Theoretical Calculation And Experimental Analysis Of The .
WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICSYongjun Jin, Jianwu Zhang, Xiqiang GuanTheoretical calculation and experimental analysis of the rigid bodymodes of powertrain mounting systemYONGJUN JIN, JIANWU ZHANG, XIQIANG GUANSchool of Mechanical EngineeringShanghai Jiao Tong University800 Dong Chuan Road, Shanghai, 200240China13816691189@139.com http://www.sjtu.edu.cnAbstract: -The performance of dynamic property of the mount is influenced by multiple factors and stronglydepends on the working conditions. This means that the modal parameters of powertrain mounting systemwould make changes under different operating conditions. A novel approach to simulate the actual workingcondition is proposed in the testing of dynamic stiffness. Then the mechanical model of powertrain mountingsystem based on dynamic stiffness is established in this paper. In order to examine the rationality and accuracyof the computational model based on dynamic stiffness; experimental modal analysis is performed by multiplemeans and methods in this paper. Through the contrast analysis, advantages and disadvantages of these methodsare illustrated and it is shown that using the method of Operational Modal Analysis could obtain more accurateand more reliable results. Based on the experimental and evaluation results, it is shown that there is smallerrelative error and higher fitting degree between the calculation results based on dynamic stiffness and theresults obtained from operational modal analysis. Moreover, the proposed method also enjoys satisfactoryconsistency with the actual working condition.Key-Words: - Powertrain mounting system; Dynamic stiffness; Static stiffness; Operational Modal Analysis;Rigid body modes; Experimental modal analysis; Impact hammer testingactual conditions, the mount is working under certainfrequency and amplitude of excitation, which isclosely related to the dynamic stiffness [5].Therefore, computational results from the modelbased on static stiffness are inconsistent with theactual situation. Taking the idle speed condition ofengine as an example in this paper, a novel approachto simulate the actual working condition is proposedin the process of dynamic stiffness testing. Then themechanical model of powertrain mounting systembased on dynamic stiffness is established. Bycomparison the computational results respectivelyfrom the model based on static stiffness and dynamicstiffness, the differences between two methods aredemonstrated.In order to examine the rationality and theaccuracy of the computational model based ondynamic stiffness, multiple means and methods areused to obtain the natural frequencies of mountingsystem through experimental modal analysis in thispaper. First, the traditional hammer impact testing iscarried out. In the process of the experiment, it’sfound that applying different mass of hammer couldcause quite different results. This is due to thenonlinear characteristics of the stiffness of mounts.To avoiding such deficiency of traditional hammer1 IntroductionOther than road-tire excitation, powertrain is one ofthe major sources of vibration in the vehicle. Ittransmits vibration energy caused mostly by theunbalanced engine disturbances to the passengercompartment through powertrain mounting system.The vehicle powertrain mounting system, generally,consists of a powertrain and several mountsconnected to the vehicle structure [1]. Besidessupporting the powertrain weight, the major functionof powertrain mounts is to isolate the unbalancedengine disturbance force from the vehicle structure.Accurate understanding of the dynamic properties ofpowertrain mounting system is a main reference ofvehicle vibration controlling and NVH enhancement[2]. Thus, establishing a feasible mechanical modelof powertrain mounting system or accuratelyacquiring its dynamic properties throughexperiments play an important role in system designand optimization [3]. The main purpose of this paperis to present a comprehensive theoretical analysis andexperimental research on the dynamic properties ofpowertrain mounting system.In some studies, the stiffness matrix in the modelof powertrain mounting system is constructed basedon static stiffness of mounts [4]. However, in mostE-ISSN: 2224-3429193Issue 3, Volume 8, July 2013
WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICSYongjun Jin, Jianwu Zhang, Xiqiang Guan[q] [ ximpact method, operational modal analysis isadopted to identify the modal parameters in the idlespeed condition of engine.Furthermore, the modal parameters, whichcalculated from mechanical model based on staticstiffness and that based on dynamic stiffness, arecompared respectively with the identification resultsobtained under different experimental modal analysismethod. Through the contrast analysis, advantagesand disadvantages of these methods are illustratedclearly.yz θ x θ y θ z ]T .(1)2 Theoretical analysis and calculationIn this section, the equations of motion of powertrainmounting system are presented. An approach toestablish the mechanical model based on dynamicstiffness and based on static stiffness is discussed.2.1 Mechanical model of vehicle powertrainmounting systemsModes are characterized as either rigid body orflexible body modes. Up to six rigid body modes,three translational modes and three rotational modescould exist in all structures [6]. If the structurebounces on some soft springs, its motionapproximates a rigid body mode [7].Because the powertrain’s natural frequencies ofthe flexible modes are much higher than that of themounting system, it can be modeled as a rigid bodywhich has six degrees of freedom: three translationalmotions and three rotational motions [8]. Thepowertrain is supported on vehicle frame by somemounts. Each mount can be represented by threemutually perpendicular sets of spring and viscousdamper in each of the three principal directions,which defined are as u , v and w . With thethree-point mounting system as an example, thesimplified mechanical model is shown in Figure 1.Define a Cartesian coordinate system G0 XYZas system coordinate, the origin of the coordinatesystem G0 is in static equilibrium position ofpowertrain’s center of gravity. X axis points tothe backward direction of vehicle, Y axis isparallel to the direction of engine crankshaft, andZ axis is vertically upward. Moving coordinatesystem G0 xyz moves with the powertrain [9].The generalized coordinates of the system is defined asthree translational coordinates x, y, z relativeto X , Y , Z , and three rotational coordinates θ x ,θ y ,θ zFig.1 Mechanical model of mounting systemThe dynamic equations of powertrain mountingsystem can be written as the matrix form:(2)[ M ][q ] [C ][q ] [ K ][q ] [ F ] ,where [M ] is mass matrix of the system, [C ] isdamping matrix of the system, [K ] is stiffnessmatrix of the system and [F ] is a 6 1 vector ofexcitation forces and moments.Ignoring damping and external force, thedifferential equations of the system’s free vibrationcan be obtained. Analysis equation of system’sinherent characteristics could be presented as（3）[ M ][q ] [ K ][q ] 0 .Mass matrix of the system [M ]0 m 0 0 0 m 00 0 0 m0[M ] 0 0 0 Jxx 0 0 0 J xy 0 0 0 J zxaround X , Y , Z . It can be written asE-ISSN: 2224-3429194can be written as00 00 00 J xy J zx J yy J yz J yz J zz .Issue 3, Volume 8, July 2013
WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICSwhich is used to construct the stiffness matrix in themechanical model. Static stiffness is the ratiobetween the static load variation and thedisplacement variation, which could be calculatedby the equation: F,k Swhere F and S can be tested by usingElectronic universal testing machine. The schematicdiagram of testing is shown in Figure 2.where m is the total mass of the powertrain, J ij isthe powertrain’s moment of inertia around each axis.Stiffness matrix of the system [K ] can be writtenasn[ K ] [ Bi ] [Ti ] [ ki ][Ti ][ Bi ] ,TTi 1where [ki ] is stiffness matrix of mount i : kui [ ki ] 0 00kvi00 0 k wi Mount[ Bi ] refers to the position matrix of mount i :zi yi 1 0 0 0 [ Bi ] 0 1 0 zi 0 xi 0 0 1 yi xi0 where xi , yi , zi is the coordinate of mount i .