Driveline components connected to internal combustion engines can be critically loaded by dynamic forces due to motion irregularity. In particular, flexible couplings used in engine test rig are usually subjected to high levels of torsional oscillations and timevarying torque. This could lead to premature failure of the test rig. In this work an effective methodology for the estimation of the dynamic behavior of highly flexible couplings in real operational conditions is presented in order to prevent unwanted halts. The methodology addresses a combination of numerical models and experimental measurements. In particular, two mathematical models of the engine test rig were developed: a torsional lumpedparameter model for the estimation of the torsional dynamic behavior in operative conditions and a finite element model for the estimation of the natural frequencies of the coupling. The experimental campaign addressed torsional vibration measurements in order to characterize the driveline dynamic behavior as well as validate the models. The measurements were achieved by a coderbased technique using optical sensors and zebra tapes. Eventually, the validated models were used to evaluate the effect of design modifications of the coupling elements in terms of natural frequencies (torsional and bending), torsional vibration amplitude, and power loss in the couplings.
Flexible couplings enable the transmission of torque from a driver to a driven part of rotating equipment, by accommodating a certain amount of shaft misalignment. This is obtained by reducing the reaction forces due to axial, lateral, angular displacements that are usually present between the coupled shafts. Flexible couplings with torsional compliance (also known as highly flexible couplings) are used to reduce the transmission of shock loads from one shaft to another and/or to alter the elastodynamic characteristics of the driveline by controlling the natural frequencies of the rotating units. The literature is rich of research works regarding highly flexible couplings, with particular reference to installation in test bench for automotive applications, as in the work of Rabeih and Crolla [
In this work, the dynamic analysis of the coupling elements in internal combustion (IC) engine test rigs is accounted from both the numerical and experimental standpoints. The goal of the research is to define an effective methodology which aims at foreseeing the dynamic behavior of highly flexible couplings in real operational conditions in order to prevent a number of problems, such as high level of torsional oscillations, whirling of coupling shafts, damage of driven or driver components, and catastrophic failure of couplings or shafts. An industrial application is used as operative framework. The test rig under investigation consists of a few main components: the engine, gears, a transmission shaft, a highly flexible coupling, and a break. Despite the peculiarity of the case study described, this paper aims at proving the complexity of a complete dynamic analysis performed on this kind of mechanical system. In particular, the full understanding of the dynamic behavior requires more than one modeling approach besides an experimental activity.
The paper is organized as follows: Section
The proposed methodology is schematically draft in the block diagram of Figure
Block diagram of the proposed methodological approach.
The used methodology and the obtained results have a general meaning from a qualitative point of view. Thus, the adopted approach could be generalized to provide an effective procedure to obtain improvements in the dynamic behavior of IC engine test rig drivelines.
A schematic of the test rig being studied is depicted in Figure
Schematic of the system under study.
Schematic of the highly flexible coupling with rubber elements.
An experimental campaign was carried out, with the aim of characterizing the current system dynamic behavior, determining the response signature, and detecting the source of critical problems, as proposed by Troncossi et al. [
Test rig and optical sensor setup.
Torsional vibration measurements on the coupling were carried out by using two optical sensors (Optel Thevon), acquiring TTL signals from zebra tapes with line width of 2 mm. The two sensors were equipped with two different probes (probe Optel Thevon MULTI TBYO 6M HM6X100 SURG, and probe Optel Thevon MULTI SLIT YO 6M HM6X80 SURG), both fixed to a stiff bracket. The zebra tapes were mounted in the two endsections of the coupling, at the brakeside and engineside (in Figure
The system was tested in different operational conditions, namely, runup and stationary regimes, and for different engine speeds. Runup tests were conducted in order to find out resonant bandwidths of the system for a continuous change of engine speed. In addition, stationary tests were performed to have more precise information about the natural frequencies for a number of different constant regimes. Two test campaigns were performed, corresponding to rubber elements of the highly flexible coupling having different hardness, namely, 45 Sh and 70 Sh in the Shore scale, respectively. For the sake of data reliability, three different runs were performed for each test condition. The torsional oscillations of the two coupling ends were evaluated in terms of their relative velocity, denoted as
A FE model was developed in order to estimate the natural frequencies and mode shapes of the driveline in the bandwidth of interest (Figure
3D mesh features.
Element type  Number of nodes  Number of elements  

