Analysis of start-up transient for a powertrain system with a nonlinear clutch damper

Mechanical Systems and Signal Processing 62-63 (2015) 460–479

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Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/ymssp

Analysis of start-up transient for a powertrain system with a nonlinear clutch damper Laihang Li, Rajendra Singh n Acoustics and Dynamics Laboratory, Ssmart Vehicle Concept Center, Department of Mechanical and Aerospace Engineering, The Ohio State University, Columbus, OH 43210, USA

a r t i c l e in f o

abstract

Article history: Received 30 December 2013 Received in revised form 23 February 2015 Accepted 3 March 2015 Available online 2 April 2015

The transient vibration phenomenon in a vehicle powertrain system during the start-up (or shut-down) process is studied with a focus on the nonlinear characteristics of a multistaged clutch damper. First, a four-degree-of-freedom torsional model with multiple discontinuous nonlinearities under flywheel motion input is developed, and the powertrain transient event is validated with a vehicle start-up experiment. Second, the role of the nonlinear torsional path on the transient event is investigated in the time and time
frequency domains; interactions between the clutch damper and the transmission transients are estimated by using two metrics. Third, the harmonic balance method is applied to examine the nonlinear characteristics of clutch damper during a slowly varying non-stationary process in a simplified and validated single-degree-of-freedom powertrain system model. Finally, analytical formulas are successfully developed and verified to approximate the nonlinear amplification level for a rapidly varying process. & 2015 Elsevier Ltd. All rights reserved.

Keywords: Torsional systems Transient vibration amplification Time
frequency domain methods Non-stationary process Harmonic balance method

1. Introduction The engine start-up (or shut-down) process is receiving attention recently for both conventional internal combustion (IC) engines [1–4] and hybrid electric vehicles [5–9]. Transient torsional vibration phenomena are more noticeable for reasons related to the enhancement of fuel consumption efficiency. First, downsizing of the IC engine, including fewer cylinders and lightweighting, has led to higher torque fluctuations. Second, the IC engine in some modern vehicles frequently starts up and shuts down [5–9]. The engine instantaneous firing frequency passes through the natural frequencies of the powertrain torsional modes, creating a vibration amplification and thus human discomfort. Such instantaneous frequency driven problems have been studied primarily for a linear single-degree-of-freedom (SDOF) mechanical system [10–14]. However, the transient vibration in a multidegree-of-freedom powertrain system is yet be analyzed because of multiple discontinuous nonlinearities in the clutch damper and the transmission [15,16]. Therefore, this article intends to develop powertrain system models and examine the non-stationary process using numerical, experimental, and semi-analytical methods. The effect of a multi-staged clutch damper on the stationary periodic vibro-impacts processes (such as gear rattle) has been studied [16–20]. In addition, the discontinuous nonlinearities, such as piecewise linear stiffness, hysteresis, and preload, have been studied in the frequency and time domains under harmonic excitation with a time-invariant excitation frequency [21–27]. For the non-stationary process with an instantaneous excitation frequency, the transient envelope of the

n

Corresponding author. Tel.: þ1 614 292 9044. E-mail address: [email protected] (R. Singh).

http://dx.doi.org/10.1016/j.ymssp.2015.03.001 0888-3270/& 2015 Elsevier Ltd. All rights reserved.

L. Li, R. Singh / Mechanical Systems and Signal Processing 62-63 (2015) 460–479

461

vibratory response is normally used to evaluate the severity of the transient amplification. For instance, Sen et al. [13,14] utilized the Hilbert transform [28] to find the transient envelope of a nonlinear SDOF system with a clearance; Markert and Seidler [29] and Hok [30] analytically found the transient envelope in the closed form of a linear SDOF system. However, the effects of the discontinuous nonlinearities on the transient envelope have not been adequately studied in depth.

2. Problem formulation Based on the void seen in previous studies, a simplified powertrain system of a conventional vehicle with a clutch damper and manual transmission is considered as shown in Fig. 1(a). A four-degree-of-freedom (4DOF) nonlinear torsional model with instantaneous motion input θ0 ðtÞ; θ_ 0 ðtÞ will be developed where a bar over a symbol indicates a dimensional parameter. Specific objectives are as follows: 1. Develop a new nonlinear 4DOF powertrain model excited by the flywheel motion
and
  validate it by using a vehicle start-up experiment; 2. Estimate the role of the multi-staged clutch damper on θ01 t (where θ01 t ¼ θ0 ðtÞ
θ1 ðtÞ is the relative torsional displacement between the flywheel and the clutch hub) and study its interaction with transient gear rattles in the

Fig. 1. Example case. (a) 4DOF nonlinear powertrain model given motion input from the flywheel; (b) Asymmetric nonlinear characteristics of a multistaged clutch damper.

462

L. Li, R. Singh / Mechanical Systems and Signal Processing 62-63 (2015) 460–479

transmission (given by δ_ 23 ðtÞ and δ_ 34 ðtÞ) during the engine start-up process; 3. Examine the nonlinear characteristics of the multistaged clutch damper and estimate the transient envelope by using the harmonic balance method (HBM) and approximation formulas. 3. Nonlinear model The proposed 4DOF nonlinear model is developed under the assumption that the flywheel motion should not be affected by the motion of the clutch hub and other elements due to its massive inertia, and therefore it can be used as an input to the system. The downstream driveline components including the propeller shaft, axles, and the differential are decoupled from the transmission during the start-up process. Here, I i , C ij , and K ij indicate the torsional inertia, viscous damping, and stiffness of the lumped element, respectively. Two gear backlashes with the backlash size b (mm) between the headset gear pair and the unloaded gear pair are present. An asymmetric multi-staged clutch damper is located between the flywheel and the clutch hub; its characteristic curve is shown in Fig. 1(b) where T 01 is the torque transmitted through the clutch damper. The nonlinear function of T 01 ðθ01 Þ is given as below, where uðxÞ is a unit step function which is normally replaced by ðtanhðσ xÞ þ 1Þ=2, and sgnðxÞ is a sign function which is replaced by tanhðσ xÞ; σ (say of the order of 106–109) is a regularizing factor that is used to smoothen the discontinuities to facilitate numerical calculation [31]:       T 01 θ01 ¼ T s θ01 þ T H θ01 ; θ_ 01 ; ð1aÞ             T s θ01 ¼ T 1 θ01 þT 2 θ01 þT 3 θ01 þT 4 θ01 þT 5 θ01 ;

ð1bÞ

    T 1 θ01 ¼ K III θ01 þ φII_c þ φI_c
K I φI_c
K II_c φII_c
T p ;

ð1cÞ

    n
o  T 2 θ01 ¼ u θ01 þ φII_c þ φI_c K II_c
K III θ01 þ φII_c þ φI_c ;

