Sage Journals: Discover world-class research

Abstract

An ideal inerter has been widely used in various vibration isolation fields. Due to its mechanical construction, some nonlinear properties of the inerter need to be taken into consideration. This article concerns the nonlinearities of the fluid inerter and its impact on vehicle suspension performance. A complete nonlinear dynamic model of the fluid inerter is introduced, and the sensitive analysis of the designed parameters is studied. Comparing to the traditional passive suspension, the vehicle suspension performances of three different suspension layouts incorporating the nonlinear fluid inerter are investigated at different velocities and different spring stiffness. Furthermore, the influences of the nonlinearities in frequency domain are also noted. Results show that the nonlinear properties of the fluid inerter have a great effect on vehicle suspension performance except S2 suspension. The impact is more significant in low frequency domain of the vibration.

Keywords

Vehicle suspension fluid inerter nonlinear performance

Vehicle suspension is the general term of power transmission device between wheels and vehicle body. The performance of suspension has a great impact on the vehicle ride comfort, handle, and stability.¹ Active suspension² and semi-active suspension³ have been widely known and used in high-class passenger cars because a passive suspension system cannot preserve the two desired aims which are vehicle handling and passenger comfort very well. Recently, a new kind of passive vehicle suspension system employing the inerter, a new mechanical element, has been heatedly discussed.

Inerter was first proposed by Professor Smith⁴ in Cambridge University for the appreciation of the gap in the old analogy between mechanical and electrical networks. It was called “The missing mechanical circuit element,”⁵ in that the mass in mechanical circuit element cannot completely correspond to the capacitor in electric circuit element in “force-current” analogy for its one-terminal property. Some research has shown that the performance of traditional spring-damper network can be significantly improved by involving the inerter.

Until now, the inerter has been widely applied to various fields, such as passive network synthesis,^6–8 mechanical vibration isolation system including vehicle suspension,^9–13 train suspension,¹⁴ building suspension,¹⁵ and the steering compensation for high-performance motorcycles.¹⁶ Furthermore, the influence of inerter on the natural frequencies of vibration systems was investigated in Chen et al.¹⁷ to demonstrate that the inerter can reduce the natural frequencies of the vibration system. In Hu and Chen,¹⁸ the inerter-based dynamic vibration absorbers (IDVA) were proposed by replacing the damper in the traditional dynamic vibration absorber with some inerter-based mechanical networks and significant improvement for both the H_∞ and H₂ performances was obtained in the numerical simulation. The semi-active inerter was proposed and physically verified in Hu et al.¹⁹ to promote the development of inerter.

Inerter can be mechanically realized in various ways. Mechanical types of the inerter involve the rack-and-pinion inerter and the ball-screw inerter. In Papageorgiou and Smith,²⁰ the ball-screw inerter and the rack-and-pinion inerter were designed, and experiments were carried out to test their performance. Furthermore, the nonlinearities including friction and the elastic effect were taken into consideration in Wang and Su,²¹ the impact on vehicle suspension control was further investigated in time domain. A fluid inerter was presented in Glover and colleagues^22,23 and was pointed out that it had the advantage of simpler design than the former ones. Total damping force was analyzed due to the effect of helical channel, port losses, and the viscous shear friction. A nonlinear fluid inerter model with parasitic damping was built and variable orifices and valves were also included to provide series or parallel damping. It means that a fluid inerter can be functioned as an inerter, which is in parallel or in series with a damper. Therefore, a fluid inerter is more suitable for the application in vehicle suspension system because of the limited working space. It is of great interest to analyze the nonlinear properties and their impact on vehicle suspension performance.

In this article, the nonlinear model involving the friction and nonlinear damping force of the fluid inerter is introduced. On the basis of the experimental verification, the nonlinear model is applied to three different vehicle suspensions, and its influences on the vehicle suspension performance in both time domain and frequency domain are discussed. The article is hereafter structured as follows.