Transmission shaft  TETRA 4  23850  106553 
Rubber element  TETRA 4  18844  80412 
FE model.
A specific procedure was carried out in order to evaluate the input parameters to be included in the FE model. The coupling manufacturer provided the global torsional stiffness of the coupling, but the 3D FE model requires Young’s Modulus of the rubber elements. Therefore, an iterative procedure based on a static FE model of the coupling was performed in order to estimate suitable Young’s Modulus. A unitary torque was applied to the FE model of the coupling and the corresponding rotational displacement was computed. The torsional stiffness was thus calculated by means of a static analysis [
The 3D FE model requires the complete inertia tensor of the coupling flanges, whereas only the moment of inertia around the rotational axis was available from the component technical manual. In order to estimate the missing parameters, an experimental technique based on frequency response functions (FRFs) measurements was performed for the indirect measurement of the rigid body inertia properties; such a methodology is based on the wellknown Inertia Restrain Method, a technique suitable for a wide range of applications when the mass distribution of components or assemblies is not known (e.g., Mucchi et al. used it in medical [
The FE model has been used in the presented research in order to simulate three design modifications (namely, MOD 1, MOD 2, and MOD 3) conceived for moving the resonances outside the band of the engine main excitation (i.e., the band of interest). The shifting in frequency of the resonances outside the band of interest and the reduction of the number of excited modes were the tasks for improving the system dynamic behavior. At the same time, a number of design constraints had to be respected, as geometrical dimensions and final weight. The used methodology is as follows. The resonances close to the lower threshold of the bandwidth were decreased in frequency, by adding mass or reducing stiffness in specific zones, depending on the mode shape involved. The resonances close to the upper threshold were increased in frequency by reducing mass or increasing stiffness of the transmission shaft and rubber elements. In Section
A detailed elastodynamic model has been developed in order to capture the local modes of the coupling and to simulate the working torsional behavior, proving the effectiveness of the suggested design corrections. The elastodynamic model of the driveline was focused on the torsional dynamics only through a lumpedparameter torsional model.
The main elements of the driveline are the IC engine, the highly flexible coupling, the brake, and the shafts linking them to each other. Since the focus of the model was the coupling, it was further divided into three main parts: the two halves facing the IC engine and the brake, respectively, and the middle flange connected to the halves by means of the rubber elements. These macroelements of the driveline are indeed the same that were used in the FE model in Section
Usually, a key point of engine modeling is the loss torque due to friction associated with the piston assembly [
The analysis was focused on the oscillations around steady working conditions only. In particular, the brake system was not taken into account, since the brakeside shaft is affected by negligible oscillations (see also Section
Schematic drawing of the physical model.
Time data of the normalized velocity signals acquired during a runup test: (a) 45 Sh rubber elements and (b) 70 Sh rubber elements.
Inertia
The computation of inertia
Each linking element (
The characteristics of the transmission shaft (link
The dynamic behavior of rubber could be very complex, due to nonlinearity of the material response. Different models have been proposed in the literature. Qi and Boyce [
While the viscous damper model is correct for the steel material (in the elastic domain), for the rubber material an equivalent viscous coefficient was computed starting from a hysteretic damping model [
The only nonzero external torque is
A detailed description of the formula in (
The equations of torsional motion of the three DOFs system were arranged in matrix form:
Hereafter, the results relative to the current system (with 45 Sh rubber elements of the coupling) will be firstly discussed, with the aim of highlighting the dynamic effects that likely led to the early collapse of the rubber elements. Then, the main data resulting from the substitution of the 45 Sh rubber elements with harder ones (70 Sh) will be shown and compared with the previous ones. Due to confidentiality agreement with the industrial partner, no data related to the shaft velocities can be explicitly reported. Therefore, data relative to runup tests will be shown as normalized to
A preliminary comparison among the data acquired in the three different runs confirmed the extreme repeatability of results, being negligible any difference. The following results correspond to the second runs performed (for both the 45 Sh and 70 Sh rubber element cases). In Figure
Timefrequency analysis of the relative velocity
The further test campaign carried out with the coupling carrying on the 70 Sh rubber elements led to significantly different results (Figure
Stationary tests at different velocities were performed. The results of a limited, significant selection are presented and discussed in the paper and the corresponding runs are conventionally referred to as Regime A, Regime B,…, Regime F (for confidentiality reasons), where Regime A and Regime F are about 50% and 94%, respectively, of the maximum speed achieved in the runup tests and that will be kept as reference in the paper. After resampling and synchronously averaging the velocity signals based on the rotation of the crankshaft, the angle and order analyses as well as the time statistics are available. In particular, a more accurate estimation of the system natural frequencies with respect to the runup results was achieved.
Starting again from the analysis of the 45 Sh results, Figures
Crankshaftanglebased trend (reported over five thermodynamic cycles) and orderbased spectral analysis of
Table
Statistical parameters of

