1. Introduction
The strict regulations for reduced emissions have driven the automotive industry towards develo** downsized, turbo-charged engines with higher fuel efficiency. The drawback is the generation of aggressive drivetrain torsional oscillations due to the increased power to weight ratio [
1]. A common by-product of drivetrain oscillations that penetrate the transmission is gear rattling noise [
2,
3], which is predominantly radiated due to the impacts of the unloaded (non-engaged) gear pairs. To overcome the adverse effects of gear rattling noise on vehicle quality perception, expensive and sizeable palliatives tuned to certain frequency ranges (such as dual-mass flywheels, clutch pre-dampers and centrifugal pendulum absorbers) [
4,
5] are employed by automotive manufacturers. Nevertheless, because of the tuning conditions, these palliatives do not act in a broadband manner over the powertrain response.
Recent studies [
6,
7,
8,
9,
10,
11] have revealed the potential of using nonlinear vibration absorbers (nonlinear energy sinks (NES)) with low mass and inertia properties for effectively attenuating the generation and propagation of powertrain torsional oscillations. The NES possesses essentially nonlinear stiffness and, thus, instead of being tuned to a preferential frequency, it is activated by the energy input. This is the concept of targeted energy transfer (TET) (also known as energy pum**), whereby the oscillatory energy is transferred in a nearly irreversible manner from a primary system donor (drivetrain) to a secondary system receiver (NES) [
12], where it is then either redistributed within the modal domain or locally dissipated. Numerous publications in the literature have been dedicated to understanding, designing and optimising NES vibration absorbers by studying the underlying dynamics [
13,
14,
15,
16,
17,
18]. The vast majority of works concern absorbers that are acting in translational motion and comprise investigations where an NES was implemented on reduced-order primary system models [
13,
14]. The dynamics of energy pum** in systems equipped with an NES have been systematically investigated by parametric studies of the system’s main properties [
15], whereas studies have considered both grounded and ungrounded NES absorbers [
16,
17] to understand their nonlinear dynamics and the NES effectiveness. The activation of nonlinear normal modes (NNMs) has been also discussed [
18] to illustrate how the NES may interact with the primary system.
More recently, NES mechanisms have been studied for passive structural control purposes and were shown to effectively dissipate energy from structural systems during oscillatory operating conditions. Time-domain numerical methods have been utilised due to the geometric nonlinearities of the NES [
9,
19]. An NES with an inerter (NESI) for passive vibration control has been presented [
20,
21], converting relative translational motion to rotational motion. The small physical mass of the inerter is an advantage, potentially enabling the NESI to provide comparable vibration attenuation to an NES that possesses a larger physical mass. An alternative NES configuration has been proposed utilising bi-stability (BNES) with buckled beams to suppress the oscillations of unbalanced rotor systems [
22]. The numerical simulations have been verified by experimental work to confirm the performance of the BNES. The results showed that the BNES can effectively deal with varied energy quantities.
Nevertheless, a limited number of studies involving rotational NES have been reported in the literature. A torsional NES was proposed in [
23] to stabilise a rotational drilling system. The dynamics of a rotational NES mounted within a linear oscillator were studied in [
24]. The suppression of torsional aeroelastic instabilities on a flexible wing was examined experimentally using a rotational NES that was housed on the tip of the wing [
25]. Steel wire was employed as a mechanism for generating cubic force nonlinearity. The effectiveness of using individual and paired NES in suppressing broadband torsional vibrations in automotive powertrains connected to a turbo-charged engine was examined theoretically in [
6,
7]. The capability of NES to attenuate automotive powertrain torsional oscillations by energy redistribution and dissipation was demonstrated in a numerical study [
8]. The design of NES for attenuating the torsional oscillations of propulsion systems was shown experimentally [
9]. A theoretical analysis involving a spur gear pair coupled with a rotary NES has been presented [
26], showing that targeted energy transfer (TET) can be activated, introducing quasiperiodic motion. The effect of NES mounted on a spur gear pair that operates at low, harmonically varying operating loads (resembling rattling conditions of transmission unloaded gears) has been studied theoretically to demonstrate the performance of the NES in attenuating the gear’s torsional oscillations [
27]. In this paper, a rotational NES prototype utilising thin shims is experimentally tested. The device is shown to effectively reduce the output speed fluctuations of a lightly loaded spur gear pair.
2. Materials and Methods
The numerical model that is used in this paper has been presented in detail in Friskney et al. [
27]. For completeness, we reiterate some of the key aspects and findings from this previous work.
Figure 1 shows a sketch of the meshed gears with an NES attached on the gear.