In section “Nonlinear fluid inerter,” a more complete nonlinear fluid inerter which involves the friction and nonlinear damping force is introduced and the sensitive analysis of the designed parameters is studied. Three different vehicle suspension layouts employing the nonlinear fluid inerter are built in section “Suspension model incorporating nonlinear inerter.” In section “Analysis of the nonlinearities of inerter,” the impact of the nonlinear properties on suspension performance at different velocities and different stiffness are investigated numerically. The frequency analysis is also carried out in this section, which differs from the other research. Finally, we draw some conclusion in the final section.

Nonlinear fluid inerter

The fluid inerter is similar with the rack-and-pinion inerter and the ball-screw inerter because the fluid flowing in the helical channel can be taken as a “fluid flywheel” to provide the inertia force. As detailed in Swift et al.,²³ the damping force is the main nonlinear property of the fluid inerter. The inertance and the damping coefficient will be synchronously changed when there is a variation of the designed parameters. It was noted that the viscous damping force which considered the secondary flow in helical channel was very close to the result of using Hagen–Poisseau formula for a straight tube of circular cross. Since the friction between the piston and cylinder cannot be ignored in low frequency,²¹ a more complete nonlinear model of the fluid inerter was proposed in Shen et al.²⁴ and illustrated in Figure 1.

Figure 1.

Nonlinear model of fluid inerter.

The force between the two terminals of the fluid inerter is given by

\begin{matrix} F = f + f_{c} + f_{b} \\ f = f_{0} sign ({\overset{\cdot}{z}}_{1} - {\overset{\cdot}{z}}_{2}) \\ f_{c} = C_{1} ({\overset{\cdot}{z}}_{1} - {\overset{\cdot}{z}}_{2})^{2} + C_{2} ({\overset{\cdot}{z}}_{1} - {\overset{\cdot}{z}}_{2}) \\ f_{b} = b ({\overset{\cdot\cdot}{z}}_{1} - {\overset{\cdot\cdot}{z}}_{2}) \end{matrix}

(1)

where z₁, z₂ are the displacements of the two nodes of the fluid inerter, f is the friction between the piston and the hydraulic cylinder, f_c is the nonlinear damping force, f_b is the inertia force, f₀ is the amplitude of the friction, and sign(v) is a symbolic function.²¹ The direction of friction f is determined by the velocity of the piston. b is the inertance, and C₁ and C₂ are the damping coefficients and can be calculated by the following equations²⁴

b = \frac{ρ S_{2} l}{1 + {(h / (2 π r_{4}))}^{2}} {(\frac{S_{1}}{S_{2}})}^{2}

(2)

C_{1} = \frac{3 ρ S_{1}^{3}}{4 S_{2}^{2}}

(3)

C_{2} = \frac{8 μ {lS}_{1}^{2}}{r_{3}^{2} S_{2}}

(4)

where S₁ is the effective area of the hydraulic cylinder, S₂ is the section area of the helical channel, ρ is the oil density, l is the length of helical channel, r₃ and r₄ are the radius of the helical channel and the helix radius of channel, h is the pitch of helical channel, and μ is the viscosity of fluid. In this article, S₁ is 0.002 m², S₂ is 0.0000785 m², ρ is 800 kg m⁻³, l is 8.99 m, r₃ is 0.005 m, r₄ is 0.1 m, h is 0.012 m, and μ is 0.027 Pa s. The fluid inerter was manufactured and the bench tests were carried out in Jiangsu University. The nonlinear model of the fluid inerter was verified and details are introduced in Shen et al.²⁴ The scheme of the fluid inerter and bench tests are depicted in Figure 2.

Figure 2.

Prototype of fluid inerter: (a) scheme of fluid inerter and (b) bench test.