45 Sh  70 Sh  

RMS [rpm]  IR [%]  RMS [rpm]  IR [%]  
Regime A  115  15.4  125  17.0 
Regime B  183  21.2  63  8.0 
Regime C  245  25.2  42  5.2 
Regime D  236  18.8  34  4.0 
Regime E  173  13.2  31  3.2 
Regime F  94  6.8  72  5.2 
From the analysis of all the data retrieved from both runup and stationary tests performed on the current test rig (with 45 Sh rubber elements), it can be concluded that five natural frequencies were likely present in the bandwidth 0–
the mediumhigh natural frequencies
Table
Crankshaftanglebased trend and orderbased spectral analysis of
In order to analyze if the stiffening effect of the harder rubber induced secondary effects in other parts of the driveline, the IAS
Crankshaft velocity oscillations RMS value,

45 Sh  70 Sh  


IR 

IR 

Regime A  177  7.8  239  10.6 
Regime B  153  6.0  205  7.8 
Regime C  193  7.0  182  6.2 
Regime D  187  6.4  159  4.8 
Regime E  176  5.0  144  4.0 
Regime F  144  3.6  139  3.2 
Timefrequency analysis of the crankshaft velocity
It is worth recalling that the performed experimental analysis did not permit determining the vibration modes associated with the mentioned natural frequencies. In other words, it is not possible to state that only torsional modes were excited, since it cannot be excluded that a flexural mode of the transmission shaft could induce coupled oscillations in the IAS.
Table
Input data for FE analysis.
Rubber properties  Value 

Rubber density (456075 Shore)  1000 kg/m^{3} 
Poisson’s ratio  0.49 
Young’s Modulus rubber 45 Shore (60°C) 

Young’s Modulus rubber 60 Shore (60°C) 

Young’s Modulus rubber 70 Shore (60°C) 



Coupling inertial properties  


Mass 







The simulation results obtained through a numerical modal analysis (Sol 103 in MSC.Nastran) regarding 45 Sh and 70 Sh rubber configurations were compared with measurements (Sections
45 Sh configuration. (a) List of experimental resonances. (b) List of numerical (FE) natural frequencies, in percentage of
45 Sh: experimental frequencies
Mode 1  ~3% 
Mode 2  14%–16% 
Mode 3  32%–38% 
Mode 4  ~68% 
45 Sh: numerical results
Mode 1  2.9% (1st global torsional mode [1st TG]) 
Mode 2  7% (1st local mode of coupling [1st LC]) 
Mode 3  14.9% (2nd global torsional mode [2nd TG]) 
Mode 4  19.7% (2nd local mode of coupling [2nd LC]) 
Mode 5  36.8% (3rd local mode of coupling [3rd LC]) 
Mode 6  44% (3rd global torsional mode [3rd TG]) 
Mode 7  50.6% (1st axial mode middle flange [1st AF]) 
Mode 8  69.7% (1st bending mode output shaft [1st BS]) 
70 Sh configuration. (a) List of experimental resonances. (b) List of numerical (FE) natural frequencies, in percentage of
70 Sh: experimental frequencies
Mode 1  6%–8% 
Mode 2  24%–30% 
70 Sh: numerical results
Mode 1  5.1% [1st TG] 
Mode 2  13% [1st LC] 
Mode 3  25.7% [2nd TG] 
Mode 4  37.2% [2nd LC] 
Mode 5  60.2% [3rd TG] 
Mode 6  60.4% [3rd LC] 
Mode 7  83.6% [1st AF] 
Mode 8  85.1% [1st BS] 
Experimental results clearly showed that, within the frequency band of interest, that is, 18%–60% of
The first design modification (MOD 1) regarded the 45 Sh rubber element configuration; MOD 1 addressed an increased weight of the middle flange that reduced at 17.8% the frequency of the third local mode of the coupling. Furthermore, the steel transmission shaft was replaced by a stiffer titanium shaft keeping the third torsional mode outside the band of interest. Eventually, a flywheel was introduced on the engine shaft in order to keep the second torsional frequency at low frequency. The comparison between Tables
Numerical natural frequencies (in % of
Simulation results 45 Shore MOD 1
Mode 1  2.5% [1st TG] 
Mode 2  6.1% [1st LC] 
Mode 3  8% [2nd TG] 
Mode 4  13.8% [2nd LC] 
Mode 5  17.8% [3rd LC] 


