Post on 21-Jan-2023
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The Hover-Ride Suspension System
Chad H. SuttonCollege of Engineering and Applied Sciences
University of Cincinnati
A.Abstract
THE purpose of this Design Study Summary is to present thedetailed technical analysis and trade study results for theproposed Hover-Ride Suspension System with the purpose ofelevating the Cincinnati Car Company (CCC) above and beyond thereach of the competition in the eyes of the consumer. As abackground to this Design Study, it was decided to employ an MRdamper to create a semi-active suspension system based on thetheoretical Skyhook control system, and then improve upon thistheoretical model to optimize it for use in the field. Abaseline theoretical analysis has been performed on the existinglegacy suspension design in order to provide a point of referencefrom which to compare each subsequent design. From there, theoriginal Skyhook Control Algorithm was applied to the MRsuspension design and all performance characteristics wereassessed. The Skyhook Control Algorithm was then improved uponto give CCC a competitive edge over the competition by means ofan optimization technique being applied to the yield force in thedamper for various road input frequencies. Since the automobileindustry is very well optimized and hosts products of closesimilarity in today’s market, even a marginal change in the ridequality that the driver can noticeably feel will cause ourproduct to be held superior to the competition. It has been shownin this document that the Hover-Ride Suspension System,consisting of the legacy suspension design augmented with an MRdamper and modified Skyhook Control Algorithm, has proven toreduce the velocity of the car body in the vertical direction inresponse to various road input frequencies by as much as 85%.This translates to a much improved comfort in ride for thecustomer. This report outlines the detailed design, assumptionsmade, and all system response characteristics that prove this
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point. Therefore the recommendation of this Design Study Summaryis to proceed with the optimized and improved Skymod ControlAlgorithm for the new aged MR suspension system, Hover-Ride forimplementation into the CCC’s next generation of mid-sizedautomobile.
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Table of Contents
I. Abstract.................................................................1II. Nomenclature.............................................................3III.........................................................Introduction
4IV. Analysis.................................................................5V. Conclusions.............................................................27VI. References..............................................................28VII............................................................Appendix
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B.Nomenclature
m = mass of the lumped automobile system [slugs]W = weight of the lumped automobile system [lb] g = acceleration due to gravitational forces [ft/ss] k = stiffness of car suspension [lb/ft]c = damping coefficient of conventional damper used in suspension [lb-sec/ft]cmr = damping coefficient of MR fluid damper with no magnetic field applied [lb-sec/ft]FYield = maximum force that can be applied by the MR damperv = velocity of the car to be used in road input generation [ft/s]λ = wavelength of the road to be used in road input generation [ft]u = control parameter used to turn on or off the Skyhook control lawyo = amplitude of the road input to be used in road input generation [ft]y = vertical displacement of the road [ft]x = vertical displacement of the car [ft]
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C.Introduction
The purpose of this Design Study Summary is to present thedetailed technical analysis and trade study results for theproposed Hover-Ride Suspension System with the purpose ofelevating the Cincinnati Car Company (CCC) above and beyond thereach of the competition in the eyes of the consumer. As abackground to this Design Study, it was decided to employ an MRdamper to create a semi-active suspension system based on thetheoretical Skyhook control system, and then improve upon thistheoretical model to optimize it for use in the field. The CCCautomobile model that was proposed for implementation of thisproduct has the descriptive system characteristics as portrayedin Table 1 below. All subsequent analysis will be presentedbased on this vehicle.
Table 1: Typical Values for Mid-Sized CCC Model
W 3220 lbM 100 slugC 134 lb-sec/ftK 4000 lb/ft
This analysis will walk through the derivation of all relevantequations of motion related to the system and will discuss allrelevant algorithm methodologies related to the modeling of thissystem. As a baseline case, the standard suspension design(consisting of only a passive spring and damper element) will bemodeled first to check for accurate solutions and to prove thealgorithm is correct and what is expected. Then the theoreticalSkyhook algorithm will be applied to the algorithm and the MRdamper will be included in the equations of motion to determinethe theoretical optimization that can be achieved if this designis used in comparison to our standard suspension design.Finally, the Skyhook algorithm will be optimized and modified topresent that most sophisticated design that is being proposed foruse in our consumer market.