The equations of motion of the system shown in
Figure 1 are given in Equations (1)–(3) below. The reader is referred to previous work [
27] for a comprehensive description and detailed documentation of the model parameters. Note that the model parameters are informed by the experimental setup that is used in the current work, and therefore the interested reader is referred to [
27] for the gear pair setup details.
where
,
,
denote the angular acceleration, velocity and position of the gear (subscript G), the pinion (subscript P) and the NES (subscript N), respectively.
is the pinion radius,
is the gear radius,
is the pinion average torque,
is the pinion fluctuating torque of angular frequency
and
is the gear torque. The time-varying meshing stiffness is a function of the angular position of the pinion and the applied torque and it is denoted by
. The mesh displacement
is described by a piecewise linear function with backlash [
27]. The dam** coefficient in the pinion–gear contact is
, and the NES dam** coefficient is denoted by
, whereas the NES nonlinear stiffness coefficient is
, where an essentially cubic nonlinearity is assumed.
Rich information for establishing targeted energy transfer in a primary system with an NES is contained in the NNMs of the coupled nonlinear system of oscillators. However, the piecewise linear profile of the gear teeth deflection induced by the presence of backlash introduces difficulties in attaining the NNMs. In order to mitigate computational difficulties, the backlash is ignored in the calculation of the NNMs, leading to a constant average meshing stiffness, without, as will be found in the experimental results, affecting the conclusions drawn.
Figure 2 shows the NNMs of the gear pair with an NES possessing a cubic stiffness of
Nm/m
3, as informed by optimising the NES properties using the presented model [
27].
The NNM corresponding to the primary system (gear pair) is the one resulting from the meshing stiffness. Due to the very high value of this stiffness, the resulting NNM exists at very high frequencies (solid red line in
Figure 2a), far away from the frequency ranges typically observed in the speed fluctuations of automotive driveshafts. On the other hand, the essential nonlinearity of the NES stiffness sweeps through the entire frequency range depending on the vibration energy. Closer observation of this NNM in the frequency range of interest (up to 200 Hz) in
Figure 2b reveals the NES frequency coverage that motivates this experimental study to reduce gear speed fluctuations by the action of an NES attachment. Previous modelling work [
27] has shown that the inclusion of the NES in the gear pair results in a promising reduction in the gear fluctuations that is linked to the gear rattling root cause.
Figure 3 shows the modelled reduction in the gear speed amplitude when the NES is included (“active” result) with respect to the corresponding amplitude without the NES (“locked” result) across the speed range that is achievable in the experimental gear pair setup. Note that the NES is attached to the gear in both scenarios, but it is restrained from oscillating in the locked case in order to eliminate an inertia bias in the comparison of the results.
The NES assembly has been designed in order to physically realise the predicted force–deflection profile. Different views of the NES assembly are shown in
Figure 4. The mechanism to achieve the desired stiffness nonlinearity is by the deflection of steel shims as the rocker oscillates about its mean position, producing a reactionary torque. The latter is rigidly mounted on the rotating shaft that is connected to the primary (host) system. As the rocker deflects the shims, the NES inertia disc (made of aluminium) is engaged, accepting energy from the primary system. The NES disc hosts the shim mounts and itself is mounted on the rotating shaft (of the primary system) through a bearing connection so that it can freely rotate with respect to the shaft. Deep groove ball bearings have been used to reduce friction between the shaft and the NES disc. Each shim is fixed at its end mounts and assembled so that it is in contact with a brass roller. The rocker rotation moves the rollers, deflecting the shims.
The force that is deflecting the shims is calculated using beam bending theory, considering that the shims are fixed at both ends and eccentrically loaded. The key assumptions are as follows:
The beam is straight (i.e., it has no curvature);
The beam material elasticity is the same in both tension and compression;
The stress in the beam is below the elastic limit;
The deflections are small;
The beam undergoes pure bending;
The friction between the rocker and shim is small (the NES dam** will be identified experimentally in a later section);
It is assumed that the rocker touches the shim at a single point of contact.
The configuration of the shim is shown in
Figure 5 for unloaded and loaded scenarios. The total length of the rocker arm is 2 h (see
Figure 5a), the length of the shim between the mounts is L and, initially, the roller is situated at the shim’s midpoint (unloaded scenario). The shims have a rectangular cross-section with width (w) and thickness (t). Following some counter-clockwise rotation of the rocker by angle θ, the new rocker position is as shown in
Figure 5b as A’B’. Using beam bending theory, the force applied at each shim is given by:
where
,
,
,
and
is the is modulus of elasticity for steel, while
I is the second moment of area
. Values for the abovementioned parameters are provided in
Table 1.