[Ti ] is angle matrix between the elastic principalaxis of mount and system coordinates, which can bewritten as cos α ui [Ti ] cos α vi cos α wicos βuicos β vicos β wiSpecialized fixturecos γ ui cos γ vi . cos γ wi Fig.2 Static stiffness testing of the mountThe distinction is made between the static andthe dynamic stiffness on the basis of whether theapplied load has enough acceleration in comparisonto the structure's natural frequency. A static load isone which varies sufficiently slowly. Static stiffnesscan be obtained in this process.Take static stiffness values of each mount intothe equation (6), and the natural frequencies ofpowertrain mounting system can be obtained. Theresults calculated are listed in Table 1.For equation (3), assuming the theoreticalsolution is(4)[q ] [ X i ] sin(ωi t α ) .Take equation (4) into (3), and the followingequation can be obtained as(5)[ K ][ X i ] ω i2 [ M ][ X i ] .Then(6)[ K ] ωi2 [ M ] 0 .And the natural frequency of the system is(7)f i ωi / 2π .The circular frequency of the system is theeigenvalues of the matrix [ M ] 1[ K ] . And thecorresponding shape of vibration is the eigenvectorof the matrix [ M ] 1[ K ] . Thus, the naturalfrequency and the shape of powertrain mountingsystem can be obtained by solving eigenvalues andeigenvectors of the matrix [ M ] 1[ K ] .Table 1 Calculated modal parameters of mountingsystem based on the static stiffnessModes orderNatural frequency (Hz)16.0628.1738.59412.71516.07620.412.3 Calculations of modal parameters basedon the dynamic stiffness of the mounts2.2 Calculations of modal parameters basedon the static stiffness of the mountsA dynamic load is one which changes with timefairly quickly in comparison to the structure'snatural frequency. In general, dynamic stiffnessdepends on three factors: preload, excitationThe performance of powertrain mounting systemdepends on the stiffness characteristic of the mount,E-ISSN: 2224-3429Yongjun Jin, Jianwu Zhang, Xiqiang Guan195Issue 3, Volume 8, July 2013
WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICSYongjun Jin, Jianwu Zhang, Xiqiang Guanschematic diagram of dynamic testing is shown inFigure 4.frequency and the amplitude of dynamic load [10].For powertrain mounting system supported by threemounts, the value of preload equals to the total massof powertrain, which is distributed to each mount atvertical direction. This is a space force problem,which can be easily calculated. The excitationfrequency and amplitude of dynamic load aredetermined by the corresponding workingconditions [11]. The value of excitation frequencyand amplitude of dynamic load at certain operatingcondition can be obtained through vibrationmeasurement. Then exerting the load of equivalentfrequency and amplitude on the mount usingvibration testing equipment, the value of dynamicstiffness can be acquired properly. This simulationof testing condition is consistent with the actualsituation.In typical idle speed condition of the engine, forexample, acceleration response signals can bemeasured by accelerometer fixed on the side ofmount [12]. The excitation frequency on the mountcan be obtained through using the Fast FourierTransform (FFT) of acceleration response signals[13]. And the amplitude of dynamic load can be gotby twice integration of acceleration response signals.A sample of frequency spectrum response in thevertical direction from left mount under idle speedcondition can be seen in Figure 3. The diagramshows that the vibration energy of left mount ismainly concentrated in 27Hz. It means that thedominant frequency of excitation is 27Hz. So theinput signals of 27Hz can be excited on the mount inthe dynamic stiffness testing.Specialized fixtureMountForce sensorAcceleration sensorSinusoidal excitationFig. 4 Dynamic stiffness testing of the mountTake dynamic stiffness values of each mountacquired through the approach above into theequation (6), the natural frequencies of powertrainmounting system in the idle speed condition can beobtained. The results calculated are listed in Table 2.Table 2 Calculated modal parameters of mountingsystem based on the dynamic stiffnessModes orderNatural frequency (Hz)8.05111.85212.58313.16415.57516.5260. 08gAmpl i t ude2.4 Comparison between two groups ofcalculation resultsBased on dynamic stiffnessBased on static stiffness0. 000. 027. 0HzFrequency (Hz)2064. 0Fig. 3 Frequency spectrum of left mount in the idlespeed conditionUsing the approach described above, the preload,dominant excitation frequency and amplitude ofinput signals can be obtained. Then the dynamicstiffness of all mounts at all directions can bemeasured by vibration testing machine [14]. TheE-ISSN: 2224-3429151051234Modes order56Fig.5 Comparison between two groups ofcalculation results196Issue 3, Volume 8, July 2013
WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICSYongjun Jin, Jianwu Zhang, Xiqiang GuanDifferent sized hammers can provide differentimpact forces. Here, the test is performedrespectively using two kinds of impact hammerswith different sizes. The magnitude of impact forceproduced by big hammer is about ten times of thesmall hammer.Natural frequencies calculated from the mechanicalmodel based on static stiffness are compared withthe results based on dynamic stiffness, shown inFigure 5. Obvious differences between the twogroups of results could be found out in the graph.The relative error of the third order frequencyreached 46.4%, which indicates that calculationmodel based on static stiffness is not consistent withthe actual working condition. As for the methodbased on dynamic stiffness, it’s required to befurther verified by the experimental modal analysis.Right mountLeft mount3 Experimental modal analysis ofpowertrain mounting systemIn this section, the experimental modal analysis isused to identify the modal parameters of powertrainmounting system. Through contrastive analysis ofthe experiment and the theoretical calculation, therationality of calculation model based on dynamicstiffness can be verified.Rear mountFig. 7 Impact hammer testingSmall hammer3.1 Experimental modal analysis based on theimpact hammer testingImpact testing is the most classical modal testingmethod used today. According to the theory ofexperimental modal analysis, the modal parametersof the system can be identified through the curvefitting of the Frequency Response Function (FRF),which is the ratio between a response (outputacceleration) signal and a reference (input force)signal expressed in frequency domain [15]. Impulseforce can be provided by the impact hammer whichhas a force transducer attached to its head to measurethe input force. Acceleration response signals can bemeasured using accelerometers fixed on the surfaceof the system [16]. The process of impact hammertesting is depicted in Figure 6 and Figure 7.ImpulseBig hammerFig. 8 Different size of impact hammerTable 3 lists the experimental results identifiedthrough these two impact testings.Table 3 Experimental modal results of mountingsystem based on the impact hammer testingModesNatural frequency (Hz)Relativeorder Small hammer Big esponseFFTFRFCurveFitModalParametersAccording to the characteristics of the FrequencyResponse Function, the magnitude of impact forceshould not influence the results in experimentalmodal analysis. Table 4 shows that there are obviousFig. 6 Method of experimental modal analysisE-ISSN: 2224-3429197Issue 3, Volume 8, July 2013
WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICSYongjun Jin, Jianwu Zhang, Xiqiang Guanpowertrain, Measure the vibration response signalsof the powertrain mounting system in idle speedcondition and the modal parameters can be obtainedby using identification methods of operationalmodal analysis. Testing and analysis procedures areshown in Figure 9, Figure 10 and Figure 11.differences between the modal parameters obtainedfrom the testing using small hammer and thoseusing big hammer. This phenomenon is due to thenon-linear relationship between the dynamicstiffness of the mount and the frequency andamplitude of the excitation force. The stiffnessperformance of the mount varies under differentimpact forces.These experiments indicate that the modalparameters will vary with working conditions due tothe change of dynamic stiffness values of mounts.Therefore, It is unable to obtain accurate modalparameters of mounting system under actualworking conditions using traditional experimentalmethod based on impact hammer testing.3.2 Operational modal analysisTraditional Experimental modal identificationmethod and procedure such as impact hammertesting is based on forced excitation laboratory testsduring which Frequency Response Functions aremeasured. However, the real loading conditions towhich the mounting system is subjected often differconsiderably from those used in laboratory testing.The method of operational modal analysis is toextract modal frequencies, damping and modeshapes from data taken under operating conditions[17]. This means that environmental and its naturalexcitation influences on system behavior, such aspreload of mounts and dynamic-load-inducedstiffening, will be taken into account [18]. Theidentification results under the actual working statuswill more accurately reflect the actual dynamiccharacteristics of the mounts [19].Fig. 10 Test environment and real operating statusof powertrain mounting systemFig. 