ð1dÞ

   n
 o  T 3 θ01 ¼ u θ01 þ φI_c K I
K II_c θ01 þ φI_c þ T p ;

ð1eÞ

    n
 o  T 4 θ01 ¼ u θ01
φI_d K II_d
K I θ01
φI_d þ T p ;

ð1fÞ

    n
o  T 5 θ01 ¼ u θ01
φI_d
φII_d K III
K II_d θ01
φI_d
φII_d ;

ð1gÞ

TH



 θ01 ; θ_ 01 ¼ 



( )  H
H  H
H   H II_c I I II_c II_d þ u θ01 þ φI_c þ u θ01
φI_d sgn θ_ 01 : 2 2 2

ð1hÞ

Here, T s θ01 represents the torque from the multi-staged and asymmetric springs with a preload; the stiffness K I is in the pre-damper stage I, K II_c and K II_d correspond to the main spring (in stage II), and K III refers to the stopper (in stage III); preload T p is between stages I and II, and  subscripts  c and d imply coast and driving sides, respectively; also, the asymmetric length of each stage is given in φ; T H θ10 ; θ_ 10 describes the multi-staged and asymmetric hysteresis caused by the dry friction between the spring and clutch shell [15,16]. The governing equations under the flywheel motion input θ0 ðtÞ; θ_ 0 ðtÞ are given as follows: 
  
 






 I 1 θ€ 1 t þ C 01 θ_ 1 t
θ_ 0 t
T 01 θ01 t ; θ_ 01 t þC 12 θ_ 1 t
θ_ 2 t þ K 12 θ1 t
θ2 t ¼ 0; ð2Þ 
 
 
 



 I 2 θ€ 2 t þ C 12 θ_ 2 t
θ_ 1 t þ K 12 θ2 t
θ1 t þ F head δ23 t ; b r h_i þ C 23 δ_ 23 t r h_i ¼ T D3 ;

ð3Þ


  
 


 I 3 θ€ 3 t þ F head δ23 t ; b r h_o þC 23 δ_ 23 t r h_o þ F unload δ34 t ; b r u_i þC 34 δ_ 34 t r u_i ¼ T D2 ;

ð4Þ


 

 I 4 θ€ 4 t þ F unload δ34 t ; b r u_o þ C 34 δ_ 34 t r u_o ¼ T D1 :

ð5Þ

Here, T Di (i¼1, 2, and 3) are the drag torques 
on thegears and shafts 
inthetransmission which are caused by the lubricant and bearings and assumed to be constants; F head δ23 t ; b and F unload δ34 t ; b represent the nonlinear gear mesh forces from the two gear pairs with backlashes. The nonlinear functions are given below, where K g ¼ K 23 ¼ K 34 is the translational gear mesh stiffness, b ¼ 0:1 mm is the net backlash (7 0:5b), r h_i , r h_o , r u_i , r h_i are the gear radii for the headset gear pair and the unloaded





 gear pair, respectively, and δ23 t ¼ θ2 t r h_i þ θ3 t r h_o , δ34 t ¼ θ3 t r u_i þ θ4 t r u_o : 
 
  
 
 1 0 
 
 sgn δ23 t
ðb=2Þ δ23 t
ðb=2Þ
sgn δ23 t þðb=2Þ δ23 t þðb=2Þ A; ð6Þ F head δ23 t ; b ¼ K g @δ23 t þ 2

L. Li, R. Singh / Mechanical Systems and Signal Processing 62-63 (2015) 460–479

463


  F unload δ34 t ; b 0


 
  
 
 1 sgn δ34 t
ðb=2Þ δ34 t
ðb=2Þ
sgn δ34 t þ ðb=2Þ δ34 t þ ðb=2Þ A: ¼ K g @δ34 ðt Þ þ 2

ð7Þ

4. Experimental validation of 4DOF nonlinear model A vehicle start-up experiment is conducted on a medium-duty pickup truck with a 4-cylinder gasoline engine and a 6-speed manual transmission to validate the nonlinear model. The schematic of the test rig is given in Fig. 2; note that other downstream driveline components including the propeller shaft, axles, and the differential are not shown as they are decoupled during the

 engine start-up. The absolute torsional velocities of the flywheel (θ_ t ) and clutch hub (θ_ t ) are measured by two magnetic 0

1

sensors, and the key system parameters are estimated from the physical dimensions or vehicle measurements [16,19]. First, the nonlinear model is linearized by assuming that b ¼ 0 and only one stage of the clutch damper is activated. The eigensolutions are
 calculated as reported in Table 1. Second, the measured flywheel motion (θ_ 0 t ) is applied to the nonlinear 4DOF model as an


 input. Third, the relative velocity through the clutch damper (θ_ 01 t ¼ θ_ 0 t
θ_ 1 t ) is calculated in both time and time
frequency domains. The comparisons of Fig. 3a and b and Table 2 suggest that the 4DOF nonlinear model yields accurate , θ_ , θ_ , and t . In addition, the time
frequency domain results, by a short-time predictions, especially for θ_ 01_ max

01_ min

01_pp

max

Fourier transform (STFT) algorithm, in Fig. 3c and d indicate that the 4DOF nonlinear model accurately captures first two dominant orders of the instantaneous frequency and the instant (around 0.4 s) when the maximum amplification (θ_ or θ_ ) occurs 01_ max

01_ min

(dashed circle). Also, Fig. 3c and d shows that the clutch mode with stage I stiffness (12 Hz) is excited leading to an amplification. The discrepancies in the time or time
frequency domains may be due to other damping mechanisms in the vehicle experiment which are beyond the scope of this article. 5. Effect of multi-staged clutch damper on amplification The role of the multi-staged clutch damper and its interaction with the transmission are found in both the time and time
frequency domains. Fig. 4b
d shows the transient responses calculated by the validated nonlinear 4DOF model with

Fig. 2. Schematic of start-up experiment for a medium-duty truck with a 4-cylinder gasoline engine and 6-speed manual transmission. Table 1 Eigensolutions of the linearized 4DOF powertrain system model. Mode (Stage I activated)

Mode (Stage IId activated)

Natural frequency (Hz)

1 12

2 446

3 2172

4 2515

1 52

2 460

3 2172

4 2515

Flywheel Clutch hub Headset gear and input shaft Counter shaft and “welded” gears Output gear

