Design and modelling of a fluid inerter

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International Journal of Control

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Design and modelling of a fluid inerter

S. J. Swift a , M. C. Smith a , A. R. Glover b , C. Papageorgiou a , B. Gartner c & N. E.

Houghton a

a Department of Engineering , University of Cambridge , Cambridge , CB2 1PZ , UK

b McLaren Automotive , Chertsey Road, Woking, Surrey , GU21 4YH , UK

c Penske Racing Shocks , 150 Franklin Street, Box 1056, Reading , PA , 19603 , USA

Accepted author version posted online: 18 Sep 2013.Published online: 14 Nov 2013.

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International Journal of Control ,2013

V ol.86,No.11,2035–2051,https://www.doczj.com/doc/a02962256.html,/10.1080/00207179.2013.842263

Design and modelling of a ?uid inerter

S.J.Swift a ,M.C.Smith a ,?,A.R.Glover b ,C.Papageorgiou a ,B.Gartner c and N.E.Houghton a

a

Department of Engineering,University of Cambridge,Cambridge,CB21PZ,UK;b McLaren Automotive,Chertsey Road,W oking,

Surrey,GU214YH,UK;c Penske Racing Shocks,150Franklin Street,Box 1056,Reading,P A 19603,USA

(Received 6January 2013;accepted 4September 2013)

Mechanical spring-damper network performance can often be improved by the inclusion of a third passive component called the inerter.This ideally has the characteristic that the force at the terminals is directly proportional to the relative acceleration between them.The ?uid inerter presented here has advantages over mechanical ball screw devices in terms of simplicity of design.Furthermore,it can be readily adapted to implement various passive network layouts.Variable ori?ces and valves can be included to provide series or parallel damping.Test data from prototypes with helical tubes have been compared with models to investigate parasitic damping effects of the ?uid.Keywords:design;modelling;control;mechanical device;inerter

1.Introduction

In Smith (2002),an ideal mechanical modelling element called the inerter was introduced with the property that the applied force at the terminals is directly proportional to the relative acceleration between them.Embodiments using a rack,pinion and gears,ball screw or gear pump driving a ?ywheel were described in Smith (2001).Such devices have been successfully used in Formula 1race cars where they have been given the name ‘J-damper’(Chen,Papageorgiou,Scheibe,Wang,&Smith,2009).It was also shown in Smith (2002)that general passive mechan-ical impedances can be realised using mechanical circuits comprising springs,dampers and inerters only.This re-port describes a new inerter implementation (Gartner &Smith,2010;Glover,Smith,Houghton,&Long,2009)which uses the mass of a ?uid ?owing through a heli-cal channel to provide the inertance.Durability and sim-plicity are the main advantages of this new implementa-tion,with additional bene?ts from the straightforward ad-dition of ?ow restrictions to incorporate series or parallel damping effects into the device.This paper presents the mechanical design of the new implementation and some variants and investigates their modelling and experimental behaviour.

A schematic diagram of the new implementation is shown in Figure 1.The cylinder body and the piston rod are the two device terminals,their relative motion driving ?uid through the helical channel.The channel ?uid velocity is scaled up from the piston velocity by the ratio of the areas of the channel and the piston.Thus,the device inertance can be increased by reducing the area of the channel or

?

Corresponding author.Email:mcs@https://www.doczj.com/doc/a02962256.html,

increasing the area of the piston,both of which increase the ?uid velocity for a given rate of strut movement.

Prototypes have been constructed with tightly wound helical channels inside the piston head and alternatively outside the piston cylinder.This ?rst type results in a slim design which is relatively long due to the length of the pis-ton head,the latter style has a shorter and wider shape with a larger helix diameter.These have been used in experiments with low and medium viscosity automotive oils to deter-mine frequency response,parasitic damping and inertance.The parasitic damping is due to viscous effects in the ?uid which act to resist the ?ow through the channel.This be-haviour appears to be well modelled by a damper in parallel with the inertance where the damping force is proportional to a power of the piston velocity which is a little less than 2.Increasing the helix diameter reduces the parasitic damping due to channel curvature,and hence this paper will focus upon designs using an external-helical channel.2.Fluid inerter modelling