From the nonlinear model of the fluid inerter, it can be seen that the friction is determined by the materials between the cylinder and the piston, and the damping coefficient C₁ is related to S₁ and S₂. When the device is completed, both of the f and C₁ cannot changed by ignoring the variation of the oil density and the viscosity of fluid. But for b and C₂, they all related to l. So, the relationship of b and C₂ can be given by

\frac{b}{C_{2}} = \frac{ρ r_{3}^{2}}{8 μ} \frac{1}{1 + {(h / 2 π r_{4})}^{2}}

(5)

It can be seen from equation (5) that b and C₂ will be synchronously changed by the variation of h and r₄. Different inertance and the corresponding C₂ can be obtained by tuning h and r₄ to build up the nonlinear model of the fluid inerter.

In order to show the different influences of h and r₄ on b and C₂, sensitive analysis is carried out and the results are shown in Figures 3 and 4. As h is usually larger than the twice of r₃, the range from −10% to +50% of r₄ and h are considered.

Figure 3.

Sensitive analysis of r₄.

Figure 4.

Sensitive analysis of h.

From the above figures, the variation of r₄ has a dramatic effect on the parameters b and C₂. But for h, there is little change in both b and C₂ when changing h. So, it can be concluded that the helix radius of channel r₄ is the most important factor to determine the inertia force and the damping force. Different parameters can be achieved by means of tuning r₄ when the device is completed.

Suspension model incorporating nonlinear inerter

The general quarter car model is illustrated in Figure 5, with dynamic equations as follows

\begin{matrix} m_{s} {\overset{\cdot\cdot}{z}}_{s} - F = 0 \\ m_{u} {\overset{\cdot\cdot}{z}}_{u} + K_{t} (z_{u} - z_{r}) + F = 0 \end{matrix}

(6)

where m_s is the sprung mass set as 320 kg, m_u is the unsprung mass set as 45 kg, K_t is the stiffness of the tire set as 190 kN m⁻¹, and F is the suspension force. z_s and z_u are the vertical displacements of the sprung mass and the unsprung mass. z_r is the road roughness displacement and can be gained by the equation

{\overset{\cdot}{z}}_{r} (t) = - 0.111 [v z_{r} (t) + 40 \sqrt{G_{q} v} w (t)]

(7)

where z_r(t) is the road roughness displacement, G_q is the road roughness coefficient, when the car is driving on a B-grade road, it is 64 × 10⁻⁶ m³ cycle⁻¹. v is the velocity, w(t) is the integral white noise.

Figure 5.

General quarter car model.

In order to analyze the effect of nonlinearities on vehicle suspension performance, four different suspension layouts are shown in Figure 6.

Figure 6.

Four configurations of suspension.

Note that S0 is the traditional passive suspension, the force F₀ is shown in equation (8). S1 is a basic parallel arrangement, and the force F₁ is shown in equation (9). S2 is a basic series arrangement, and the force F₂ is shown in equation (10). The S3 suspension, known as a TID system in Lazar et al.,²⁵ was also investigated in Shen et al.²⁶ in vehicle suspension. The force in the S3 structure F₃ is shown in equation (11), where z_b is the vertical displacement of the inerter

F_{0} = K (z_{u} - z_{s}) + c ({\overset{\cdot}{z}}_{u} - {\overset{\cdot}{z}}_{s})

(8)

\begin{array}{l} F_{1} = K (z_{u} - z_{s}) + c ({\dot{z}}_{u} - {\dot{z}}_{s}) + f_{0} sign ({\dot{z}}_{u} - {\dot{z}}_{s}) \\ + b ({\overset{\cdot\cdot}{z}}_{u} - {\overset{\cdot\cdot}{z}}_{s}) + C_{1} {({\dot{z}}_{u} - {\dot{z}}_{s})}^{2} + C_{2} ({\dot{z}}_{u} - {\dot{z}}_{s}) \end{array}

(9)