Eigenvector 0.00 1.00 1.00 1.00 1.00

0.00 1.00
0.19
0.22
0.23

0.00 0.01
0.19 0.01 1.00

0.00
0.02 0.80
0.33 1.00

Eigenvector 0.00 0.93 1.00 0.93 0.93

0.00 1.00
0.18
0.20
0.20

0.00 0.01
0.20 0.01 1.00

0.00
0.02 0.86
0.37 1.00

464

L. Li, R. Singh / Mechanical Systems and Signal Processing 62-63 (2015) 460–479



 Fig. 3. Experimental validation of the nonlinear powertrain model with an instantaneous flywheel motion input. (a) θ_ 01 t from the experiment; (b) θ_ 01 t

 from the 4DOF model; (c) Spectrogram of θ_ 01 t from the experiment where the magnitude has a reference value of 1.0 rad/s; (d) Spectrogram of θ_ 01 t from the 4DOF model where the magnitude has a reference value of 1.0 rad/s. Table 2


 Validation of the 4DOF nonlinear powertrain system model given measured flywheel motion input θ_ 0 t . Symbol (Units)

Experiment

4DOF nonlinear model

θ_ 01_ max (rpm)

299

298

0.3%


215


232

8.2%

θ_

01_ min (rpm)

Prediction error

θ_ 01_pp (rpm)

513

531

3.3%

t max (s)

0.44

0.44

0%


 the parameters of Section 4 and with the measured flywheel motion input θ_ 0 t that is displayed in Fig. 4 (a). The time domain predictions indicate that θ_ 0 ðtÞ rapidly increases up to 1600 rpm within the first 1.0 s and starts to decrease from 1.5 s
 to the idle speed (in the neutral status). Accordingly, θ01 t exhibits significant vibration between 0.3 and 0.5 s which induces vibro-impacts at the transition between stages I and II on both the driving and the coast sides. Similarly, δ23 ðtÞ and

δ34 ðtÞ show severe transient gear rattle phenomenon (single- and double-sided impacts) over the same duration. Short time

Fourier transform is utilized to conduct the time
frequency analysis for a clear explanation. In Fig. 5 (a), a spectrogram of
 θ€ 0 t shows three dominant firing orders. The acceleration of the instantaneous firing frequency at each order is significantly high before 1.0 s. Observe the following from the spectrograms of θ_ ðtÞ, δ_ ðtÞ and δ_ ðtÞ in Fig. 5b
e: First, 01

23

34

Mode 1 (clutch mode in Table 1) is excited by the instantaneous firing frequencies (at 2nd, 4th, and 6th orders) through the natural frequencies of the clutch modes around 0.5 s (for 12 Hz) and 3.0 s (for 52 Hz) though the 12 Hz mode is more

L. Li, R. Singh / Mechanical Systems and Signal Processing 62-63 (2015) 460–479

465


 Fig. 4. Time domain responses of the nonlinear 4DOF system given the measured transient input θ_ 0 ðt Þ. (a) Measured transient input θ_ 0 t from flywheel;

 (b) Relative torsional displacement θ01 t between flywheel and clutch hub; (c) Relative translational displacement δ23 t between headset input gear and
 counter shaft; (d) Relative translational displacement δ34 t between unloaded gear and counter shaft.

dominant since most of the vibration occurs within stage I (Fig. 5(a)); Second, the resonant amplification within the clutch damper induces significant vibro-impacts near 0.5 s and 3.0 s both within the clutch damper and across the gear backlashes, and these vibro-impacts excite higher modes (say 460 Hz, 2170 Hz, and 2500 Hz). Therefore, the clutch damper has a decisive effect on the transient events. Evaluation metrics are proposed next to correlate the transient vibration within the clutch damper and transmission from the numerical results. Within the start-up time window (say from 0.0 s to 1.5 s), discrete response vectors of dimension n are defined as θ_ ðjÞ, δ ðjÞ, and δ ðjÞ, 1 rj r n given the time vector t ¼ ½t t t :::t . In addition, two new response vectors δ ðkÞ and 01

23

34

1

2

3

n

23_loss

δ34_loss ðlÞ are defined when the gear pairs lose contact over 1 r k r n23_loss r n,1 r l r n34_loss r n respectively; these are

    numerically extracted from δ23 ðjÞ and δ34 ðjÞ. The first metric is the peak to peak value θ_ 01_pp ¼ max θ_ 01 ðjÞ
min θ_ 01 ðjÞ , which can be also experimentally measured (as shown in Fig. 3). Then two new metrics δ23_rattle (%) and δ34_rattle (%) are developed as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 Pn23_loss   δ ð k Þ
ðb=2Þ =n ðn23_loss =nÞ   23_loss 23_loss k¼1

δ23_rattle ¼

δ34_rattle ¼

b

 100%;

ð8Þ

 100%:

ð9Þ

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 Pn34_loss   δ ðlÞ þðb=2Þ =n34_loss ðn34_loss =nÞ l ¼ 1  34_loss b

The stiffness of stage I (K I ) is defined as βI K r , where K r is the reference value of stage I stiffness estimated from the vehicle K K experiment, and βI denotes the variation. Two metrics θ_ 01_pp and δ23_rattle are calculated when βI is varied from 0 to 2 as an example. Similar trends between θ_ 01_pp and δ23_rattle in Fig. 6 suggest that these two metrics successfully correlate the transient vibration events within two components. Given that it is difficult to measure the relative motion in the transmission, this correlation would be helpful in evaluating the transient gear rattle severity based on measured transient amplification (θ_ 01_pp ) in the clutch damper. Further, the nonlinear 4DOF model may be reduced to allow a focus on the clutch damper characteristics. A semi-definite 2DOF torsional model with flywheel motion input is developed by lumping the clutch hub
and the  gears
together,  and eliminating the gear backlash elements. Eqs. (2)–(5) are simplified as follows, where T D ¼ T D3
T D2 r h_i =r h_o þT D1 r h_i =r u_o ,
2
2 I ¼ I 1 þI 2 þI 3 r h_i =r h_o þ I 4 r h_i =r u_o :     I θ€ 1 ðtÞ þ C 01 θ_ 1 ðtÞ
θ_ 0 ðtÞ
T 01 θ01 ðtÞ; θ_ 01 ðtÞ ¼ T D : ð10Þ K

Comparisons between the semi-definite 2DOF (or definite SDOF) nonlinear model and the vehicle start-up experiment in both the time and time
frequency domains in Fig. 7 demonstrate the accuracy and utility of the reduced order nonlinear model.

