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Estimation of Acceleration Amplitude of Vehicle by Back Propagation Neural Networks

Estimation of Acceleration Amplitude of Vehicle by Back Propagation Neural Networks Hindawi Publishing Corporation Advances in Acoustics and Vibration Volume 2013, Article ID 614025, 7 pages http://dx.doi.org/10.1155/2013/614025 Research Article Estimation of Acceleration Amplitude of Vehicle by Back Propagation Neural Networks 1 2 Mohammad Heidari and Hadi Homaei Mechanical Engineering Group, Aligudarz Branch, Islamic Azad University, P.O. Box 159, Aligudarz, Iran Faculty of Engineering, University of Shahrekord, P.O. Box 115, Shahrekord, Iran Correspondence should be addressed to Mohammad Heidari; moh104337@yahoo.com Received 5 April 2013; Accepted 19 May 2013 Academic Editor: Emil Manoach Copyright © 2013 M. Heidari and H. Homaei. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. This paper investigates the variation of vertical vibrations of vehicles using a neural network (NN). eTh NN is a back propagation NN, which is employed to predict the amplitude of acceleration for different road conditions such as concrete, waved stone block paved, and country roads. In this paper, four supervised functions, namely, newff, newcf, newelm, and newd, fft have been used for modeling the vehicle vibrations. eTh networks have four inputs of velocity ( 𝑉 ), damping ratio (𝜁 ), natural frequency of vehicle shock absorber (𝑤 ), and road condition (R.C) as the independent variables and one output of acceleration amplitude (AA). Numerical data, employed for training the networks and capabilities of the models in predicting the vehicle vibrations, have been verified. Some training algorithms are used for creating the network. The results show that the Levenberg-Marquardt training algorithm and newelm function are better than other training algorithms and functions. This method is conceptually straightforward, and it is also applicable to other type vehicles for practical purposes. 1. Introduction was the phase control on vibration transmission in hydraulic engine methods. The other was the vector synthesis approach Recently, improveing comfort and safety conditions for vehi- in treating multiple vibrations input to the vehicle body. cles considering disturbances due to road roughness has been A new method for predicting vibration characteristics of studied by several researchers. To minimise the disturbing a structure that is considered to undergo a design change effects of vibration, optimum damping factor has been inves- has been presented [6]. Methodologies for determining the tigated. In the case of den fi ite road profile, that is, for the case vibration characteristics of the modified structure have also been discussed. A vehicle-subgrade model of vertical coupled of definite vibration with single, two, and three degrees of freedom systems, physical and mathematical models can be system has been presented, and the interactions between the established. However, in practice, vehicle vibrations arising vehicle tuning quality and the subgrade design parameters from road roughness possess random character. Vibration have been investigated in systematic concept and from the analysis for such systems can be accomplished by random viewpointofsystematicmatching[7]. theory basedonstatistics. Amethodwhich cansimulatethe A method for the analysis and simulation of nonstation- set vibrations of vehicle has been developed by Guclu and ary random vibrations has been presented by Rouillard and Gulez [1]. In their investigation, neural network control for a Sek [9]. Their method pays particular attention to the nonsta- tionary nature of vibrations generated by transport vehicles. nonlinear full vehicle model was defined by using permanent magnet synchronous motor. Chaos and bifurcation in non- The limitations of current methods used for analysing and linear vehicle model have been studied by Li et al. [2], Zhu simulating nonstationary random vehicle vibrations were and Ishitobi [3], and Litak et al. [4]. A solving method of low- also demonstrated. Yildirim and Uzmay used a radial basis frequency vehicle vibration problems has been presented by neural network to predict the amplitude of acceleration of Ishihama et al. [5]. Two ideas have been employed. eTh rfi st vehicle under different road conditions [ 10, 11]. 2 Advances in Acoustics and Vibration Table 1: Parameters depending on road conditions [ 8]. −1 −1 −1 Road 𝐴 𝐴 𝑎 (m ) 𝑎 (m ) 𝑏 (m ) 𝜎 (m) 1 2 1 2 2 𝑥0 0.85 0.15 0.2 0.05 0.60 0.0080–0.0126 As. (R1) 0 1 0.22 0.44 0.12 ESBP (R2) 1 0 0.45 — — 0.0135–0.0225 WSBP (R3) 0.85 0 0.45 — — 0.0250–0.0380 BP (R4) 0 1 — 0.32 0.64 0.017 0 1 — 0.47 0.94 0.019 CR (R5) 0 1 — 0.11 0.146 0.067–0.227 CO (R6) 1 0 0.15 — — 0.005–0.0124 Co.: concrete; As.: asphalt; ESBP: even stone block paved; WSBP: waved stone block paved; BP: boulder paved; CR: country road. 𝑆 (𝑤 ) In this study, vertical vehicle vibrations are studied using 𝑥 random theory, and some back propagation artificial neural 2 2 2 2 2𝜎 𝐴 𝛼 (𝑤 +𝛼 +𝛽 ) 𝐴 𝛼 networks (ANNs) with four functions such as newff, newelm, 𝑥 2 2 2 2 0 1 1 [ + ] |𝑤 |<𝑤 2 2 2 2 2 2 2 newcf, and newfftd are also employed to predict amplitudes = 𝜋 𝑤 +𝛼 (𝑤 −𝛼 −𝛽 )+4𝛼 𝑤 1 2 2 2 of accelerations of vehicles for different road conditions. 