Research on dynamic response of centrifugal pendulum vibration absorber based on hybrid damping model

Abstract

The application of the centrifugal pendulum vibration absorber (CPVA) has expanded from the aerospace sector to the automotive sector. To date, in most previous studies, viscous damping has been assumed to be present between the absorber and rotor, and damping has been neglected in other studies. To reflect and control the dynamic behaviour of the CPVA in vehicular applications realistically, a hybrid damping model incorporating viscous damping and rolling resistance was developed in this study and validated by conducting tests. Under the combined action of the centrifugal force, gravity, viscous resistance, and rolling resistance, an equation of motion of the CPVA was established using the Lagrangian function equation of the second type. The wear state of the kinematic pair between the absorber and rotor, which is common in practical applications, was included into a mathematical model in which the rolling resistance coefficient changes with the travel of the absorber, whereas the viscous resistance coefficient remains unchanged. A model was established to simulate the response of the absorber under a wide range of working conditions, and corresponding tests were performed. Compared with the results obtained using only viscous damping as reported in other studies, those of the proposed hybrid damping model are more consistent with the experimental results. This work fills the existing research gap and lays a foundation for further control of the dynamic behaviour of CPVAs in the gravitational field, particularly at low rotational speeds.

Keywords

centrifugal pendulum vibration absorber viscous damping rolling resistance gravity hybrid damping modelling

1. Introduction

To reduce the torsional vibration of drivelines, methods such as torsional dampers (Rahnejat, 1998), dual-mass flywheels (Kelly et al., 2010; Theodossiades et al., 2006), tuned absorbers (Denman, 1992; Hausner and Zink, 2010; Newland, 1964; Shaw et al., 1997), nonlinear energy sinks (Haris et al., 2017; Motato et al., 2017) parameter optimisation (Wu and Zhao, 2022), and modern control theory (Wu et al., 2023) can be used. The centrifugal pendulum vibration absorber (CPVA) was one type of tuned absorbers invented in the early 1920s and has since been applied in aviation (Denman, 1992; Newland, 1964). It was used in the automobile industry in the 1980s (Hausner and Zink, 2010; Shaw et al., 1997). A CPVA with a circular path exhibits softening nonlinear characteristics (Newland, 1964), whereas a cycloidal path CPVA exhibits hardening nonlinear effect characteristics (Alsuwaiyan and Shaw, 2002). A landmark discovery is the tautochronic bifilar CPVA; the path of the absorber is epicycloid and has a weak nonlinear effect regardless of the amplitude of the absorber (Denman, 1992).

The consideration of damping varies depending on the research purpose. Chao et al. (1997b) studied two types of friction between the absorber and its path: viscous and hysteretic. If the damping of the absorber is viscous, the number of absorbers will influence the system stability. Lee and Shaw (1996) proposed creating a nonlinear response of a CPVA system by varying the absorber path in an ideal undamped system of one and/or two absorbers to achieve an accurate offset of the torque, which in a realistic environment requires the absorber to have the lowest friction. Lee and Shaw (1997) found that a pair of absorbers can cancel two oscillation torque harmonics at small absorber damping levels. Vidmar et al. (2012) found that Coulomb friction dominates the viscous damping effect under smaller detuning and low-amplitude conditions of the absorber. Issa and Shaw (2015) studied the bifurcations of the synchronous response under different numbers of absorbers and showed that an undamped CPVA system has a degenerate response but can be made robust by introducing appropriate damping. Gomez et al. (2021) implemented constant damping along with normal-force-related damping in the equation of motion (EOM) under the premise of no viscous damping during absorber movement. The EOM was verified under test speeds of 900 and 1500 rpm and further studied in heavy-truck powertrains (Gomez et al., 2022). However, damping has been ignored in some studies (Demeulenaere et al., 2005; Shi and Parker, 2012).