L. Li, R. Singh / Mechanical Systems and Signal Processing 62-63 (2015) 460–479

Frequency [Hz]

466

6 th order 4 th order 2 nd order

Frequency [Hz]

460 Hz

12 Hz

Frequency [Hz]

Frequency [Hz]

Frequency [Hz]

t [s]

2500 Hz

Dominant firing frequency 52 Hz

2170 Hz

t [s]

t [s]



 Fig. 5. Predicted spectrograms of the nonlinear 4DOF system given measured motion input θ_ 0 t . (a) Spectrogram of θ€ 0 t with a reference value of 1.0 rad/s2;

 (b) Spectrogram of θ_ 01 t with a reference value of 1.0 rad/s; (c) enlarged view of (b); (d) Spectrogram of δ_ 23 t with a reference value of 1.0 m/s;
 (e) Spectrogram of δ_ 34 t with a reference value of 1.0 m/s.

25 δ 23_rattle [% ]

θ 01_pp [rpm]

800 600 400 200 0

1 K βI

2

20 15 10 0

1

2

K βI

Fig. 6. Correlation between the transient events at the clutch damper and the gearbox, as a function of βKI using two metrics θ_ 01_pp and δ23_rattle .

θ01(t) [rpm]

θ01(t) [rpm]

L. Li, R. Singh / Mechanical Systems and Signal Processing 62-63 (2015) 460–479

467

200 0 -200 0

0.5

0

0.5

1

1.5

1

1.5

200 0 -200

12 Hz

Frequency [Hz]

Frequency [Hz]

t [s]

12 Hz

t [s]
 Fig. 7. Experimental validation of the nonlinear SDOF powertrain model with an instantaneous flywheel motion input. (a) θ_ 01 t from the experiment;


 (b) θ_ 01 t from the SDOF model; (c) Spectrogram of θ_ 01 t from the experiment with a reference value of 1.0 rad/s; (d) Spectrogram of θ_ 01 t from the 4DOF model with a reference value of 1.0 rad/s.

Although the start-up operation is of primary interest in this article, the proposed 4DOF and SDOF nonlinear models with instantaneous motion input are also applicable to other stationary and non-stationary vibration issues that have been discussed in the literature [5–9,17–20]; these include engine shut-down and idle gear rattle problems. For such torsional vibration issues, the flywheel speed should be measured and utilized as an excitation as successfully illustrated in this article. 6. Nonlinear characteristics of multi-staged clutch damper Eq. (10) is further simplified from a semi-definite 2DOF system to a definite SDOF system with normalized unity torsional inertia as given below where the effect from the drag torque is not considered: 

 θ€ ðtÞ þ2ζω1 θ_ ðt Þ þ T θ t ; θ_ t ¼ T e ðtÞ: ð11Þ Here, ω1 is the natural frequency of the clutch mode with stage IId activated, and ζ is the associated damping ratio. An

 equivalent torque T
e t (corresponding to the flywheel motion input) is applied, and it consists of a mean part T m t and an alternating part T a t [32–34] which are given as

 Te t ¼ Tm þTa t ; ð12aÞ
 T m t ¼ ω1 2 γ m ;
 T a t ¼ ω21 γ a sin

ð12bÞ 

1 2





Ω 0 þ αt t þ ϕ :

ð12cÞ

468

L. Li, R. Singh / Mechanical Systems and Signal Processing 62-63 (2015) 460–479

Table 3 System parameters for numerical and analytical studies of the multi-staged clutch damper. A. Numerical studies of nonlinear characteristics η

Case

1 0.1 2 1 3 0.1 4 1 5 1 B. Verification of the HBM for a Case

η

Th

Tp

Tp

0.05 0.1 0 0 0 0 0.05 0 0 0.1 slowly varying process

Th

γa

γm

6 0.3 0 0 0.2 ϕη 7 0.3 0 0 0.2 ϕη 8 1 0.05 0 0.2 ϕη 9 1 0 0.1 0.2 ϕη C. Verification of the proposed formulas for Emax and Ωp

γa

γm

0.2 0.2 0.2 0.2 0.2

ϕη ϕη ϕη ϕη ϕη

jα j

ζ

nmax

0.001 0.015 8 0.001 0.075 8 0.001 0.03 8 0.001 0.03 5 for a rapidly varying process

Cases

η

Tp

Th

γa

γm

ζ

10 11 12 13

1 1 0.1, 0.3, 0.5, 1.0 0.1, 0.3, 0.5, 1.0

0 0, 0.025, 0.05 0 0

0, 0.05, 0.1 0 0 0

0.2 0.2 0.2 0.2

ϕη ϕη ϕη ϕη

0.03 0.03 0.015 0.075



 Here, T m t is assumed to be time-invariant, and a single order instantaneous excitation frequency in T a t is assumed to linearly vary with a constant rate (α) with a time-invariant phase ϕ though it is assumed to be zero without losing any generality. For the run-up, α 40 and for the run-down, α o0. The assumption of a constant α is based on the constant slope of the
 instantaneous firing frequency over the focused region as given in the STFT result of θ€ 0 t in Fig. 5 (a). Also, γ m and γ a determine

 the amplitudes of T m t and T a t , respectively. The nonlinear torque transmitted through a symmetric clutch damper 

 T θ t ; θ_ t is simplified from the asymmetric expressions of Eq. (1) as follows, where ϕ ¼ ϕ ¼ ϕ , T ¼ H ¼H ¼H : 
 
 
 

 T θ t ; θ_ t ¼ T 1 θ t þ T 2 θ t þT 3 θ_ t ;

c

d

h





  
  
  
 i
 1
η h
 T 1 θ t ¼ ω21 θ t þ θ t
φ sgn θ t
φ
θ t þ φ sgn θ t þ φ ; 2 
  
  
 sgn θ t
φ þsgn θ t þ φ ; T2 θ t ¼ Tp 2

II_d

II_c

I

ð13Þ

ð14Þ

ð15aÞ


 T 
 T 3 θ_ t ¼ h sgn θ_ t : ð15bÞ 2 
 Here, T 1 θ t represents the torque from the piecewise linear damper which has two symmetric stages of stiffness (ηω21 and ω21 ); here, ηω21 represents the stiffness of pre-damper with a compliant spring, and ω21 is the main stiffness (for

η is normally less than 1 as the pre-damper is relatively more 
 compliant than the main spring, 0 r η r1 is of interest. The T 2 θ t term represents the symmetric preload at 7 φ, and 
 term describes one stage and symmetric hysteresis. The dimensionless form of Eq. (11)
(15) are found by the T 3 θ_ t transmitting the torque to the driveline) [15–16]. Since

defining the following:

θ φ

t ¼ ω1 t; θ ¼ ; α ¼

Tp α Ω0 γ γ T ω1 ; Ω0 ¼ ; γ ¼ a; γ ¼ m; T p ¼ ; T h ¼ h 2 ; ω1 ¼ ¼ 1; ϕ ¼ ϕ; ω1 a φ m φ ω1 ω1 2 φω21 φω1

φ φ ¼ ¼ 1; φ


θ t ¼ θðt Þ ¼ θðt Þφ;

