血管造影機(jī)C-臂設(shè)計【醫(yī)療器材】

血管造影機(jī)C-臂設(shè)計【醫(yī)療器材】,醫(yī)療器材,血管造影機(jī)C-臂設(shè)計【醫(yī)療器材】,血管,造影,設(shè)計,醫(yī)療,器材 Dynamic analysis of automotive clutch dampers C.L. Gaillard, R. Singh* Acoustics and Dynamics Laboratory, Department of Mechanical Engineering and Center for Automotive Research, The Ohio State University, 206 West 18th Avenue, Columbus, OH 43210-1107, USA Received 16 July 1997; received in revised form 17 January 1999; accepted 24 January 1999 Abstract The torsional dynamic characteristics of an automotive clutch are simulated by five lumped parameter linear or nonlinear models. Each nonlinear model includes visco-elastic and dry friction elements. Dynamic stiC128ness and energy dissipation spectra clearly show excitation amplitude and frequency dependent behavior. Also, dynamic hysteresis curves are predicted and analyzed. The proposed models compare well with limited experimental data. Finally, the utility of such models is illustrated via a transmission rattle simulation program. # 2000 Elsevier Science Ltd. All rights reserved. Keywords: Vehicle driveline; Dynamic hysteresis; Nonlinear models 1. Introduction Vibro-impacts in manual transmissions are of critical concern to vehicle manu- facturers based on noise, vibration and reliability considerations. This phenomenon is often perceived as the gear rattle problem, and some practical solutions have been suggested that may either reduce the noise or eliminate the likelihood of its occur- rence 17. Proper selection of clutch parameters such as multi-valued spring and hysteresis rates is necessary to solve the problem 5, 6. Several computer simulation models, based on the linear and nonlinear analyses, have been proposed 19. Most often, these are lumped parameters torsional models that describe nonlinear and linear components 2,712; comparable multibody dynamics approaches have also been suggested 4. Fig. 1 shows a simplified 4 degree of freedom (DOF) model with flywheel, clutch hub, input gear and output gear inertial elements, each characterized by torsional displacement . Dampers on the gears account for the drag torques and Applied Acoustics 60 (2000) 399424 0003-682X/00/$ - see front matter # 2000 Elsevier Science Ltd. All rights reserved. PII: S0003-682X(00)00005-0 * Corresponding author. Tel.: +1-614-292-9044; fax: +1-614-292-3163. E-mail address: singh.3osu.edu (R. Singh). the input shaft is represented by a linear torsional spring. The interfacial contact between the gears is represented by the mesh stiC128ness and a backlash. The clutch (C) is located between the flywheel and the input shaft. Assume that the flywheel velo- city C10 1 and torque T are given in terms of mean (m) and fluctuating (p) compo- nents; these may be related to engine, transmission and vehicle parameters and conditions: C10 1 t C10 1m C10 1p t and T e t T 1m T ep t : The clutch characteristics in terms of torque (T c ) vs. relative angular displacement ( C ) are invariably based on static measurements. Practical design evidence and limited experimental measurements suggest otherwise 1,5,10. It is seen that dynamic characteristics are frequency and amplitude dependent. Since this issue is not well understood, it is the subject of this article. 2. Problem formulation Fig. 2 shows typical static and dynamic characteristics of clutch dampers. Dual staged curve T C C is shown in Fig. 2a that is similar to those measured under static (zero frequency) conditions. This may be described by two stiC128nesses and a constant hysteresis; it is also a source of clearance nonlinearity since the stiC128ness changes abruptly from one value to the other. Fig. 2b compares the static and dynamic curves when the clutch is being operated in the first stage. Observe that stiC128ness under dynamic excitation diC128ers from the one measured under the static case. Also, dynamic hysteresis deviates from the static curve. Fig. 2c shows the typical energy dissipation curve it will be discussed later. As a component, the clutch may be modeled as shown in Fig. 3a and b where the harmonic excitation may be in the form of harmonic displacement c t at frequency Fig. 1. Vehicle gearbox model used for gear rattle analysis. 400 C.L. Gaillard, R. Singh/Applied Acoustics 60 (2000) 399424 Fig. 2. Typical clutch properties. (a) Static torque vs. displacement. (b) Dynamic torque vs. displacement. (c) Energy dissipation per unit amplitude for various displacement amplitudes C . Here, f is the frequency in Hz. C.L. Gaillard, R. Singh/Applied Acoustics 60 (2000) 399424 401 Fig. 3. (a) Free body diagram of the clutch in torsional mode (b) Dynamic clutch testing concept. (c) Dynamic parameters extracted from the experiment. 