Although the damping mechanism has not been comprehensively examined in most studies, useful results have been obtained using only viscous damping. Chao et al. (1997a) studied the post-bifurcation dynamic behaviour of CPVAs, called mode localisation. Subsequently, Chao et al. (1997b), Alsuwaiyan and Shaw (2002, 2003, 2014), Vidmar et al. (2013), and Nishimura et al. (2016) systemically studied the localisation behaviour. Shaw and Geist (2010), Monroe et al. (2011), Mayet and Ulbrich (2014), and Newland (2020) explored detuning methods to avoid localisation. Mayet et al. (2022) noted that the detuning level and absorber damping will affect the occurrence of asynchronous responses, similar to the conclusions drawn by Issa and Shaw (2015) and Alsuwaiyan and Shaw (2003). Focusing on localisation, Mahé et al. (2022) proposed new design guidelines. Cirelli et al. (2020) and Cera et al. (2021a, 2021c) developed a novel approach to build an EOM using higher-order curvature ratios of the absorber path, with the epicycloid as a special case. Overall, viscous damping was assumed in most of studies (Alsuwaiyan and Shaw, 2002, 2003, 2014; Cera et al., 2021a, 2021c; Chao et al., 1997a, 1997b; Cirelli et al., 2020; Issa and Shaw, 2015; Lee and Shaw, 1997; Mahé et al., 2022; Mayet and Ulbrich, 2014; Mayet et al., 2022; Monroe et al., 2011; Nishimura et al., 2016; Newland, 2020; Shaw and Geist, 2010; Tchokogoué et al., 2021; Vidmar et al., 2012, 2013; Zhao et al., 2022; Zhang et al., 2022a, 2022b, 2023a, 2023b; Zhang and Wu 2023), whereas damping was ignored in others (Demeulenaere et al., 2005; Denman, 1992; Lee and Shaw, 1996; Newland, 1964; Shaw et al., 1997; Shi and Parker, 2012).

The continuous promotion of hybrid vehicles causes internal combustion engines to work frequently under start-stop and idle charging conditions. At low rotor speeds, gravitational effects will significantly affect the response of the absorber, adding further complexity to the overall dynamics, such as the noise issue (Kim, 2020; Zhang et al., 2022b, 2023b). Controlling the response of the absorber without hitting the absorber motion-limiting device can help solve this problem. Assuming viscous damping and incorporated gravitational effects, Tchokogoué et al. (2021) systematically predicted the response of an absorber under gravity using the cyclic symmetry of a CPVA. Zhao et al. (2022) studied the overall performance of a bifilar epicycloid CPVA as well as its stability (Zhang et al., 2022a; Zhang and Wu, 2023) [38], [39], response at low engine speeds (Zhang et al., 2022b), and knocking power during engine start-stop (Zhang et al., 2023b). The experimental and theoretical analysis results were found to be well correlated but differed in some aspects. To overcome this drawback, as a follow-up to Zhang et al. (2022b, 2023b), we propose a hybrid damping model along with an EOM, which lays a solid foundation for further investigations into the response control of absorbers at the time of engine start-up and ignition and provides an accurate modelling method for other structural types of CPVAs. Thus, we would like to define the previously widely used viscous damping as ‘equivalent viscous damping’, where the rolling resistance is equivalent to viscous damping.

The remainder of this paper is organised as follows. Section 2 explains why a hybrid damping model is required, in terms of the CPVA structure and operation conditions. Section 3 provides the EOMs for the CPVA. Section 4 presents a comparison of the numerical simulation results with the test results. Section 5 summarizes the conclusions of the study.

2. CPVA structure and damping model selection

The damping mechanism is important for the accurate mathematical modelling of the torsional dynamics of reciprocating engines (Pennestrì et al., 2016; Newland, 2020). In real-world scenarios, describing and controlling the influence of damping in terms of its type and magnitude are difficult (Chao et al., 1997b). Nevertheless, some practical applications have provided good reference data. Viscous damping is a simple damping model widely used in structural dynamic analyses. The Coulomb friction force is greater than the viscous damping at low absorber oscillation amplitudes (Vidmar et al., 2012), which does not dominate the large oscillation amplitude of the CPVA when it starts in a gravitational field.

Figure 1 (Zhang et al., 2023a; Zhao, 2022) shows a CPVA (© Schaeffler AG) disassembled from a vehicle. The study of this CPVA revealed that the response of the absorber in the test was inconsistent with the numerical simulation results in some special cases (Zhang et al., 2022b; 2023a; 2023b). To investigate the superposition effect of the oscillation torque and gravitational field excitation at lower engine speeds and under dynamic working conditions, such as during start-up or ignition, the reason for the differences between the test and simulation should be systematically analysed. One reason is that the controllability of the test equipment is not efficient due to open-loop control, and the real torque phase should be obtained by actual observation. The other reason is that the CPVA was obtained from the aftermarket. Although the trajectories of the two pendulums were measured to be identical, some visible imperfections were observed, as shown in Figure 1.

A small layer of grease on the surface of the CPVA with appropriate clearance between the spacer and rotor ensure relative motion between them. Owing to vehicle or engine vibration and flywheel wobbling, the absorber and spacer contact the rotor (Kim, 2020) and the grease produces viscous resistance. Thus, a viscous damping coefficient for the absorber along its path must be assumed. Almost all previous researchers have made the same assumption.