ð16a–kÞ


θ_ t ¼

dθ dt _ ¼ θðt Þω1 ¼ θ_ ðt Þφω1 ; dt dt

ð17a; bÞ

L. Li, R. Singh / Mechanical Systems and Signal Processing 62-63 (2015) 460–479




θ€ t ¼

469

2  2 d θ dt ¼ θ€ ðt Þω21 ¼ θ€ ðt Þφω21 ; dt 2 dt



ð18Þ 

θ€ ðt Þφω21 þ 2ζω1 θ_ ðt Þφω1 þT θðt Þ; θ_ ðt Þ ¼ ω21 γ m þ ω21 γ a sin 



θ€ ðt Þ þ2ζ θ_ ðt Þ þ T θðt Þ; θ_ ðt Þ ¼ T e ðt Þ ¼ γ m þ γ a sin





1 2

1 2





Ω 0 þ αt t þ ϕ ;



ð19Þ



Ω0 þ αt t þ ϕ ;

ð20Þ

   

 T θðt Þ; θ_ ðt Þ ¼ T 1 θðt Þ þ T 2 θðt Þ þ T 3 θ_ ðt Þ ;

ð21Þ





 


 1
η
θðt Þ
φ sgn θðt Þ
φ
θðt Þ þ φ sgn θðt Þ þ φ ; T 1 θðt Þ ¼ θðt Þ þ 2

ð22Þ




 sgn θðt Þ
φ þsgn θðt Þ þ φ ; T 2 θðt Þ ¼ T p 2

ð23aÞ

  T   T 3 θ_ ðt Þ ¼ h sgn θ_ ðt Þ : 2

ð23bÞ

7. Numerical examination of nonlinear characteristics Eq. (20) is solved by using a numerical integration method (MATLAB [35]), and the transient envelope E(t) is then estimated via Hilbert transform [28]. For the envelope study, two metrics are of particular interest: the maximum amplification Emax and the peak frequency Ωp at which Emax is reached [3]. Note that the operating point θm is assumed to be γ m =η ¼ ϕ (transition point on the driving side) to activate all nonlinearities. The comparison between nonlinear (Case 1 of Table 3A) and linear (Case 2 6 5 4

3

E (t)

E (t)

4

2

2 1 0 0.5

1

0

1.5

0

-5

1

1.5

5

θ (t)

θ (t)

5

0.5

-4

-2

0

2

4

6

0

-5 -4

-2

0

2

4

6

Fig. 8. Comparison of transient envelopes E (t) and phase planes between nonlinear and linear SDOF systems; refer to Table 3A for Cases 1 and 2. Figures (a) and (b) are for the run-up process and (c) and (d) are for the run-down process where Ω0 þ αt is the instantaneous excitation frequency. Key: , nonlinear SDOF system; , linear SDOF system.

ω

ω ω

L. Li, R. Singh / Mechanical Systems and Signal Processing 62-63 (2015) 460–479

ω

470

Fig. 9. Comparison of spectrograms of θ(t) between nonlinear and linear SDOF systems with a reference value of 1.0; refer to Table 3A for Cases 1 and 2. Figures (a and b) are for the run-up process and (c and d) are for the run-down process; (a, c) are for a nonlinear system with jαj ¼ 0:001; ζ ¼ 0:015; (b, d) are for a linear system. Here, ω is the instantaneous frequency of response θ (t) and Ω0 þ αt is the instantaneous excitation frequency.

Fig. 10. Spectrograms of θ (t) of nonlinear SDOF system with a reference value of 1.0; refer to Table 3A for Case 1. Figures (a and b) are for the run-up process and (c and d) are for the run-down process; (a, c) are with jαj ¼ 0:005; ζ ¼ 0:015; (b, d) are with jaj ¼ 0:001; ζ ¼ 0:075. Here, ω is the instantaneous frequency of response θ (t) and Ω0 þ αt is the instantaneous excitation frequency.

L. Li, R. Singh / Mechanical Systems and Signal Processing 62-63 (2015) 460–479

471

of Table 3A) systems in Fig. 8 where Ω0 þ αt is the instantaneous excitation frequency (speed) indicates that the nonlinear case has direction dependent transient envelops Eðt Þ with respect to ω1 ¼ 1 (especially for Emax and Ωp ) for both run-up and run-down processes. In addition, the STFT results of θðt Þ for Cases 1 and 2 in Fig. 9 suggest that the linear system (Fig. 9b and d) only has the primary order (Ω0 þ αt), and the horizontal line at 1 indicates the natural frequency (ω1) of the linear system.  Conversely,  multiple orders qðΩ0 þ αt Þ exist in θðt Þ for the nonlinear case, where q is a real and positive constant. Therefore, T θðt Þ; θ_ ðt Þ introduces both the super-harmonic (q41) and sub-harmonic (qo1) responses though the q¼1 and q41 terms are dominant. The effect of α and ζ on the nonlinear responses are examined for Case 1 as an example. The STFT results of θðt Þ in Figs. 9 and 10 clearly show that the nonlinear transient responses are more sensitive to a change of α than ζ, and a higher α suppresses the formation of super- and sub-harmonic components. In addition, the contribution of each nonlinearity on the multiple harmonic response components is also numerically investigated by three cases with jαj ¼ 0:001; ζ ¼ 0:015. As shown in Fig. 11, only primary and minor super-harmonic components are found for preload and hysteresis cases; and piecewise linear stiffness curve generates most of the super- and sub-harmonic components.

Fig. 11. Spectrograms of θ (t) of nonlinear SDOF system with jαj ¼ 0:001; ζ ¼ 0:015 with a reference value of 1.0. Figures (a
c) are for the run-up process and (d
f) are for the run-down process; (a, d) are for Case 3; (b, e) are for Case 4; (c, f) are for Case 5; refer to Table 3A for Cases 3
5. Here, ω is the instantaneous frequency of response θ (t) and Ω0 þ αt is the instantaneous excitation frequency.