Consider a piston and cylinder driving ?uid through a he-lical tube surrounding the cylinder,as shown in Figure 1.Let A 1be the annular area of the main cylinder,namely the working area of the piston face.Let A 2be the channel cross-sectional area, be the channel length and ρbe the ?uid density.Let F be the equal and opposite force applied to the terminals and x be the relative displacement between them.An ideal inerter is described by the following equation:

F =b ¨x,

(1)

C 2013Taylor &Francis

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Figure 1.Schematic of external-helix ?uid inerter.

where b is the inertance in kg.We ?rst neglect any dissi-pative effects due to ?uid viscosity and density and regard the ?uid as incompressible.If u is the mean velocity of the ?uid in the helical path,then

˙xA

1=uA 2(2)

by conservation of volume.The stored energy of the ?uid

in the helical path is given by

ρA 2 u 2/2

and the stored energy in an ideal inerter is b ˙x

2/2,which suggests the following approximate value for the device inertance:

b =ρ A 2

1A 2

.

(3)

This calculation only considers the inertia of the ?uid ?ow-ing in the channel and neglects the inertia of the ?uid in the

piston chamber and the inertia of the piston https://www.doczj.com/doc/a02962256.html,rge inertance values are possible as the mean ?uid velocity in this channel is scaled by the ratio of the piston area to the channel area A 1/A 2.Energy losses in the ?uid give rise to a departure from the force law (1).We will now consider these effects separately.

The main contribution to the pressure drop p across the piston comes from viscous effects in the channel.These effects depend upon the properties of the ?uid and the chan-nel and must be determined from experimental results by the use of appropriate dimensionless quantities.In the follow-ing sections,relevant dimensionless quantities and models for these properties are described.Other contributions to the pressure drop result from energy losses at the ends of the channel where the ?ow transitions between the main cylinder and the narrow channel.In addition,shear fric-tion between the piston and cylinder walls is found to be negligible for typical clearance tolerances.

2.1Channel cross-sectional shape

Following the work in Rodman and Trenc (2002),the hy-draulic diameter of the channel will be used for the esti-mation of the damping forces.The hydraulic diameter is de?ned to be four times the ratio of the cross-sectional area of the channel to the wetted perimeter of the channel:

D h =4×

cross-sectional area

wetted perimeter

.

In the case of a circular cross-sectional channel,this is the same as the channel diameter.The hydraulic diameter is four times the hydraulic mean depth or hydraulic radius (Massey,1997),which is a common alternative term.This enables channels of various cross-sectional shapes to be modelled.As the cross-sectional shape deviates further from circular and the ratio of area to circumference decreases,then the resistance to ?ow increases.

2.2Reynolds number

In a straight tube,the Reynolds number (Re)is used to determine whether the ?uid ?ow will be laminar or turbu-lent,with this transition occurring around Re =2×103(Massey,1997).Calculation of ?uid friction is dependent upon the Reynolds number in the laminar and transitional

?ow regimes.Given that A 1˙x

=A 2u ,the Re for the tube is equal to

Re =

2ρD h

μu =2ρD h A 1μA 2

˙x,

where μis the dynamic viscosity in Pa s.2.3Dean number

Fluid travelling along a curved pipe is affected by cen-trifugal force which sets up a pattern of movement called secondary ?ow (Massey,1997)as shown in Figure 2.This has been partly characterised by the Dean number (De)(Dean,1928),which is dependent upon the ?uid inertia and viscosity and upon the curvature ratio of the bend D h /R ,

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Figure 2.Secondary ?ow in a curved channel with circular cross

section.

where R is the bend radius,

De =Re

D h R

1/2

.

Secondary ?ow is an example of a stable pattern,and was observed to increase the critical Reynolds number for tur-bulent ?ow with increasing Dean number (White,1929).This was investigated further in Ito (1959)for helical pipes with a moderate bend radius.

2.4Pressure drop in helical channels

A wider characterisation of ?ow in circular cross-sectional curved pipes is presented in Ali (2001),using the Euler number Eu = p /(ρu 2),Reynolds number and a new dimensionless quantity based upon the coil dimensions.This characterisation was ?tted using experimental data and displayed four regimes of ?ow.These regimes were characterised in terms of Reynolds number and termed low laminar ,laminar,mixed and turbulent,with the criti-cal Reynolds numbers 500,6300and 10,000,respectively,separating these regions.Though this model was shown to be in good agreement with previous works (Ito,1959),it is based upon a limited range of coil dimensions and was not found to match the inerter experimental data as closely as the approach in Rodman and Trenc (2002)which will be described next.

