4218-19208-1-PB

March 22, 2018 | Author: antonioulbra | Category: Suspension (Vehicle), Probability Density Function, Statistics, Monte Carlo Method, Random Variable
Share
Report this link


Comments



Description

Mecánica Computacional Vol XXXI, págs.2745-2755 (artículo completo) Alberto Cardona, Paul H. Kohan, Ricardo D. Quinteros, Mario A. Storti (Eds.) Salta, Argentina, 13-16 Noviembre 2012 DYNAMIC ANALYSIS OF MODELS FOR SUSPENSION SYSTEMS OF GROUND VEHICLES WITH UNCERTAIN PARAMETERS Claudio Gattia and Marcelo T. Piovana,b a Centro de Investigaciones de Mecánica Teórica y Aplicada, Universidad Tecnológica Nacional FRBB. 11 de Abril 461, B8000LMI, Bahía Blanca. Argentina, {cdgatti,mpiovan}@frbb.utn.edu.ar, http:// www.frbb.utn.edu.ar b CONICET Keywords: Uncertainty quantification, platform dynamics, suspension systems. Abstract. Land transportation vehicles have suspension systems to limit the vibrations caused by the movement. The main function of these systems is to maintain the welfare of people (in vehicles or cars) or the integrity and functionality of installed equipment (in robotic platforms). Suspension systems are constituted by damping and elastic components whose properties may have a wide variation due to the geometrical configuration, manufacturing process and assembly protocols. This entails that the dynamic response of the system is not the expected one, which implies a degree of uncertainty although the properties of the components appear to be correctly specified. In this context it is important to characterize the propagation of uncertainty of the response of the system, caused by the uncertainty of its mechanical components: dampers, dash-pots, springs, anti-roll bars, etc. In this paper, the stochastic dynamic response of a suspension platform is analyzed. The mathematical model of the platform, characterized by 7 degrees of freedom, is derived by means of newtonian dynamics of rigid bodies. This mathematical model is deterministic and it is adopted as the mean model for the further stochastic studies. The parameters of the system: mass, springs and dampers are considered uncertain. Second order random variables are defined for every uncertain parameter involved. The probability density functions of the random variables are derived using the Maximum Entropy Principle. Once the probabilistic model is constructed, the Monte Carlo method is employed to perform the statistical simulations. Numerical studies are carried out to show the main advantages of the modeling strategies employed, as well as to quantify the propagation of the uncertainty in the dynamics of suspension systems. Copyright © 2012 Asociación Argentina de Mecánica Computacional http://www.amcaonline.org.ar bell-cranks. such as cars. 2009). 1995). etc). springs and/or elastic components such as anti-roll bars. PIOVAN INTRODUCTION The suspension system of a vehicle. In more advanced systems. trucks. SUV. to analyze and to simulate the dynamics of a suspension system. the effective elastic constant of the tires and the constants of the damping elements. GATTI. multibody method. push-rods. taking into account that the randomness of the system parameters can substantially modify the dynamic response of suspension. A basic conceptual suspension system is constituted by buffers and/or shock absorbers. is an important part to control the motion. and the method of condensed parameters among others (Thomsen and True. However. Consequently the design. etc. This means that stability.e. the type of forces involved. The parametric probabilistic approach is a method normally employed to quantify the uncertainty. study and complete analysis of the suspension system is quite important. the components and/or parameters of real systems eventually can have random features (depending on