{\begin{matrix} F_{2} = K (z_{u} - z_{s}) + c ({\overset{\cdot}{z}}_{u} - {\overset{\cdot}{z}}_{b}) \\ c ({\overset{\cdot}{z}}_{u} - {\overset{\cdot}{z}}_{b}) = f_{0} sign ({\overset{\cdot}{z}}_{b} - {\overset{\cdot}{z}}_{s}) + b ({\overset{\cdot\cdot}{z}}_{b} - {\overset{\cdot\cdot}{z}}_{s}) + C_{1} {({\overset{\cdot}{z}}_{b} - {\overset{\cdot}{z}}_{s})}^{2} + C_{2} ({\overset{\cdot}{z}}_{b} - {\overset{\cdot}{z}}_{s}) \end{matrix}

(10)

{\begin{matrix} F_{3} = K (z_{u} - z_{s}) + K_{2} (z_{u} - z_{b}) + c ({\overset{\cdot}{z}}_{u} - {\overset{\cdot}{z}}_{b}) \\ K_{2} (z_{u} - z_{b}) + c ({\overset{\cdot}{z}}_{u} - {\overset{\cdot}{z}}_{b}) = f_{0} sign ({\overset{\cdot}{z}}_{b} - {\overset{\cdot}{z}}_{s}) + b ({\overset{\cdot\cdot}{z}}_{b} - {\overset{\cdot\cdot}{z}}_{s}) + C_{1} {({\overset{\cdot}{z}}_{b} - {\overset{\cdot}{z}}_{s})}^{2} + C_{2} ({\overset{\cdot}{z}}_{b} - {\overset{\cdot}{z}}_{s}) \end{matrix}

(11)

Analysis of the nonlinearities of inerter

In order to evaluate the vehicle suspension performance, three performance indexes are considered as follows.

J ₁ represents the root mean square (RMS) of the vehicle body acceleration

J_{1} = \sqrt{E [{\overset{\cdot\cdot}{z}}_{s}^{2}]}

(12)

J ₂ represents the RMS of the suspension deflection

J_{2} = \sqrt{E [{(z_{s} - z_{u})}^{2}]}

(13)

J ₃ represents the RMS of the dynamic tire load

J_{3} = \sqrt{E [K_{t} {(z_{r} - z_{u})}^{2}]}

(14)

For the vehicle suspension system, a spring is usually used to support the weight of vehicle body. In order to analyze the different effect of the nonlinearities of the fluid inerter on vehicle suspension performance, the spring stiffness is fixed. A traditional passive suspension from a passenger car is made as a comparison, and the parameters of the linear suspension S1, S2, and S3 are optimized by means of genetic algorithm, which are detailed in Shen et al.²⁶

In the optimization, the multi-objective is changed to single-objective, the objective function is set as the linear combination of the performance indexes of the vehicle suspension employing inerter and the passive suspension as shown in equation (15)

min J = \frac{BA}{B A_{pas}} + \frac{SWS}{SW S_{pas}} + \frac{DTL}{DT L_{pas}}

(15)

where BA, SWS, and DTL are the RMS of the body acceleration, suspension deflection, and dynamic tire load of the suspension employing inerter, respectively. BApas, SWSpas, and DTLpas are the RMS of the body acceleration, suspension deflection, and dynamic tire load of the passive suspension, respectively.

According to equations (2)–(4), it is noted that b and C₂ will be synchronously changed by the variation of l, which is related to h and r₄ from Shen et al.²⁴ The friction f₀ and C₁ are fixed, and the different b and C₂ are obtained according to the relationship from equation (5). The parameters of the several suspensions are shown in Table 1.

Table 1.

Suspension parameters.

