Start-Up Transient Vibration Analysis of a Vehicle Powertrain … · 2019-12-18 · cylinders,...

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INTRODUCTIONTransient events during the engine start-up (or shut-down) process are receiving attention recently for both conventional internal combustion (IC) engines [1-2] and hybrid electrical vehicles [3, 4, 5, 6, 7]. Such vibration issues are now more noticeable due to two reasons. First, downsizing of the IC engine including the fewer cylinders, cylinder deactivation, and light-weighting has resulted in higher pulsating torques. Second, new start-up/shut-down methods induce more frequent resonant transient vibration amplifications [3, 4, 5, 6, 7]. The engine instantaneous firing frequency passes through the natural frequencies of the powertrain torsional modes, resulting in a vibration amplification and thus increasing human discomfort. Due to multiple discontinuous nonlinearities in the clutch damper and the transmission [8-9], the transient vibration in the nonlinear powertrain subsystem is yet to be fully analyzed. Thus this article intends to develop powertrain system nonlinear models and examine the non-stationary process using numerical, experimental, and analytical approximation methods.

PROBLEM FORMULATIONThe effects of discontinuous nonlinearities, such as piecewise linear stiffness, hysteresis, and preload of a multi-staged clutch damper on the stationary periodic vibro-impacts events have been studied in frequency and time domains [10, 11, 12, 13, 14, 15, 16, 17, 18]. For a non-stationary process, the transient envelope of oscillations is normally utilized to evaluate the peak amplification level. Li and Singh [2] and Sen et al. [19-20] utilized the Hilbert transform [21] to establish the transient envelope of nonlinear single-degree-of-freedom system. Earlier, indirect analytical methods were developed by Markert and Seidler [22] and Hok [23] to estimate transient envelope of linear SDOF system. Also, closed form solutions of the transient envelope of a linear oscillator have been developed in the

instantaneous frequency domain by Li and Singh [24, 25, 26]. However, the transient events occurring in the nonlinear multi-degree-of-freedom (MDOF) torsional system have not been quantitatively interpreted. Therefore, a MDOF powertrain system of a conventional vehicle with a clutch damper and manual transmission is considered in this article. As shown in Figure 1(a), a 4DOF nonlinear torsional model excited by instantaneous flywheel torsional

displacement and torsional velocity will be developed where a bar over a symbol indicates a dimensional parameter. This MDOF model and its reduced order version will be experimentally validated and their utility will be discussed.

NONLINEAR MDOF MODEL OF POWERTRAIN SYSTEMThe 4DOF nonlinear model is developed based on the assumption that the flywheel motion (displacement and velocity) can be employed as the system input since the flywheel motion should not be affected by the motion of other downstream elements due to its massive inertia. The downstream driveline components including the propeller shaft, axles, and the differential are decoupled from the transmission during the engine start-up. Here, , , and indicate the torsional inertia, viscous damping, and stiffness of the lumped element, respectively. Two gear backlashes with the backlash size (mm) between the headset gear pair and the unloaded gear pair are present. An asymmetric multi-staged clutch damper is placed between the flywheel and clutch hub; its characteristics curve is given in Figure 1(b), where is the torque transmitted through the clutch damper, and represent the stiffness and hysteresis, and I, II, and III indicate the pre-damper, main spring, and stopper stages; and subscripts c and d imply coast and driving sides, respectively. The

detailed nonlinear function , where the hysteresis is

Start-Up Transient Vibration Analysis of a Vehicle Powertrain System Equipped with a Nonlinear Clutch Damper

Laihang Li and Rajendra SinghOhio State University

ABSTRACTThe transient vibration phenomenon in a vehicle powertrain system during the start-up (or shut-down) process is studied with focus on the development and experimental validation of the nonlinear powertrain models. First, a new nonlinear four-degree-of-freedom torsional powertrain model for this transient event, under instantaneous flywheel motion input, is developed and then validated with a vehicle start-up experiment. Second, the interactions between the clutch damper and the transmission transients are established via transient metrics. Third, a single-degree-of-freedom nonlinear model, focusing on the multi-staged clutch damper, is developed and its utility is then verified.