0 |𝑤 | <𝑤 , { 1 eTh organization of the paper is as follows. Section 2 (2) describes the theory of random vibration for vehicles. Overview of neural network is presented in Section 3.More where 𝐴 +𝐴 =1, 𝛼 =𝑎 𝑉, 𝛼 =𝑎 𝑉, 𝛽 =𝑏 𝑉, 𝑤 = 1 2 1 1 2 2 2 2 1 details of modeling of vehicle vibrations using neural net- Ω𝑉 .Asshown in (2), if the spectral density of the road works are given in Section 4. eTh simulation results obtained roughness is explained in terms of excitation frequency 𝑤 , 2 2 form BP are given in Section 5. eTh paper is concluded with namely, 𝑤 , it is described as 𝑥 /𝑤 = 𝑚 𝑠 ,orifitiswritten in Section 6. terms of the length frequency Ω (1/m), it is also described as 2 3 𝑥 /Ω = 𝑚 . Some parameters depending on road conditions are shown in Table 1. es Th e given parameters are the results 2. Random Vibration Theory of an experimental investigation [12]. The frequency of vehicle’s shock absorber must be chosen Vehicle vibrations due to road roughness have no definite between body frequency of 1–1.5 Hz and axle frequency of character, and system dynamics depends on the profile 10–15 Hz. eTh refore, damping ratio has to be selected so of roughness. eTh refore, statistical basis random theory is that the frequency of shock absorber is in the range of 4– employed in determining roughness character. Assuming 6 Hz. Consequently, the damping factor for absorber may be such vehicle vibrations to be linear, dynamic model of these taken as a value <0.5. In this damping ratio interval (0.1– systems can be represented as [12, 13] 0.5), vehicle body accelerations decrease for different road conditions [14]. 𝑥+2 ̈ 𝑤𝜁 𝑥+𝑤 ̇ 𝑥=− 𝑥 ̈ (𝑡 ) , (1) 𝑛 𝑛 0 3. Overview of Neural Networks where 𝑥 is the relative displacement of vehicle body; 𝑥 (𝑡) A neural network is a massive parallel system comprised is the amplitude over a specific level of the road roughness of highly interconnected, interacting processing elements, on which vehicle’s tyre moves at a definite time 𝑡 ; 𝜁 is the or nodes. Neural networks process through the interactions damping ratio, and 𝑤 also denotes natural frequency of of a large number of simple processing elements or nodes, vehicle’s shock-absorber system. also known as neurons. Knowledge is not stored within By determining amplitudes over a reference plane level on individual processing elements, rather represented by the a certain road condition by means of repeated measurements, strengths of the connections between elements. Each piece statistical roughness features are obtained. The road rough- of knowledge is a pattern of activity spread among many ness can be determined with enough approximation by some processing elements, and each processing element can be measurements accomplished for different road conditions. In involved in the partial representation of many pieces of order to describe the influence of the road roughness, the information. In recent years, neural networks have become most appropriate statistical parameter is its spectral density, a very useful tool in the modeling of complicated systems which is a mean square value of road roughness in a definite becausetheyhaveanexcellent abilitylearn andtogeneralize frequencyrange.Whenavehiclemovesatvelocity 𝑉 ,theroad (interpolate) the complicated relationships between input roughness spectral density can be written as follows [12]: and output variables. Also, the ANNs behave as model free ··· Advances in Acoustics and Vibration 3 Bias =1 k−1 1 k w =b w j0 j j1 k−1 j2 . . . . . . . . . Sum Activation . k function function ji Output (Σ) (f) k−1 Input layer First layer hidden layer Figure 1: Back propagation neural network with two hidden layers. jn k−1 estimators; that is, they can capture and model complex Figure 2: Architecture of an individual PE for BP network. input-output relations without the help of a mathematical model [15]. In other words, training neural networks, for example, eliminates the need for explicit mathematical mod- Before practical application, the network has to be eling or similar system analysis. This property of ANNs is trained. To properly modify the connection weights, an extremely useful in a situation where it is hard to derive a error-correcting technique, oen ft called as back propagation mathematical model. As a result, neural networks can provide learning algorithm or generalized delta rule [16], is employed. an eecti ff ve solution to solve problems that are intractable or Generally, this technique involves two phases through dieff r- cumbersome with mathematical approaches. ent layers of the network. The first is the forward phase, which occurs when an input vector is presented and propagated 3.1. Back Propagation (BP) Neural Network. The back prop- forwardthrough thenetwork to computeanoutputfor each agation network (Figure 1)iscomposedofmanyintercon- neuron. During the forward phase, synaptic weights are all nected neurons or processing elements (PEs) operating in fixed. eTh error obtained when a training pair (pattern-“ 𝑝 ”) parallel and are oen ft grouped in different layers. consists of both input and output given to the input layer of As shown in Figure 2, each articfi ial neuron evaluates the network is expressed by the following equation: the inputs and determines the strength of each through its weighing factor. In the artificial neuron, the weighed inputs 𝐸 = ∑ (𝑇 −𝑂 ) , are summed to determine an activation level. That is, 𝑝 (5) 𝑘 𝑘 𝑘−1 net = ∑ 𝑤 𝑜 , 𝑗 𝑗𝑖 𝑖 (3) where 𝑇 is the 𝑗 th component of the desired output vector, and 𝑂 is thecalculatedoutputof 𝑗 th neuron in the output layer. The overall error of all the patterns in the training set is where net is the summation of all the inputs of the 𝑗 th denfi edasmeansquareerror (MSE)and is givenby neuron in the 𝑘 th layer, 𝑤 is the weight from the 𝑖 th neuron 𝑗𝑖 𝑘−1 to the 𝑗 th neuron, and 𝑜 is the output of the 𝑖 th neuron in 𝐸= ∑ 𝐸 , (6) the (𝑘−1 )th layer. 𝑝 𝑝=1 eTh output of the neuron is then transmitted along the weighed outgoing connections to serve as an input where 𝑛 isthenumberofinput-outputpatternsinthetraining to subsequent neurons. In the present study, a hyperbolic set. eTh second is the backward phase which is an iterative tangent, log-sigmoid, and linear functions (𝑓( net ))witha error reduction performed in the backward direction from bias 𝑏 are used as an activation function of hidden and output the output layer to the input layer. In order to minimize neurons. Therefore, output of the 𝑗 th neuron 𝑜 for the 𝑘 th the error, 𝐸 , as rapidly as possible, the gradient descent layer can be expressed as method adding a momentum term is used. Hence, the new incremental change of weight Δ𝑤 (𝑚 + 1) can be 𝑗𝑖 𝑘 𝑘 (net +𝑏 ) −(net +𝑏 ) 𝑗 𝑗 𝑗 𝑗 𝑒 −𝑒 𝑘 𝑘 𝑜 =𝑓( net )= (tansig), 𝑘 𝑘 𝑗 𝑗 (net +𝑏 ) −(net +𝑏 ) 𝑘 𝑘 𝑗 𝑗 𝑗 𝑗 𝑒 +𝑒 Δ𝑤 (𝑚+1 ) =−𝜂 +𝛼Δ𝑤 (𝑚 ) , (7) 𝑗𝑖 𝑗𝑖 𝜕𝑤 𝑗𝑖 (4) 𝑘 𝑘 𝑜 =𝑓( net )= (logsig), 𝑗 𝑗 𝑘 −net +𝑏 1+𝑒 where 𝜂 is a constant real number between 0.1 and 1, called learning rate, 𝛼 is the momentum parameter usually set to 𝑘 𝑘 𝑘 𝑜 =𝑓( net )= net +𝑏 (linear) . 