This paper introduces a new active hydro pneumatic suspension system (HFD) and examines

the dynamic behavior of the system. The HFD is developed at the Technische Universität

Darmstadt within the Collaborative Research Centre (SFB) 805, supported by Deutsche

Forschungsgemeinschaft (DFG). Unlike other active suspension systems, this system is

characterized by a reduced complexity. This reduced complexity is succeeded by the integration

of the actuator inside the system. Hence, pumps, hoses, filters or tanks can be omitted. Thereby

a control of uncertainties is intended by the reduction of components fraught with uncertainty. In

addition to this research focus the design of the HFD leads to new functions like active vibration

control, stiffness control and the separation of hardware and function. The latter one means that

it is possible to adapt the HFD to varying customer demands (such as sport or comfort set up)

without any modifications of the hardware

Bedarff, Thomas

Pelz, Peter F.

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An active suspension with reduced complexity

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An active suspension with reduced complexity Thomas Bedarff Peter Pelz Institut für Fluidsystemtechnik, Technische Universität Darmstadt, Darmstadt, Germany ABSTRACT This paper introduces a new active hydro pneumatic suspension system (HFD) and examines the dynamic behavior of the system. The HFD is developed at the Technische Universität Darmstadt within the Collaborative Research Centre (SFB) 805, supported by Deutsche Forschungsgemeinschaft (DFG). Unlike other active suspension systems, this system is characterized by a reduced complexity. This reduced complexity is succeeded by the integration of the actuator inside the system. Hence, pumps, hoses, filters or tanks can be omitted. Thereby a control of uncertainties is intended by the reduction of components fraught with uncertainty. In addition to this research focus the design of the HFD leads to new functions like active vibration control, stiffness control and the separation of hardware and function. The latter one means that it is possible to adapt the HFD to varying customer demands (such as sport or comfort set up) without any modifications of the hardware. NOMENCLATURE A Area m² tA pressure exposed area m² vA Fluid displacing area m² wA Surface for heat exchange m² fb Diameter of the bellow convolution m dcc, Stiffness, stiffness at design point N/mm  Licence: CC BY 4.0 / Creative Commons Attribution 4.0 InternationalD  Diameter m ff ,  Frequency, cut-off frequency 1/s dFFFˆ,ˆ,  Force, Force amplitude, Force amplitude at design point N g  gravity acceleration m/s² k  Heat transfer coefficient W/m²K l  Length of the bellow in contact with the piston m ,n  Polytrophic, isentropic  exponent - 0, pp  Pressure, pressure at initial state Pa P  Power W vq  Volume flow rate due to the deflection m³/s fq  Volume flow rate due to the piston widening m³/s aq  Volume flow rate into the hydraulic accumulator m³/s rr ˆ,  Radius of the piston, radius amplitude m mr  Piston radius plus half of the bellow convolution diameter m tr  Bearing radius m ar  Radius of the surrounding tube m R  Specific gas constant J/(kg K) ss ˆ,   Deflection, deflection amplitude m T  Temperature K u  Velocity m/s 0,VV  Volume, volume at initial state m³   ratio of radii -   compressibility m/Pa   Density of the fluid kg/m³    1 INTRODUCTION Every technical system is affected by uncertainties, during the processes of construction, manufacturing and usage. To handle these uncertainties the Deutsche Forschungsgemeinschaft (DFG) is supporting a special research project (Sonderforschungsbereich SFB805, www.sfb805.tu-darmstadt.de) at the Technische Universität Darmstadt. Within this project an active suspension system will be developed, being a hydro pneumatic spring-damper-system with reduced complexity. On the one hand, the reduced complexity leads to fewer uncertainties, on the other hand the innovative design gives a chance to better functions compared with today’s suspension components. To achieve the reduced complexity, hydraulic infrastructure systems like pumps, hoses, accumulators and tanks are being replaced by an integrated actuator. This actuator implements the functions of leveling as well as the active reaction on dynamic loads by widening the piston radius.  