Helical channels of rectangular cross section with vari-ous aspect ratios were characterised in Rodman and Trenc (2002)for a range of Dean numbers 100

p =f F Re

2 μu

D 2h

,(4)

where f F is the Fanning friction factor (dimensionless).In

Rodman and Trenc (2002),the following approximation

of

Figure 3.

f F Re versus De from Equation (5).

f F Re in terms of De was proposed:

f F Re =2.4629De 1/2(1?18.553De ?1/2

+275.38De ?1?1015.9De ?3/2)

(5)

based on curve ?https://www.doczj.com/doc/a02962256.html,ing Equation (4)with this curve we can calculate an estimate of the damping force experienced in the ?uid inerter due to the helical channel:

F hc

=f F Re 2 μ

D 2

h

A 21A 2

˙x.

The curve of f F Re versus De is shown for 100

f F Re =0.03426De +17.54,

(6)

and hence

p =0.034262ρ u 2√D h R

+17.542μ u

D 2

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The ?rst term in this approximation is dependent upon the density and the square of the ?uid velocity,and the second term is dependent upon the viscosity and the ?uid velocity.2.5Inlet and outlet losses

At either end of the helical channel,the ?uid velocity changes where it connects with the main cylinder.A ?uid ?ow entering a channel experiences an energy loss which is dependent upon the shape of the inlet.The empirical formula for the resulting pressure drop is given in Massey (1997),where the dimensionless loss coef?cient is approx-imately 0.5for a sharp inlet from a reservoir.The pressure drop across the inlet is thus estimated to be

p inlet

=0.5ρu 2

2

,and hence contributes an estimated force resisting motion of

F inlet =A 1 p inlet =0.25A 1ρ

A 1A 2

2

˙x

2.Where the ?ow exits the channel,the kinetic energy of

the ?uid is lost;in this case by causing turbulence in the body of ?uid which it ?ows into.This outlet pressure drop is

p outlet =ρu 2/2,

and the resulting contribution of force resisting motion of the piston is

F outlet =A 1 p outlet =0.5A 1ρ

A 1A 2

2

˙x

2.2.6Piston viscous shear friction

The thin ?lm shear friction force between the side wall of the piston and the surrounding cylinder casing can be estimated as

F shear =

μA f ˙x r

,where μis the dynamic viscosity in Pa s, r is the clearance between the piston head and the cylinder wall,r is the piston radius,the friction area is A f =2πr piston and piston is the length of the piston head in contact with the cylinder wall.A typical clearance is r =0.1mm.This makes a small contribution at low strut velocities,but negligible at working velocities in the range 0.1–1m/s.An internal-helix design has a long piston with a much larger shear area than an external-helix design,though this calculation indicates that the use of ?uids with moderate viscosity does not give rise to large forces in either case.

Table 1.

External-helix prototype details.

Description Value Piston area A 11.11×10?3m 2Channel area A 228.2×10?6m 2

Helix radius r 438.1mm Channel length 1.68m Damper stroke

54.4mm

2.7Total damping force

The total predicted damping force is given by the summa-tion of the forces due to the helical channel,the port losses and the viscous shear friction between the piston and the cylinder:

F damping =F hc +F inlet +F outlet +F shear .

(8)

The predicted total damping force for an external-helix

prototype inerter with dimensions from Table 1is shown in Figure 4using Pro-RSF Silkolene oil at 26?C.Note that F shear is negligible,and F inlet and F outlet are small compared to F hc .Therefore,F hc is a useful approximation to F damping and this reduces to the formula:

F d (˙x

)=0.034262ρ A 1√D h R A 1A 2

2˙x 2

+17.542μ A 1D 2h

A 1

A 2

˙x (9)

using Equation (6).The second term representing the vis-cous damping reduces to the form 8.77πμ A 1A 2

2

˙x for a circular cross-sectional channel.It is interesting to note that the assumption of laminar ?ow and the use of the Hagen–Poisseau formula for

a straight tube of circular cross

Figure https://www.doczj.com/doc/a02962256.html,position of the total parasitic damping force for the external-helix prototype inerter with inertance =60kg.