	K (kN m⁻¹)	K ₂ (kN m⁻¹)	C (N s m⁻¹)	b (kg)	C ₁ (N s² m⁻²)	C ₂ (N s m⁻¹)	f ₀ (N)
S0	22	–	1000	–	–	–	–
S1	22	–	2800	80	790	874	400
S2	22	–	1500	515	790	5566	400
S3	22	10	1067	225	790	2437	400

The nonlinearities of a ball-screw inerter, namely dry friction and elastic effect, were separately discussed in Wang and Su,²¹ and the different values of the friction and elastic effect are taken into consideration. For the fluid inerter, the effect of the friction and the nonlinear damping force are also investigated separately.

When the vehicle is driving on a B-grade road at a velocity of 20 m s⁻¹, Table 2 shows the performance indexes of the different suspensions. The ideal suspensions are the linear system without considering the nonlinearities. Then, the nonlinearities shown in Table 1 are added in the systems to show the influence of them on the performance indexes. Three scenarios, namely inerter with the friction (f_b + f), referred to equation (1), inerter with the damping force (f_b + f_c) and inerter with both the friction and the damping force (f_b + f_c + f), are taken into consideration to evaluate its impact on the suspension performance.

Table 2.

Performance indexes.

	J ₁ (m s⁻²)	Impro (%)	J ₂ (m)	Impro (%)	J ₃ (N)	Impro (%)
S0	1.4641	–	0.0146	–	1007	–
S1 (ideal)	2.5540	−74.44	0.0086	41.10	1222	−21.35
S1 (friction)	2.7996	−91.22	0.0072	50.68	1194	−18.57
S1 (damping)	2.6496	−80.97	0.0073	50.00	1181	−17.28
S1 (both)	2.9776	−103.40	0.0064	56.16	1213	−20.46
S2 (ideal)	1.4639	0.013	0.0123	15.75	888	11.82
S2 (friction)	1.5160	−3.55	0.0120	17.81	889	11.72
S2 (damping)	1.5080	−3.00	0.0126	13.70	896	11.02
S2 (both)	1.5167	−3.60	0.0119	18.49	888	11.82
S3 (ideal)	1.3855	5.37	0.0111	23.97	945	6.16
S3 (friction)	1.7213	−17.57	0.0140	4.11	992	1.49
S3 (damping)	1.5576	−6.39	0.0131	10.27	975	3.18
S3 (both)	1.7403	−18.86	0.0140	4.11	994	1.29

Comparing with the passive suspension, there is 41.10% decrease in J₂ value for ideal S1 suspension, while for the body acceleration and the dynamic tire load, it is usually larger than the passive suspension when the spring stiffness is not very large. It can be seen that the nonlinearities of the fluid inerter sharply increase the J₁, and the friction plays a more important role than the nonlinear damping force. But for J₂ and J₃, they are decreased by the nonlinearities.

For S2 analyses, there is a slight increase in J₁ and a negligible variation in J₃ when considering the nonlinearities of the fluid inerter. A decrease in J₂ can also be seen except the scenario of involving the damping force only.

For S3 layout, all the performance indexes J₁, J₂, and J₃ are increased. Comparing with the passive suspension, the performance improvements are changed from 5.37% to −18.86% of J₁, from 23.97% to 4.11% of J₂, and from 6.16% to 1.29% of J₃. It can be also inferred that the friction has a more effective influence than the nonlinear damping force.

It is noted that the friction cannot be ignored in the nonlinear model, and the three suspensions employing a nonlinear fluid inerter are all greatly influenced by the nonlinearities except S2, on which the impact is quite slight.

Performance evaluation at different velocities

In this section, the suspension performances are analyzed at different velocities. This article assumes that the vehicle is driving on a B-grade road, Figures 7 –9 show the performance indexes and the improvement comparing to the passive suspension S0 at velocities from 5 to 50 m s⁻¹, where S1(N), S2(N), and S3(N) represent for the suspensions involving a nonlinear fluid inerter.

Figure 7.

(a) J₁ at different velocities and (b) improvement of J₁ at different velocities.

Figure 8.

(a) J₂ at different velocities and (b) improvement of J₂ at different velocities.

Figure 9.