The force generated by the relative motion between the rotor and absorber is transmitted to the rotor through the roller. The roller bears the centrifugal force created by the absorber, which is similar to the contact between a train wheel and wheel track (Ma et al., 2020). Particularly in a gravitational field, the centrifugal force and gravity interact with each other and the load on the roller constantly varies. In addition, because of manufacturing errors, wear, or deformation, differences must exist in the rolling resistance caused by the roller and the roller track, equivalent to the differences in the rolling resistance coefficient. As the rolling resistance is related to the pressure and surface condition, it should not be neglected.

3. Equation of motion of CPVA

The absorbers of a parallel bifilar CPVA can be treated kinematically as a point mass (Monroe et al., 2011; Mayet and Ulbrich, 2014; Newland, 2020; Wu and Zhang, 2023). The centre of mass of the absorber follows an epicycloid path, as shown by the arc in Figure 2. Both ends of the arc are equipped with end stops of stiffness $k_{s}$ . The mass of the absorber is denoted by $m_{i}$ , and $i$ refers to the serial number of the absorbers ( $i$ = 1, 2, 3,…N). Each absorber has its own $x_{i} - y_{i}$ coordinate system, which is fixed on the rotor. The $X - Y$ coordinate system is fixed to the engine, and $x_{i} - y_{i}$ rotates about $X - Y$ at a nominal speed of $Ω$ .

The torque applied to the rotor can be expressed as $T = T_{0} + T_{α} \sin (n α + δ)$ . The order of the oscillation torque is denoted by $n$ . The rotational angle of the rotor relative to the $X - Y$ axis is denoted by $α$ , $δ$ is the phase of the torque, and $α_{m i}$ is the rotational angle of $x_{i} - y_{i}$ relative to the $X - Y$ axis.

$J$ Is the inertia of the rotor, and the rotation centre of the rotor is marked as $O$ . The roller effects were neglected, and the roller mass is equivalent to a part of the absorber mass $m_{i}$ . $O_{p}$ is the virtual suspension point of $m_{i}$ , which runs along a predetermined epicycloid path that is symmetric about the x-axis, and has its midpoint $O_{m}$ on the x-axis. $L_{i 0}$ is the distance from the midpoint to the rotation centre of the rotor. $S_{i}$ is the distance moved by the $i^{t h}$ absorber from the midpoint. $S_{p i}$ is the distance from the midpoint to the end stop.

The radius of curvature $ρ_{i}$ is different for each point on the path, and the angle $ϕ_{i}$ between the $x_{i}$ axis and the radius of curvature changes accordingly. We write $ρ_{i}$ as ${ρ_{i}}_{0}$ when $ϕ_{i}$ equals zero. The relationship between $ρ_{i}$ and ${ρ_{i}}_{0}$ can be expressed as $ρ_{i} = {ρ_{i}}_{0} \cos (λ_{i} ϕ_{i})$ . $r_{i}$ is the value obtained by subtracting ${ρ_{i}}_{0}$ from $L_{i 0}$ .

A Lagrangian equation of the second type is used to formulate the EOM of the CPVA:

\frac{d}{d t} (\frac{\partial E}{\partial {\dot{q}}_{i}}) - \frac{\partial E}{\partial q_{i}} + \frac{\partial V}{\partial q_{i}} + \frac{\partial U}{\partial q_{i}} = Q_{i} .

(1)

The kinetic energy of the CPVA is denoted by $E$ , gravitational potential energy is denoted by $V$ , and elastic potential energy of the rubber end stop is denoted by $U$ . Here, $α$ and $ϕ_{i}$ are generalised coordinates.

The viscous damping forces applied to the absorber are $- μ_{s i} {\dot{S}}_{i}$ , where the dots above $S_{i}$ indicate time derivatives. According to the principle of virtual work, the generalised force converted into $ϕ_{i}$ coordinates is as follows:

- μ_{s i} {ρ_{i}}_{0}^{2} \cos^{2} (λ_{i} ϕ_{i}) {\dot{ϕ}}_{i} .

(2)

Figure 3 shows the centrifugal and gravitational forces acting on $m_{i}$ . The components of the gravitational and centrifugal forces in the normal direction were $m_{i} g \cos (90^{o} - α_{m i} - ϕ_{i})$ and $m_{i} Ω^{2} r_{i} \cos (ϕ_{i} - θ_{i})$ , respectively. The rolling resistance expressed by equation (3) is the product of the absolute value of the difference between these two components and the rolling resistance coefficient, that is, $μ_{r i}$ . The roller was assumed not to experience any slippage during the operation. When $θ_{i}$ increases, the rolling resistance is in the tangential direction of the absorber path, which is the same direction as the viscous damping force:

sgn ({\dot{ϕ}}_{i}) μ_{r i} | m_{i} g \cos (\frac{π}{2} - α_{m i} - ϕ_{i}) - m_{i} Ω^{2} r_{i} \cos (ϕ_{i} - θ_{i}) | .