a number of factors such as the manufacturing process. the condensed parameters method. Dixon. M. such as finite element method. enables the implementation of various strategies associated with the calculation of the dynamic response when the model parameters have variations or dispersions. the calculation and analysis of the suspension dynamics would not have any complexity. anti roll-bars. If the parameters of the suspension system were clearly identified (i. maneuverability or comfort of a vehicular platform is subjected mainly to the features of the components of the suspension system (Gillespie. robotic platform or any other ground vehicular platform. Now.T. anti-roll bars and dash-pots as well as the elastic coefficients of the pneumatics.) and others are modeled as rigid elements (arms.2746 1 C. Each support is composed by a suspended mass (whose vertical displacement adds a degree-of freedom to the system) and the elastic spirals. etc). 1996).ar . pneumatics or tires. These parameters are effective on the hubs. The variable parameters of the system are the elastic constants of the spiral. lower and upper arms. even if it is done with highly sophisticated programs in which the several components of the suspension are modeled as elastic elements (springs. although it is not very common (Dixon. The seven degrees-of-freedom are the vertical displacement and two rotations of a rigid platform suspended by four elastic supports at each corner. depend strongly on the manufacturing process and the manufacturer occasionally supplies the data of that parameter. the deterministic problem). like the ones employed in racing cars some extra elements are included such as pull-push rods and rockers among others. The elastic/damping coefficients of the tires. In this paper a basic model of seven degrees-of-freedom representing the suspension of a vehicle platform is derived by means of newtonian mechanics (Williams. though simple in concept as it leads to mathematical models of a few degrees of freedom. parameters not easy to be measured directly. which makes the model independent of the suspension geometry. The first two methods are general and highly sophisticated although they are quite expensive in computational time. among others. Some configurations of suspension systems can have more than one damper or spring/elastic elements. 2009). springs. then it is important to establish a way to quantify the uncertainty associated with the suspension parameters and its propagation to the dynamic response. 2010. Condensed parameter models for suspension systems require many degree of freedom (involving mechanical components) in order to be more representative.amcaonline. There are different approaches to model.org. in which the uncertain parameters of the model or system are associated to random variables characterized by appropriate probability distribution functions (PDF) constructed ac- Copyright © 2012 Asociación Argentina de Mecánica Computacional http://www. On the other hand. the system assembling. which lead to a stochastic problem. ar .org. K is the matrix of effective spring elements. Then computational studies are made to show the features of the modeling strategies employed. among others (Soize. M is the matrix of effective inertia parameters. 2 DETERMINISTIC MODEL OF THE VEHICULAR PLATFORM Figure 1 shows a basic mechanic model of the vehicular platform. In this article the Maximum Entropy Principle (Shannon. Jaynes. After that. (1). The Monte Carlo method is used to simulate the response of a number of realizations to guaranty the convergence of a stochastic response. The effective mass. 1995) by a suspended mass that can move vertically connected to a pair of springs and a damper: One spring identifies the elastic interaction between the suspended mass and the ground. Then. can be considered the given mean response. 