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Figure 5.Pro-RST Silkolene oil density versus temperature (as-suming a linear extrapolation).

section gives the expression 8πμ (A 1A 2

)2˙x for the damping force.

3.Model ?tting to experimental data

Experiments performed with a prototype inerter device on a test rig have given data on the parasitic damping for constant strut velocities and on the frequency response of the device by using sinusoidal excitation.Manufacturer’s data for the test ?uid have been used to calculate the density and dynamic viscosity at the recorded temperature.These parameters are used with the models for comparison with the experimental results.

The thermal expansion coef?cient for damper oil is taken to be αhc =0.00090K ?1,giving the density versus temperature curve shown in Figure 5for Pro-RST

Silkolene

Figure 6.Prototype inertance versus temperature due to chang-ing oil density as in Figure 5

.

Figure 7.Pro-RST Silkolene oil dynamic viscosity versus tem-perature based on manufacturer’s data and interpolation.

oil from the known density of 816kg m ?3at 20?C,

ρ=ρ1/(1+αhc (T ?T 1)).

With the inertance calculated as ρ A 21/A 2,we hence have the inertance versus temperature relationship shown in Figure 6for the data of Table 1.

The manufacturer’s data for Pro-RST Silkolene oil gives the kinematic viscosity at 40?C and 100?C as 13.6and 5.83cSt,respectively,and using the above density curve and the Walther equation (American Society for Testing Materials viscosity–temperature chart ASTM D341,Sta-chowiak &Batchelor,2005),log 10(log 10(ν+0.7))=A ?B log 10(T ),the dynamic viscosity relationship with temper-ature is plotted in Figure 7.The Walther equation uses the kinematic viscosity νin centiStokes and the temperature in Kelvin.The parameters A and B are obtained from ?tting the two known data points ν1and ν2at temperatures T

1

Figure 8.Prototype parasitic damping versus strut velocity.

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Figure 9.Experimental and simulated results using identi?ed values for inertance 61.6kg and bulk modulus 48.1

MPa.

Figure 10.Prototype frequency response from sampled data (symbols)compared with model (continuous lines),using the observed damping characteristic and identi?ed values for inertance and effective bulk modulus.The ?gure shows the variation with the sine wave amplitude.

and T 2as

B =

log 10 log 10(ν2+0.7) ?log 10 log 10(ν1+0.7) / log 10(T 1)?log 10(T 2)

A =log 10

log 10(ν1+0.7) +B log 10T 1).

Estimating the damping from the Walther equation for the Silkolene oil at a temperature of 26?C gives a viscosity of 0.0144Pa s and the curve shown in Figure 8.The damping curve estimated from the constant velocity experiments in this ?gure is 35%greater than predicted,possibly because the viscosity of the oil is greater than the rated value at this

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Figure 11.Predicted damping force versus ?uid viscosity for a strut velocity of 1m/s in the prototype inerter (inertance =60kg)using the methods of Rodman and Trenc (2002)and Ali (2001).

relatively low operating temperature.Attempting to match

the observed curve by increasing the viscosity in the model yields a very good match for a viscosity of 0.033Pa s,which seems somewhat high,perhaps indicating that a discrepancy in oil viscosity is not suf?cient to explain the difference.

For effective model ?tting,it was necessary to consider the effect of compliance due to compressibility of the ?uid.The compliance was calculated as a function of the effective bulk modulus of the oil and the dimensions of the device,including the position of the piston.The bulk modulus b m is de?ned by the equation

p =b m

v v ,

where v is a volume of ?uid, v is the change in volume and p is the corresponding change in pressure.Let L t be the length of the cylinder,L t ? be the available travel of the piston and x be the position of the piston,so that x ∈[?(L t ? )/2,(L t ? )/2].Then,the change in pressure on each side of the piston due to a small change δx only

in

Figure 12.Model of ?uid inerter with parasitic

damping.Figure 13.Schematic of ?uid inerter with channel ?ow restric-

tion.

the piston position is given by

δp 1=b m

?δx L t /2?x δp 2=b m

δx

L t /2+x

(10)

with the corresponding force due to the compliance given by δF =A 1(δp 2?δp 1).This leads to the following expression for the effective series spring stiffness:

k f (x )=b m A 1

1L t /2?x +

1

L t /2+x

.