Table 2 provides the characteristics of the MR damper that is tobe used in this design. The full effect of the damping (Yield)
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force may or may not be used in the end result algorithm. Notethat when this damper is added to the system, the vibrationalcharacteristics will be affected regardless of the presence ofthe yield force since the fluid in the damper will always have adamping coefficient as shown in Table 2.
Table 2: Characteristics of Selected MR Damper
CM R 21 lb-sec/ftM ax FYield 784 lb
The next section will walk the reader through the methodology andassumptions used in the design of this new semi-activelycontrolled automotive suspension system known as the Hover-RideSuspension. All relevant results needed to make an accurateassessment of this new technology as compared to its predecessorwill be presented. Any other relevant information used orcreated will be posted in the Appendix section of this DesignStudy Report.
D.Analysis
The idea behind the Hover –Ride Suspension system (as well asthe original Skyhook Control Algorithm) is to give the illusionthat an automobile ride can be so smooth, it would feel as if onewas floating on a cloud when driving. In order to accomplishthis, it becomes necessary to create an improved system that willeffectively minimize the kinetic energy of the vehicle due to aninput to the suspension system. In order to model how thissystem will operate, the following assumptions were made forsimplification purposes necessary to create an initialsimulation:
Assumptions1. The automobile can be approximated by a Single Degree of
Freedom (SDOF), single lumped mass supported by a wheel and massless suspension system.
2. The wheel always remains in contact with the road surface.
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3. The MR control system is fast enough to provide an instantaneous force.
4. The stiffness, damping, and dynamics of the tire can be combined with the existing properties of the automobile suspension.
5. The road inputs that will be considered are constant and sinusoidal.
6. Only forces in the vertical direction will be considered. That is, there will be no analysis on the pitch and roll of the automobile, rather only the vertical movement of the single lumped mass as a function of the input.
7. The velocity of the car defines only the lateral direction and does not account for tracing of the sinusoidal road input.
Road Inputs and Vehicle Speeds to be consideredManagement has provided the following sinusoidal road inputs(Assumption 5) that are to be considered in analyzing theperformance of the new suspension technology and comparing to thestandard suspension design. Table 3 below shows these road inputconditions, where yo is the amplitude of the road to beconsidered, λ is the wavelength of the road, and v is the lateralvelocity of the car at steady state.
Table 3: Car Speed and Road Input Characteristics
yo 0.118 ftλRoad 10, 20, 88 ftvCar 5, 13.726, 40, 60 m ph
All of the possible combinations of the road input frequency willbe considered in this analysis. That is, if Equation 1 belowrepresents the assumed sinusoidal input that is the road, thenall possible combinations of the wavelength of the road and thevehicle speed will be considered.
y(t)=yosin (ωt )=yosin (2πft )=yosin(2πvλ t) (1)
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As will be observed when the equations of motion are derived forthis system, an expression will be needed to describe the rate ofchange (or velocity) of the road input. This can be obtained bytaking the first derivative with respect to time of the functiongiven in Equation 1. This is shown below:
y (t)=2πvλ
yosin(2πvλ t) (2)
It should also be noted that that by Assumption 7, the velocityof the car is assumed to be in the lateral direction only, anddoes not take into account the pitching and rolling that wouldoccur as the car follows the sinusoidal road input for longerwavelengths. This is illustrated in Figure 1 below.