CITATION: Li, L. and Singh, R., "Start-Up Transient Vibration Analysis of a Vehicle Powertrain System Equipped with a Nonlinear Clutch Damper," SAE Int. J. Passeng. Cars - Mech. Syst. 8(2):2015, doi:10.4271/2015-01-2179.

2015-01-2179Published 06/15/2015

Downloaded from SAE International by Rajendra Singh, Wednesday, June 17, 2015

determined by and , can be found in [26]. The system

governing equations under the flywheel motion input , are given as follows:

Figure 1. Powertrain torsional system model. (a) 4DOF nonlinear powertrain model given motion input from the flywheel; (b) Asymmetric nonlinear characteristics of a multi-staged clutch damper.

Here, (i = 1, 2, and 3) represents the drag torques on the gears and shafts in the transmission that are caused by the lubricant and

bearings and assumed to be constants; and

are the nonlinear gear mesh forces from the headset gear pair and unloaded gear pair with the backlashes, respectively. Detailed nonlinear function expressions can be found in [26].

EXPERIMENTAL VALIDATIONIn order to validate this nonlinear model, a vehicle start-up experiment is conducted on a medium-duty truck with a 4-cylinder gasoline engine and a 6-speed manual transmission. The schematic of the test rig is shown in Figure 2; other downstream driveline components including the propeller shaft, axles, and the differential are not shown since they are decoupled during the engine start-up with manual transmission. Two magnetic sensors are utilized to

measure the absolute torsional velocities of the flywheel ( ) and

clutch hub ( ), and the key system parameters are estimated from the physical dimensions or vehicle measurements. For the sake of comparison, modal properties of the linearized model are calculated as reported in Table 1. In the linearization, net backlash size is assumed to be zero and only one stage of the clutch damper is activated. Then, the measured instantaneous flywheel motion ( ) is applied to the nonlinear 4DOF model as an input. Finally, the

relative velocity through the clutch damper is predicted via the nonlinear model and compared with test measurements in both time and time-frequency domains. The comparison shown in Figure 3(a), (b) and Table 2 clearly indicate that the 4DOF nonlinear model yields accurate predictions, especially for

, , , and . Also, the time-frequency domain results, calculated by short-time Fourier transform algorithm, in Figure 3(c), (d) suggest that the 4DOF nonlinear model is able to accurately capture first two dominant orders of the instantaneous frequency and the instant (around 0.4 s) when the maximum amplification ( or ) occurs (dashed circle). In addition, the nonlinear model shows that the clutch mode with stage I activated (12 Hz) results in the amplification, which is consistent with the test measurement. The minor discrepancies in the comparison may be caused by other damping mechanisms; a discussion of these is beyond the scope of this article.

Figure 2. Schematic of start-up test for a medium-duty truck with a 4-cylinder gasoline engine and 6-speed manual transmission.

Li et al / SAE Int. J. Passeng. Cars - Mech. Syst. / Volume 8, Issue 2 (July 2015)

CORRELATION BETWEEN CLUTCH VIBRATION AMPLIFICATION AND GEAR RATTLEBy utilizing the validated 4DOF nonlinear model with the same parameters and measured flywheel input ( in Figure 4(a)), the correlation between the transient vibration amplification at the multi-staged clutch damper and the gear rattle in the transmission are

found. As shown in Figure 4(b), (d), while flywheel speed rapidly

increases up to 1600 rpm within the first 1.0 s, exhibits significant vibration between 0.3 and 0.5 s, which induces vibro-impacts at the transitions between clutch stages I and II on both driving and coast sides. Over the same duration, and exhibit severe transient gear rattle events (single- and double-sided impacts).

Figure 3. Experimental validation of the 4DOF nonlinear model given instantaneous flywheel motion input. (a) from the test; (b) from the 4DOF model; (c) Spectrogram of from the test; (d) Spectrogram of

from 4DOF model.