Mode 7  72.2% [3rd TG] 
Mode 8  98.6% [1st BS] 
Simulation results 70 Shore MOD 2
Mode 1  4.5% [1st TG] 
Mode 2  13.4% [1st LC] 




Mode 5  60.9% [3rd LC] 
Mode 6  73.1% [1st AF] 
Mode 7  80.4% [3rd TG] 
Simulation results 60 Shore MOD 3
Mode 1  3.7% [1st TG] 
Mode 2  11.5% [1st LC] 




Mode 5  62.2% [3rd LC] 
Mode 6  73.1% [1st AF] 
Mode 7  77.6% [3rd TG] 
The second design modification (MOD 2) concerned the 70 Sh rubber element configuration, where the middle flange was lightened and a flywheel on the engine shaft and a stiffer transmission shaft were placed. Table
The last design modification (MOD 3) took into consideration the use of rubber elements with 60 Sh hardness. The standard middle flange was lightened by milling some parts and by replacing steel screws with titanium ones. The transmission shaft was included in titanium and a flywheel was mounted on the engine shaft. Table
It has to be highlighted that since in the frequency range of interest only puretorsional modes occur (as well as notexcited axial and local modes, see Section
The validation of the elastodynamic model described in Section
The natural frequencies computed by the LP model are in good accordance with those from FE model and experimental activity. The comparison is shown in Table
Comparison of resonance frequencies in (a) experimental measurements, (b) FE model, and (c) LP model for the 70 Sh rubber element configuration with reference to
Experimental resonances
Mode 1  6%–8% 
Mode 2  24%–30% 
FE results
Mode 1  5.1% [1st TG] 
Mode 2  13% [1st LC] 
Mode 3  25.7% [2nd TG] 
Mode 4  37.2% [2nd LC] 
Mode 5  60.2% [3rd TG] 
Mode 6  60.4% [3rd LC] 
Mode 7  83.6% [1st AF] 
Mode 8  85.1% [1st BS] 
LP results
Mode 1  4.5% 
Mode 2  24.9% 
Mode 3  58.2% 
Comparison between simulation results and experimental data, as IR values.
Simulation  Regime A  Regime B  Regime C  Regime D  Regime E  Regime F 


4.5  3.5  2.9  2.5  2.1  1.3 

4.9  2  2.2  2.7  3.0  3.4 

10.3  3.3  2.0  1.7  1.4  4.0 


Experiment  Regime A  Regime B  Regime C  Regime D  Regime E  Regime F 



5.4  4.3  3.5  2.9  2.5  2.1 

7.9  4.0  2.9  2.4  1.8  3.2 
Simulation results are in good accordance with the experimental data, both clearly identifying a higher oscillation at Regime A that decreases as the speed increases from A to E, and then it increases again at Regime F. The advantage of the elastodynamic model is that the causes of the increased oscillation could be easily investigated through the simulation. In particular, the analysis of the resonance frequencies of the system shows that mode shape 2 is particularly burdensome for coordinate
Normalized mode shapes of the driveline.
The elastodynamic model of the driveline was used to simulate the dynamic behavior of the system in three different design changes. The same designation of the modifications discussed in Section
Simulation results of MOD 1.
MOD 1  Regime A  Regime B  Regime C  Regime D  Regime E  Regime F 