2001. elastic and dashpot components of each subsystem are condensed in the center of the corresponding supporting tires. The system dynamics is characterized by 7 degrees-of-freedom: the vertical displacement of the four suspended masses. 2745-2755 (2012) 2747 cording to given statistical information such as mean.Mecánica Computacional Vol XXXI. (2008) provided some statistical measures such as the mean value and the dispersion coefficient of the random variables. The equations of motion of the vehicular platform can be derived by means of newtonian mechanics. 2008). R is the matrix of the damping components. After that the probabilistic model is constructed. and they can be written in the following compact matrix form: ˙ + KU = F ¨ + RU MU U(0) = U0 ˙ ˙ U(0) =U 0 (1) In Eq. a statistical analysis is carried out with the outcome in order to evaluate the magnitude of the dispersion in the response and the dispersion of the response provided the features of the random parameters of the vehicular suspension system. Copyright © 2012 Asociación Argentina de Mecánica Computacional http://www. whereas F is the vector of applied ˙ are the initial conditions of the forces and U is the vector of kinematic variables. the two rotation angles of the rigid plate and the vertical displacement of the gravity center of the rigid plate. Each suspension subsystem is characterized (Gillespie. The article is structured in the following form: the first part is devoted to the description and derivation of the deterministic model with 7 degrees-of-freedom that simulates the vehicular platform. standard deviation. 1948. The platform consists of a rigid plate connected to the ground by four subsystems at each corner of the plate. The construction of the PDFs follows the methodology indicated in the works of Soize (2001) and Ritto et al. 2003) is employed in order to define the PDFs of the random variables. the probabilistic model is constructed on the mathematical structure of the deterministic model employing the random variables.amcaonline. and the other spring together with the damper alleviate the vibration to the platform. Ritto et al. as well as to quantify the propagation of the uncertainty in the dynamics of the suspension systems of a vehicular platform. págs.. The deterministic mathematical model of seven degrees-of-freedom. U0 and U 0 differential system. The rigid plate can move vertically and can rotate around x-axis and y-axis. The matrix of effective inertia parameters can be written in the following form:   Mdd 0 0 0 0 0 0  Mdi 0 0 0 0 0     Mtd 0 0 0 0     M 0 0 0 M= ti    Ms 0 0     sym Ixx 0  Iyy (2) where Mdd and Mdi are the effective suspended masses at the right front wheel and left front wheel respectively. Rxx = b2dd Rdd + b2di Rdi + b2td Rtd + b2ti Rti Rsx = bti Rti − bdd Rdd + bdi Rdi − btd Rtd . Mtd and Mti are the effective suspended masses at the right rear wheel and left rear wheel respectively. Ms is the mass of the platform and finally Ixx and Iyy are the rotary inertias of the platform in their corresponding directions. Rxy = bdd ed Rdd − bdi ed Rdi − btd et Rtd + bti et Rti . R4 = Rti + Rnti . GATTI. Rss = Rdd + Rdi + Rtd + Rti . Ryy = e2d Rdd + e2d Rdi + e2t Rtd + e2t Rti (4) Copyright © 2012 Asociación Argentina de Mecánica Computacional http://www.ar . M.org. R3 = Rtd + Rntd .amcaonline. Rsy = et Rti − ed Rdd − ed Rdi + et Rtd .2748 C. PIOVAN Figure 1: Platform model: basic scheme of the 7 dof model. R2 = Rdi + Rndi . The matrix R of effective damping elements can be written in the following form:   R1 0 0 0 −Rdd bdd Rdd ed Rdd  R2 0 0 −Rdi −bdi Rdi ed Rdi      R 0 −R b R −e R 3 td td td t td    R4 −Rti −bti Rti −et Rti  R= (3)    Rss Rsx Rsy    sym Rxx Rxy  Ryy in which the following