(3)

According to the principle of virtual work, the generalised rolling resistance converted into $ϕ_{i}$ coordinates is as follows:

sgn ({\dot{ϕ}}_{i}) μ_{r i} | m_{i} g \cos (\frac{π}{2} - α_{m i} - ϕ_{i}) - m_{i} Ω^{2} r_{i} \cos (ϕ_{i} - θ_{i}) | {ρ_{i}}_{0} \cos (λ_{i} ϕ_{i}) .

(4)

Here,

r_{i} = \sqrt{x_{i}^{2} + y_{i}^{2}}; θ_{i} = \tan^{- 1} (\frac{x_{i}}{y_{i}}) .

(5)

The $x_{i}, y_{i}, X_{i}, and Y_{i}$ coordinates of $m_{i}$ are given by equations (6)–(9), where $λ_{i}$ is the shape factor of the epicycloid path and ${\tilde{n}}_{i}$ is the tuning order of the $i^{t h}$ absorber (Chen and Wu, 2018; Wu and Wu, 2017):

x_{i} = L_{i 0} + \frac{{ρ_{i}}_{0}}{1 - {λ_{i}}^{2}} (\cos ϕ_{i} \cos λ_{i} ϕ_{i} + λ_{i} \sin λ_{i} ϕ_{i} \sin ϕ_{i} - 1)

(6)

y_{i} = \frac{{ρ_{i}}_{0}}{1 - {λ_{i}}^{2}} (\sin ϕ_{i} \cos λ_{i} ϕ_{i} - λ_{i} \sin λ_{i} ϕ_{i} \cos ϕ_{i})

(7)

X_{i} = - y_{i} \sin α_{i} + x_{i} \cos α_{i}

(8)

Y_{i} = y_{i} \cos α_{i} + x_{i} \sin α_{i}

(9)

λ_{i} = \sqrt{\frac{{\tilde{n}}_{i}^{2}}{{\tilde{n}}_{i}^{2} + 1}} = \sqrt{\frac{r_{i}}{L_{i 0}}} {; \tilde{n}}_{i} = \sqrt{r_{i} / ρ_{i 0}} .

(10)

The rotational viscous damping coefficient of the rotor is denoted by $μ_{α}$ . The viscous damping forces applied to the rotor are $- μ_{α} \dot{α}$ , where the dots above $α$ indicate time derivatives. Therefore, the generalised forces applied to the rotor can be expressed as ${- μ}_{α} ω + T$ .

Based on the assumption that the two absorbers have the same mass, geometric properties, and tuning order, subscript $i$ is removed from $m_{i}$ , $r_{i}$ , $ρ_{i 0}$ , $S_{p i}$ , and $λ_{i}$ . Substituting equations (2)–(10) into equation (1) yields

\frac{d}{d t} (\frac{\partial E}{\partial \dot{α}}) - \frac{\partial E}{\partial α} + \frac{\partial V}{\partial α} + \frac{\partial U}{\partial α} + μ_{α} \dot{α} - T = 0

(11)

[\begin{array}{l} \frac{d}{d t} (\frac{\partial E}{\partial {\dot{ϕ}}_{1}}) - \frac{\partial E}{\partial ϕ_{1}} + \frac{\partial V}{\partial ϕ_{1}} + \frac{\partial U}{\partial ϕ_{1}} + μ_{s 1} {ρ_{0}}^{2} {[\cos (λ ϕ_{1})]}^{2} {\dot{ϕ}}_{1} \\ + sgn ({\dot{ϕ}}_{1}) μ_{r 1} m ρ_{0} \cos (λ ϕ_{1}) | g \sin (α_{m 1} + ϕ_{1}) - Ω^{2} r \cos (ϕ_{1} - θ_{1}) | \end{array}] = 0

(12)

[\begin{array}{l} \frac{d}{d t} (\frac{\partial E}{\partial {\dot{ϕ}}_{2}}) - \frac{\partial E}{\partial ϕ_{2}} + \frac{\partial V}{\partial ϕ_{2}} + \frac{\partial U}{\partial ϕ_{2}} + μ_{s 2} {ρ_{0}}^{2} {[\cos (λ ϕ_{2})]}^{2} {\dot{ϕ}}_{2} \\ + sgn ({\dot{ϕ}}_{2}) μ_{r 2} m ρ_{0} \cos (λ ϕ_{2}) | g \sin (α_{m 2} + ϕ_{2}) - Ω^{2} r \cos (ϕ_{2} - θ_{2}) | \end{array}] = 0 .

