its location and fault degree can still be effectively identified by the combined analysis of MED. In conclusion, the front and rear MEDs in bounce and pitch modes can serve as a vector to indicate the location and severity of the fault that occurred in the dump truck suspension. It shows that the proposed average correlation signal based stochastic subspace identification (ACS-SSI) method can identify the changes in suspension modal parameters effectively with respect to different
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spring stiffness and damping ratio conditions, which paves a promising way for the application of vibration-based online fault diagnosis to commercial vehicles operating in poor working environments and complex operating conditions. Author Contributions: The contributions of the individual authors to this research work are as follows: B.L. and T.W. conceived and designed the experiments; Z.J. and Z.T. performed the experiments; B.L. and G.L. analyzed the data; B.L. reviewed previous research work and wrote the paper. Funding: This research received no external funding. Conflicts of Interest: The authors declare no conflicts of interest.
Acronyms and Notations ACS-SSI COV-SSI DOF SNR SVD MED MAC A B C D w(k) v(k) k y(t) rk (τ) rk (τ ) λi ψi Zb Θ Φ ZA . . . F θC , θD kE , kF cE , cF M Z C K Ct Kt Qg x u dEbf dEbr
average correlation signal based stochastic subspace identification covariance-driven stochastic subspace identification degrees of freedom signal-to-noise ratio singular value decomposition modal energy difference modal assurance criterion state matrix input matrix output matrix feedthrough matrix random input signal measurement noise discrete time value matrix of l-channel measurement signal correlation function of original signals average correlation function eigenvalue of matrix A eigenvector of matrix A bounce motion of vehicle body pitch motion of vehicle body roll motion of vehicle body bounce motion of axles pitch motion of balanced suspension separate stiffness of leaf spring for balanced suspension separate damping of leaf spring for balanced suspension mass matrix vector of vehicle dynamic responses matrix of vehicle damping matrix of vehicle stiffness matrix of tire damping matrix of tire stiffness vector of displacement inputs intermediate variable input variable MED between the left and right suspensions at the front of vehicle in bounce mode MED between the left and right suspensions in rear of vehicle in bounce mode
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