definitions are introduced: R1 = Rdd + Rndd .T. amcaonline.e. K3 = Ktd + Kntd . are distances from the gravity center as it can be seen in Figure 1. et . Ldi . i. Recall that the differential system given Eq. ed . bdd . The functions z0dd (t). (1) are given in the following expressions: F=                    Fdd (t) − Mdd g + Kndd [Lndd − z0dd (t)] − Kdd Ldd Fdi (t) − Mdi g + Kndi [Lndi − z0di (t)] − Kdi Ldi Ftd (t) − Mtd g + Kntd [Lntd − z0td (t)] − Ktd Ltd Fti (t) − Mti g + Knti [Lnti − z0ti (t)] − Kti Lti Fss (t) − Ms g + Kdd Ldd + Kdi Ldi + Ktd Ltd + Kti Lti Mxx (t) − bdd Kdd Ldd + bdi Kdi Ldi − btd Ktd Ltd + bti Kti Lti Myy (t) − ed Kdd Ldd − ed Kdi Ldi + et Ktd Ltd + et Kti Lti            udd     udi      utd uti . Kxy = bdd ed Kdd − bdi ed Kdi − btd et Ktd + bti et Kti . respectively. right front. Rntd and Rnti are damping constants associated to the material properties of the wheels. whereas Ldd . respectively. Ltd and Lti are the equivalent free length of springs of the elastic elements that support the platform. z0td (t) and z0ti (t) are related to disturbances in the ground that perturb the motion of the suspended masses and consequently the whole platform.e:   K1 0 0 0 −Kdd bdd Kdd ed Kdd  K2 0 0 −Kdi −bdi Kdi ed Kdi      K 0 −K b K −e K 3 td td td t td    K4 −Kti −bti Kti −et Kti  K= (5)    Kss Ksx Ksy    sym Kxx Kxy  Kyy in which the following definitions are introduced: K1 = Kdd + Kndd . págs. right rear and left rear wheels. utd . (1) and handling the remaining equations Copyright © 2012 Asociación Argentina de Mecánica Computacional http://www. 2745-2755 (2012) where. Rndd . Kntd and Knti are spring constants associated to the elastic properties of the wheels. K2 = Kdi + Kndi . Lntd and Lnti are the equivalent free length of springs related to the elastic properties of the wheels. uss is the vertical displacement of the platform.org. z0di (t). etc. left front. Kxx = b2dd Kdd + b2di Kdi + b2td Ktd + b2ti Kti Ksx = bti Kti − bdd Kdd + bdi Kdi − btd Ktd . whereas Ktd and Kti are the effective spring constants of the right rear and left rear suspension springs.ar . Kdd and Kdi are the effective spring constants of the right front and left front suspension springs. φxx and φyy are the rotations of the platform.2749 Mecánica Computacional Vol XXXI. respectively. right front. Kndd . right rear and left rear wheels. by substituting U = US + UD in Eq. i. Kndi .e. The force vector F and displacement vector U introduced in Eq. (1) can be split into a static system and a dynamic counterpart. K4 = Kti + Knti . left front. Kyy = e2d Kdd + e2d Kdi + e2t Ktd + e2t Kti (6) where. uti . Kss = Kdd + Kdi + Ktd + Kti . Lndi . Rdd and Rdi are the damping constants of the right front damper and left front damper. Ksy = et Kti − ed Kdd − ed Kdi + et Ktd . i. are the vertical displacements of the right rear and left rear wheels. Lndd . are the vertical displacements of the right front and left front wheels. respectively. udi .U =     u   ss       φ      xx  φyy           (7)          where udd . The matrix K of effective elastic coefficients and spring constants is written in a similar form as the previous expression. whereas Rtd and Rti are the damping constants of the right rear damper and left rear damper. Rndi . Knti }. Kdi . The Maximum Entropy Principle is used to derive the probability density function of the random variables associated to the uncertain parameters.ar . consequently the support is Λ =]0. Thus the probability functions of the random variables are obtained solving the following optimization problem: p∗V = arg max pV ∈C p S (pV ) (8) where p∗V is the optimal probability density function such that ∀ pV ∈ C p . δVi and V i are the dispersion parameter and the mean value of the random variable Vi .T. (b) the mean value of each random variable