To examine the fundamental behavior and causal relations of such a system, first studies were done with analytic calculations, 0-D dynamic simulations (Modelica/Dymola), and 3-D Finite Element Method (ABAQUS). With the analytic calculations, the correlation between the piston radius, the pressure and the related force is investigated. The 0-D dynamic model deals with the dynamic behavior of the system; the 3-D FEM model is used to determine the forces during the piston widening. 2 THE HYDROPNEUMATIC SPRING-DAMPER SYSTEM – OVERVIEW The HFD (Figure 1) is made up of the four main parts: bellow, piston rod with two pistons, restrictor and hydraulic accumulator. Either one or both pistons have an integrated actuator to increase or decrease the piston radius. The bellow is fixed at both pistons and is in contact with the surrounding tube which significantly increases the burst pressure (/Pel07/). With deflection, the upper piston displaces fluid, the lower one opens space for fluid. The displacement surface for each piston is the piston area minus the piston rod cross sectional area plus half of the surface projected by the bellow (membrane) convolution. The total displacement surface is the difference between these two displacement surfaces. Since the diameter of the lower piston is smaller than the upper one, a volume is displaced. This displaced volume results in a pressure increase inside the hydraulic accumulator. The pressure acts on the effective bearing surface and results in a force. The damping force is either the result of a restriction of the cross sectional area or, in further project phases, the result of an applied potential or current when using electrorheological or magnetorheological Fluids (ERF or MRF). 3 STIFFNESS Each passive or semi active suspension system has three main functions: Load carrying, energy conservation and energy dissipation. This chapter deals with the dynamic stiffness (0.01Hz…20Hz) and the eigen frequency considering the piston widening and varying loads. All calculations assume one piston with fixed radius (piston 2) and one with variable radius (piston 1).  Figure 2 shows the used parameters for the following calculations. The variable parameter is the radius of piston 1, 1r . In the initial state, all parameter are set in order to obtain a pressure-resulted force of 7500N at a deflection of 76mm. The radii 21  , aa rr  of the supporting tube remain constant, in the initial state 21  , aa rr  are 12mm (= bf) larger than 21,rr . As shown in /Pel01/, the fluid displacing area vA  is given by the piston area plus half of the projected area of the bellow convolution s1vq 1fq2vqaq1r1ar1tr1mr2r2tr2arfb Figure 2: Notations. piston with variable radiushydraulic accumulatorrestrictorFFF d ~AAA d  11~UAAA d  22~ Figure 1: Schematic diagram of the HFD.  222122212rrrrA aav . (1) For cross layer bellows, the radius of the pressure exposed area tA  is approximately the piston radius plus 1/3 of the bellow convolution diameter fb       222211 229rrrrA aat , (2) with constrrr aa 221 ,, . In general, the stiffness is given by dsdFc   with tAdpdF  . Using the total derivative of the polytrophic relation 0)( 1   dVnpVdpVpVd nnn , the differential deflection vAdVds   and the pressure dp  at the design point (deflection dss  ) with the vehicle load F , the stiffness becomes a function of the initial states 0p  and 0V  and the cross sectional areas vA  and tA which both are functions of 1r :   tvvdvtvtAApVnFAsVAAVnpAAVnpdsdFc0022000 , (3) for a cylindrical l piston. By doing so, we assume that the vibration’s frequency is higher than the cut-off frequency f  which marks the change between isothermal and adiabatic behavior. At f , the loss angle has its first maximum, as shown in chapter 4 Hzf 1.0 . Assuming the load F is due to a heavy mass with gmF   the system’s eigen frequency ( )2/(/ Fgcf  ) is shown in Figure 3 for three different cases as a function of the stiffness and the eigen frequency ( )2/(/ Fgcf  ) of F  after leveling: leveling with constant gas mass (mgas = const, typical hydro pneumatic),  constant gas volume (Vgas = const, typical air spring) (/Eul03/) and with variable piston radii (HFD). Obviously, the behavior of the HFD is similar to an air spring, with tendency to a hydro pneumatic system with preferred constant remaining eigen frequency. It has to point out, that the piston radius remains constant during deflection at these studies. A variable piston radius during deflection is investigated in chapter 4.  