Figure 4. Time domain responses of the nonlinear 4DOF system given the measured instantaneous input .

Table 1. Modal analysis of the linearized 4DOF powertrain system model.

Table 2. Validation of the 4DOF nonlinear powertrain system model given measured flywheel motion .

Evaluation metrics are proposed next to further describe the correlation observed in Figure 4. First, is utilized, and then two new metrics δ23_rattle (%) and δ34_rattle (%), as originally developed in [26], are used to evaluate the rattle severities of headset and unloaded gear pairs respectively. The stiffness of the pre-damper ( ) is redefined as , where is the reference stiffness (base design), and denotes the variation. Similar trends

between the metrics and δ23_ rattle in Figure 5, as varies from 0 to 2, suggest that these two metrics quantitatively correlate the transients between two powertrain components. Given the fact that it

is difficult to measure the transient rattle events within the transmission, this correlation would be helpful in evaluating the

transient rattle severity based on measured at clutch damper.

Figure 5. Correlation between the transient events at the clutch damper and the transmission, as a function of using two metrics and δ23_rattle.

SIMPLIFIED NONLINEAR SDOF MODEL: DEVELOPMENT AND VALIDATIONGiven the correlation between the clutch damper and the transient gear rattle, the validated nonlinear 4DOF model may be further reduced to allow a focus on the clutch damper characteristics. A definite SDOF torsional model is developed by lumping the transmission elements with the clutch hub. Eqs.(1), (2), (3), (4) are simplified as follows, where

Comparisons between the SDOF nonlinear model and the vehicle test in both time and time-frequency domains (in Figure 6) demonstrate the accuracy and utility of the reduced order nonlinear model. Then, Eq. (7) is further simplified with normalized unity torsional inertia as given below where the effect of drag torque is not considered:

Figure 6. Experimental validation of the SDOF nonlinear model given instantaneous flywheel motion input. (a) from the test; (b) from the SDOF model; (c) Spectrogram of from the test; (d) Spectrogram of

from SDOF model.

Here, is the natural frequency of the clutch mode with stage II activated, and ζ is the associated damping ratio. An equivalent engine torque (corresponding to the flywheel motion input) is applied, and it is assumed to consist of a mean part and an alternating part [27-28] which may be expressed as:

(9 a, b, c)

Here, is assumed to be time-invariant, and a single order firing frequency in linearly varies with a constant rate ( ) with a constant phase though it is assumed to be zero without losing any generality. Run-up and run-down processes are represented by and , respectively. The assumption of a constant is based on the constant slope of the firing frequencies over the interested start-up window between 0 s and 1.5 s as shown in Figure 4(a). Also, and

determine the amplitudes of and , respectively. The

nonlinear torque through the symmetric clutch damper is simplified from the asymmetric expressions and the details are discussed in [26]. For further studies, the dimensionless form of Eq. (8) is found by defining the following:

And the dimensionless governing equation is given as:

Here, T1(θ(t)) is the torque through the piecewise linear damper that has two symmetric stages of stiffness (ηω1

2 and ω12), and η represents

the ratio of the stiffness between pre-damper and main spring, and 0 ≤ η ≤ 1 is of interest due to the nature of the pre-damper. The T2(θ(t)) term is the symmetric preload at the bounds of two stages (±φ), and

describes symmetric hysteresis caused by the main spring.

ILLUSTRATIVE DESIGN STUDIESFor the transient envelop E(t) of , two metrics are of particularly interest: the maximum amplification Emax and the peak frequency ΩP at which Emax is reached; these are discussed in [26, 29],. The parametric studies using Eq. (11) suggest that Emax is mainly determined by the hysteresis (Th) among the nonlinearities in the multi-staged clutch damper; in addition, the stiffness ratio η imposes a significant effect on Ωp. These trends are consistent with α up to 0.03 for both run-up and run-down processes. In order to approximate the transient amplification severity (Emax and ΩP) associated with these nonlinearities, some practical approximation formulas were proposed in [26, 29] based on the prior expressions for a linear system which is reported in [22]. One of them is given as follows, where Ta = ω1

By using the following parameters: η = 1, Tp = 0, γa = 0.2 γm = ϕη, ζ = 0.03, the amplification level Emax is approximated via using Eq. (15) and the approximation is compared with numerical prediction. As shown in Figure (7), Eq. (15) is able to approximate the transient amplification severity with a reasonable accuracy.