(13)

The energy-related term can be expressed as follows (Chen and Wu, 2018; Zhang et al., 2023b):

E = \frac{1}{2} J {\dot{α}}^{2} + \frac{1}{2} \sum_{i = 1}^{2} m ({\dot{X}}_{i}^{2} + {\dot{Y}}_{i}^{2})

(14)

V = \sum_{i = 1}^{2} m g Y_{i}

(15)

U = \frac{1}{2} \sum_{i = 1}^{2} k_{s} {(| \frac{ρ_{0}}{λ} \sin (λ ϕ_{i}) | - S_{p})}^{2} .

(16)

Equations (14)–(16) were substituted into equations (11)–(13) for the subsequent numerical simulations. The CPVA’s equation of motion is explicitly shown in the appendix.

4. Comparison of numerical solutions with experimental results

To verify the hybrid friction model, the numerical simulation results were compared with experimental results. Figure 4 shows the layout of the test equipment. An electrical motor with a hydraulic servo unit can output a second-order sinusoidal oscillation component, similar to that of an internal combustion engine. To facilitate the testing data analysis, two external Hall sensors were installed to record the speed signals of the electrical motor and rotor. A code wheel with polar coordinate rays was mounted on the rotor, and the centre of mass of the absorber was marked to compare the movement of the absorber with the polar coordinate rays on the code wheel (Zhang et al., 2022b). A high-speed camera captured the motion posture of the CPVA, and a Python program was used to convert the captured video into frame-by-frame images. A MATLAB program identified the relative positions of the absorber and rotor in each image.

Figure 5 shows the model established for the numerical simulation of the test equipment using the CPVA. A constant rotational speed $Ω_{d y n o}$ and sinusoidal speed fluctuation with amplitude of $A_{d y n o}$ were used as the power source of the system. The sine wave displacement can be expressed as $Ω_{dyno} + A_{dyno} * \sin (2 * π * f * t + δ_{dyno})$ . The spring and damping devices were connected directly to the power source, and the output of the second block diagram can be expressed as $T = T_{0} + T_{α} \sin (n α + δ)$ . The EOM of the CPVA was embedded in the third block diagram, and the drivetrain inertia was equated and added to the rotor inertia.

Unless otherwise specified, the parameters used in this study are listed in Table 1. The masses of the absorber, roller, and rotor were obtained by establishing a 3D model (Zhang et al., 2022b, 2023a; Zhao, 2022) through reverse engineering. The mass of the absorbers

m

used in the numerical simulation was the sum of the absorber mass

m_{a}

, roller mass

m_{r}

, and spacer mass

m_{s}

. The stiffness of the end stop was tested by a cupping machine, and the positions of the defect on the roller track were measured by a coordinate measuring machine.

Table 1.

Parameters of CPVA shown in Figure 1.

Parameter	Symbol	Value	Units
Number of absorbers	$N$	2
Order of oscillation torque	$n$	2
Absorber mass	$m_{a}$	0.168	kg
Roller mass	$m_{r}$	0.009	kg
Spacer mass	$m_{s}$	0.015	kg
Pendulum length	$L_{i} (0)$	0.089	m
Tuning order of $i^{t h}$ absorber	${\tilde{n}}_{i}$	2.01
Acceleration due to gravity	$g$	9.800	m・s⁻²
Inertia of rotor	$J$	0.064	kg・m⁻²
End stop position	$S_{p}$	0.69	rad
End stop stiffness	$k_{s}$	350,000	N/m
Viscous damping coefficient of absorber 1	$μ_{s 1}$	1.8
Rolling resistance coefficient of absorber 1 (The Engineering Toolbox, 2022)	$μ_{r 1}$	0.002
Viscous damping coefficient of absorber 2	$μ_{s 2}$	1.8
Rolling resistance coefficient of absorber 2 without imperfection (The Engineering Toolbox, 2022)	$μ_{r 2}$	0.002

The equivalent viscous damping coefficient between the absorber and rotor is normally 3.2–3.5 (Zhang et al., 2022b). In current research and previous investigations (Zhang et al., 2022b; 2023b),parameter studies have been conducted with different equivalent viscous damping coefficients from 1.5 to 5.0 in failed attempts to correlate experimental results with and numerical simulation results. Therefore, we assumed that the rolling resistance and viscous damping should play the same role in actual applications and adopted 1.8 as the viscous damping coefficient. The rolling resistance coefficients of the two absorbers are the same except for the imperfections on the roller track of absorber 2, which are shown in Figure 1, and can be expressed as

μ_{r 2} = {\begin{cases} 0.002 - 0.690 \leq ϕ_{2} < - 0.449 \\ 0.08 - 0.449 \leq ϕ_{2} < - 0.262 \\ 0.002 - 0.262 \leq ϕ_{2} < 0.262 \\ 0.03 0.262 \leq ϕ_{2} < 0.400 \\ 0.002 0.400 \leq ϕ_{2} < 0.690 \end{cases}} .