0.8 1 1.2 1.4 1.60.90.9511.051.11.151.21.25F / Fdf / f d  HFDmgas= constVgas = const0.8 1 1.2 1.4 1.60.511.522.5F / Fdc / cd  HFDmgas= constVgas = const Figure 3: Stiffness ratio (left) and eigen frequency ratio for the HFD compared with common leveling systems. Figure 4 shows the general effect of the piston widening. For a given fixed radii ratio 21 rr  of the two piston radii and a given force F , a change of the ratio to 21~~~ rr  results in a force F~.  If for example the ratio   is 1.1 a widening about 5% of this ratio (   251~  E ) causes a force F~  that is 25% larger than F , i.e. it is possible to vary the force in a wide range with slight changes of the piston radius. The most important task of research is the design of the adjustable piston. Since only slight changes of the piston radius are necessary, it might 10-310-210-10.911.11.210-310-210-10.911.11.21.011.021.051.11.251.52FF~21 rr  1~  Figure 4: Correlation between piston widening and Force  be possible to use piezo-actuators for the widening. Figure 5 (left) shows an early design study for such a system (see also /Pel08/): the piston consists of an elastomer ring with inlaid piezo-actuators. If the actuators increase in length, they extrude the elastomer radially and the piston radius increases. In Figure 5, the elastomer ring affects on radially ordered lamellae. These lamellae form a flexible surface on which the bellow lies on.  To examine the resulting forces during the piston widening, a complex multi-layer FE-model of the bellow was created, see Figure 5 (right).   12r11 2~2 rr  Figure 5: Piston widening with piezo actuator (left) and corresponding FE model (right). 4 DYNAMIC SIMULATION By using the equation of mass conservation with consideration to the yielding of the bellow, we get 01  favb qqqVdtdp , (4) whereby bV  is the volume inside the bellow in its deflected position,   is the yieldingness of the bellow (determined by using the FE-model) and vvvv Asqqq  21 . By definition, the volume flow rate for inflowing fluid is negative. The volume bV  is calculated by integrating ab qdtdV  . To estimate the volume flow rate fq  due to the widening of the piston, we examine the change of the area A  and the corresponding volume V  covered by the bellow convolution.  Because the volume increases due to the widening of the piston radius ( r ) and the increase of the spring deflection ( s ), we have to develop an equation for the flow rate at constant spring deflections. To obtain this equation, we make some assumptions to simplify the model. The radius r  shall be constant along the piston; the bellow convolution shall be a semicircle. Because the bellow is fixed with a rigid ring (see Figure 6) the first assumption can hardly be achieved. In fact, this assumption represents the upper limit for the volume flow rate due to the piston widening.  With the definitions in Figure 6, the change of the cross sectional area A  becomes by neglecting terms of higher order ldrdrrAdA  . (5) Hence the volume ArV m2  and the volume flow rate fq  can be calculated by fmmmconstsconstsqlrArπ drdArdrdrA rπ dtdrdrdVdtdV 2122  . (6) The required values for mrlr ,,  and fb  are only functions of the parameters spring deflection s  , initial piston radius dr  and r , determined by geometry: A drrAAr drr ldll ff dbb mr mm drr fbrigid fixation Figure 6: Enlargement of the piston radius      .       ,21       ,42      , rrbrrrrrslrrr afamdd  (7) The hydraulic accumulator is described with the laws of mass conservation and energy conservation (/Pel01/). This leads to a system of algebraic differential equations for the density , the pressure p  and the temperatureT : 010  )Tk (T)A(γγ pqp,      VρqρVρRT,      p uw , (8) (with the volume flow rate q , the volume of the accumulator V , the ideal gas constant R , the isentropic exponent  , and the heat transfer coefficient k ). The advantage of this model compared to the polytrophic relation used in Eq. (3) is that no assumption about the polytrophic exponent is needed. The restrictor can be described with Bernoulli’s law for unsteady flow. Assuming a rigid pipe with constant cross sectional area, incompressible fluid and negligible potential difference, Bernoulli’s law results in  vpuLpp  21  (9) with the length of the pipe L , the flow velocity