Figure 1: Sinusoidal Road Input Courtesy of (Wikipedia: Wavelength)
System SchematicThe assumed schematic of the automobile lumped mass andsuspension system is shown below in Figure 2. It can be observedthat the vehicle to be considered follows Assumption 1 as asingle lumped mass with a supportive massless suspension system.The road input will be measured at the base as an input, making
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this a typical Single Degree of Freedom (SDOF) system foranalysis. Taking a closer look at the suspension system, theappearance of two other elements is noticeable in addition to thetypical passive spring and damper components. The conceptualapplication of the MR augmentation requires that an additionaldamping actuator be added to the system, where the damping force(defined in this analysis as FYield) will be varied as a functionof the car velocity. Also, the addition of the sensor element(could be an accelerometer or some other sort) will be used toobtain the relative velocity of the car with respect to the baseas a function of time. This input would be used in the actualsystem to determine when to apply the control law to actuate theMR suspension system. However, due to the fact that the roadinputs to be analyzed for comparison are defined for this systemas sinusoidal inputs and the characteristics of these wave inputsare defined, there is no need for a sensor in this preliminaryconfiguration, and it will not affect the application of Newton’s2nd Law.
Figure 2: Schematic of simplified SDOF system
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It is also important to note that this analysis denotes y(t) asthe input (forcing function) of the road imposed on theautomobile suspension system, and x(t) as the resultingdisplacement of the supported vehicle. Since conventionalsuspension design suggests that the weight of the automobile willforce the spring into compression as it provides a supportiveforce, it must be noted that the initial deflection of the systemsitting in equilibrium with no external forces applied will haveto be subtracted from the system in order to accuratelycharacterize all resulting responses in the system. This initialdisplacement of the vehicle is denoted as xo.
Free Body DiagramIt is now necessary to create a free body diagram of the systemshown if Figure 2 above in order to provide insight on how towrite the equations of motion for this system. As statedpreviously, we will consider the conventional automotivesuspension system augmented with the suggested semi-active MRdamper addition for adding additional kinetic energy mitigation.Figure 3 below shows the associated free body diagram with theforces drawn for the vehicle only, since the outputs of concernare about this mass. It is assumed that the displacement andvelocity of the car are of greater magnitude and in the samedirection for the purpose of creating this diagram. That is,Equation 3 below is true.
x>y∧x> y(3)
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Figure 3: Free Body Diagram of SDOF System
It can be seen from Figure 3 that the displacement of the mass isassumed to be greater than the displacement of the base as inEquation 3. This allows the internal forces in the spring,damper, and MR damper to be drawn in the direction shown. If wewere to fix the base and pull the mass upward, the force in thespring would be wanting to pull the mass back into its originalequilibrium position. Therefore, the spring force is drawn inthe opposite direction of displacement. We know from Hooke’s lawthat the stress in the spring is directly proportional to thestrain as shown by Equation 4 below where k is the springconstant. This shows that the relative motion of the car withrespect to the base is driving what force will be in the spring.Also note that the initial displacement of the vehicle due to theweight is also subtracted from the relative spring force.
Fs=k (x−y)−kxo (4)
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The main purpose of the damper elements are to resist the motionof the car, so the dampers will want to slow down the mass as itmoves away from the base. It is known that the damper force isproportional to the relative velocity of the car body withrespect to the base and the damping coefficient c. Therefore, itis trivial that the force in the conventional damper is simplystated in Equation 5.
Fd=c (x−y ) (5)
In order to write an equation for the force in the MR actuator,some knowledge of how the MR damper operates is needed. Sincethis is not as common of knowledge as the conventional damper, ashort conceptual introduction to the physics behind MR actuationis given courtesy of (Srinivasan and McFarland 88).
An MR (Magnetorheological) fluid is one that can vary inviscosity with the application of a magnetic field. This is madepossible because of the particles that are suspended within thefluid (usually ferric or of some other magnetic type) which areon the order of 1-10 micro-meters and called fabrils. When nomagnetic field is applied to the fluid, the suspended particlesare randomly oriented and have no preferred direction, so thefluid flows normally with normal viscous properties. However,when a magnetic force is applied, the fabrils align with themagnetic field (N-S like a magnet) and become much more difficultto move because the flow becomes “plug like”, or moves similar toa bullet in a barrel. This suggests that the shear layers nearthe wall are conducive for motion. The viscosity of the fluidsubstantially increases and becomes almost solid if the field isgreat enough. This is illustrated in Figure 4 along with theassociated velocity profiles that would result from turning on oroff the magnetic field.