(17)

where the value ranges of

ϕ_{2}

are measured by CMM on the CPVA shown in Figure 1.

Figures 6 –9 compare the numerical simulation results of the absorber response with the test results. To obtain Figures 6 and 7, four rounds of tests with different sinusoidal excitation amplitudes and phase combinations were performed under the most severe low-speed test condition: 200 rpm. Figure 8 displays the results for $Ω_{dyno} = 300$ rpm with different sinusoidal oscillation amplitudes. Figure 9 presents the results for $Ω_{dyno} = 400$ rpm with a specific sinusoidal oscillation phase. The test and simulation results correlate more closely at 300 and 400 rpm than at 200 rpm, because the centrifugal force increases and the influence of imperfections decreases gradually. Furthermore, Figure 10 shows the results of further tests conducted at $Ω_{d y n o} = 600$ rpm, where the numerical simulation results are very consistent with the test results.

In the analytical results predicted by linear theory, case 2: n = 2, N = 2 (Tchokogoué et al., 2021) indicates that the two absorbers will have the same waveform but that it will be shifted due to a cyclic phase. These characteristics were indeed observed in the numerical simulation results using only viscous damping. However, when using the hybrid damping model, these features do not occur in the response of the two absorbers due to the presence of rolling resistance.

Overall, the results of the mathematical model with the hybrid damping model agreed more closely with the test results than those of the viscous damping model reported in Zhang et al. (2022b; 2023a). As an example, the experimental results and simulation results using viscous damping only are given in Figure 11 (Zhang et al., 2023a) for comparison with Figure 7(b).

Table 2 summarizes the root-mean-square error integrated over one period of travel of two absorbers. Significant improvements are achieved by using hybrid damping model in these scenarios.

Table 2.

Root-mean-square error of the travel of two absorbers (mm).

Scenario	Figure 6(a)		Figure 6(b)		Figure 7(a)		Figure 7(b)
Scenario	Absorber1	Absorber2	Absorber1	Absorber2	Absorber1	Absorber2	Absorber1	Absorber2
Hybrid damping	0.0020	0.0010	0.0010	0.0009	0.0018	0.0014	0.0014	0.0015
Viscous damping	0.0021	0.0015	0.0018	0.0013	0.0021	0.0024	0.0081	0.0086
Improvement	4.76%	33.33%	44.44%	30.77%	14.29%	41.67%	82.71%	82.55%

Scenario	Figure 8(a)		Figure 8(b)		Figure 9		Figure 10
Scenario	Absorber1	Absorber2	Absorber1	Absorber2	Absorber1	Absorber2	Absorber1	Absorber2
Hybrid damping	0.0004	0.0007	0.0016	0.0017	0.0014	0.0020	0.0014	0.0022
Viscous damping	0.0006	0.0013	0.0025	0.0021	0.0018	0.0022	0.0022	0.0023
Improvement	33.33%	46.15%	36.00%	19.05%	22.22%	9.09%	36.36%	4.34%

5. Conclusions and future work

Considering only viscous damping in the mathematical model cannot explain the particular response of the pendulum that occurs under specific experimental conditions. This inability exists because the rolling resistance cannot be replaced by equivalent viscous damping because the former is related to the positive pressure between the absorber and rotor, whereas the latter is related to the contact between the spacer and rotor. The former changes with the square of the rotor speed and absorber position, whereas the latter is related to the relative speed between the spacer and rotor. When studying the dynamic performance of CPVAs, careful consideration of rolling resistance and imperfections on the roller track, particularly in the lower rotational speed range, is important.

The inconsistencies between the test results and hybrid damping model results need to be studied in the future. For example, in Figure 6(b), inconsistencies occur at an amplitude of −0.005 m in absorber 2; and in Figure 7(b), inconsistencies occur at an amplitude of 0 m in absorber 1. In addition to the test errors mentioned in Section 2, the static friction and stick-slip effects play key roles when the acceleration ${\dot{S}}_{i}$ of the absorber is close or equal to zero, which is not represented in the hybrid damping model. Furthermore, the parameters used for hybrid damping model come from empirical value and experimental fitting, these parametric errors should be should be further reduced in subsequent research. Linear and random vibration damping test, resonant resistance test, and impact damping test can help researchers more directly determine the true value of damping.