(a) J₃ at different velocities and (b) improvement of J₃ at different velocities.

From Figures 7 –9, J₁ of S1(N), S2(N), and S3(N) are all increased by considering the nonlinearities of the fluid inerter. But the variation of S2(N) is smaller than that of S1(N) and S3(N). For J₂, all the performance indexes are smaller than S0. S3(N) is larger than S3, while the results for S1(N) and S2(N) are the contrary. Compared with the variation of S1(N) and S3(N), the change of S2(N) is not obvious. For J₃, there is a slight decrease in S1(N), and a dramatic increase in S3(N). At the same time, the variation of S2(N) is negligible.

Performance evaluation at different stiffness

There is a wide range of the spring stiffness from a passenger car to a sports car. The suspension performance at different stiffness by considering the nonlinear effect will also be discussed. Note that there is another spring K₂ in S3 suspension. The effect of the K₂ stiffness on S3 suspension will be investigated first. The performance indexes are shown in Figures 10 –12 with the variation of K₂.

Figure 10.

J ₁ at different K₂ stiffness.

Figure 11.

J ₂ at different K₂ stiffness.

Figure 12.

J ₃ at different K₂ stiffness.

It can be seen that J₁ will decrease at first and then dramatically increase. The same trend can be seen for J₂ value. But for J₃, it will continuously decrease with the increase in the K₂ stiffness. Note that when K₂ stiffness is closed to the main spring stiffness K = 22,000 N m⁻¹, a relative small value of J₂ and J₃ can be obtained. However, a large J₁ value, which is even larger than the passive suspension, is achieved. The results show that the K₂ stiffness plays an important role when selecting the parameters of the S3 suspension. Therefore, the K₂ stiffness is determined by the optimization and fixed when analyzing the suspension performance at different main spring stiffness.

Figures 13 –15 show the performance indexes and the improvement comparing to S0 at stiffness from 10 to 50 kN m⁻¹.

Figure 13.

(a) J₁ at different stiffness and (b) improvement of J₁ at different stiffness.

Figure 14.

(a) J₂ at different stiffness and (b) improvement of J₂ at different stiffness.

Figure 15.

(a) J₃ at different stiffness and (b) improvement of J₃ at different stiffness.

At different main spring stiffness, the J₁ values of nonlinear suspensions are all larger than the linear suspension. But for J₂, the nonlinear suspensions S1(N) and S2(N) are smaller than the linear suspensions S1 and S2, and also the same case of S3(N) when the spring stiffness is very small. With the increase in the spring stiffness, the J₂ value of S3(N) is larger than that of S3. Meanwhile, all the J₂ values are smaller than that of S0 except S3 and S3(N) with a small stiffness. For J₃ analyzes, the S1(N) and S2(N) are smaller than S1 and S2 with a small stiffness, while the results are contrary with a large stiffness. But for S3, the J₃ value of S3(N) is always larger than that of S3 considering the nonlinearities’ effect. It is noted that, for S2(N), all of the variations in J₁, J₂, and J₃ are relatively small compared with that in S1(N) and S3(N).

To conclude, the influences of the fluid inerter nonlinearities on the suspension performance indexes J₁, J₂, and J₃ are quite complex. The J₁ values are all increased by taking the nonlinear effect into consideration. For J₂, the nonlinear factors will decrease the suspension deflection of S1 and S2 at different velocities and different spring stiffness. However, the J₂ of S3 will become larger except with the small spring stiffness. For J₃, it will dramatically increased in S3(N), while there is a slight effect on S1 and S2 at different velocities and different stiffness. Results show that the S2 suspension’s performances are rarely affected by the nonlinearities of fluid inerter compared with S1 and S3.