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Figure 4: Magnetorheological Fluids (Source Unknown)
We can therefore define the yield force as the force necessary toovercome the viscous resistance created in the MR fluid.Assuming that a system for generating a reliable magnetic fieldcan be heuristically controlled by the brains of the automobilesystem, (Srinivasan and McFarland) gives us Equation 6 below thattreats the suggested MR damper the same as a conventional damperbut with the addition of the yield force created. This force iscontrolled (turned on or off) by the sensed value of the relativevelocity of the material.
Fa=Fyield (u )∗sgn (x−y )+cMR (x− y) (6)
The term u is part of a magnitude control law that will turn theforce (additional viscosity due to the magnetic field) on or offbased on logic governed in the Skyhook and Skymod controlalgorithms that will presented later in this section. The sgnfunction is simply the sign function that will govern thedirection of the yield force when it is applied to the free bodydiagram in Figure 3 for all cases. That is, it checks thedirection of the relative velocity vector of the car with respectto the base, and applies a 1, 0 or -1 if the value of therelative velocity it positive, negative or zero respectively.With all of the forces in the free body diagram of Figure 3
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quantified, it is now time to derive the governing equation ofmotion to appease Newton’s Second Law.
Deriving the Equations of MotionFrom the free body diagram in Figure 3, it is time to applyNewton’s Second Law to obtain a force balance for the systembeing analyzed. Equation 7 shows that the vector sum of theforces in the vertical direction is equal to the lumped mass ofthe automobile system multiplied by the acceleration of the carbody in the absolute reference frame.
∑Fy=mx (7)
Applying the force balance corresponding to the vector sum of allforces shown in Figure 3, Equation 8 is obtained. Note thatsince the displacement of the car is assumed in the positivevertical direction that all of the forces acting on the body areopposing the direction of motion.
−Fs−Fd−Fa−W=mx (8)
Substituting Equations 4, 5 and 6 into the above force balanceyields Equation 9 in terms of the known and unknown valuescharacterizing this suspension system.
−[k (x−y)−kxo ]−[c ( x−y) ]−[Fyield (u )∗sgn (x−y )+cMR ( x−y) ]−W=m x (9)
For readability, the negatives are distributed into Equation 9and shown in Equation 10. It is also important at this time torecall that the initial deflection of the car body denoted as kxo
is due to the weight of the car sitting on the suspension system.Therefore, we can say that at the zero condition (x and y areboth zero and the car is not moving) all of the terms in Equation10 will cancel except for the weight of the car (W) and theforce in the spring due to the initial spring displacement (kxo).Therefore, these two terms will cancel and are no longer trackedafter Equation 10 below:
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−k (x−y )+kxo−c (x−y )−Fyield (u)∗sgn ( x−y )−( x−y )−W=mx (10)
Rearranging, Equation 10 can be put into a more recognizable formof a second order linear differential equation where the input tothe system is seen as a function of the road displacement y onthe right hand side.
mx+(c+cMR) x+kx=ky+(c+cMR) y−Fyield (u )∗sgn (x−y ) (11)
This suggests the use of the general solution of a second order,linear non-homogeneous differential equation using thecomplimentary and particular solution would be trivial. However,the non-homogeneous solution to this equation (right hand side)cannot be solved analytically due to the control law termsassociated with the yield force term. This suggests the use ofnumerical integration to allow computation of this equation ofmotion.
Newmark-Beta Numerical IntegrationIn order to numerically integrate the governing equation ofmotion generated for this SDOF automotive suspension system, itbecomes necessary to employ a numerical algorithm that isprogrammable into the program of choice for this analysis,Matlab. Other numerical methods such as the Newton-Raphsonmethod could also be used, but from (Mark Shulz) the Newmark-Betaintegration method has proven to be the most accurate and simplewhen integrating equations of motion in the past. (Mark Shulz)provides information on how to integrate the obtained forcebalance equation of motion using this technique provided in thereferenced material. Equations 14, 15 and 16 show the Newmark-Beta method as applied to the Standard Automotive Suspensionsystem that is currently in use on automobiles (consisting ofonly a spring and conventional damper). As a numerical baselinealgorithm, this case will be analyzed prior to the application ofthe MR fluid damper to the system.