𝑗 𝑗 𝑗 𝑗 a number between 0 and 1, and 𝑚 is the index of iteration. 𝜕𝐸 𝑝𝑗 𝑝𝑗 𝑝𝑗 𝑝𝑗 4 Advances in Acoustics and Vibration eTh refore, the recursive formula for updating the connection weights becomes 𝑘 𝑘 𝑘 𝑤 (𝑚+1 ) =𝑤 (𝑚 ) +Δ𝑤 (𝑚+1 ) . (8) V 𝑗𝑖 𝑗𝑖 𝑗𝑖 These corrections can be made incrementally (aer ft each pattern presentation) or in batch mode. In the latter case, R.C the weights are updated only aer ft the entire training pattern AA set has been applied to the network. With this method, the order in which the patterns are presented to the network does w Output layer not influence the training. This is because of the fact that adaptation is done only at the end of each epoch. And thus, we have chosen this way of updating the connection weights [17]. Input layer 4. Modeling of Vehicle Vibrations Using Hidden layer Neural Networks Figure 3: General ANN topology. Modeling of vehicle vibrations with BP neural network is composed of two stages: training and testing of the networks with numerical data. eTh training data consisted of velocity Table 2: eTh variable training methods. (𝑉 ), damping ratio (𝜁 ), natural frequency of vehicle shock absorber (𝑤 ), road condition (R.C), and the corresponding Acronym Description acceleration amplitude. A total of 90 data sets were used, LM Levenberg-Marquardt of which 80 were selected randomly and used for training BFG BFGS Quasi-Newton purposes whilst the remaining 10 data sets were presented to the trained networks as new application data for verification RP Resilient back propagation (testing) purposes. u Th s, the networks were evaluated using SCG Scaled Conjugate Gradient data that had not been used for training. Before the ANN CGB Conjugate Gradient with Powell/Beale Restarts could be trained and the mapping learnt, it is important to CGF Fletcher-Powell Conjugate Gradient process the numerical data into patterns. Training/testing CGP Polak-Ribier ´ e Conjugate Gradient pattern vectors are formed, each formed with an input OSS One Step Secant condition vector GDX VariableLearningRatebackpropagation velocity (𝑉 ) [ ] damping ratio (𝜁 ) [ ] 𝑃 = (9) [ ] natural frequency (𝑤 ) 5. Numerical Results of BP Neural road condition R.C ( ) [ ] Network Model and the corresponding target vector The size of hidden layer(s) is one of the most important 𝑇 =[amplitude of acceleration (AA)]. (10) 𝑖 considerations when solving actual problems using multi- layer feed-forward network. However, it has been shown that Mapping each term to a value between −1and 1, we use BP neural network with one hidden layer can uniformly the following linear mapping formula: approximate any continuous function to any desired degree of accuracy givenanadequatenumberofneurons in the (𝑅 − 𝑅 )∗(𝑁 −𝑁 ) min max min 𝑁= +𝑁 , (11) min hidden layer and the correct interconnection weights [18]. (𝑅 −𝑅 ) max min Therefore, one hidden layer was adopted for the BP model. where 𝑁 is normalized value of the real variable; 𝑁 = To determine the number of neurons in the hidden layer, min −1 and 𝑁 =1 are minimum and maximum values of a procedure of trial and error approach needs to be done. max normalization, respectively; 𝑅 is real value of the variable; As such, attempts have been made to study the network 𝑅 and 𝑅 are minimum and maximum values of the performance with a different number of hidden neurons. min max real variable, respectively. These normalized data was used as Hence, a number of candidate networks are constructed, each the inputs and output to train the ANN. Figure 3 shows the of trained separately, and the “best” network was selected general network topology for modeling vehicle vibration. basedonthe accuracy of thepredictions in thetesting phase. The names of training algorithms used in this paper are It should be noted that if the number of hidden neurons shown in Table 2. is too large, the ANN might be overtrained giving spurious In what follows, the use of four neural networks will be values in the testing phase. If too few neurons are selected, discussed and the results are presented. en, Th the best model the function mapping might not be accomplished due to is picked based on the accuracy of AA in the verification stage. undertraining [19]. Table 3 shows 10 numerical data sets, used Advances in Acoustics and Vibration 5 Table 3: Vibration conditions for verification analysis. Table 6: Comparison of AA desired and predicted by the BP neural network model and newcf function. Natural Acceleration Test Velocity Damping Road Test no. Desired AA (cm) BP model AA (cm) Error (%) frequency amplitude no. (m/sec) ratio (𝜁) condition (Hz) (cm) 1 2.35 2.54 8.34 2 3.85 4.36 13.26 1 12 0.20 R1 10 2.35 3 5.28 5.76 9.10 2 15 0.33 R2 12 3.85 4 4.44 4.80 8.15 3 24 0.45 R5 15 5.28 5 2.90 3.23 11.56 4 35 0.50 R3 15 4.44 6 1.27 1.29 1.64 5 18 0.60 R6 8 2.90 7 4.18 4.94 18.29 6 50 0.65 R1 10 1.27 8 2.41 2.54 5.65 7 60 0.85 R3 13 4.18 9 5.47 5.66 3.49 8 40 0.25 R4 12 2.41 10 3.04 3.38 11.36 9 27 0.55 R4 10 5.47 10 19 0.75 R6 10 3.04 Table 7: Comparison of AA desired and predicted by the BP neural network model and newfftd function. Table 4: eTh eeff ct of different number of hidden neurons on the BP Test no. Desired AA (cm) BP model AA (cm) Error (%) network performance. 1 2.35 2.58 9.86 No. of hidden Epoch with LM Average error in AA (%) 2 3.85 4.75 23.39 neurons method training with newelm function 3 5.28 5.80 9.90 414390 9.80 4 4.44 5.09 14.67 5 5170 11.87 5 2.90 3.11 7.49 6 1753 4.95 6 1.27 1.47 16.24 7 1028 8.48 7 4.18 4.54 8.79 8 2.41 2.45 2.07 8739 15.45 9 5.47 6.04 10.47 10 3.04 3.08 1.39 Table 5: Comparison of AA desired and predicted by the BP neural network model and newff function. Table 8: Comparison of AA desired and predicted by the BP neural Test no. Desired AA (cm) BP model AA (cm) Error (%) network model and newelm function. 