Frequency analysis

The analysis above are all in time domain, but an interesting question is, whether the nonlinearities of the fluid inerter will have a great impact on the suspension performance in low frequency or in high frequency. In order to solve this problem, the performance indexes which are greatly influenced by the nonlinearities, namely J₁ and J₂ of S0, S1, S1(N), and S3, S3(N), are selected to analyze the frequency changes. Figures 16 and 17 show the power spectrum density (PSD) of vehicle body acceleration and suspension deflection.

Figure 16.

PSD of body acceleration.

Figure 17.

PSD of suspension deflection.

From the PSD of body acceleration and suspension deflection, it shows that the nonlinearities of the fluid inerter do have a great impact on the vibration isolation performance of the suspension in the frequency domain. Note that, for the vehicle body acceleration, the PSD of linear S1 and S3 suspensions are all less than the nonlinear model S1(N) and S3(N) suspension, especially in the low frequency domain comparing to the passive suspension S0. For the suspension deflection, the PSD of S1, S1(N), S3, and S3(N) are all less than S0, especially the peak value in the low frequency. While the PSD of linear S3 suspension is less than that of S3(N) suspension, the PSD of linear S1 suspension is much larger than that of S1(N). All of the impacts are more significant in the low frequency regardless of it is becoming larger or smaller.

On the basis of the analysis, it is noted that, for S1 suspension, there is a dramatic decrease in J₂ value at different velocities and different stiffness, while J₁ and J₃ values will become larger at the same time. It is not a good choice to improve the suspension deflection with the cost of degrading the body acceleration and dynamic tire load. For S2 suspension, it is the best one to be applied to the vehicle suspension design because it is rarely influenced by the nonlinearities of the fluid inerter. At last, for S3 suspension, all the performance indexes will be increased by the nonlinearities of fluid inerter except the J₂ when the stiffness is quite small. So the nonlinear model of the fluid inerter should be seriously taken into consideration when S3 is applied to the vehicle suspension.

Conclusion

In this article, the nonlinear properties of the fluid inerter, namely the friction and the nonlinear damping force, were taken into consideration to analyze their impact on vehicle suspension performance. The nonlinear model of the fluid inerter was introduced and sensitive analysis of the designed parameters was carried out to study the relationship between inertance and the damping coefficient. Considering the nonlinear effects, three measurements of the vehicle suspensions at different velocities and different spring stiffness were discussed. Results showed that J₁ values were all increased by the nonlinear effect. But for J₂, it was decreased in S1(N) and S2(N), while increased in S3(N) except small spring stiffness setting. For J₃, there was a slight variation in S1(N) and S2(N), but a dramatic increase in S3(N). In general, the nonlinearities’ impact on S2 suspension was not obvious, compared to that on S1 and S3. When a fluid inerter is applied to the vehicle suspension, S2 structure is a better choice to avoid the influence of the nonlinearities. Furthermore, frequency domain analysis was also investigated and it was shown that the effects of nonlinearities of fluid inerter were more significant in low frequency.

Footnotes

Handling Editor: Anand Thite

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 supported by the National Natural Science Foundation of China (Grant No. 51405202). Yujie Shen is also supported by the Scientific Research Innovation Projects of Jiangsu Province (Grant No. KYLX15_1081) and the China Scholarship Council.

References

Rajamani

. Vehicle dynamics and control. New York: Springer, 2012.

Pan

Sun

Gao

et al . Finite-time stabilization for vehicle active suspension systems with hard constraints. IEEE T Intell Transp 2015; 16: 2663–2672.

Chen

Long

Yuan

et al . Non-linear modelling and control of semi-active suspensions with variable damping. Vehicle Syst Dyn 2013; 51: 1568–1587.

Smith

. Synthesis of mechanical networks: the inerter. IEEE T Automat Contr 2002; 47: 1648–1662.

Chen

MZQ

Papageorgiou

Scheibe

et al . The missing mechanical circuit element. IEEE Circ Syst Mag 2009; 9: 10–26.

Chen

MZQ

Wang

Zou

et al . Realization of three-port spring networks with inerter for effective mechanical control. IEEE T Automat Contr 2015; 60: 2722–2727.