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The Newmark-Beta method uses the constant acceleration principlesto step the displacement and velocity (first derivative ofdisplacement) variables through time and test them with theequilibrium of the governing force balance derived in the lastsection. To see the relationship between the Newmark-Betaintegration equations and the generic constant accelerationprinciples, Equations 12 and 13 are shown below.
x2=x1+x ∆t+x ∆t2 (12)
It follows that the change in displacement is equal to thevelocity multiplied by the elapsed time added to the change invelocity (acceleration) multiplied by the square of the elapsedtime.
x2=x1+x ∆t (13)
Similarly, the change in velocity can be calculated as theacceleration multiplied by the elapsed time in the system. Theseconcepts are used along with the rearranged equation of motion inEquation 11 (solved for the velocity of the car) to iterate thesolution with respect to time.
Equations 14, 15 and 16 show similarity to Equations 12 and 13with a little more complexity as an attempt to step throughreference matrix variables is made. Equation 14 is used tocalculate the displacement of the automobile in the verticaldirection due to the input of the road at the current discretetime step as a result of the previous time step displacement,velocity and acceleration values and the current time step valueof acceleration from the previous nth iteration. That is, thesubscripts correspond to the time step variables and thesuperscripts correspond to the nth iteration of the convergenceloop.
xtn=xt−1+xt−1∆t+[(12−β)xt−1+βxt
n−1]∆t2 (14)
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Equation 15 is used to calculate the velocity of the automobileusing the previous time step velocity and acceleration, alongwith the current time step accelerating from the previous nth
iteration.
xtn=xt−1+[ (1−λ ) xt−1+λxt
n−1 ]∆t (15)
The constants β and λ correspond to 0.25 and 0.5 respectively forthe application of constant acceleration to provide a stableiteration convergence according to (Mark Shulz). Equation 16recalculates the acceleration for the current time step as everynth iteration passes such that equilibrium is met for theapplication of Newton’s Second Law. This fact has shown in pasthistory that the Newmark-Beta numerical integration method hasimproved accuracy over similar methods since an element ofphysics in the system in question is applied (Mark Shulz).
xtn=
−1m [k (xt
n−yt)+c (xtn−yt)] (16)
It is important to note again that Equation 16 only holds for theStandard Automotive Suspension System. The modificationnecessary for applying the full equation of motion including theMR damper will require replacing Equation 16 with the following:
xtn=−1
m [k (xtn−yt)+(c+cmr ) (xt
n−yt )+Fyield (u )∗sgn (xtn−yt )] (17)
Note that the only terms that are different in Equation 17 fromEquation 16 are the addition of the MR damping coefficient, cmr,and the FYield term based on the control parameter, u, and the signof the relative velocity, sgn (xt
n−yt), which will return 1, 0 or -1depending on the positive, zero, or negative sign.
In order to perform the Newmark-Beta numerical integration methodin Matlab, two loops using “for” logic are used. (Please seeAppendix A.1-A.4 for reference). The outer loop is used to stepterm by term through a discretized version of the road inputevaluated over a time of 15 seconds with 10,000 terms. The
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purpose of using this many terms is to make sure that goodresolution is observed in the input and that the numericalintegration has a sufficiently small enough time step to minimizethe error in each calculation. This loop will step the tsubscript in Equations 14-16 (replace 16 with 17 for MR solution)term by term through the integration. However, the way thisintegration scheme works is to iterate within each time step forthe acceleration term at the current time step, and apply to theconstant acceleration equations shown above. This is performedwith the second “for” loop that was referenced earlier. It issafe to assume that this inner iteration loop will convergewithin 10 iterations according to (Mark Shulz).