1 2.35 2.78 18.69 Test no. Desired AA (cm) BP model AA (cm) Error (%) 2 3.85 3.99 3.66 1 2.35 2.48 5.77 3 5.28 5.31 0.70 2 3.85 3.90 1.37 4 4.44 4.57 3.1 3 5.28 5.44 3.14 5 2.90 3.25 12.16 4 4.44 4.47 0.9 6 1.27 1.30 2.44 5 2.90 3.21 10.88 7 4.18 4.55 8.95 6 1.27 1.29 1.80 8 2.41 2.57 6.79 7 4.18 4.64 11.15 9 5.47 5.94 8.70 8 2.41 2.57 6.79 10 3.04 3.71 22.27 9 5.47 5.69 4.17 10 3.04 3.15 3.62 for verifying or testing network capabilities in modeling the vehicle vibration. eTh refore, the general network structure is supposed to performances were examined for these networks. As the error be 4-𝑛 -1, which implies 4 neurons in the input layer, 𝑛 criterion for all networks was the same, their performances neurons in the hidden layer, and 1 neuron in the output layer. are comparable. As a result, from Table 4,the best network Then, by varying the number of hidden neurons, different structure of BP model is picked to have 6 neurons in the network configurations are trained, and their performances hidden layer with the average verification errors of 4.95% in are checked. eTh results are shown in Table 4. amplitude acceleration over the 10 numerical verification data For training problem, equal learning rate and momentum sets. Tables 5, 6, 7 and 8 show the comparison of desired and constant of 𝜂 = 𝛼 = 0.85 were used [16]. Also, error stopping predicted values for amplitude acceleration in verification criterion was set at 𝐸 = 0.01 , which means that training cases with different functions. epochs continued until the mean square error fell beneath Figure 4 illustrates the convergence of the output error this value. Both the required iteration numbers and mapping (mean square error) with the number of iterations (epochs) 6 Advances in Acoustics and Vibration Table 9: e Th ( 𝑅 ) values for AA with various neurons in the hidden layer. Acronym of training method Number of hidden neurons LM BFG RP SCG CGB CGF CGP OSS GDX 0.9977 0.9888 0.9819 0.9957 0.9738 0.9918 0.9905 0.9913 0.9891 0.9981 0.9676 0.992 0.9909 0.9942 0.9864 0.9846 0.9969 0.9916 6 0.9999 0.9981 0.9942 0.9578 0.9986 0.9978 0.9681 0.9963 0.9945 0.9985 0.9998 0.9967 0.9996 0.9996 0.9995 0.9595 0.9890 0.9856 0.9967 0.9928 0.9994 0.9998 0.9998 0.9698 0.9898 0.9898 0.9881 Table 10: The results of the variable training methods in the BPN Table 12: The results of the variable training methods in the BPN with newelm function. with newff function. Epoch in Epoch in Acronym Error goal Train time (s) Test time (s) Acronym Error goal Train time (s) Test time (s) goal goal LM Met 28.7240 0.054526 LM Met 40.1450 0.054789 1753 1293 BFG 2528 Met 77.4824 0.046583 BFG 1528 Met 88.1024 0.056673 RP Not met 67.6773 0.046466 RP Met 100.6773 0.056906 3000 1210 SCG Not met 101.3281 0.044047 SCG Not met 97.9497 0.065237 3000 4000 CGB Not met 118.1020 0.052230 CGB Not met 138.7890 0.062230 2932 3745 CGF 2125 Not met 88.8947 0.046125 CGF 3450 Not met 112.8964 0.053565 CGP Not met 94.6653 0.046852 CGP Not met 100.1333 0.056907 2302 3162 OSS Not met 112.0710 0.046677 OSS Not met 160.1489 0.066677 3000 3400 GDX Not met 61.2435 0.046046 GDX Not met 43.4465 0.056044 3000 2000 Table 13: The results of the variable training methods in the BPN Table 11: The results of the variable training methods in the BPN with newcf function. with newfftd function. Epoch in Epoch in Acronym Error goal Train time (s) Test time (s) Acronym Error goal Train time (s) Test time (s) goal goal LM Met 38.7261 0.057241 LM Met 43.1808 0.061301 1251 2167 BFG Met 64.1920 0.058573 BFG 2018 Met 97.4814 0.042678 19018 RP Met 99.7452 0.064916 RP Not met 49.1627 0.048146 1890 SCG Not met 33.1907 0.042047 SCG 3786 Not met 39.1877 0.061937 CGB Not met 128.1590 0.055130 CGB Not met 134.3904 0.067220 3232 4150 CGF Not met 102.5594 0.057345 CGF 3025 Not met 108.1528 0.056985 1129 CGP Not met 100.1333 0.055897 CGP Not met 67.1503 0.044152 2001 OSS Met 59.0112 0.058677 OSS 4744 Not met 80.9358 0.060127 GDX Not met 67.1205 0.056046 GDX Not met 38.2065 0.059391 3230 3190 In Tables 10, 11, 12,and 13, the results of training the during training of the chosen 4-6-1 BP network. After 1753 network using nine different training algorithms by 6 neurons epochs, the MSE between the desired and predicted outputs in the hidden layer and logsig-purlin activation function are becomes less than 0.01. At the beginning of the training, the output from the network is far from the target value. summarized. Each entry in the table represents 10 different trials, where different random initial weights are used in each However, the output slowly and gradually converges to the trial. target valuewithmoreepochsand thenetwork learns the input/output relation of the training samples. eTh regression value (𝑅 ) of theoutputvariablevaluesfor thetestdataset 6. Conclusions and Summary for various neurons in hidden layer is shown in Table 9.It should be noted that these data were completely unknown to In this paper, four supervised neural networks have been thenetwork.Thecloserthisvalue is to unity, thebetteristhe used forthe vehiclevibrations. Basedonthe test resultsof prediction accuracy. eTh best (𝑅 ) value obtained is 0.9999, each network with some data set, different from those used and it is obtained from the LM algorithm by 6 neurons in in the training phase, it was shown that newelm function hidden layer. neural model has superior performance than newff, newcf, Advances in Acoustics and Vibration 7 Performance is 0.0099982; goal is 0.01 [5] T. Ishihama, H. Masao, and M. Seto, “Vehicle vibration reduc- tion by transfer function phase control on hydraulic engine mounts,” JSME International Journal C,vol.37, no.3,pp. 536– 541, 1994. [6] T. Y. Yi and P. E. Nikravesh, “A method of identify vibra- tion characteristics of modified structures for flexible vehicle −1 dynamics,” Proceedings of the Institution of Mechanical Engineers D,vol.216,no. 1, pp.55–63,2002. [7] B. Liang, D. Zhu, and Y. 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Estimation of Acceleration Amplitude of Vehicle by Back Propagation Neural Networks