472

L. Li, R. Singh / Mechanical Systems and Signal Processing 62-63 (2015) 460–479

8. Semi-analytical solutions for a slowly varying process The general nonlinear characteristics of the clutch damper and its specific effect on engine start-up are analytically examined for slowly and rapidly varying processes, respectively. Since the nonlinear characteristics are more dominant for smaller values of α (say jαj ¼ 0:001), a slowly varying process is considered first to analytically estimate the transient envelope E(t). Given the extremely slow rate α, the nonlinear response θðt Þ is assumed to be quasi-periodic which indicates that the harmonic balance method (HBM) can be utilized for such a quasi-stationary or slowly varying process. First, a new variable is introduced as

τ ¼ αt:

ð24Þ

Therefore, the governing equation is written as follows:    1 θ€ ðt Þ þ2ζ θ_ ðt Þ þT θðt Þ; θ_ ðt Þ ¼ γ m þ γ a sin Ω0 þ τ t : 2

ð25Þ

Since primary and super-harmonic components are more dominant than sub-harmonic components, θðt Þ is assumed as follows where θan ðτÞ is the amplitude of the nth harmonic given τ and n is a positive integer to describe primary (n ¼1) and super harmonic (n 41) components:

θðt Þ ¼ θm þ

n max X

 

θan ðτÞ sin n Ω0 þ

n¼1

τ 2

 t þψn :

ð26Þ

The strong form residual is defined as below at a given τ:   r ðt Þ ¼ T e ðt Þ
θ€ ðt Þ
2ζ θ_ ðt Þ
T θðt Þ; θ_ ðt Þ -0 8 t:

ð27Þ

Since the system excitation T e ðt Þ and the nonlinear response θðt Þ are assumed to be quasi-periodic due to a slowly varying τ, the frequency domain analysis is conducted by applying the Fourier transform on both sides of strong form residual as follows:        ℑðr ðt ÞÞ ¼ ℑðT e ðt ÞÞ
ℑ θ€ ðt Þ
2ζ ℑ θ_ ðt Þ
ℑ T θðt Þ; θ_ ðt Þ -0 8 t: ð28Þ
 Then, the residual is minimized in the Newton-Raphson form as below where Θ ðτÞ ¼ ℑ θðt Þ is the Fourier component vector of θðt Þ at a given τ: R ðτ Þ ¼ R 0 ðτ Þ þ

∂R ðτÞ ΔΘ ðτÞ: ∂Θ ðτÞ

ð29Þ



1 Here, R 0 ðτÞ is the residual from the initial guess, and ΔΘ ðτÞ ¼
∂R ðτÞ=∂Θ ðτÞ R ðτÞ is the correction factor. The Newton–Raphson method is employed to find Θ ðτÞ during an iterative process [25–26]. The Fourier transformation of the quasi-periodic θðt Þ is expressed as follows, where is the discrete Fourier transform matrix:
 ℑ θ ðt Þ 8 9 2 1 θðt 0 Þ > > > > > > > > > > 6 1 > < θðt 1 Þ > = 6 6 ⋮ ⋮ ¼6 6 > > > > 61 > > θðt N
2 Þ > > 4 > > > > : 1 θðt N
1 Þ ;



  sin Ω0 þ 2τ t 0

  τ sin Ω0 þ 2 t 1



  cos Ω0 þ 2τ t 0

  τ cos Ω0 þ 2 t 1

⋮

  sin Ω0 þ 2τ t N
2

  sin Ω0 þ 2τ t N
1



  sin 2 Ω0 þ 2τ t 0

  τ sin 2 Ω0 þ 2 t 1



  cos Ω0 þ 2τ t N
2

  cos Ω0 þ 2τ t N
1



  cos 2 Ω0 þ 2τ t 0

  τ cos 2 Ω0 þ 2 t 1

⋮

  sin 2 Ω0 þ 2τ t N
2

  sin 2 Ω0 þ 2τ t N
1

9 38 ⋯ > Θ0 ðτÞ > > > > > > > 7 > > ⋯ 7> Θ1 ðτÞ > < = 7 7 ⋮ ⋮ ⋮ ⋮ ¼ Ξ Θ ðτÞ 7>

  > > 7> > cos 2 Ω0 þ 2τ t N
2 ⋯ > Θ2nmax ðτÞ > > 5 > >

  > > :Θ ; cos 2 Ω0 þ 2τ t N
1 ⋯ 2nmax þ 1 ðτÞ

ð30Þ  T is quasi-periodic as well, Since θðt Þ is assumed to be quasi-periodic at a given τ, the vector θt ¼ θðt 0 Þ θðt 1 Þ:::θðt N
1 Þ where T indicates the transpose of a vector. The symbol N represents the number of sampling points of θðt Þ within one period. In addition, the column vector Θ ðτÞ contains the Fourier amplitude of each component which is calculated by the

  Discrete Fourier Transform (DFT) algorithm. A new dimensionless time variable
is introduced as ϑ ¼ Ω0 þ τ=2 t and

 0 _ € _ ϑ A ½0; 2π Þ. Then, θðt Þ is represented as θðt Þ ¼ d
θ=d ϑ dϑ=dt ¼ ðΩ þ τÞθ ϑ . Similarly, θðt Þ is described as
 

0 θ€ ðt Þ ¼ ðΩ0 þ τÞ2 θ″ ϑ . Thus the vector presentations of θ ϑ , θ0 ϑ , and θ″ ϑ within one period are given as below: 

θϑ ¼ Ξ Θ ðτÞ; θ1ϑ ¼ Ξ Λ Θ ðτÞ; θ″ϑ ¼ Ξ Γ Θ ðτÞ; 2

1

61 6 6 ⋮ Ξ ¼6 6 61 4 1

ð31Þ


 sin ϑ0
 sin ϑ1


 cos ϑ0
 cos ϑ1


 sin 2ϑ0
 sin 2ϑ1


 cos 2ϑ0
 cos 2ϑ1

⋮

⋮

⋮

⋮


 sin ϑN
2
 sin ϑN
1


 cos ϑN
2
 cos ϑN
1


 sin 2ϑN
2
 sin 2ϑN
1


 cos 2ϑN
2
 cos 2ϑN
1

⋯

3

⋯7 7 7 ⋮ 7: 7 ⋯7 5 ⋯

ð32Þ

L. Li, R. Singh / Mechanical Systems and Signal Processing 62-63 (2015) 460–479

2 6 6 6 6 6 6 6 Λ ¼6 6 6 6 6 6 4

0



0


1

1

0

3



0


2

2

0



⋱ "

0
nmax

2 6 6 6 6 6 6 6 Γ ¼
6 6 6 6 6 6 6 4

0 "

1

2

1

2

473

7 7 7 7 7 7 7 7; 7 7 #7 7
nmax 7 5 0

ð33aÞ

3

#

7 7 7 7 7 " # 7 7 22 7: 2 7 2 7 7 ⋱ " 7 # 7 2 nmax 7 5 n2max

ð33bÞ

Therefore, the vector presentations of θðt Þ, θ_ ðt Þ, and θ€ ðt Þ are given as follows:

θt ¼ Ξ Θ ðτÞ; θ_ t ¼ ðΩ0 þ τÞΞ Λ Θ ðτÞ; θ€ t ¼ ðΩ0 þ τÞ2 Ξ Γ Θ ðτÞ:

ð34Þ

Regarding the system excitation, the Fourier transform is described as follows where T e is the vector form of T e ðt Þ within one period: ℑðT e ðt ÞÞ-T e ¼ Ξ E:

ð35Þ

The column vector E represents the excitation torque amplitude vector. Since only one mean part and one harmonic T component are considered,  E ¼ γ m γ a 0 0 0⋯ . In addition, the column vector Χ ðτÞ, the Fourier components of the nonlinear term T θðt Þ; θ_ ðt Þ , is defined as T ¼ Ξ Χ ðτÞ:

ð36Þ

Then, the residual in the time domain is described in the discrete form by using the Fourier series expansion as below þ where Ξ is the pseudo-inverse of Ξ :

Ξ R ðτÞ ¼ Ξ E
ðΩ0 þ τÞ2 Ξ Γ Θ ðτÞ
2ζ ðΩ0 þ τÞΞ Λ Θ ðτÞ
Ξ Χ ðτÞ-0;

ð37Þ

R ðτÞ ¼ E
ðΩ0 þ τÞ Γ Θ ðτÞ
2ζ ðΩ0 þ τÞΛ Θ ðτÞ
Ξ þ T -0:

ð38Þ

2

In order to utilize the Newton
Raphson algorithm, the Jacobian matrix is calculated as below:   ∂ Ξ þT ∂R ðτÞ 2 ¼
ðΩ0 þ τÞ Γ
2ζ ðΩ0 þ τÞΛ
: J ðτ Þ ¼ ∂Θ ðτÞ ∂ Θ ðτ Þ     þ The term ∂ Ξ T =∂Θ ðτÞ is calculated as follows:   þ ∂ Ξ T ∂T ∂T ∂θt ∂T ¼Ξ þ ¼Ξ þ ¼Ξ þ Ξ ; ∂ Θ ðτ Þ ∂Θ ðτÞ ∂θt ∂Θ ðτÞ ∂θt  ∂T ∂T ¼ diag ðt 0 Þ ∂θt ∂θt

∂T ðt 1 Þ ∂θ t

⋯

∂T ðt N
2 Þ ∂θt

∂T ðt N
1 Þ ∂θ t

:

ð39Þ

ð40Þ

ð41Þ

The next prediction (for iteration kþ 1) is adjusted by the following:

Θ ðτ Þk þ 1 ¼ Θ ðτ Þk


R ðτÞk ¼ Θ ðτÞk þ ΔΘ ðτÞk : J ðτ Þk

ð42Þ

This iteration process stops when the error criterion is satisfied, as defined below, where ε is an extremely small value:   R ðτÞ   ð43Þ   o ε:  J ðτ Þ 

474

L. Li, R. Singh / Mechanical Systems and Signal Processing 62-63 (2015) 460–479

Therefore, at given τ ¼ αt, column vector Θ ðτÞ is found as follows:

Θ ðτÞ ¼ Θ ðαt Þ ¼ Θ ðΩ0 þ αt Þ:

ð44Þ

This equation implies that for a slowly varying excitation frequency Ω0 þ αt, the amplitude of each harmonic (Θ ) from the nonlinear response θðt Þ is successfully found. Further, the transient envelope Eðt Þ in the time domain is calculated as follows where Θn ðαt Þ is the nth component of the column vector Θ ðαt Þ, and Θ0 ðαt Þ is the mean (DC) part of θðt Þ: vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u2nmax þ 1 u X Eðt Þ ¼ t Θn 2 ðαt Þ þ Θ0 ðαt Þ: ð45Þ n¼1

The proposed HBM for jαj ¼ 0:001 is utilized to examine the transient envelope of each nonlinearity by using the parameters for Cases 6
9 of Table 3B. Note that a higher value of ζ is needed to ensure the numerical convergence. Fig. 12 compares E(t) estimated by the HBM and the Hilbert transform method, and the results indicate that piecewise linear stiffness curve introduces a direction dependent envelope, and the HBM accurately captures primary and super harmonic amplifications from these nonlinearities. 9. Approximate formulas for a rapidly varying process

8

2.5

6

2

E(t)

E(t)

A rapidly varying process with a much higher jαj (say up to 0.03) is considered next since this rate is closer to the actual engine start-up or shut-down process [2]. Three nonlinearities are separately examined with a focus on Emax and Ωp by using the parameters of Table 3C. Case 10 results in Fig. 13 suggesting that a higher value of hysteresis (Th) decreases Emax, and this trend is consistent with α up to 0.03 for both run-up and run-down processes (Fig. 13a and b). However, hysteresis does not have a significant effect on Ωp (Fig. 13c and d). A change in preload Tp does not influence Emax or Ωp as seen in Fig. 14 with Case 11. The effect of the stiffness (η) is examined by Cases 12 and 13 for run-up and run-down processes, respectively. Although η does not affect Emax, it has a dominant effect on Ωp (Fig. 15c and d). In order to approximate Emax and incorporate the effects of Th, the following formula is proposed based on a prior expression for a linear system that is

4 2

1 0

0.5

1

0.5

1.5

4

4

3

3

E(t)

E(t)

0

2 1 0

0

0.5

1

1.5

0

0.5

1

1.5

2 1

0

0.5

1

0

1.5

3.5

3.5

3

3

2.5

2.5

E(t)

E(t)

1.5

2 1.5 1

2 1.5

0

0.5

1

1.5

1

0

0.5

1

1.5

Fig. 12. Verification of the HBM for a slowly varying process with jαj ¼ 0:001. Figures (a
c) are for the run-up process and (d
f) are for the run-down process; (a) is for Case 6; (d) is for Case 7; (b, e) are for Case 8; (c, f) are for Case 9; refer to Table 3B for Cases 6
9. Key: numerical predication using Hilbert transform.

, HBM;

,

L. Li, R. Singh / Mechanical Systems and Signal Processing 62-63 (2015) 460–479

reported in [29], where T a ¼ ω1 2 γ a :
 T a
ðT h =2Þ pffiffiffiffiffiffiffiffiffiffiffiffi þ θm : Emax ¼ 2ζ þ ðα=2Þ Likewise, the formulas for

475

ð46Þ

Ωp are as follows based on the stiffness sensitivity for the run-up or run-down process:

For run
up process: Ωp ¼

rffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þη 2 2 1
ζ þ 2ðπ
1Þα þ ζ : 2

For run
down process: Ωp ¼

ð47Þ

qffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1þη 2 2 1
ζ
2ðπ
1Þjαjþ ζ : 2

ð48Þ

4.5

1.3

3.5

Ωp

Emax

4 3

1.1

2.5 2

1 0

0.01

0.02

0.03

4.5 4 3.5

0

0.01

0.02

0.03

0

0.01

0.02

0.03

1 0.9

Ωp

Emax

1.2

3 2.5

0.8 0.7

2 0

0.01

0.02

0.03

Fig. 13. The effect of hysteresis (Th) on Emax and Ωp for a rapidly varying process. Figures (a, c) are for the run-up process and (b, d) are for the run-down , Th ¼ 0;

, Th ¼0.05;

4.5

1.4

4

1.3

3.5

1.2

Ωp

Emax

process; refer to Table 3C for Case 10. Key:

3

, Th ¼ 0.1.