Chen

MZQ

Wang

Shu

et al . Realizations of a special class of admittances with strictly lower complexity than canonical forms. IEEE T Circuits: I 2013; 60: 2465–2473.

Chen

MZQ

Wang

Zou

et al . Realization of a special class of admittances with one damper and one inerter for mechanical control. IEEE T Automat Contr 2013; 58: 1841–1846.

Smith

Wang

. Performance benefits in passive vehicle suspensions employing inerters. Vehicle Syst Dyn 2004; 42: 235–257.

10.

Scheibe

Smith

. Analytical solutions for optimal ride comfort and tyre grip for passive vehicle suspensions. Vehicle Syst Dyn 2009; 47: 1229–1252.

11.

Soong

Ramli

Mahadi

WNLW

. Vehicle suspensions with parallel inerter: effectiveness in improving vibration isolation. J Vibroeng 2014; 16: 256–265.

12.

Zhang

Ahmadian

Guo

. On the benefits of semi-active suspensions with inerters. Shock Vib 2012; 19: 257–272.

13.

Liang

Wang

et al . Vibration suppression using two-terminal flywheel. Part II: application to vehicle passive suspension. J Vib Control 2012; 18: 1353–1365.

14.

Wang

Liao

et al . The performance improvements of train suspension systems with mechanical networks employing inerters. Vehicle Syst Dyn 2009; 47: 805–830.

15.

Wang

Hong

Chen

. Building suspensions with inerters. Proc IMechE, Part C: J Mechanical Engineering Science 2010; 224: 1605–1616.

16.

Evangelou

Limebeer

DJN

Sharp

et al . Steering compensation for high-performance motorcycles. In: Proceedings of the 43rd IEEE conference on decision and control, Nassau, Bahamas, 14–17 December 2004, pp.749–754. New York: IEEE.

17.

Chen

MZQ

Huang

et al . Influence of inerter on natural frequencies of vibration systems. J Sound Vib 2014; 333: 1874–1887.

18.

Chen

MZQ

. Performance evaluation for inerter-based dynamic vibration absorbers. Int J Mech Sci 2015; 99: 297–307.

19.

Chen

MZQ

et al . Semiactive inerter and its application in adaptive tuned vibration absorbers. IEEE T Contr Syst T 2017; 25: 294–300.

20.

Papageorgiou

Smith

. Laboratory experimental testing of inerters. In: Proceedings of the 44th IEEE conference on decision and control, and the European control conference, Seville, 15 December 2005, pp.3351–3356. New York: IEEE.

21.

Wang

. Impact of inerter nonlinearities on vehicle suspension control. Vehicle Syst Dyn 2008; 46: 575–595.

22.

Glover

Smith

Houghton

. Force-controlling hydraulic device. International patent application no: PCT/GB2010/001491 (US Patent), 2010.

23.

Swift

Smith

Glover

et al . Design and modelling of a fluid inerter. Int J Control 2013; 86: 2035–2051.

24.

Shen

Chen

Liu

et al . Optimized modeling and experiment test of a fluid inerter. J Vibroeng 2016; 18: 2789–2800

25.

Lazar

Neild

Wagg

. Using an inerter-based device for structural vibration suppression. Earthq Eng Struct D 2014; 43: 1129–1147.

26.

Shen

Chen

Yang

et al . Improved design of dynamic vibration absorber by using the inerter and its application in vehicle suspension. J Sound Vib 2016; 361: 148–158.

Influence of fluid inerter nonlinearities on vehicle suspension performance

Abstract

Keywords

Nonlinear fluid inerter

Suspension model incorporating nonlinear inerter

Analysis of the nonlinearities of inerter

Performance evaluation at different velocities

Performance evaluation at different stiffness

Frequency analysis

Conclusion

Footnotes

Declaration of conflicting interests

Funding

References