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Copyright © 2013 Mohammad Heidari and Hadi Homaei. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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Hindawi Publishing Corporation Advances in Acoustics and Vibration Volume 2013, Article ID 614025, 7 pages http://dx.doi.org/10.1155/2013/614025 Research Article Estimation of Acceleration Amplitude of Vehicle by Back Propagation Neural Networks 1 2 Mohammad Heidari and Hadi Homaei Mechanical Engineering Group, Aligudarz Branch, Islamic Azad University, P.O. Box 159, Aligudarz, Iran Faculty of Engineering, University of Shahrekord, P.O. Box 115, Shahrekord, Iran Correspondence should be addressed to Mohammad Heidari; moh104337@yahoo.com Received 5 April 2013; Accepted 19 May 2013 Academic Editor: Emil Manoach Copyright © 2013 M. Heidari and H. Homaei. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. This paper investigates the variation of vertical vibrations of vehicles using a neural network (NN). eTh NN is a back propagation NN, which is employed to predict the amplitude of acceleration for different road conditions such as concrete, waved stone block paved, and country roads. In this paper, four supervised functions, namely, newff, newcf, newelm, and newd, fft have been used for modeling the vehicle vibrations. eTh networks have four inputs of velocity ( 𝑉 ), damping ratio (𝜁 ), natural frequency of vehicle shock absorber (𝑤 ), and road condition (R.C) as the independent variables and one output of acceleration amplitude (AA). Numerical data, employed for training the networks and capabilities of the models in predicting the vehicle vibrations, have been verified. Some training algorithms are used for creating the network. The results show that the Levenberg-Marquardt training algorithm and newelm function are better than other training algorithms and functions. This method is conceptually straightforward, and it is also applicable to other type vehicles for practical purposes. 1. Introduction was the phase control on vibration transmission in hydraulic engine methods. The other was the vector synthesis approach Recently, improveing comfort and safety conditions for vehi- in treating multiple vibrations input to the vehicle body. cles considering disturbances due to road roughness has been A new method for predicting vibration characteristics of studied by several researchers. To minimise the disturbing a structure that is considered to undergo a design change effects of vibration, optimum damping factor has been inves- has been presented [6]. Methodologies for determining the tigated. In the case of den fi ite road profile, that is, for the case vibration characteristics of the modified structure have also been discussed. A vehicle-subgrade model of vertical coupled of definite vibration with single, two, and three degrees of freedom systems, physical and mathematical models can be system has been presented, and the interactions between the established. However, in practice, vehicle vibrations arising vehicle tuning quality and the subgrade design parameters from road roughness possess random character. Vibration have been investigated in systematic concept and from the analysis for such systems can be accomplished by random viewpointofsystematicmatching[7]. theory basedonstatistics. Amethodwhich cansimulatethe A method for the analysis and simulation of nonstation- set vibrations of vehicle has been developed by Guclu and ary random vibrations has been presented by Rouillard and Gulez [1]. In their investigation, neural network control for a Sek [9]. Their method pays particular attention to the nonsta- tionary nature of vibrations generated by transport vehicles. nonlinear full vehicle model was defined by using permanent magnet synchronous motor. Chaos and bifurcation in non- The limitations of current methods used for analysing and linear vehicle model have been studied by Li et al. [2], Zhu simulating nonstationary random vehicle vibrations were and Ishitobi [3], and Litak et al. [4]. A solving method of low- also demonstrated. Yildirim and Uzmay used a radial basis frequency vehicle vibration problems has been presented by neural network to predict the amplitude of acceleration of Ishihama et al. [5]. Two ideas have been employed. eTh rfi st vehicle under different road conditions [ 10, 11]. 2 Advances in Acoustics and Vibration Table 1: Parameters depending on road conditions [ 8]. −1 −1 −1 Road 𝐴 𝐴 𝑎 (m ) 𝑎 (m ) 𝑏 (m ) 𝜎 (m) 1 2 1 2 2 𝑥0 0.85 0.15 0.2 0.05 0.60 0.0080–0.0126 As. (R1) 0 1 0.22 0.44 0.12 ESBP (R2) 1 0 0.45 — — 0.0135–0.0225 WSBP (R3) 0.85 0 0.45 — — 0.0250–0.0380 BP (R4) 0 1 — 0.32 0.64 0.017 0 1 — 0.47 0.94 0.019 CR (R5) 0 1 — 0.11 0.146 0.067–0.227 CO (R6) 1 0 0.15 — — 0.005–0.0124 Co.: concrete; As.: asphalt; ESBP: even stone block paved; WSBP: waved stone block paved; BP: boulder paved; CR: country road. 𝑆 (𝑤 ) In this study, vertical vehicle vibrations are studied using 𝑥 random theory, and some back propagation artificial neural 2 2 2 2 2𝜎 𝐴 𝛼 (𝑤 +𝛼 +𝛽 ) 𝐴 𝛼 networks (ANNs) with four functions such as newff, newelm, 𝑥 2 2 2 2 0 1 1 [ + ] |𝑤 |<𝑤 2 2 2 2 2 2 2 newcf, and newfftd are also employed to predict amplitudes = 𝜋 𝑤 +𝛼 (𝑤 −𝛼 −𝛽 )+4𝛼 𝑤 1 2 2 2 of accelerations of vehicles for different road conditions. 