1.1

2.5 1

2

0

0.01

0.02

0.03

0

0.01

0.02

0.03

0

0.01

0.02

0.03

4.5

1

3.5

Ωp

Emax

4

3

0.8

2.5 2

0.9

0.7 0

0.01

0.02

0.03

Fig. 14. The effect of preload (Tp) on Emax and Ωp for a rapidly varying process. Figures (a, c) are for the run-up process and (b, d) are for the run-down process; refer to Table 3C for Case 11. Key:

, Tp ¼ 0;

, Tp ¼ 0.025;

, Tp ¼0.05.

476

L. Li, R. Singh / Mechanical Systems and Signal Processing 62-63 (2015) 460–479

The Emax calculation with Eq. (46) is compared with numerical predictions by using the parameters of Case 10. Fig. 16 clearly shows that Eq. (46) is able to approximate Emax reasonably well when jαj is up to 0.03 for both run-up and run-down processes. Also, Eqs. (47) and (48) are verified in Fig. 17 with the parameters of Cases 12 and 13 of Table 3C for run-up and run-down respectively. Again, a relatively close match between calculations using Eqs. (47) and (48) and numerical predications is achieved. 10. Conclusion This article has examined the transient vibration amplification in a nonlinear powertrain system during the start-up or shutdown process. Numerical, analytical, and experimental methods are used to investigate the role of nonlinear characteristics of a 8 1.4 1.2

4

Ωp

Emax

6

2 0

1 0.8

0

0.01

0.02

0.6

0.03

0

0.01

0.02

0.03

0.01

0.02

0.03

1

3

Ωp

Emax

0.8 2 1

0

0.6 0.4

0

0.01

0.02

0.2

0.03

0

Fig. 15. The effect of piecewise linear stiffness (η) on Emax and Ωp for a rapidly varying process. Figures (a, c) are for the run-up process (Case 12) and (b, d) , η ¼ 0.1;

are for the run-down process (Case 13); refer to Table 3C for Cases 12 and 13. Key:

, η ¼0.3;

, η ¼0.5;

4

4

3.5

3.5 Emax

Emax

, η ¼ 1.0.

3

2.5

2.5

2

2 0

0.01

0.02

0.03

3.5

3.5

3

3

Emax

Emax

3

2.5

2.5

2

2

0

0.01

0.02

0.03

0

0.01

0.02

0

0.01

0.02

0.03

0.03

Fig. 16. Verification of the proposed Emax formula for a rapidly varying process. Figures (a and b) are for the run-up process and (c and d) are for the rundown process; (a, c) are for Th ¼0.05; (b, d) are for Th ¼ 0.1; refer to Table 3C for Case 10. Key: prediction.

, proposed formula;

, numerical

L. Li, R. Singh / Mechanical Systems and Signal Processing 62-63 (2015) 460–479

477

0.8

1.4 0.6

Ωp

Ωp

1.2 1

0.2

0.8 0.6

0.4

0

0.01

0.02

0

0.03

0

0.01

0.02

0.03

0

0.01

0.02

0.03

0

0.01

0.02

0.03

1

1.4 0.8

Ωp

Ωp

1.2 1

0.4

0.8 0.6

0.6

0

0.01

0.02

0.2

0.03

1 1.4

0.8

Ωp

Ωp

1.2 1

0.4

0.8 0.6

0.6

0

0.01

0.02

0.03

0.2

Fig. 17. Verification of the proposed Ωp formula for a rapidly varying process. Figures (a
c) are for the run-up process (Case 12) and (d
f) are for the rundown process (Case 13); (a, d) are for η ¼ 0.1; (b, e) are for η ¼ 0.3; (c, f) are for η ¼ 0.5; refer to Table 3C for Cases 12 and 13. Key:

, proposed

, numerical prediction.

formula;

multi-staged clutch damper. 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 process. Second, the vibro-impacts within the multistaged clutch damper are correlated with those in the transmission by using new transient metrics in the context of both 4DOF and SDOF models. Third, the transient amplification envelope for a slowly varying process is successfully estimated by utilizing the harmonic balance method; it is also well approximated for a rapidly varying process by extending formulas based on prior work [29]. The main limitation of this article is that only a minimal order system of the vehicle powertrain is utilized and the effect of the transmission temperature on the transient vibration events is not considered.

Acknowledgement The 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. Appendix A List of symbols b

ζ

C

backlash size damping ratio viscous damping coefficient

478

η E

θ E

ϕ F

φ H

ω1 I

Ωp J

L. Li, R. Singh / Mechanical Systems and Signal Processing 62-63 (2015) 460–479

stiffness ratio between clutch stages transient envelope torsional displacement excitation torque amplitude vector transition point of clutch damper gear mesh force phase of torque excitation clutch damper hysteresis natural frequency torsional inertia excitation frequency at which Emax is reached Jacobian matrix

Subscripts K torsional or gear mesh stiffness 0, 1, 2, 3 index of torsional elements q order of harmonics I, II, III index of clutch damper stages r strong form residual a alternating part T torque c coast side of clutch damper α constant acceleration rate d driving side of clutch damper β variation of design parameters D drag torque γ torque amplitude e torsional excitation δ relative translation displacement within a gear pair g gear mesh stiffness DFT discrete Fourier transform h_i, h_o headset input/output gear HBM harmonic balance method h hysteresis m mean part min minimum value max maximum value p clutch damper preload pp peak to peak value r stiffness reference value u_i, u_o unloaded input/output gear Superscripts K þ


clutch torsional spring stiffness pseudo-inverse of a matrix dimensional parameter

Abbreviations DOF STFT

degree-of-freedom short-time Fourier transform

References [1] R. Ledesma, C. Keeney, S. Shih, Transient analysis of an integrated powertrain and suspension system, in: Proceedings of the European ADAMS User Conferences, Marburg, Germany, November 18–19, 1997. [2] L. Li, R. Singh, Analysis of speed-dependent vibration amplification in a nonlinear driveline system using Hilbert transform, SAE Int. J. Passeng. Cars – Mech. Syst. 6 (2) (2013) 1120–1126, http://dx.doi.org/10.4271/2013-01-1894.

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