Within these loops, a set of dummy variables are used in order tostep the iteration along while their values at each time step aresaved to a pre-allocated set of matrix vectors. This can be seenin the Appendix A.1-A.4 reference section as well. While it isnot necessary to outline the algorithm construction in thisDesign Study Report, it is important to note that the way theinitial conditions had to be handled for the initial Newmark-BetaIntegrations could have a small effect on the answer obtained. It is important to note at this time that in all proceedingalgorithms the initial conditions of the system are set equal tozero for acceleration, velocity, and displacement when the timeintegration begins. The dummy variables that were used to stepthe iteration along in the code are shown below with theirrespective components of Equations 14-16.
xtn=x (18)
xtn∧xt
n−1=xdot (19)
xtn∧xt
n−1=xddot (20)
x0=x0 (21)
x0=xdot0 (22)
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x0=xddot0 (23)
The algorithms presented in Appendix A.1-A.4 are now used togenerate the outputs in the following sections. Detailedanalysis will be performed and presented on the outputs of thesealgorithms, but the compilation of all plots output from thesealgorithms was too large to put entirely in the Analysis body.For this reason, an example of each output will be shown once forthe v = 5 mph, λ = 10 ft case for each algorithm and all otherswill be located in Appendix A.5, A.6 and A.7 for the Baseline,Theoretical Skyhook and Modified Skyhook cases respectively. Thecase of this road wavelength and vehicle speed combination willbe referred to as the “Reference Case” from here forward.
Standard Automotive SuspensionThe algorithm presented in Appendix A.1 is used to develop thesubsequent results presented in this section. The purpose ofthis section is to provide a Baseline case to check the algorithmand to provide a point of comparison for the Skyhook algorithmsand their performance. For the baseline standard suspension casewith the “Reference Case” for the road input, the Matlab ouput ofresults is shown in Figure 5 with the Road Input, CarDisplacement, Car Velocity, and Car Acceleration overlaid as afunction of time. Note that while all calculations wereperformed for 15 seconds, the amount of oscillations shown hasbeen scaled down to make the output plots more legible.
It can be seen for this case as well as all of the other casesanalyzed for the standard suspension design in the Appendix thata startup transient occurs when the road input is first appliedbefore the vehicle settles into a steady-state period ofoscillation.
Note: It has been suggested to neglect this first transientperiod of oscillation for the purpose of comparing designshowever, this was against my better judgment and will beconsidered for all cases. This is because the quality of ride
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and safety of a vehicle is not only important when the vehicle isoperating at steady-state. More times than none, it is in factthis first transient overshoot that causes the most amount ofdiscomfort when an uneven portion of road is first struck.
Figure 5: Standard Suspension V = 5 mph, λ = 10 ft
It is noticeable that there is a finite amount of phase lag inthe input-output system from the timing of the road input to thetiming of the displacement of the car. At this first condition,there is about 4 inches of automobile displacement due to theinput of the road at steady-state. When we compare all of theother road input conditions side by side, it will becomenoticeable that there is indeed a considerable amount of room forimprovement in this system. The velocity of the car is also ofsignificant magnitude. When the Skyhook algorithm is introducedin the next section, this will be the mechanism of action that we
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try to mitigate to ensure a smooth ride in this automobile. Itis also noticeable that the force in the spring tends to fightagainst the relative displacement of the car body. It does notmatch the absolute displacement of the car body because thespring force is proportional to the base displacement as well.Similarly, the damping force also seems to counteract velocity,which is what it is designed to do, resist motion. This force isalso directly equal to the relative velocity of the car body withrespect to the base instead of the absolute reference frame.
It should also be mentioned that this solution was checked withthe general solution for the second order non-homogeneousdifferential equation presented in (Palm) with near identicalresults, giving confidence to the algorithm that has beencreated.
It also follows that by considering the rest of the output graphsin Appendix A.5, any change in λ or v will have an associatedeffect on the frequency of the input system as depicted inEquation 1. This can be explained by the fact that the automotivesuspension system can be thought of as an input-output transferfunction from the world of controls, and that varying the inputfrequency will vary the output in the system proportional to thesecond order, homogeneous equation of motion that is the leftside of Equation 11. From all of the combinations of the roadinput wavelengths and vehicle velocities, the maximum values ofthe absolute value of the resultant output [max(abs(x(t)))] havebeen tabulated and are shown below if Figure 6 for comparison.