402 C.L. Gaillard, R. Singh/Applied Acoustics 60 (2000) 399424 !underagivensetofconditionssuchasspeed C10 1 andtheboundaryconditionmaybe fixed. Only the steady state (and not the transient state) is studied in this paper. The resulting dynamic torque T C t of the whole system is measured or calculated. In this study, simplified lumped parameter models are developed in order to analyze the clutch behavior in terms of T C vs. C at given !: Internal damping phenomena such as viscous and dry-friction damping are not evaluated from a microscopic view- point. Instead, an assumption is made that both viscous and dry-friction damping components are of interest and that the cyclic energy dissipation d is proportional to the bounded area of the hysteretic loop. Therefore, each nonlinear visco-elastic model will include elastic stiC128ness, viscous damping and a dry-friction element. March and Powell 11 have suggested an empirical approach where the clutch damper stiC128ness is obtained from measured static data, and the hysteresis and damping values are calculated from dynamic measurements; however, no details are provided in this article. Their model 11 appears to show an excellent correlation with the measured static properties, but not with the measured dynamic response that is reported only at 50 Hz with one excitation amplitude (1.0 degree). To the best of our knowledge, no dynamic models of clutch dampers are currently available. Further the literature on the measured dynamic behavior of clutches is virtually non-existent 5,10,15. Nonetheless, prior studies on elastomers and constrained layer damping treatments suggest that this approach is viable 1618. Specific objectives of this study are, therefore, to develop five lumped parameter dynamic models based on theory, study these models in terms of dynamic stiC128ness k ; loss factor and cyclic energy dissipation d ; and then compare with available experi- mental results 10,13. Finally some of these models are included in a gear rattle simulation program 9,19. Results of experimental tests on several clutches were made available to this study 10,13. The clutch was tested first with a locked shaft, which means that the clutch assembly did not rotate and was dynamically excited by a torsional actuator 14. Then a rotating shaft rig test was carried out with a mean velocity : Cm C10 1m : Reference 14 describes the test methodology. Results show that the rotational speed C10 1 has an influence only beyond 2000 rpm. This is believed to be outside the range of concern for gear rattle type problems 13. Tests have been run at four diC128erent excitation frequencies: 1 (quasi-static), 50, 75 and 100 Hz. For the quasi-static case, the complete cycle has been obtained. At the other frequencies, cycles of 2 degrees of peak-to-peak amplitude, centered at 3 degrees have been obtained. Also, for each curve, the signal is filtered using a low pass filter with a cut-oC128 frequency of 400 Hz. Fig. 4 shows sample results for 50 and 100 Hz excitation frequencies. These curves may be defined in terms of storage modulus and hysteresis. For an ideal linear system, the curve would be a perfect ellipse, and the storage modulus (or torsional stiC128ness) line would be determined by the points of maximum strain on each side of the curve. In our case, the curves seem to be a combination of a parallelogram due to dry-friction, and an ellipse due to viscous damping. Still, we can roughly estimate the storage modulus slope by ignoring the oC128set created by dry friction. The storage stiC128ness values are 20.3, 22.11, 19.4 and 18.9 Nm/deg at 1, 50, 75 and 100 Hz, respectively. C.L. Gaillard, R. Singh/Applied Acoustics 60 (2000) 399424 403 Experiments suggest that the dynamic hysteresis may be defined in terms of cyclic energy dissipated per unit torsional displacement amplitude ( ). It seems that drops with an increase in frequency up to a certain frequency and increases after this frequency 13. Figs. 2c and 5 show this eC128ect. Also, observe that these variations may be more noticeable with higher excitation amplitudes. Some other system parameters such as operating temperature or C10 1m do not seem to aC128ect dynamic properties. Therefore we may conclude that the frequency and excitation amplitude are the two main factors that need to be examined in this study. 3. Dynamic modeling concepts 3.1. Physical system The damping in the clutch assembly is not easily identifiable since there are several sources of energy dissipation such as internal damping of the steel or facing mate- rial, the damping caused by the joints between the clutch plates and the friction facings, and the friction of springs when they rub against the clutch plates. Also, experiments show that the trends can be diC128erent from one clutch to another in terms of loss factors, spring rates, etc. probably because of the use of diC128erent types of material, and diC128erent mechanical designs 1,5,15. Our approach is then quite intuitive, yet theoretical, as we decompose the total damping into two component types, dry-friction and viscous. By including these two in our dynamic model and varying their respective influence we hope to be able to model the observed damping Fig. 4. Measured torque vs. displacement curve at excitation frequencies 50 and 100 Hz (with 2deg. peak- to-peak amplitude and 3deg. mean). 