E{Vi } = V i is known and it coincides with the nominal value of the corresponding parameter.amcaonline. S(p∗V ) ≥ S(pV ). V16 } the random variables associated to the parameters {Rdd . Rndi . however the most relevant for the suspension system are the effective spring coefficients and effective damping coefficients of each suspension subsystem in the plate corners. Kntd . V2 .. Under this context and employing the Maximum Entropy Principle the following expression of the probability density function can be derived: 1 pVi (vi ) = 1]0. {V5 . Copyright © 2012 Asociación Argentina de Mecánica Computacional http://www.. {V9 . Ktd . V10 . whereas Γ(z) is the gamma function defined for z > 0. V8 } the random variables associated to the parameters {Kndd . Let introduce {V1 .e. It should be noted that √ the dispersion coefficient is constrained to be δVi ∈ [0. 2008). V14 . Rntd . The available information to construct consistently the probability density functions is the following: (a) random variables should be positive. Depending on the case to be analyzed the whole differential system is used. PIOVAN appropriately (Lee et al. 1/ 3] due to the condition that Vi is a second order random variable. V3 . 3 PROBABILISTIC MODEL The Maximum Entropy Principle (Jaynes. V12 } the random variables associated to the parameters {Rndd . V6 . V i δV2i ) can be used to generate realizations. 16 should be second order random variable to guarantee finite dispersion in the displacements (Ritto et al. V11 . which is the measure of entropy and C p is the set of the admissible probability density functions that satisfy the information available.∞] (vi ) Vi  1 δV2i 1/δV2 i 1  Γ 1/δV2i  vi Vi 1/δV2 i −1   vi exp − 2 δVi V i (10) where 1]0. Rdi . The deterministic model developed in the previous section has many parameters that can be uncertain.2750 C. In the case in which scarce information is available about the dependency among random variables. i = 1. where US and UD are the static and dynamic parts that comprise the solution. Rti }. (1) or only the dynamic part. 2003) is used to construct the probabilistic models taking into account the uncertain parameters. The entropy is defined as (Shannon. (c) Vi . V15 . 1948): Z pV ln (pV ) dv (9) S (pV ) = Λ where Λ is the support of the probability distributions.∞] (vi ) is the support. V7 . M. V4 } as the random variables associated to the parameters {Kdd . . Kti }. The Matlab function gamrnd(1/δV2i . Kndi . Rnti } and {V13 .. the Maximum Entropy Principle states that the random variables involved have to be independent. GATTI. Rtd .org. i... Eq. 2008). ∞[. Mecánica Computacional Vol XXXI.m2 . Rtd = Rti = 12566 N s/m. It is possible to see that a good convergence is obtained with NM S = 1000 or more. btd = bti = 1 m. which is similar to all the preliminary cases tested.2 m.05.ar . Rdd = Rdi = 12566 N s/m.org. Kdd = Kdi = 157913 N/m. Fig. Rntd = Rnti = 16500 N s/m. Ixx = 10000 Kg. which implies the release of the platform from initial conditions are U(0) = 0 and U(0) a situation with uncompressed springs. the parameters of the model shown in Fig. Fig. and vector U and its time derivatives are the random response.. ed = 0. Iyy = 16000 Kg. 4 COMPUTATIONAL STUDIES For the computational studies of uncertainty quantification in the dynamics of the platform. Kndd = Kndi = 1. z0td (t) = 0 and z0ti (t) = 0. Fig.207 106 N s/m. the convergence of the stochastic response U is evaluated appealing to the following function: v u M S Z t1 u 1 NX t Uj (t) 2 dt conv (NM S ) = (12) NM S j=1 t0 where NM S is the number of Monte Carlo samplings. In order to control the quality of the simulation. Lntd = Lnti = 0.207 106 N/m.e. 3 shows an example of the convergence pattern. g = 9..2 m. which implies the integration of a deterministic system for each independent realization of random variables Vi . Ktd = Kti = 157913 N/m. For these studies no perturbation in the ground is involved. However with NM S = 500 or even less it is possible to reach an acceptable solution convergence. Several dispersion coefficients have been tested.8 m. (11) the math-blackboard font is used to identify the random entities. et = 1. 