0 |𝑤 | <𝑤 , { 1 eTh organization of the paper is as follows. Section 2 (2) describes the theory of random vibration for vehicles. Overview of neural network is presented in Section 3.More where 𝐴 +𝐴 =1, 𝛼 =𝑎 𝑉, 𝛼 =𝑎 𝑉, 𝛽 =𝑏 𝑉, 𝑤 = 1 2 1 1 2 2 2 2 1 details of modeling of vehicle vibrations using neural net- Ω𝑉 .Asshown in (2), if the spectral density of the road works are given in Section 4. eTh simulation results obtained roughness is explained in terms of excitation frequency 𝑤 , 2 2 form BP are given in Section 5. eTh paper is concluded with namely, 𝑤 , it is described as 𝑥 /𝑤 = 𝑚 𝑠 ,orifitiswritten in Section 6. terms of the length frequency Ω (1/m), it is also described as 2 3 𝑥 /Ω = 𝑚 . Some parameters depending on road conditions are shown in Table 1. es Th e given parameters are the results 2. Random Vibration Theory of an experimental investigation [12]. The frequency of vehicle’s shock absorber must be chosen Vehicle vibrations due to road roughness have no definite between body frequency of 1–1.5 Hz and axle frequency of character, and system dynamics depends on the profile 10–15 Hz. eTh refore, damping ratio has to be selected so of roughness. eTh refore, statistical basis random theory is that the frequency of shock absorber is in the range of 4– employed in determining roughness character. Assuming 6 Hz. Consequently, the damping factor for absorber may be such vehicle vibrations to be linear, dynamic model of these taken as a value <0.5. In this damping ratio interval (0.1– systems can be represented as [12, 13] 0.5), vehicle body accelerations decrease for different road conditions [14]. 𝑥+2 ̈ 𝑤𝜁 𝑥+𝑤 ̇ 𝑥=− 𝑥 ̈ (𝑡 ) , (1) 𝑛 𝑛 0 3. Overview of Neural Networks where 𝑥 is the relative displacement of vehicle body; 𝑥 (𝑡) A neural network is a massive parallel system comprised is the amplitude over a specific level of the road roughness of highly interconnected, interacting processing elements, on which vehicle’s tyre moves at a definite time 𝑡 ; 𝜁 is the or nodes. Neural networks process through the interactions damping ratio, and 𝑤 also denotes natural frequency of of a large number of simple processing elements or nodes, vehicle’s shock-absorber system. also known as neurons. Knowledge is not stored within By determining amplitudes over a reference plane level on individual processing elements, rather represented by the a certain road condition by means of repeated measurements, strengths of the connections between elements. Each piece statistical roughness features are obtained. The road rough- of knowledge is a pattern of activity spread among many ness can be determined with enough approximation by some processing elements, and each processing element can be measurements accomplished for different road conditions. In involved in the partial representation of many pieces of order to describe the influence of the road roughness, the information. In recent years, neural networks have become most appropriate statistical parameter is its spectral density, a very useful tool in the modeling of complicated systems which is a mean square value of road roughness in a definite becausetheyhaveanexcellent abilitylearn andtogeneralize frequencyrange.Whenavehiclemovesatvelocity 𝑉 ,theroad (interpolate) the complicated relationships between input roughness spectral density can be written as follows [12]: and output variables. Also, the ANNs behave as model free ··· Advances in Acoustics and Vibration 3 Bias =1 k−1 1 k w =b w j0 j j1 k−1 j2 . . . . . . . . . Sum Activation . k function function ji Output (Σ) (f) k−1 Input layer First layer hidden layer Figure 1: Back propagation neural network with two hidden layers. jn k−1 estimators; that is, they can capture and model complex Figure 2: Architecture of an individual PE for BP network. input-output relations without the help of a mathematical model [15]. In other words, training neural networks, for example, eliminates the need for explicit mathematical mod- Before practical application, the network has to be eling or similar system analysis. This property of ANNs is trained. To properly modify the connection weights, an extremely useful in a situation where it is hard to derive a error-correcting technique, oen ft called as back propagation mathematical model. As a result, neural networks can provide learning algorithm or generalized delta rule [16], is employed. an eecti ff ve solution to solve problems that are intractable or Generally, this technique involves two phases through dieff r- cumbersome with mathematical approaches. ent layers of the network. The first is the forward phase, which occurs when an input vector is presented and propagated 3.1. Back Propagation (BP) Neural Network. The back prop- forwardthrough thenetwork to computeanoutputfor each agation network (Figure 1)iscomposedofmanyintercon- neuron. During the forward phase, synaptic weights are all nected neurons or processing elements (PEs) operating in fixed. eTh error obtained when a training pair (pattern-“ 𝑝 ”) parallel and are oen ft grouped in different layers. consists of both input and output given to the input layer of As shown in Figure 2, each articfi ial neuron evaluates the network is expressed by the following equation: the inputs and determines the strength of each through its weighing factor. In the artificial neuron, the weighed inputs 𝐸 = ∑ (𝑇 −𝑂 ) , are summed to determine an activation level. That is, 𝑝 (5) 𝑘 𝑘 𝑘−1 net = ∑ 𝑤 𝑜 , 𝑗 𝑗𝑖 𝑖 (3) where 𝑇 is the 𝑗 th component of the desired output vector, and 𝑂 is thecalculatedoutputof 𝑗 th neuron in the output layer. The overall error of all the patterns in the training set is where net is the summation of all the inputs of the 𝑗 th denfi edasmeansquareerror (MSE)and is givenby neuron in the 𝑘 th layer, 𝑤 is the weight from the 𝑖 th neuron 𝑗𝑖 𝑘−1 to the 𝑗 th neuron, and 𝑜 is the output of the 𝑖 th neuron in 𝐸= ∑ 𝐸 , (6) the (𝑘−1 )th layer. 𝑝 𝑝=1 eTh output of the neuron is then transmitted along the weighed outgoing connections to serve as an input where 𝑛 isthenumberofinput-outputpatternsinthetraining to subsequent neurons. In the present study, a hyperbolic set. eTh second is the backward phase which is an iterative tangent, log-sigmoid, and linear functions (𝑓( net ))witha error reduction performed in the backward direction from bias 𝑏 are used as an activation function of hidden and output the output layer to the input layer. In order to minimize neurons. Therefore, output of the 𝑗 th neuron 𝑜 for the 𝑘 th the error, 𝐸 , as rapidly as possible, the gradient descent layer can be expressed as method adding a momentum term is used. Hence, the new incremental change of weight Δ𝑤 (𝑚 + 1) can be 𝑗𝑖 𝑘 𝑘 (net +𝑏 ) −(net +𝑏 ) 𝑗 𝑗 𝑗 𝑗 𝑒 −𝑒 𝑘 𝑘 𝑜 =𝑓( net )= (tansig), 𝑘 𝑘 𝑗 𝑗 (net +𝑏 ) −(net +𝑏 ) 𝑘 𝑘 𝑗 𝑗 𝑗 𝑗 𝑒 +𝑒 Δ𝑤 (𝑚+1 ) =−𝜂 +𝛼Δ𝑤 (𝑚 ) , (7) 𝑗𝑖 𝑗𝑖 𝜕𝑤 𝑗𝑖 (4) 𝑘 𝑘 𝑜 =𝑓( net )= (logsig), 𝑗 𝑗 𝑘 −net +𝑏 1+𝑒 where 𝜂 is a constant real number between 0.1 and 1, called learning rate, 𝛼 is the momentum parameter usually set to 𝑘 𝑘 𝑘 𝑜 =𝑓( net )= net +𝑏 (linear) . 