404 C.L. Gaillard, R. Singh/Applied Acoustics 60 (2000) 399424 trends. With reference to Fig. 3, T facings and T transmission are the input and output torques, where T facings is the torque transmitted by the flywheel to the clutch and T transmission is the torque transmitted by the clutch to the transmission. The other torques such as the friction damping, viscous damping, and elastic (spring) torques are internal to the clutch. The eC128ect of the clutch inertia I c is described by accelera- tion C127 C : Taking all these elements into account, we obtain the equation of motion: I C C127 C T facings T friction damper T viscous damper T spring T transmission 1 3.2. Dry friction model The dry-friction is modeled using the Coulomb friction model. As shown in Fig. 6, it is defined by two system parameters: (1) shear stiC128ness (k f ) between the facings and the pressure plate or the flywheel, and (2) saturation friction torque (T f ), i.e. torque transmitted during slipping. Some other models were also investigated to simulate the dry-friction phenomenon 18. In particular, a model that included a speed-dependent friction coe cient has been considered but the rounded shape of the hysteresis cycle such as Fig. 2 is mostly determined by the viscous damping. Thus the main contribution of the dry-friction is the parallelogram-like shape of the loop. Another reason why our model should not be too complex is that the data extracted from experimental results may not be very accurate. Consequently, we intend to simulate trends and not try to match the exact numbers. The behavior of the dry-friction element can be divided into three diC128erent states, namely stick state, positive state and negative state. The positive state is defined by a positive slip speed : f 0 during which the transmitted torque is T f : The transition Fig. 5. Measured energy dissipation per unit amplitude ( ) of clutch damper vs. frequency (f), as extracted from 10. Here the excitation amplitudes ( C ) vary from 0.1 to 1.5 degrees. C.L. Gaillard, R. Singh/Applied Acoustics 60 (2000) 399424 405 to the stick state occurs when the decreasing slip speed reaches : f 0: During this state, there is no relative slipping between 1 and f ; therefore : f : 1 : If 1 1 cos !t ; this occurs for !t 0 2k ; k2N , i.e. when 1 1 : The negative slip state is defined by : f 0 and during this state, the transmitted torque is T f : The transition to the stick state occurs when the increasing slip speed reaches : f 0; this is when !t 2k ; k2N; i.e. when 1 1 : The stick state is defined by : f 0: During this state, the transmitted torque is dictated by the spring stiC128ness, T k f 1 f 0 where f 0 is the value of f at the end of the preceding state and during the stick state. If the positive slip is the preceding state, then f 0 1 T f =k f ; and if negative slip is the preceding state, then f 0 1 T f =k f : The transition to the negative slip state occurs when the transmitted torque reaches T f ;this is when 1 1 2T f =k f :The transition to the positive slip state occurs when the transmitted torque reaches T f ; this is when 1 1 2T f =k f : Fig. 6. Torque vs. excitation displacement 1 for the dry friction element. 406 C.L. Gaillard, R. Singh/Applied Acoustics 60 (2000) 399424 3.3. Visco-elastic models Initially, we assume an inertialess linear spring-damper model that should describe the internal damping that does not come from interfacial friction as well as the inter- nal stiC128ness of the clutch plates. As shown in Table 1, Model (A) is a combination of one spring and one dashpot in parallel (Voight model), Model (B) is a combination Table 1 Lumped parameter visco-elastic models, without and with dry-friction Model Without dry-friction With dry-friction Parameters A k 13 ;c 13 ;J 2 Dry-friction k f ;T f B k 12 ;c 23 ;J 2 Dry-friction k f ;T f C k 12 ;k 23 ;c 23 ;J 2 Dry-friction k f ;T f D c 12 ;k 23 ;c 23 ;J 2 Dry-friction k f ;T f E k 12 ;c 23 ;J 2 ; k 14 ;c 43 ;J 4 Dry-friction k f ;T f C.L. Gaillard, R. Singh/Applied Acoustics 60 (2000) 399424 407 of one spring and one dashpot in series (Maxwell model), Model C is a Voight model in series with a spring, and Model D is a Voight model in series with a dash- pot. One might yet come up with more visco-elastic elements 16 but let us limit the choices for the sake of simplicity. Finally, one needs to investigate the possibility of including clutch inertia in the model. If it is included, where should it be located? One may lump them at location 1 of Table 1 where it would not aC128ect the stiC128ness and damping properties of the clutch as a component but it would aC128ect the drive- line system dynamics. Or one may lump them at location 2 where it may aC128ect the clutch dynamic properties. All of the nonlinear lumped parameter models combine the dry-friction damping element and one of the visco-elastic elements (Table 1). 