CONCLUSIONThis paper has examined the transient vibration amplification in a nonlinear powertrain system. The role of a multi-staged clutch damper is examined via numerical and experimental methods. Specific contributions are as follows. First, a new 4DOF nonlinear model is successfully developed and experimentally validated for a transient event during the engine start-up. Second, with the focus on the multi-staged clutch damper, a new SDOF nonlinear model is further developed and validated. In addition, the utility of the validated SDOF nonlinear model is discussed via illustrative design studies.

Figure 7. Verification of the proposed Emax formula. (a-b) are for run-up process and (c-d) are for the run-down process; (a, c) are for Th = 0.05; (b, d) are for Th = 0.1.Key: , proposed formula; , numerical prediction.

Figure 7. (cont.) Verification of the proposed Emax formula. (a-b) are for run-up process and (c-d) are for the run-down process; (a, c) are for Th = 0.05; (b, d) are for Th = 0.1.Key: , proposed formula; , numerical prediction.

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Integrated Powertrain and Suspension System,” 1997 European ADAMS User Conferences, Marburg, Germany, 1997.

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3. Sugimura, H., Takeda, M., Takei, M., Yamaoka, H. et al., “Development of HEV Engine Start-Shock Prediction Technique Combining Motor Generator System Control and Multi-Body Dynamics (MBD) Models,” SAE Int. J. Passeng. Cars - Mech. Syst. 6(2):1363-1370, 2013, doi:10.4271/2013-01-2007.

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CONTACT INFORMATIONProfessor Rajendra SinghAcoustics & Dynamics LaboratoryNSF I/UCRC Smart Vehicle Concepts CenterDept. of Mechanical and Aerospace EngineeringThe Ohio State Universitysingh.3@osu.eduPhone: 614-292-9044www.AutoNVH.orghttp://svc.engineering.osu.edu/

ACKNOWLEDGMENTSThe authors acknowledge the Eaton Corporation's Clutch Division for providing support for applied research. Individual contributions from L. Pereira, B. Franke, P. Kulkarni, J. Dreyer, and M. Krak are gratefully appreciated. The Smart Vehicle Concepts Center (www.SmartVehicleCenter.org) and the National Science Foundation Industry/University Cooperative Research Centers program (www.nsf.gov/eng/iip/iucrc) are acknowledged for providing partial support for basic work.

DEFINITIONS/ABBREVIATIONSC - viscous damping coefficient

E - transient envelope

H - hysteresis

F - gear mesh force

I - torsional inertia

K - torsional/translation stiffness

T - torque

t - time

α - acceleration rate

γ - excitation amplitude

δ - relative gear mesh displacement

ζ - damping ratio

η - piecewise linear stiffness ratio

θ, , - angular displacement, velocity, and acceleration

φ - angular travel of clutch damper

ϕ - angular phase

ω - natural frequency

Ω - instantaneous excitation frequency

SUBSCRIPTS1, 2, … - indices used to denote torsional elements

I, II, III … - stages of clutch damper

a - alternating part

h - hysteresis element

m - mean part

p - preload

pp - peak to peak value

max - maximum value

SUPERSCRIPTS. - first derivative with respect to time

.. - second derivative with respect to time

- - dimensionless value

ABBREVIATIONS4DOF - four-degree-of-freedom

MDOF - multi-degree-of-freedom

SDOF - single-degree-of-freedom

STFT - short-time Fourier transform

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Start-Up Transient Vibration Analysis of a Vehicle Powertrain … · 2019-12-18 · cylinders,...

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