2.4  2.1  1.8  1.3  0.9  0.7 

3.4  4.2  1.9  1.6  3.2  1.4 

0.30  0.19  0.13  0.07  0.04  0.03 
Simulation results of MOD 2.
MOD 2  Regime A  Regime B  Regime C  Regime D  Regime E  Regime F 


2.4  2.1  1.8  1.3  1.0  0.7 

2.8  2.7  2.2  1.4  1.8  1.1 

3.24  1.55  1.14  1.04  2.09  0.35 
Simulation results of MOD 3.
MOD 3  Regime A  Regime B  Regime C  Regime D  Regime E  Regime F 


2.4  2.0  1.7  1.3  1.0  0.7 

2.7  4.4  2.0  1.5  4.0  1.2 

2.82  1.43  1.16  1.19  1.51  0.32 
The power loss was used as a further indicator. According to standard DIN 740 the relative damping is the ratio between the power loss of one vibration cycle and elastic deformation energy. The elastic deformation energy depends on the main frequency in the oscillation spectrum of the coupling and can be easily computed, while the relative damping is usually given in the manufacturer’s catalog (it is the
The power loss of one vibration cycle for each design modification.
[W]  Regime A  Regime B  Regime C  Regime D  Regime E  Regime F 

MOD 1  1.03  1.23  0.72  0.99  0.98  0.87 
MOD 2  31.87  13.18  4.80  4.14  2.90  1.88 
MOD 3  19.14  9.76  3.83  3.54  2.66  1.89 
Modification MOD 1 keeps torsional vibrations at low level, with a stable and limited oscillation around the reference speed. Consequently, the power loss has low values compared to the other modifications. As a drawback, the FE analysis shows that all the resonance frequencies are shifted to lower values; for example, five resonances lay in the band (0%–18% of
Modifications MOD 2 and MOD 3 are in the opposite direction of MOD 1. They shifted the resonance frequencies at higher values out of the selected frequency bandwidth. Torsional vibrations are still acceptable: the speed oscillation is less than 5% with respect to regime. In MOD 2 the FE model shows that the 3rd local mode frequency is outside the limit of 60% of
The paper presents an effective methodology based on numerical models and experimental measurements, which aims at foreseeing the dynamic behavior of highly flexible couplings of IC engine test rigs in real operational conditions. The main goal is to prevent high level of torsional oscillations, whirling of coupling shafts, and early failures of any components. A procedure to optimize the dynamic behavior of the driveline is thus proposed and applied to an industrial application, useful to show that fully understanding of the dynamic behavior of a real mechanical system requires more than one modeling approach besides an experimental activity. In the test bench being studied, the output shaft of the engine is connected to an electromechanical brake through a transmission shaft, which hosts a highly flexible coupling with rubber elements.
The results of the reported case study led to draw the following conclusions:
the experimental activity showed the presence of a resonance close to the working condition of the coupling, which led to an early breaking of the rubber elements of the joint;
the FE model enabled the characterization of the shape of the mode determining such a resonance and the suggestion of three different design modifications to avoid the resonance in working conditions;
the LP model allowed choosing the design modification less burdensome in terms of torsional vibration of the driveline.
Pressure in
Bore of the piston
Length of the crank
Mass of the piston
Center of gravity
Mass of the piston rod (p.r.)
Length of the p.r.
Inertia of the p.r.
Distance COG and head of p.r.
Distance COG and foot of p.r.
The authors declare that they have no conflicts of interest.
This work has been developed within the Advanced Mechanics Laboratory (MechLav) of Ferrara Technopole, realized through the contribution of Regione EmiliaRomagnaAssessorato Attività Produttive, Sviluppo Economico, Piano telematicoPORFESR 2007–2013, Activity I.1.1.