11 Online acquisition and analysisThe experimental modal parameters ofpowertrain mounting system identified byoperational modal analysis at the idle speedcondition are listed in Table 4. Evaluation of theresults will be given in the next section.Acceleration sensorPowertrainTable 4 Experimental modal results of mountingsystem based on operational modal analysisMode orderNatural frequency (Hz)8.50112.34213.38313.85416.92517.766LMS data acquisition systemOperational Modal analysisFig. 9 Testing and analysis procedures ofoperational modal analysis3.3 Results comparison ofexperimental modal analysesAttach the accelerometers on the surface of theE-ISSN: 2224-3429198thethreeIssue 3, Volume 8, July 2013
WSEAS TRANSACTIONS on APPLIED and THEORETICAL MECHANICSExperimental modal analyses of powertrainmounting system have been performed describedabove using three different test methods and means.Thus, three groups of natural frequency of themounting system have been obtained already. Thecomparison of these experimental identificationresults is shown in Figure 12.Experimental (Operational modal analysis)Calculated (Based on dynamic stiffness)Calculated (Based on static stiffness)20Frequency (Hz)Small hammerOperationalYongjun Jin, Jianwu Zhang, Xiqiang GuanBig hammer151020Frequency (Hz)51234Modes order1556Fig. 13 Results Comparison between theoreticalcalculation and experimental analysis10Obviously, there is smaller relative error andhigher fitting degree between the calculation resultsbased on dynamic stiffness of mounts and the resultsobtained from operational modal analysis, and therelative error between them of every orderfrequency is less than 8% as shown in Figure 14.While, there is large error between the calculationresults based on static stiffness and the resultsobtained from operational modal analysis, forexample, the relative error between them of the thirdorder frequency is even more than 55%.51234Modes order56Fig. 12 Results comparison of the threeexperimental modal analysesThe graph shows that there are obviousdifferences among the three groups of results. Intheory, the natural frequency of a linear system isthe inherent property of the system itself, and shouldnot vary
obtained under different experimental modal analysis method. Through the contrast analysis, advantages and disadvantagesof these methods are i llustrated clearly. 2 Theoretical analysis and calculation . In this section, the equations of motion of powertrain mounting system are presented. An approach to
THEORETICAL ANALYSIS AND REDUCTION TECHNIQUES OF DC CAPACITOR RIPPLES AND REQUIREMENTS IN INVERTERS By . The comparison between the calculation result and experimental result shows that they are in close agreement. . derivation of the dc capacitance theoretical calculation, and teaching me how to debug and built inverters. Many thanks go to .
The 2D and 3D theoretical equations for the thermal conductivity of pebble bed are derived, and compared with the modelling results using COMSOL as a numerical tool [3]. The effective thermal conductivity of Li2TiO3 pebble bed can be preliminarily obtained by analysis modelling or theoretical calculation under the lack of experimental set-up at .
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From the above theoretical response time calculation, we can see that the response time decreases as the diameter of µ-tubes increases and the response time increases as the aspect ratio of µ-tubes s increases for same diameter of µ-tubes. 3. EXPERIMENTAL ANALYSIS Experimental Pressure measurement system with
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Keywords: Power analysis, minimum detectable effect size, multilevel experimental, quasi-experimental designs Experimental and quasi-experimental designs are widely applied to evaluate the effects of policy and programs. It is important that such studies be designed to have adequate statistical power to detect meaningful size impacts, if they .
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STM32 MCUs listed in Table 1. Outsourcing of product manufacturing enables original equipment manufacturers (OEMs) to reduce their direct costs and concentrate on high added-value activities such as research and development, sales and marketing. However, contract manufacturing puts the OEM's proprietary assets at risk, and since the contract manufacturer (CM) manipulates the OEM's intellectual .