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Figure 6: Max(ABS(x(t))) for Standard Suspension Design
It is immediately noticeable that two peaks stand out asastronomically high, weighing in at over 6 inches based on theroad input. These correspond to a vehicle velocity of 13.726 mphand road wavelength of 20 ft, and a vehicle velocity of 60 mphand road wavelength of 88 ft. This demands the question ofwhether or not some form of resonance or near resonance behavioris occurring at these locations. A simple check with the systemcharacteristics provided in Table 1 is performed below inEquation 24.
ωo=√km=√ 4000 lbft
3220lb/32.2 fts2
≈6.325 rads(24)
Now that we know where the natural (resonant) frequency is forthe standard suspension system, we can inspect all of the roadwavelength and vehicle velocity combinations for their inputfrequency in relation to this to check for resonant behavior.Recall from Equation 1 that the input frequency of the road can
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be calculated as a function of the road wavelength and vehiclevelocity as:
ω=2π vλ
[rad /s] (25)
This allows the calculation of the input frequency to be laidatop each of the input combination bars in Figure 7 below.
Figure 7: Resonant Behavior Check
In Figure 7 above, a red dotted line has been overlain at theexact point of the natural frequency of the system from thisvisualization of the input frequencies, it is apparent that ourconcerns are true and that for the inputs of a vehicle velocityof 13.726 mph and road wavelength of 20 ft, and a vehiclevelocity of 60 mph and road wavelength of 88 ft, we indeedobserve near resonant behavior in the system. This isunacceptable from a ride comfort perspective, and will beattenuated with the new design. We now have an acceptablebaseline case for comparison with the subsequent designs to bepresented.
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Skyhook Control Algorithm and MR SuspensionThe theoretical Skyhook control algorithm is the basis for howthe governing equation for the MR suspension design will operate.Note that everything referenced in this section will refer toAppendix A.2 for the Skyhook Control Algorithm written in Matlab.From this section forward, the Newmark-Beta integration methodwill incorporate Equation 17 in place of Equation 16 to encompassthe effects of the MR damper in the suspension design. Recallfrom the section about MR fluids that the magnitude of the yieldforce used in Equation 6 will be determined using a heuristiccontrol law that will define the variable, u, with eachiteration.
It is now that the Skyhook Control Algorithm will be introducedfrom the reference (Skyhook Control Algorithm) and will bediscussed in the sense that it was used in the Skyhook ControlAlgorithm in Appendix A.2. The objective of this semi-activecontrol algorithm is to minimize kinetic energy imparted intoautomobile due to road input to suspension system. That is, thevelocity of the car in the vertical direction due to theoscillation of the road is desired to be minimized in comparisonto the case with the standard suspension design presented in theprevious section. There are four overall control laws from theSkyhook Control Algorithm (Shulz), 2 for a positive car verticalvelocity and two for a negative car vertical velocity due to theroad input.
The first pair of control laws govern the scenario when theautomobile is vertically displacing in the positive upwarddirection. We first consider the scenario where the car body ismoving faster in the upward direction faster than the base of thesuspension system (stuck to the road by Assumption 2). In thiscase the base of the vehicle suspension could be moving in eitherthe positive or negative direction. That is, in either case, thevehicle body is moving away from the base of the suspension, sowe want to turn the MR damper on such that we slow the car bodydown. Then we can define the first control law as follows:
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ifx>0∧x> ythenu=1 (26)
The second control law states that if the vehicle is moving inthe positive upward direction and the base of the vehiclesuspension is moving faster than the car body (that is, it isclosing in on the body) then we turn the MR suspension off so asto prevent the excess transmission of force into the vehicle bystiffening the MR damper. This is shown in Equation 27 below:
ifx>0∧x< ythenu=0 (27)
The second pair of control laws govern the scenario in which thecar body is displacing in the downward and negative direction inits period of oscillation. This set of control laws is not asintuitive as the first, because the negative sign has to be takeninto account possibly on both x and y. We consider the first ofthese two laws when the car body is moving downward (in thenegative direction) slower than the road (also moving in thenegative downward direction). In this case, we want to turn offthe MR damper so that the car body is not forced downward withthe contour of the road as the base is forced to follow. This isshown in Equation 28 below.