𝑗 𝑗 𝑗 𝑗 a number between 0 and 1, and 𝑚 is the index of iteration. 𝜕𝐸 𝑝𝑗 𝑝𝑗 𝑝𝑗 𝑝𝑗 4 Advances in Acoustics and Vibration eTh refore, the recursive formula for updating the connection weights becomes 𝑘 𝑘 𝑘 𝑤 (𝑚+1 ) =𝑤 (𝑚 ) +Δ𝑤 (𝑚+1 ) . (8) V 𝑗𝑖 𝑗𝑖 𝑗𝑖 These corrections can be made incrementally (aer ft each pattern presentation) or in batch mode. In the latter case, R.C the weights are updated only aer ft the entire training pattern AA set has been applied to the network. With this method, the order in which the patterns are presented to the network does w Output layer not influence the training. This is because of the fact that adaptation is done only at the end of each epoch. And thus, we have chosen this way of updating the connection weights [17]. Input layer 4. Modeling of Vehicle Vibrations Using Hidden layer Neural Networks Figure 3: General ANN topology. Modeling of vehicle vibrations with BP neural network is composed of two stages: training and testing of the networks with numerical data. eTh training data consisted of velocity Table 2: eTh variable training methods. (𝑉 ), damping ratio (𝜁 ), natural frequency of vehicle shock absorber (𝑤 ), road condition (R.C), and the corresponding Acronym Description acceleration amplitude. A total of 90 data sets were used, LM Levenberg-Marquardt of which 80 were selected randomly and used for training BFG BFGS Quasi-Newton purposes whilst the remaining 10 data sets were presented to the trained networks as new application data for verification RP Resilient back propagation (testing) purposes. u Th s, the networks were evaluated using SCG Scaled Conjugate Gradient data that had not been used for training. Before the ANN CGB Conjugate Gradient with Powell/Beale Restarts could be trained and the mapping learnt, it is important to CGF Fletcher-Powell Conjugate Gradient process the numerical data into patterns. Training/testing CGP Polak-Ribier ´ e Conjugate Gradient pattern vectors are formed, each formed with an input OSS One Step Secant condition vector GDX VariableLearningRatebackpropagation velocity (𝑉 ) [ ] damping ratio (𝜁 ) [ ] 𝑃 = (9) [ ] natural frequency (𝑤 ) 5. Numerical Results of BP Neural road condition R.C ( ) [ ] Network Model and the corresponding target vector The size of hidden layer(s) is one of the most important 𝑇 =[amplitude of acceleration (AA)]. (10) 𝑖 considerations when solving actual problems using multi- layer feed-forward network. However, it has been shown that Mapping each term to a value between −1and 1, we use BP neural network with one hidden layer can uniformly the following linear mapping formula: approximate any continuous function to any desired degree of accuracy givenanadequatenumberofneurons in the (𝑅 − 𝑅 )∗(𝑁 −𝑁 ) min max min 𝑁= +𝑁 , (11) min hidden layer and the correct interconnection weights [18]. (𝑅 −𝑅 ) max min Therefore, one hidden layer was adopted for the BP model. where 𝑁 is normalized value of the real variable; 𝑁 = To determine the number of neurons in the hidden layer, min −1 and 𝑁 =1 are minimum and maximum values of a procedure of trial and error approach needs to be done. max normalization, respectively; 𝑅 is real value of the variable; As such, attempts have been made to study the network 𝑅 and 𝑅 are minimum and maximum values of the performance with a different number of hidden neurons. min max real variable, respectively. These normalized data was used as Hence, a number of candidate networks are constructed, each the inputs and output to train the ANN. Figure 3 shows the of trained separately, and the “best” network was selected general network topology for modeling vehicle vibration. basedonthe accuracy of thepredictions in thetesting phase. The names of training algorithms used in this paper are It should be noted that if the number of hidden neurons shown in Table 2. is too large, the ANN might be overtrained giving spurious In what follows, the use of four neural networks will be values in the testing phase. If too few neurons are selected, discussed and the results are presented. en, Th the best model the function mapping might not be accomplished due to is picked based on the accuracy of AA in the verification stage. undertraining [19]. Table 3 shows 10 numerical data sets, used Advances in Acoustics and Vibration 5 Table 3: Vibration conditions for verification analysis. Table 6: Comparison of AA desired and predicted by the BP neural network model and newcf function. Natural Acceleration Test Velocity Damping Road Test no. Desired AA (cm) BP model AA (cm) Error (%) frequency amplitude no. (m/sec) ratio (𝜁) condition (Hz) (cm) 1 2.35 2.54 8.34 2 3.85 4.36 13.26 1 12 0.20 R1 10 2.35 3 5.28 5.76 9.10 2 15 0.33 R2 12 3.85 4 4.44 4.80 8.15 3 24 0.45 R5 15 5.28 5 2.90 3.23 11.56 4 35 0.50 R3 15 4.44 6 1.27 1.29 1.64 5 18 0.60 R6 8 2.90 7 4.18 4.94 18.29 6 50 0.65 R1 10 1.27 8 2.41 2.54 5.65 7 60 0.85 R3 13 4.18 9 5.47 5.66 3.49 8 40 0.25 R4 12 2.41 10 3.04 3.38 11.36 9 27 0.55 R4 10 5.47 10 19 0.75 R6 10 3.04 Table 7: Comparison of AA desired and predicted by the BP neural network model and newfftd function. Table 4: eTh eeff ct of different number of hidden neurons on the BP Test no. Desired AA (cm) BP model AA (cm) Error (%) network performance. 1 2.35 2.58 9.86 No. of hidden Epoch with LM Average error in AA (%) 2 3.85 4.75 23.39 neurons method training with newelm function 3 5.28 5.80 9.90 414390 9.80 4 4.44 5.09 14.67 5 5170 11.87 5 2.90 3.11 7.49 6 1753 4.95 6 1.27 1.47 16.24 7 1028 8.48 7 4.18 4.54 8.79 8 2.41 2.45 2.07 8739 15.45 9 5.47 6.04 10.47 10 3.04 3.08 1.39 Table 5: Comparison of AA desired and predicted by the BP neural network model and newff function. Table 8: Comparison of AA desired and predicted by the BP neural Test no. Desired AA (cm) BP model AA (cm) Error (%) network model and newelm function. 