The centrifugal force generated on the shim during the assembly rotation is negligible; hence, it will be ignored. The direction of the force is parallel to the y-axis at point A’. As a result, the torque applied on the rocker due to deflection angle
θ on both shims is given by:
The above expression describes the torque as a function of the deflection angle θ and will be validated both statically and dynamically. The expansion of the torque expression in a Taylor series with the trigonometric terms considered up to the fifth order gives the following expression for the quintic (essentially) nonlinear stiffness function:
with
.
Previous modelling work that was briefly described in
Section 2 has considered purely cubic stiffness; however, practical realisations that would eliminate the quintic term were found hard to implement. Therefore, the quintic term was retained in the nonlinear NES torque design, aiming for the torque given by Equation (6) to approach the theoretical cubic torque as much as feasible. Using the values in
Table 1, the stiffness coefficients are calculated as
Nm/rad
3 and
Nm/rad
5.
Figure 6 shows a comparison of the theoretical target torque with the designed torque given by Equation (5) and with its Taylor expansion given by Equation (6). The quintic expansion shows excellent agreement with the designed torque for the examined angle range, which makes it reasonable to drop terms of higher order. Furthermore, a small deviation from the originally targeted torque profile is observed, which, nevertheless, is too small to cause any concern regarding the effect of the NES on the gear amplitude.
In order to establish robust TET conditions between the NES and the primary (host) system, the NES dam** content has to be as low as possible so that it does not contaminate the essentially nonlinear force. Thus, the friction between the dowel and roller surfaces should be low to favour pure rolling motion. To achieve this, the NES roller assembly should be lubricated. A roller bearing with a low friction coefficient is an attractive alternative and it has been employed in this work. This bearing type can exhibit a static coefficient of friction between 0.002 and 0.005, whereas the kinetic friction coefficient can vary between 0.001 and 0.0018.
4. Discussion
Based on these deductions, the numerical model is revisited to explore further agreement with the proposition than one of the shims has been malfunctioning during the experimental tests. Recalculation of the NES–gear system response in the examined frequency range with Equations (1)–(3) updated with the quintic stiffness torque given in Equation (6) leads to the amplitudes shown in
Figure 11. The reduction in the nonlinear stiffness coefficient causes a significant shift in the jump-down frequency of the NES under the same dam** conditions with the corresponding variation in the gear amplitudes. In fact, the numerical results are in good agreement with the experimentally recorded amplitudes, confirming that the NES was working with nearly half its effective stiffness. Combining this with the static identification leads to the conclusion that one of the NES shims was losing contact with the NES rocker. Nevertheless, and although the NES prototype did not follow the optimised response, the model results in
Figure 11 prove the theoretical predictability of the NES design and, by extension, the expected optimised operation, should further refining modification be allowed.
Further,
Figure 12 compares the gear time history when the NES is active, with the gear response recorded with the NES in locked state for pinion speeds of 700 rpm, 800 rpm and 900 rpm. In
Figure 12a,b, the active gear amplitude is found reduced, whereas in
Figure 12c, it is found nearly unaffected by the presence of the NES. This expected behaviour is consistent with the modelling results and it is attributed to the NES action, as
Figure 13 shows. The NES oscillation amplitude when the pinion runs at 800 rpm is significantly higher, leading to a reduction in the gear speed fluctuations. On the other hand, moving to 900 rpm, where the gear is shown to be almost unaffected, the NES is found to oscillate at very low amplitudes.
Last, the system energy is considered to further enhance the validation of the NES action. In
Figure 14a, the NNM of the system lying within the frequency range of interest is plotted according to the initial design target as informed by the model in
Section 2; the statically identified restoring torque was seen to closely match the design target, and the dynamically identified restoring torque was inferred to correspond to the partial malfunction of the prototype, with only one shim applying effective torque to the gear shaft. Juxtaposing the vibration energy of the system at different applied speeds shows that the system response is dictated by the influence of the NNM corresponding to
. Additionally, calculation of the energy damped by the NES close to the observed activation range shows the significant transfer of energy to the NES, such as in the cumulative energy dissipated by the NES when the pinion speed was 700 rpm and 800 rpm, as shown in
Figure 14b. The response of the system at these speeds corresponds to the reduced gear amplitudes that were observed in the frequency plot in
Figure 11 and the time histories in
Figure 12, with an expected drop in the dissipated energy when the NES moves out of its activation range to 900 rpm and 1000 rpm.
Considering the results from
Figure 11,
Figure 12,
Figure 13 and
Figure 14, the NES is shown to reduce the gear speed fluctuations in a manner predictable from and agreeing with the model. Despite the non-optimised operation of the physical prototype, the fundamental justification supporting the motivation of this experimental study is confirmed.