4 shows histograms of some parameters employed for this calculation. i = 1. Rndd = Rndi = 16500 N s/m. págs. This type of programs do not have the possibility to perform stochastic dynamic analysis. Rndd = Rndi = Rntd = Rnti = 0. non active dampers and non unbalanced platform.5 m. Ldd = Ldi = 0. (1) by employing Eq. As a first study a low dispersion is used in all parameters.amcaonline.m2 . The Copyright © 2012 Asociación Argentina de Mecánica Computacional http://www. Mtd = Mti = 350 Kg.5 m. bdd = bdi = 1 m. . A previous check of the quality of the deterministic model is performed by comparing its transient response with the one calculated in a commercial multibody dynamics system. 2745-2755 (2012) 2751 The probabilistic model can be constructed from Eq. Lndd = Lndi = 0. 2 shows the comparison of the dynamic responses of the present deterministic model and a computational model constructed in the Multibody dynamic analysis system called Working Model 3D. 1 have the following expected values: Mdd = Mdi = 350 Kg.. (10) in the assembly of damping and stiffness matrices. and the dynamic systems becomes stochastic as: ¨ + RU˙ + KU = F MU U(0) = U0 ˙ ˙ U(0) =U 0 (11) In Eq. that is: z0dd (t) = 0.81 m/s2 . that is matrices R and K are random due to the stochastic nature of the parameters involved in them. i. Also the damping effect of the lower components is neglected. In the following paragraphs a study of the dynamic stochastic response of the platform is performed. z0di (t) = 0. Ms = 16000 Kg. The ˙ = 0. Kntd = Knti = 1. The Monte Carlo method is used the simulate the stochastic dynamics. 16. Ltd = Lti = 0. that is δVi = 0.2 m. A few preliminary studies are made in order to check the convergence of the Monte Carlo Method. 5(a) and Fig.2 0 Present Model 1 2 3 4 Time. t [s] 5 Figure 2: Comparison of the deterministic model with a 3D multibody dynamics analysis system.6 0. Fig.ar .6 0.T.2 0. As expected the influence of the uncertainty is larger in the loaded suspension subsystem. platform is loaded with a sudden unitary force at each suspension subsystem on the right side..2752 C.4 0. 7 shows the response of the right front suspension subsystem with the same type of excitation adopted for the previous case. M. that is δVi = 1/sqrt3. t [s] 5 6 7 −0.org.05 φ [rad] MBD System 0. 5(b) the response of the front suspension subsystems are depicted. The results of this test are presented in frequency response functions (FRF).18 −0.4 0.2 0 1 2 3 4 Time. In Fig. −7 8 x 10 7 conv(NMS) 6 5 4 3 2 1 0 0 200 400 600 800 1000 1200 Number of simulations 1400 1600 1800 2000 Figure 3: Example of the convergence of the Monte Carlo Method.5 u ss [m] 0. although there is a increase of the size of the confidence interval in the band of 4 Hz to 20 Hz.2 0 14 0. but for maximum possible level of uncertainty in the random variables associated to the damping coefficients of the lower part of the suspension subsystems. i = 13. t [s] 10 12 2 3 4 Time. 16.19 u 0.8 udd udi utd uti φ uss 0. and the remaining random variables have Copyright © 2012 Asociación Argentina de Mecánica Computacional http://www.2 2 4 6 8 Time. .amcaonline.15 0 MBD System 0.1 Present Model y di [m] 1 Present Model 0.17 MBD System −0. t [s] 5 6 7 6 7 0 −0. PIOVAN 0. GATTI. The response of the platform is not influenced sensibly by this condition of uncertainty in the lower part of the suspension subsystems..16 0.. and applied in the corresponding concentrated masses.15 0.7 φ x y 0. 