(11)

Values for the device inertance and effective bulk modulus have been identi?ed by optimising a cost function based upon simulations of the frequency response experiments.The ?uid inerter damping characteristic observed in the constant velocity tests was used in this identi?cation.At each iteration of the optimisation,the cost function was evaluated for trial values of inertance b and effective bulk modulus b m .Evaluating this cost function requires a sim-ulation run for each sine wave experiment,as shown in Figure 9.In these experiments,the sine wave amplitude was varied across the range of frequency points due to practical considerations.The cost is given by summing up the relative errors between the complex frequency response points for the experimental and simulated results.The re-sulting estimates are 61.6kg for the device inertance and 48.1MPa for the effective bulk modulus.This inertance is close to the value of 64kg which was estimated from the device dimensions and oil density.Calculating a spring coef?cient about the centre piston position from this bulk modulus gives k f =1.96MN/m,which is comparable to the compliance of suspension

bushings.

Figure 14.Suspension network S3.

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Figure 15.Model of ?uid inerter with parasitic damping and

compliance.

Simulating the response of this identi?ed nonlinear model to sine wave inputs has given the frequency response plots in Figure 10.These curves show how the frequency response depends upon the sine wave amplitude and display a wider frequency range than can be obtained experimen-tally for any given amplitude.The effective bulk modulus can be compared to the ideal bulk modulus of oil which is around 1GPa,the factor of 20differences can be accounted for by the ?exibility of the device components and the presence of dissolved gases in the oil.If this effective bulk modulus was increased,then the resonant peak would move to a higher frequency and increase correspondingly in magnitude.

The model for the tested device has also been used to show the relationship between the predicted damping force at 1m/s and the ?uid viscosity in Figure 11.Losses due to the helical shape of the tube are dominant over the range of the model,and so the dependence of the damping force upon viscosity,and hence device temperature,is only moderate.The solid line in this ?gure predicting the damping force down to μ=1mPa s is calculated using the slightly less accurate geometric group ?t from Ali (2001)as described in Section 2.4and indicates that some reduction in damping force or device size is possible if water is used instead of oil.

The modelling accuracy shown in Figure 8gives a good degree of con?dence that the damping forces can be pre-dicted for inerter designs.Figures 7and 11indicate that choosing a less viscous ?uid,or raising the device temper-ature would only result in a slight reduction in the parasitic damping.

4.Simulation models

This section details several simulation models for ?uid inerters which are suitable for comparison with the rig experiment results and for use in vehicle models.Models including bulk modulus effects are presented for the investigation of the dependence upon ?uid and channel compliances.Devices with variable ?ow restrictions which implement parallel and serial damping components are also modelled.

4.1Inerter with parasitic damping

The parasitic damping caused by frictional losses in the ?uid can be considered to act in parallel to the device inertance,as shown in Figure 12.The force F d due to this damping is taken to be a function of the strut velocity,and the force due to inertance is considered ideal and a linear function of the acceleration:

F =F d (˙x

2?˙x 1)+b (¨x 2?¨x 1),where x 1and x 2are the positions of the device terminals and b is given by Equation (3).This is implemented in Simulink as a variable-rate damper in parallel with an ideal inerter.The relationship between strut velocity and damping force can be implemented either as a look-up table identi?ed from experimental data,or as a curve representing ?tted or predicted behaviour,or making use of the formula (9).

It is possible to increase the damping in the device at low strut velocities by including a variable ?ow restriction in the inertance channel,Figure 13.This would be in the form of a shim stack or progressive pressure relief valve where the restriction opening increases as a function of the pressure difference across it.The inertance value is unaffected but the ?ow restriction introduces a parallel damping compo-nent into the equivalent mechanical network.We can then implement the S3suspension network (Figure 14)using only this device in parallel with the main spring.4.2Inerter with parasitic damping and

series spring

Early experimental results have indicated that the high-frequency behaviour of the ?uid inerter is strongly depen-dent upon the effective compressibility of the inertance ?uid.Figure 15presents a basic model where this com-pressibility is modelled as a spring in series with the ideal inerter and parasitic damping.In the simplest representa-tion,the force due to this effective bulk modulus is

F =k f (x s ?x 1),

where k f is the ?uid spring coef?cient in N/m and is depen-dent upon the device geometry and effective bulk modulus of the ?uid.This force is also experienced by the parallel inerter and parasitic damping model:

F =F d (˙x

2?˙x s )+b (¨x 2?¨x s )as it acts in series with these components.An implementa-tion of these equations in Simulink is presented in Figure 16.