This paper proposed a hybrid damping mechanism for the absorber in a parallel bifilar CPVA, which can be considered as a one-dimensional absorber model, as it only performs translational motion. Further studies must be conducted for extension to trapezoidal bifilar CPVAs with both translational and rotational movements and to develop advanced schemes using synchro rings to force the synchronisation of all absorbers (Cera et al., 2021b; Mayet and Ulbrich, 2014). In addition, static friction and complex interactions such as direct surface interaction due to poor lubrication (Humphrey et al., 2018) should be included in complex damping characteristics studies.

Footnotes

Acknowledgements

We would like to express our heartfelt gratitude to Professor Steven Shaw of Florida Institute of Technology, who dedicated CPVA research from 1990s and published numerous papers on CPVA. He made many valuable suggestions during this study.

Author contributions

All the authors contributed to the conception and design of the study. Yi Zhang is the lead author, prepared materials, collected experimental data, and performed analyses. Jie Qiu is a co-author and gave research directions and advice of CPVA design. Guangqiang Wu is the corresponding author, supervised and guided this project, and was involved in paper writing.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was funded by the National Natural Science Foundation of China (grant number 52075388).

ORCID iD

Yi Zhang

Data Availability Statement

The data supporting the findings of this study are available from the corresponding author upon reasonable request.

Appendix

The CPVA’s equations of motion are showed as below.

[\begin{array}{l} A 1 & B 1 & C 1 & D 1 \\ A 2 & B 2 & C 2 & D 2 \\ A 3 & B 3 & C 3 & D 3 \end{array}] [\begin{array}{l} \ddot{α} \\ {\ddot{ϕ}}_{1} \\ {\ddot{ϕ}}_{2} \\ 1 \end{array}] = 0,

where

A 1 = \frac{1}{{(λ^{2} - 1)}^{2}} {\begin{cases} - m {ρ_{0}}^{2} (λ - 1) (λ + 1) \cos^{2} (λ ϕ_{1}) - 2 m ρ_{0} (r λ^{2} + λ^{2} ρ_{0} - r) \cos ϕ_{1} \cos (λ ϕ_{1}) \\ - m {ρ_{0}}^{2} (λ - 1) (λ + 1) \cos^{2} (λ ϕ_{2}) - 2 m ρ_{0} (r λ^{2} + λ^{2} ρ_{0} - r) \cos ϕ_{2} \cos (λ ϕ_{2}) \\ - 2 m λ ρ_{0} (r λ^{2} + λ^{2} ρ_{0} - r) \sin ϕ_{1} \sin (λ ϕ_{1}) \\ + m [(2 λ^{4} + 2 λ^{2}) {ρ_{0}}^{2} + (4 r λ^{4} - 4 r λ^{2}) ρ_{0} + 2 r^{2} {(λ - 1)}^{2} {(λ + 1)}^{2}] \\ + J {(λ - 1)}^{2} {(λ + 1)}^{2} - 2 m λ ρ_{0} (r λ^{2} + λ^{2} ρ_{0} - r) \sin ϕ_{2} \sin (λ ϕ_{2}) \end{cases}}

B 1 = \frac{1}{(λ^{2} - 1)} [\begin{array}{l} - m {ρ_{0}}^{2} \cos^{2} (λ ϕ_{1}) \\ + m ρ_{0} \cos ϕ_{1} (r λ^{2} + λ^{2} ρ_{0} - r) \cos (λ ϕ_{1}) \end{array}]

C 1 = \frac{1}{(λ^{2} - 1)} [\begin{array}{l} - m {ρ_{0}}^{2} \cos^{2} (λ ϕ_{2}) \\ + m ρ_{0} (r λ^{2} + λ^{2} ρ_{0} - r) \cos ϕ_{2} \cos (λ ϕ_{2}) \end{array}]