6 the response of the mass center of the platform is shown. In Fig.3 0 −0. 3 1. págs.4 −14. 5(a) and 7 and Figs.6 0 2 4 6 8 10 Frequency [Hz] 12 14 16 18 20 −28 0 2 4 6 8 10 Frequency [Hz] 12 14 16 18 20 (a) (b) Figure 5: FRFs Frequency response function suspension subsystems of the front: (a) udd (b) udi .05. 6 and 8 are compared.3 1. there is no sensible influence of the parameters uncertainty to promote an instability in the platform. The control of this type of aspects are of major concern in the dynamics of platforms.8 −24 −15 −15.4 1. 5 CONCLUSIONS This article has presented a preliminary study of the stochastic dynamics of the suspension system in a vehicular platform modeled with a 7 degree-of-freedom ordinary differential equations system. 20] when Figs.3 1.4 1.4 1.5 0 1 1.5 0 1. The stochastic aspects are related to the uncertainty of some parameters of the Copyright © 2012 Asociación Argentina de Mecánica Computacional http://www. 2745-2755 (2012) Histogram of Spring Kndd Histogram of Spring Ktd 400 400 300 300 200 200 100 100 0 1 1.org. −13.7 1.3 1.2 −20 di log(|U |) log(|Udd|) −14. 8 shows the response of the mass center of the platform.2 1.2753 Mecánica Computacional Vol XXXI.1 1.8 −18 −14 −14.6 1.1 1.6 −22 −14. However near the peak response of the platform.1 1. Fig. the same dispersion parameter δVi = 0.5 1.amcaonline.5 1.4 −15.6 4 4 x 10 x 10 Figure 4: Histograms of some parameters.2 1.2 −26 −15.4 1.9 6 5 x 10 x 10 Histogram of Damper R Histogram of Damper R dd td 400 400 300 300 200 200 100 100 0 1 1.2 1. As it is possible to see the propagation of the uncertainty associated to the damping coefficients of the lower part of the suspension system increases dramatically in the band of frequencies [4.ar .6 −16 Deterministic Model Mean Probabilistic Model 95 % Confidence Interval Deterministic Model Mean Probabilistic Model 95 % Confidence Interval −13.8 1. Although some parameters evaluated in this paper appear to scarcely affect the platform dynamics.5 −16 0 2 4 6 8 10 Frequency [Hz] 12 14 16 18 20 Figure 7: Frequency response function of udd for a higher uncertainty in the damping coefficients of the lower part of the suspension subsystems.amcaonline.2754 C. REFERENCES Dixon J. GATTI.5 Deterministic Model Mean Probabilistic Model 95 % Confidence Interval −14 log(|U dd |) −14. The Maximum Entropy Principle has been used to derive the probability density functions of the uncertain parameters which are needed to construct the probabilistic model taking into account the deterministic model as the mean model.5 −15 −15. Suspension geometry and computation. ACKNOWLEDGMENTS The authors recognize the support of Secretaría de Ciencia y Tecnología de la Universidad Tecnológica Nacional and Conicet of Argentina. Wiley and Sons LTD.T. Then the Monte Carlo Method is employed to simulate the stochastic response.ar . 2009. Copyright © 2012 Asociación Argentina de Mecánica Computacional http://www. The influence of the uncertainty of many parameters. −13. More studies should be done to evaluate and characterize the risky physical parameters in the dynamics of vehicular platforms.org. PIOVAN −13 Deterministic Model Mean Probabilistic Model 95 % Confidence Interval −14 −16 log(|U ss |) −15 −17 −18 −19 −20 0 2 4 6 8 10 Frequency [Hz] 12 14 16 18 20 Figure 6: Frequency response function of uss . model such as the effective spring coefficients or the effective damping coefficients. there is no evidence to state the same for every parameter involved in the model. on the stochastic dynamics of the vehicular platform has been tested. M. 2008. Wiley and Sons INC. págs. 28:379–423 and 623–659. Cambridge University Press. Non-smooth problems in Vehicle systems dynamics.. 2008. Williams J. and Cataldo E. 2010. Fundamentals of Vehicle Dynamics. 