More detailed nonlinear modelling of the compliance using Equation (11)is presented in Figure 17.Separately calcu-lating the pressures on each side of the piston in this way also enables the effects of cavitation to be modelled.

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Figure 16.Simulink implementation of ?uid inerter with parasitic damping and

compliance.

Figure 17.Simulink implementation of ?uid inerter with parasitic damping and nonlinear

compliance.

Figure 18.Schematic of ?uid inerter with an ori?ce in the piston.

4.3Inerter with parasitic damping and series

damper

Putting a variable ori?ce or shim stack in the piston creates a damping force which acts in series with the inertance,Figures 18and 19,without affecting the inertance value.

We

Figure 19.Model of ?uid inerter with parasitic damping and series damper.

can thus implement the S4suspension network (Figure 20)using only this device in parallel with the main spring.However,it should be noted that the parasitic damping may

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Figure 20.Suspension network S4.

not be negligible except at low suspension velocities.It may therefore be bene?cial to include the parasitic damping effects in the overall suspension design.

Suppose the force in the series damper is

F =c (˙x

s ?˙x 1),where c is the damping coef?cient due to the ori?ce in the

piston.The inerter and parallel parasitic damper experi-ences the same force:

F =F d (˙x

2?˙x s )+b (¨x 2?¨x s ).A Simulink implementation of this model is presented in

Figure 21

.

Figure 22.Model of ?uid inerter with parasitic damping,series damper and compliance.

4.4Inerter with parasitic damping,series damper

and series spring

Including the effective bulk modulus in the above model as a series spring component gives the network shown in Figure 22.

In this way,the force due to effective bulk modulus,

F =k f (x 4?x 1)

acts in series with the damping force,

F =c (˙x

3?˙x 4),and the force due to the inertance and associated parallel parasitic damping is

F =F d (˙x

2?˙x 3)+b (¨x 2?¨x 3).A Simulink implementation of these equations is presented

in Figure 23.More detailed nonlinear modelling of the compliance in this model using Equation (11)is shown in Figure 24

.

Figure 21.Simulink implementation of ?uid inerter with parasitic damping and series damper.

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Figure 23.Simulink implementation of ?uid inerter with parasitic damping,series damper and

compliance.

Figure 24.Simulink implementation of ?uid inerter with parasitic damping,series damper and nonlinear

compliance.

Figure 25.Fluid inerter model with parallel parasitic damping,series damping and compliance.

5.Identi?ed damping rates

We now consider the model shown in Figure 25which includes a linear series damping element c s to represent effects such as leakage across the piston.The measured damping rate c m (v )is taken to be the series combination of c s and a nonlinear parasitic element c p (v )where

c m (v )=

c s c p (v )c s +c p (v )

and estimates of c s (where c s >c m (v )for all test velocities v )can be used to give c p (v ):

c p (v )=

c s c m (v )

c s ?c m (v )

.

Estimates of k 1and c s for each prototype in Table 2have been obtained by ?tting frequency responses from this

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Figure 26.Frequency response plots for the external tube B inerter prototype with Silkolene 02

oil.

Figure 27.f F Re versus De for the ?uid inerter prototypes estimated parasitic damping rates c p (v )using synthetic oils.

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Figure 28.f F Re versus De for the ?uid inerter prototypes estimated parasitic damping rates c p (v )using water.

model in simulation to the experimentally measured fre-quency responses.The identi?ed parameter values are given in Table 3.Figure 26shows an example frequency response plot which was used for ?tting these parameters.Figures 27and 28show the resulting experiment data plotted as f F Re versus De for the ?uid inerter prototypes estimated parasitic damping rates c p (v ).Aside from stiction at low velocities (low Dean numbers),the estimated parameters correspond well with the model of helical tube damping de?ned through Equation

(6).

Figure 29.Simulink implementation of ?uid inerter with parasitic damping,nonlinear compliance and cavitation.

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Table 2.Device parameters.

