Study on the vibration isolation characteristics of an anti-resonant hydropneumatic suspension

: A novel anti-resonant hydropneumatic suspension system was presented in this study. The mathematical model of the presented suspension system was derived and the theoretical analysis was carried out. For lower frequency vibrations, the presented suspension system can achieve better vibration isolation performance comparing with the traditional hydropneumatic suspension system. The frequency spectrum on which the presented suspension achieves better vibration isolation performances can be widened by adjusting the length of the oil tube. Therefore, it is possible for the presented suspension system to achieve better vibration isolation performance on different road conditions.


Introduction
Nowadays, the suspension system is widely used in vehicles in order to separate the passengers from road shocks and the vibration that is arising due to the rough conditions of the road. The suspension system in any vehicle is most important because it provides ride comfort and safety towards the passenger and driver. Such a system reduces fatigue during the driving conditions. To improve the performance of the suspension system, the active or semi-active suspension control technique was used recently [1][2][3]. A hydropneumatic suspension system is a type of semi-active controlled suspension. This type of suspension has some important and good properties which are mostly used in tracked vehicles and also it improves ride comfort. Such properties are non-linear stiffness and damping, convenient turning, vertical position locking [4,5]. For traditional hydropneumatic suspension system, smaller stiffness must be adapted to isolate a vibration at a lower frequency. The smaller stiffness of the suspension system leads to larger deflection, which cannot allow them too much varying load to the vehicle. So there is a need for such a suspension system which could significantly counter the varying load problem.
Anti-resonant vibration isolation system, which uses the inertial force to cancel the spring force, can isolate a vibration at a lower frequency with higher stiffness. The description of the mechanism of the anti-resonant vibration isolator can be found in the references [6][7][8]. In the designing of anti-resonance vibration isolators, leverage was often used to adjust the effective mass of the system. Comparing with mechanical leverage, hydraulic ones have the advantage of a compact arrangement. Referring to [9,10] can get the details of the hydraulic leverage.
In this paper, the mechanism of the anti-resonance was introduced to a hydropneumatic suspension system and presented a new anti-resonant hydropneumatic suspension. The mathematic model of the anti-resonant hydropneumatic suspension was derived. The non-dimensional transformation function was used to assess the vibration isolation performance of the hydropneumatic suspension. The spectrum width of the frequency at which the presented anti-resonant hydropneumatic suspension can achieve a better performance than a traditional one was discussed. The effects of the design variables on the non-dimensional transformation function were also presented.

Mathematical model
The presented anti-resonant pneumatic suspension system contains a piston with diameter D, which is inserted in an oil cylinder. The upper part of the piston is an air chamber with height H. The oil in the oil cylinder can flow into the air chamber through an oil tube with diameter d and length l (Fig. 1).
The piston and the oil cylinder connect the vehicle body and the wheels, respectively. As the piston moves down, the oil in the cylinder will move upward through the oil tube into the air chamber so that the air in the air chamber is compressed. On the opposite side, the air in the air chamber will expand.
In the static state, the force acted on the piston by the vehicle body is denoted as G 0 and the air pressure in the air chamber p g0 can be expressed as where ρ o is the density of the oil, m p is the mass of piston and p a is the barometric pressure.
Assuming that the vehicle moves with a constant horizontal velocity U, the motion equation of piston is derived in the Cartesian coordinate system with a velocity of U. The momentum equation of the piston along the vertical direction can be written as where z cp is the vertical displacement of the piston, p gt is the relative pressure acted on the top of the air chamber by the air, p gb is the relative pressure acted on the bottom of the air chamber by the oil, p pb is the relative pressure acted on the bottom of the piston by the oil, and ΔG is the dynamic force acted on the piston by the vehicle body.
Considering the effect of the viscosity, the unsteady Bernoulli equation can be written as where v is the relative velocity of the oil in the oil tube and ζ is the viscous dissipation coefficient. As the piston vibrates, the adiabatic process can be used to describe the variation of p gt where V g is the volume of the gas in the air chamber, V g0 is the value of V g at static, and γ is the adiabatic index.
Substituting (1), (3), and (4) into (2), gives The volume of the gas in the air chamber can be calculated as where z o is the vertical displacement of the oil in the cylinder. The relative velocity of the oil in the oil tube can be calculated as Substituting (6) and (7) into (5) gives The order ratio of the second term (viscous dissipation term) to the third term in the right-hand side of (8) where F ex represents the excitation force acted on the suspension system. Imaging that a vehicle runs on the smooth surface road (z o = 0), there is no excitation force acted on the suspension system. Thus the vertical displacement of the piston and the dynamic force acted on vehicle body approach zero. Therefore, the comfortability of the vehicle can be improved by decreasing the value of F ex .

Performance of the suspension system
As a vehicle moves on a rough surface road, the vertical displacement of the oil cylinder can be expressed as where ω is the angular frequency, A(ω) is the amplitude of the variation of z o . The distribution of A(ω) is obtained by the vehicle speed and the road surface roughness power spectrum. Substituting (10) into (9) gives the expression of the excitation force acted on the suspension system where R(ω) is the transformation function. For the presented hydropneumatic suspension, spring force (calculated by the first term of the magnification ratio) and inertia force (calculated by the second term of the magnification ratio) contribute to the excitation force. For the traditional hydropneumatic suspension, only spring force contributes to the excitation force. To compare the behaviour of the presented suspension to the traditional hydropneumatic suspension, the non-dimensional transformation function R * = 16RV g0 /(γπ 2 D 4 p go ) is discussed. The non-dimensional transformation function represents the ratio of the excitation force acted on the anti-resonant hydropneumatic suspension to that acted on the traditional one. R * < 1 means that the performance of the anti-resonant hydropneumatic suspension is better than that of the traditional one. The value of R * can approaches zero at an optimal frequency ω op (p g0 D/ ρ o V g0 ) 1/2 because the spring force is canceled by the inertia force. The non-dimensional optimal frequency ω op , at which the excitation force equals zero, can be calculated as Fig. 2 gives the variation of R * as the function of non-dimensional frequency ω * (ω * = ω/(p g0 D/ ρ o V g0 ) 1/2 ). It is seen that R * < 1 as the non-dimensional frequency is less than a critical ω cr , which can be calculated by ω cr = 2ω op . Therefore, the lower frequency vibration (ω * < ω cr ) can be reduced by the anti-resonant hydropneumatic suspension.
The width of the frequency spectrum on which the anti-resonant hydropneumatic suspension achieves better performance can be expanded by increasing ω cr . Adopting a smaller l can achieve a larger value of ω cr . However, the values of R * at lower frequencies (ω * < ω op ) increase with the decreasing of l. Fig. 3 shows the

Conclusion
A novel anti-resonant hydropneumatic suspension system was presented in this paper. The non-dimensional transformation function was used to assess the performance of the presented suspension. We concluded the following points.
i. As the vibration frequency is lower than a critical value, the vibration isolation performance of the presented anti-resonant hydropneumatic suspension is better than that of the traditional one. ii. The value of critical frequency can be increased by decreasing the length of the oil tube to widen the frequency spectrum on which the presented suspension achieves better vibration isolation performances. iii. The value of non-dimensional transformation function at lower frequencies increases with the decrease of oil tube length. To achieve the best performance of the presented suspension system, the oil tube length must be adjusted according to different road conditions.