D 1 = \frac{1}{(λ^{2} - 1)} {\begin{cases} - m ρ_{0} \cos (λ ϕ_{1}) [\begin{array}{l} - 2 λ ρ_{0} \sin (λ ϕ_{1}) ({\dot{ϕ}}_{1} + \dot{α}) {\dot{ϕ}}_{1} \\ + (r λ^{2} + λ^{2} ρ_{0} - r) \sin ϕ_{1} {\dot{ϕ}}_{1} {\dot{ϕ}}_{1} \\ + 2 (r λ^{2} + λ^{2} ρ_{0} - r) \sin ϕ_{1} \dot{α} {\dot{ϕ}}_{1} \\ + g (\cos ϕ_{1} \cos α - \sin ϕ_{1} \sin α) \end{array}] \\ + m ρ_{0} \cos (λ ϕ_{2}) [\begin{array}{l} 2 λ ρ_{0} \sin (λ ϕ_{2}) {\dot{ϕ}}_{2} ({\dot{ϕ}}_{2} + \dot{α}) \\ - (r λ^{2} + λ^{2} ρ_{0} - r) \sin ϕ_{2} {\dot{ϕ}}_{2} {\dot{ϕ}}_{2} \\ - 2 (r λ^{2} + λ^{2} ρ_{0} - r) \sin ϕ_{2} \dot{α} {\dot{ϕ}}_{2} \\ + g (\cos ϕ_{2} \cos α - \sin ϕ_{2} \sin α) \end{array}] \\ - m ρ_{0} λ \sin (λ ϕ_{1}) [\begin{array}{l} (r λ^{2} + λ^{2} ρ_{0} - r) \cos ϕ_{1} {\dot{ϕ}}_{1} {\dot{ϕ}}_{1} \\ + g (\cos ϕ_{1} \sin α + \sin ϕ_{1} \cos α) \end{array}] \\ + m ρ_{0} λ \sin (λ ϕ_{2}) [\begin{array}{l} - \cos ϕ_{2} (r λ^{2} + λ^{2} ρ_{0} - r) {\dot{ϕ}}_{2} {\dot{ϕ}}_{2} \\ + g (\cos ϕ_{2} \sin α + \sin ϕ_{2} \cos α) \end{array}] \end{cases}} + μ_{α} \dot{α} - T

A 2 = \frac{1}{λ^{2} - 1} m ρ_{0} \cos (λ ϕ_{1}) [- ρ_{0} \cos (λ ϕ_{1}) + (r λ^{2} + ρ_{0} λ^{2} - r) \cos ϕ_{1}]

B 2 = m {ρ_{0}}^{2} \cos {(λ ϕ_{1})}^{2}

C 2 = 0

D 2 = {\begin{cases} \frac{m ρ_{0} [(r λ^{2} + ρ_{0} λ^{2} - r) \sin ϕ_{1} - λ ρ_{0} \sin (λ ϕ_{1})] \cos (λ ϕ_{1})}{λ^{2} - 1} {\dot{α}}^{2} \\ - \frac{m g ρ_{0} (λ - 1) (λ + 1) (\sin ϕ_{1} \sin α - \cos α \cos ϕ_{1}) \cos (λ ϕ_{1})}{λ^{2} - 1} \\ + μ_{s} {ρ_{0}}^{2} \cos^{2} (λ ϕ_{1}) {\dot{ϕ}}_{1} - m {ρ_{0}}^{2} λ \cos (λ ϕ_{1}) \sin (λ ϕ_{1}) {\dot{ϕ}}_{1}^{2} \\ + k_{s} ρ_{0} \cos (λ ϕ_{1}) {abs [\frac{ρ_{0}}{λ} \sin (λ ϕ_{1})] - S_{p}} sign [\sin (λ ϕ_{1})] \end{cases}}

A 3 = \frac{1}{λ^{2} - 1} m ρ_{0} \cos (λ ϕ_{2}) [(r λ^{2} + ρ_{0} λ^{2} - r) \cos ϕ_{2} - ρ_{0} \cos (λ ϕ_{2})]

B 3 = 0

C 3 = m {ρ_{0}}^{2} \cos^{2} (λ ϕ_{2})

D 3 = {\begin{cases} \frac{m ρ_{0} [(r λ^{2} + ρ_{0} λ^{2} - r) \sin ϕ_{2} - λ ρ_{0} \sin (λ ϕ_{2})] \cos (λ ϕ_{2})}{λ^{2} - 1} {\dot{α}}^{2} \\ + \frac{m g ρ_{0} (λ - 1) (λ + 1) (\sin ϕ_{2} \sin α - \cos α \cos ϕ_{2}) \cos (λ ϕ_{2})}{λ^{2} - 1} \\ + μ_{s} {ρ_{0}}^{2} \cos^{2} (λ ϕ_{2}) {\dot{ϕ}}_{2} - m {ρ_{0}}^{2} λ \cos (λ ϕ_{2}) \sin (λ ϕ_{2}) {\dot{ϕ}}_{2}^{2} \\ + k_{s} ρ_{0} \cos (λ ϕ_{2}) {abs [\frac{ρ_{0}}{λ} \sin (λ ϕ_{2})] - S_{p}} sign [\sin (λ ϕ_{2})] \end{cases}}

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