Density Viscosity Device

(kg/m 3)(Pa s)A 1(mm 2)A 2(mm 2)D h (mm)R (mm)l (m)ExTubeA_Pro-RSF_25C 8120.015111020.3 6.00038.1 1.68ExTubeB_Silk02_25C 8530.024111027.3 5.9027.8 1.23ExTubeB_Silk02_65C 8240.0081111027.3 5.9027.8 1.23ExTubeB_Water_25C 9970.0009111027.3 5.9027.8 1.23ExTubeB_Water_65C 9800.00044111027.3 5.9027.8 1.23ExTubeC_Silk02_25C 8530.02411108.6 3.3027.8 1.23ExTubeC_Silk02_65C 8240.008111108.6 3.3027.8 1.23ExChannel_Silk02_65C 8240.0811110 6.5 2.48822.3 2.75IntHelix_SAE10_25C 865

0.046

2758

148

11.854

28.5

0.886

Table 3.

Identi?ed device parameters (c s and k 1).

Identi?ed parameters

Calculated Measured damping c p

Mean inertance b c m at 0.1m/s at 0.1m/s c s

k 1c p /b relative Device

(kg)(kNs/m)

(kNs/m)(kNs/m)(MN/m)(Ns/kg m)?tting error ExTubeA_Pro-RSF_25C 59 3.2 3.399 2.1560.170ExTubeB_Silk02_25C 47 3.05 3.092691265.70.095ExTubeB_Silk02_65C 45 1.96 1.972781243.90.100ExTubeB_Water_25C 55 1.42 1.431901226.10.115ExTubeB_Water_65C 49 1.46 1.463151229.90.146ExTubeC_Silk02_25C 15027.929.5498131970.110ExTubeC_Silk02_65C 14515.916.4489121130.117ExChannel_Silk02_65C 4303475618.11700.053IntHelix_SAE10_25C

39

1.5

1.5

86

3.3

38

0.363

6.Pressurisation

The vapour pressure of automotive oils is very low.Hence,without pressurising the inerter,cavitation will occur for a pressure drop across the piston of approximately one atmo-sphere and above,i.e. P >100kPa.See Figure 29for a Simulink implementation including the effect of cavitation.A pressure drop of one atmosphere across the piston is given by a strut force of 110N for the 60kg external-helix proto-type inerter.This is a comparatively low strut force and will be exceeded even at low strut velocities.Pressurising the device is advisable to avoid cavitation at rated speeds.Also,the effects of air bubbles in the ?uid on the high-pressure side of the piston can be reduced by pressurisation of the oil.Such air bubbles can increase the compliance,affecting the performance.A pressure release valve across the piston may be desirable for limiting peak pressures under extreme conditions such as a kerb strike.

When cavitation is included the resulting force F =?A 1(p 2?p 1)can then be calculated from the opposing pres-sures:p 1=max

0,p pressurisation +b m x s ?x 1

L t /2?(x 2?x 1)

,

p 2=max

0,p pressurisation +b m

x 1?x s

L t /2+(x 2?x 1)

,where p pressurisation is the static offset pressure (cf.Equation (10)).

7.Conclusions

The ?uid inerter implementation introduced here is robust and durable due to its simple design and the crossover with existing damper construction.The device size is compara-ble to ball screw implementations with the additional ad-vantage that variable valves and ?ow restrictions may be included to compactly realise more suspension networks.We have shown that the device can be modelled as an ideal inerter in parallel with a nonlinear parasitic damping com-ponent.This parasitic damping can be calculated from the device geometry and ?uid properties.The inertance and de-vice compliance have been estimated from the experimental data by optimising these model parameters to ?t simulation results to the observed time series.

Funding

This work was supported by the Engineering and Physical Sci-ences Research Council [grant number EP/F062656/1],[grant number EP/G066477/1].

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2049

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2050S.J.Swift et al.

Appendix.Experimental results from ?uid inerter

tests

Figure A1.

Sinusoidal responses from the testing of the external-helix inerter with Pro-RST Silkolene at 26?C .

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2051

Figure A2.Force response of the external-helix inerter with Pro-RST Silkolene at 26?C under smooth velocity

steps.

Figure A3.Observed and simulated force response of the external-helix inerter with Pro-RST Silkolene at 26?C under smooth velocity steps.

D o w n l o a d e d b y [B e i h a n g U n i v e r s i t y ] a t 23:20 13 M a r c h 2014