1 2.35 2.78 18.69 Test no. Desired AA (cm) BP model AA (cm) Error (%) 2 3.85 3.99 3.66 1 2.35 2.48 5.77 3 5.28 5.31 0.70 2 3.85 3.90 1.37 4 4.44 4.57 3.1 3 5.28 5.44 3.14 5 2.90 3.25 12.16 4 4.44 4.47 0.9 6 1.27 1.30 2.44 5 2.90 3.21 10.88 7 4.18 4.55 8.95 6 1.27 1.29 1.80 8 2.41 2.57 6.79 7 4.18 4.64 11.15 9 5.47 5.94 8.70 8 2.41 2.57 6.79 10 3.04 3.71 22.27 9 5.47 5.69 4.17 10 3.04 3.15 3.62 for verifying or testing network capabilities in modeling the vehicle vibration. eTh refore, the general network structure is supposed to performances were examined for these networks. As the error be 4-𝑛 -1, which implies 4 neurons in the input layer, 𝑛 criterion for all networks was the same, their performances neurons in the hidden layer, and 1 neuron in the output layer. are comparable. As a result, from Table 4,the best network Then, by varying the number of hidden neurons, different structure of BP model is picked to have 6 neurons in the network configurations are trained, and their performances hidden layer with the average verification errors of 4.95% in are checked. eTh results are shown in Table 4. amplitude acceleration over the 10 numerical verification data For training problem, equal learning rate and momentum sets. Tables 5, 6, 7 and 8 show the comparison of desired and constant of 𝜂 = 𝛼 = 0.85 were used [16]. Also, error stopping predicted values for amplitude acceleration in verification criterion was set at 𝐸 = 0.01 , which means that training cases with different functions. epochs continued until the mean square error fell beneath Figure 4 illustrates the convergence of the output error this value. Both the required iteration numbers and mapping (mean square error) with the number of iterations (epochs) 6 Advances in Acoustics and Vibration Table 9: e Th ( 𝑅 ) values for AA with various neurons in the hidden layer. Acronym of training method Number of hidden neurons LM BFG RP SCG CGB CGF CGP OSS GDX 0.9977 0.9888 0.9819 0.9957 0.9738 0.9918 0.9905 0.9913 0.9891 0.9981 0.9676 0.992 0.9909 0.9942 0.9864 0.9846 0.9969 0.9916 6 0.9999 0.9981 0.9942 0.9578 0.9986 0.9978 0.9681 0.9963 0.9945 0.9985 0.9998 0.9967 0.9996 0.9996 0.9995 0.9595 0.9890 0.9856 0.9967 0.9928 0.9994 0.9998 0.9998 0.9698 0.9898 0.9898 0.9881 Table 10: The results of the variable training methods in the BPN Table 12: The results of the variable training methods in the BPN with newelm function. with newff function. Epoch in Epoch in Acronym Error goal Train time (s) Test time (s) Acronym Error goal Train time (s) Test time (s) goal goal LM Met 28.7240 0.054526 LM Met 40.1450 0.054789 1753 1293 BFG 2528 Met 77.4824 0.046583 BFG 1528 Met 88.1024 0.056673 RP Not met 67.6773 0.046466 RP Met 100.6773 0.056906 3000 1210 SCG Not met 101.3281 0.044047 SCG Not met 97.9497 0.065237 3000 4000 CGB Not met 118.1020 0.052230 CGB Not met 138.7890 0.062230 2932 3745 CGF 2125 Not met 88.8947 0.046125 CGF 3450 Not met 112.8964 0.053565 CGP Not met 94.6653 0.046852 CGP Not met 100.1333 0.056907 2302 3162 OSS Not met 112.0710 0.046677 OSS Not met 160.1489 0.066677 3000 3400 GDX Not met 61.2435 0.046046 GDX Not met 43.4465 0.056044 3000 2000 Table 13: The results of the variable training methods in the BPN Table 11: The results of the variable training methods in the BPN with newcf function. with newfftd function. Epoch in Epoch in Acronym Error goal Train time (s) Test time (s) Acronym Error goal Train time (s) Test time (s) goal goal LM Met 38.7261 0.057241 LM Met 43.1808 0.061301 1251 2167 BFG Met 64.1920 0.058573 BFG 2018 Met 97.4814 0.042678 19018 RP Met 99.7452 0.064916 RP Not met 49.1627 0.048146 1890 SCG Not met 33.1907 0.042047 SCG 3786 Not met 39.1877 0.061937 CGB Not met 128.1590 0.055130 CGB Not met 134.3904 0.067220 3232 4150 CGF Not met 102.5594 0.057345 CGF 3025 Not met 108.1528 0.056985 1129 CGP Not met 100.1333 0.055897 CGP Not met 67.1503 0.044152 2001 OSS Met 59.0112 0.058677 OSS 4744 Not met 80.9358 0.060127 GDX Not met 67.1205 0.056046 GDX Not met 38.2065 0.059391 3230 3190 In Tables 10, 11, 12,and 13, the results of training the during training of the chosen 4-6-1 BP network. After 1753 network using nine different training algorithms by 6 neurons epochs, the MSE between the desired and predicted outputs in the hidden layer and logsig-purlin activation function are becomes less than 0.01. At the beginning of the training, the output from the network is far from the target value. summarized. Each entry in the table represents 10 different trials, where different random initial weights are used in each However, the output slowly and gradually converges to the trial. target valuewithmoreepochsand thenetwork learns the input/output relation of the training samples. eTh regression value (𝑅 ) of theoutputvariablevaluesfor thetestdataset 6. Conclusions and Summary for various neurons in hidden layer is shown in Table 9.It should be noted that these data were completely unknown to In this paper, four supervised neural networks have been thenetwork.Thecloserthisvalue is to unity, thebetteristhe used forthe vehiclevibrations. Basedonthe test resultsof prediction accuracy. eTh best (𝑅 ) value obtained is 0.9999, each network with some data set, different from those used and it is obtained from the LM algorithm by 6 neurons in in the training phase, it was shown that newelm function hidden layer. neural model has superior performance than newff, newcf, Advances in Acoustics and Vibration 7 Performance is 0.0099982; goal is 0.01 [5] T. Ishihama, H. Masao, and M. Seto, “Vehicle vibration reduc- tion by transfer function phase control on hydraulic engine mounts,” JSME International Journal C,vol.37, no.3,pp. 536– 541, 1994. [6] T. Y. Yi and P. E. Nikravesh, “A method of identify vibra- tion characteristics of modified structures for flexible vehicle −1 dynamics,” Proceedings of the Institution of Mechanical Engineers D,vol.216,no. 1, pp.55–63,2002. [7] B. Liang, D. Zhu, and Y. 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