3.4. Dynamic stiC128ness concept At any frequency !; the complex dynamic stiC128ness k is as follows with respect to Fig. 3c. k T T e i !t F c e i!t T T e i F c T T c k e i K 2a k k 0 ik 00 k 0 1 i 2b Examine the linear (without friction) models of Table 1 and refer to Eq. (2) for deriving the following expressions. The cyclic energy dissipation d is given by the product of harmonic displacement and torque T T for one cycle. Since T T can be obtained from k as T T k c ; d is given by d k c d c k c d c dt dt 3 For harmonic excitation displacement c t c e i!t ; the expression becomes d k 0 c cos !t k 00 c sin !t c !sin !t dt k 00 2 c 4 4. Illustrative example (model D) 4.1. Contribution from the visco-elastic element Writing c 1 1 e j!t ; the transmitted torque is T T e j!t k 23 c 23 j! c 12 j! c 12 j! k 23 J 2 ! 2 c 23 j! 1 e j!t : 408 C.L. Gaillard, R. Singh/Applied Acoustics 60 (2000) 399424 Therefore the dynamic stiC128ness is k k 23 c 23 j! c 12 j! c 12 j! k 23 J 2 ! 2 c 23 j! : And the storage stiC128ness k 0 ; loss stiC128ness k 00 ; loss factor and energy dissipation per cycle d are as follows. k 0 c 12 ! k 23 c 12 c 23 ! k 23 J 2 ! 2 c 23 ! k 23 J 2 ! 2 ! 2 c 12 c 23 2 5a k 00 c 12 ! k 23 k 23 J 2 ! 2 ! 2 c 23 c 12 c 23 k 23 J 2 ! 2 2 ! 2 c 12 c 23 2 5b k 23 k 23 J 2 ! 2 ! 2 c 23 c 12 c 23 k 23 c 12 c 23 ! k 23 J 2 ! 2 c 23 ! 5c d c 12 ! k 23 k 23 J 2 ! 2 ! 2 c 23 c 12 c 23 k 23 J 2 ! 2 2 ! 2 c 12 c 23 2 2 1 5d Most of the other models may be deduced from model D. For example, model A would correspond to c 12 !1; model B is obtained when c 23 0; and model E includes two B models in parallel. 4.2. Contribution from the dry-friction element The contribution of the dry friction to energy dissipation is proportional to the area limited by the parallelogram. The energy dissipation d is, therefore, equal to T multiplied by over the whole cycle: d Torque d 4T f 1 T f k f : d is constant for given values of T f ; k f and 1 : Since we assumed a rate-inde- pendent dry friction mechanism, its loss factor and energy dissipation are not func- tions of !: Therefore, the inclusion of dry-friction in diC128erent viscoelastic models implies that the energy dissipation will be additive and the resulting curve will move up. As far as the storage stiC128ness is concerned, it is not an easy task to determine how the dry-friction element contributes to it in the frequency domain and how much. The friction stiC128ness is obviously time dependent; it is equal to k f for 04!t k 4acos 1 C.L. Gaillard, R. Singh/Applied Acoustics 60 (2000) 399424 409 and equal to 0 for acos 1 4!t k 4 when 1 1 cos !t (Fig. 6). On the other hand,is dependent upon k f :The higher the value of k f ;the smaller the value of : Plotted in the time domain, this corresponds to an stiC128ness impulse. In par- ticular, for the simplest Coulomb friction model, the transmitted torque jumps instantaneously from the negative value ( T f ) to the positive value (T f ) and vice- versa. This corresponds to an infinite stiC128ness over an infinitesimal instant and it can be correlated to a perfect impulse. The contribution of the friction element to the system storage stiC128ness in the frequency domain is therefore neglected in this study. 5. Results based on linear models 5.1. Typical spectral contents Figs. 7 to 10 show the typical spectra that are obtained for d ;k 0 and for selected models. One may identify three groups of models in terms of the behavior of the energy dissipation. For the Voight model (A), d increases monotonically in a linear manner. For models B and C, the energy dissipation spectrum exhibits only one peak, but for models D and E (double Maxwell model), we obtain two peaks and a valley in between. Some experimental tests show d to decrease up to a certain!and then increase afterward. Hence it is interesting to note that model D or E may represent this behavior. The storage stiC128ness spectra also show similar trends. k 0 of the Voight model (A) is constant and k 0 of models B, C and D increases mono- tonically and rapidly, then seems to follow a horizontal asymptote. For the double Maxwell model (E), k 0 also increases monotonically, but the rate of increase is lower when d is between the two peaks. Note that experimental results show a slight decrease in k 0 as ! is increased 13. Default parameter values are, however, model-dependent and reasonable values are chosen so as to match typical experimental hysteresis loops in terms of slope and hysteresis loop width. To estimate the clutch inertia, we computed the order of magnitude value by assuming a disk of radius r 100 mm, and mass M 1 kg. This leads to J 0:5Mr 2 0:005kgm 2 : We study the influence of parameters on the energy dissipation d and storage stiC128ness k 0 and how the hysteresis is aC128ected. As an example, examine model D. First consider the energy dissipation spectrum that shows the combination of a peak and a valley (Fig. 9). For a smaller value of c 12 ;the peak is broad and moves toward higher frequencies. Therefore, a high c 12 value makes the peak sharper, as well as the vall