1995. In-plane vibrational analysis of rotating curved beam with elastically restrained root. Probability Theory: The logic of Science. and Lin S. 2745-2755 (2012) 2755 −13 Deterministic Model Mean Probabilistic Model 95 % Confidence Interval −14 −15 log(|U ss |) −16 −17 −18 −19 −20 −21 0 2 4 6 8 10 Frequency [Hz] 12 14 16 18 20 Figure 8: Frequency response function of uss for a higher uncertainty in the damping coefficients of the lower part of the suspension subsystems. 1948. Fundamentals of Applied Dynamics.ar . 1996.. and True H. 2001. Berlin-Heidelberg.. Journal of Sound and Vibration.1..Mecánica Computacional Vol XXXI. Journal of Brazilian Society of Mechanical Sciences and Engineering.amcaonline. Soize C. Copyright © 2012 Asociación Argentina de Mecánica Computacional http://www.K. Jaynes E. 30:295– 303. 109(5):1979–1996. Thomsen P. Vol. Sheu J. Ritto T. A mathematical theory of communication.. 315(4). Gillespie T. Cambridge. Society of Automotive Engineers. Shannon C. Sampaio R. Lee S. U. Maximum entropy approach for modeling random uncertainties in transient elastodynamics.org. Journal of the Acoustical Society of America. 2003. Timoshenko beam with uncertainty on the boundary conditions. Bell System Tech. Springer-Verlag. Documents Similar To 4218-19208-1-PBSkip carouselcarousel previouscarousel nextTruck Shock Absorber Inspection Procedures[WBV 06] WBV Directive 200244EC- Expuneri Zilnice de VazutD37PX-21_M_EEAM024300_D31_37_39_EX_PX_21_05090crane and Tractor Semitrailers a Comparative Study of Their Dynamic Effects on a Short-span Bridge - Heywood [1]Autoduel - Accessory - Construction Sheet - Motorcycle2720501BIOSTATISTICS -1st Sem 4th Year ModuleVol2_No3_559-564_Comparing PID and Fuzzy Logic Control a Quarter-Car ...3965-11201-1-PBVehicle TechnologiesSz15rr - Cfao Motors - EnNew NavaraChassis Design WorkbookIJETT-V3I5P215.pdfWriting the abstract (part two)SAGParam OpenShock AbsorberBSC6900 GSM V900R011C00SPC700 Parameter Reference.xls4 ProbabilityDesign ConceptsVirtual Demo Case SIPART PS2Three-dimension design of EPSPiaggio Mp3 Yourban EngSCM360-CapacityPlanningEnduro Issue 002 INT704918 UkProbability Assignment 2Tyre application BrochureMercedes M-Guard Press ReleaseSamplesMore From antonioulbraSkip carouselcarousel previouscarousel nextMétodo de Eliminação de Gauss – Com PivotaçãoMesh AirfoilTablas Demo Mentos Deem Potr Amien ToEjemplo MatriciAlMatricial ProblemasPROGRAMA Analise Matricial de Estruturas AacosGENÉTICAejerciciosresueltosdeanalisisestructuraliEx Vf Analise MatricialAULA 3 - Lista de ExercíciosBloco - Alavanca de Comandoaula6-programaoGauss9_Leseprobe978-3-7315-0090-2Vol1_MatlabAnálise Preliminar Parque Eólico Hulha Negra - RSEdmond Chiu ThesisLista2Cap601015-015Mecsol Parte II J.apostila CarlosBigger Sims Win the RaceFX63-137CartesianCoordinates BoundaryLayerTrip DegradedLiftVol2_Matlab0Laminate Code Input2_propriedades Do Ar AtmosfericoBest Books About StatisticsPractical Monte Carlo Simulation with Excel Part 1by Akram NajjarStatistics II for Dummiesby Deborah J. Rumsey, PhDBusiness Statistics I Essentialsby Louise ClarkEconometricsby Fumio HayashiLearning Predictive Analytics with Rby Eric MayorStatistics for Experimentalistsby B. E. CooperFooter MenuBack To TopAboutAbout ScribdPressOur blogJoin our team!Contact UsJoin todayInvite FriendsGiftsLegalTermsPrivacyCopyrightSupportHelp / FAQAccessibilityPurchase helpAdChoicesPublishersSocial MediaCopyright © 2018 Scribd Inc. .Browse Books.Site Directory.Site Language: English中文EspañolالعربيةPortuguês日本語DeutschFrançaisTurkceРусский языкTiếng việtJęzyk polskiBahasa indonesiaSign up to vote on this titleUsefulNot usefulYou're Reading a Free PreviewDownloadClose DialogAre you sure?This action might not be possible to undo. Are you sure you want to continue?CANCELOK
Copyright © 2024 DOKUMEN.SITE Inc.