u  and the pressure loss vp . The velocity u  is given by the inflowing volume rate aq  and the pipes cross sectional area; for laminar flow, the pressure losses are given by 48 RLqp av   with the dynamic viscosity   and the pipe’s radius R  (/Spu07/). With the help of the harmonic balance nonlinear losses due to inertia can also be modeled by this approach at least for harmonic signals. The dynamic model is built up in Modelica, a standard to describe dynamic systems. To examine the dynamic behavior, a mono harmonic deflection is imposed on the system. For each frequency an individual simulation was run. Thereby, the radius of the piston is a function of the deflection:    ̂/ˆ)( srrssr dd  ( ŝ  : deflection amplitude). The amplitude of the radius r̂  varies from 0% to approx. 20% of the piston radius in the design point dr . The results of the simulation are the pressure inside the HFD and therewith the spring and damping force. These results are evaluated over one period in the steady state.    The dynamic stiffness c  and the loss angle   are here defined as sFcˆˆ: ,       czWd2ˆ:)sin(   (10) The dissipated energy dW  is the area inside the hysteresis curve.  The power P , required to widen the piston, is given by vibration frequency [Hz]c / cd10-210-11001011001010 WpassiveactiveFFˆˆ fPin Hz Figure 7: Required power to increase the force amplitude at varying frequency Figure 7 shows the results of a frequency sweep ranging from 0.01Hz to 15Hz with ẑ  = 0.01 m. Because the HFD is an active system, it is possible to influence the pqP f . (11) behavior with energy input. The force amplitude at design point (indicated by index ‘d’) dF̂  can be varied by transferring energy. For example, to increase the force amplitude at 1Hz from passiveactive FFˆˆ    to passiveactive FFˆ2ˆ  , WP 100  have to be transferred. In Figure 8, the stiffness and the loss angle are depicted as a function of the frequency. At 0.1Hz, the loss angle has its first local maximum. This is because of the change from isothermal to adiabatic behavior while compressing the gas inside the hydraulic accumulator. This proves the assumption for an adiabic change during vibration stated before.  The stiffness increases with higher amplitudes for the piston radius r̂ . For higher frequencies, the stiffness increases due to the inertia of the fluid, represented by Eq. (9). 10-210-110010110110210310-210-1100101020406080r̂ Hzin f in N/mm in c Figure 8: Stiffness (top) and loss angle (bottom) as a function of the frequency 5 FUTURE WORK On the basis of the above-mentioned general studies, the next step is the development and construction of a HFD prototype as well as the improvement of the simulation models. The main and important part of the development phase is the elaboration of robust solutions for the piston widening considering the bellow fixation. With detailed models, the snap stability will be investigated. With the depicted FE-model and experiments, the forces related to the piston widening have to be investigated. CONCLUSION In this work, analytic and simulative models were introduced to show the fundamental behavior of the hydro pneumatic suspension system. It was found that it is possible to control and vary the forces in a wide range by energy input. Although in cases of higher frequency high power is demanded, only small changes in the piston radius are required. This gives the possibility to use piezo actuators for the piston widening.  ACKNOWLEDGEMENT We like to thank the Deutsche Forschungsgemeinschaft (DFG) for funding this project within the Collaborative Research Centre (SFB) 805.  REFERENCES /Eul03/ Eulenbach, D., Nivomat, Stand und Entwicklungstrends hydropneu- matischer Niveauregelsysteme, Haus der Technik, Essen, Germany, 2003 /Pel01/ Pelz, P., Theorie des Luft-Feder-Dämpfers, Freudenberg Forschungsdienste KG, Germany, 2001 /Pel07/ Pelz, P., et al., Numerische Festigkeitsauslegung von Luftfedern, MP Material Testing, Vol 49, 447-454, 2007 /Pel08/ Pelz, P., Rösner, J., DE 10 2008 007 566 A1, Offenlegungsschrift, Deutsches Patent- und Markenamt, Patentliteratur, Germany, 2009 /Spu07/ Spurk, J. H., Aksel ,N., Strömungslehre, Einführung in die Theorie der Strömungen, Springer-Verlag, Berlin, Germany, 2007 

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An active suspension with reduced complexity

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