ifx<0∧x> ythenu=0 (28)
The second control law of this pair governs the scenario wherethe car body is moving faster in the negative direction than theroad (road could be moving in the positive or negative directionfor this case) then the damper should be switched on to reducethe downward motion of the car body. This final control law isgiven below in Equation 29.
ifx<0∧x< ythenu=1 (29)
Therefore, it can be seen that the application of this heuristiccontrol law attempts to provide feedback action to reduce thekinetic energy in the car body at any given condition.Therefore, this should (in theory) act to reduce the vibration ofthe automobile body for any given road input. This theory will
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be tested with the following outputs of the Skyhook ControlAlgorithm documented in Appendix A.2.
Figure 8 below shows the same output as Figure 5, but this timewith the application of the MR damper to the suspension designand the implementation of the Skyhook Control Algorithm. Also,the yield force in the damper is shown as a function of time asthe control law turns it on and off.
Figure 8: Skyhook MR Suspension V = 5 mph, λ = 10 ft
Again, the remaining outputs for the other 11 vehicle velocityand road wavelength combinations are tabulated in the appendix.For the Skyhook Suspension Design, see Appendix A.6. We notice avery different characteristic that was observed in the passivesuspension system output for this wavelength and velocitycombination shown in Figure 5. This time, there is evidence of a
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deliberate halting of displacement due to the control algorithm.It is interesting to see how the velocity of the car body startsto peak, and is then brought back to zero by the yield force ofthe MR damper. The acceleration curve is even more interesting,since it has to deliberately reverse directions when the MRdamper is switched on in order to slow the car body down. It isalso noticeable that the MR Force (yield force) has a “bang-onbang-off” nature that causes these abrupt changes in the vehiclebody response. Again, it makes perfect sense that the springforce always seems to fight against the relative displacement ofthe car body, while the damper does the same with the relativevelocity of the car body. This initial look at the Skyhooksuspension design has already significantly improved the responseof the vehicle for this given input combination. The followingthree Figures show a trade study that was performed to optimizethe yield force used in the MR damper for this semi-activesuspension design. Note that Figure 8 was a “cheated look” atthe optimized solution with the resulting yield force in the MRdamper applied to the Matlab code.
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Figure 10: Skyhook Suspension Design – Car Body Velocity - Trade Study
Figure 11: Skyhook Suspension Design – Car Body Acceleration - Trade Study
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It can be seen that performing a trade study by varying the yieldforce in the MR damper has a very substantial effect on themaximum response of the car body due to the various inputs incomparison with the baseline standard suspension design. We canobserve that as the yield force is backed down from its maximumvalue given in Table 2, the velocity and displacement response ofthe car body quickly approach the baseline values from theprevious analysis. It is also interesting to note that theacceleration imposed on the car body behaves to the contrary, asit is not “jerked” as hard when the MR damper yield force is notas great.
Since at this point we have a good idea for how the vehiclebehaves across 12 combinations of road input wavelengths andvehicle transverse velocities for a number of MR damping forces,it is time to perform some optimization in order to determine thebest design for the Skyhook Semi-active Suspension Design. Byconsidering the fact that it is desirable to find the best yieldforce that will minimize the velocity of the car body at ALLinput combinations a method of averaging was implemented to findthe statistical average of the car body at each yield stress forall of the input combinations. That is, for a given yield force,all of the responses for the combinations of the road wavelengthand vehicle speed are averaged together to create one averageresponse point. This leads to the creation of Figure 12 below,which clearly shows that the average “smoothness” in the ride isoptimized near a high yield force.
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Figure 12: Average Car Body Velocity vs. FYield
Further investigation of this curve in Microsoft Excel yielded anoptimized yield force shown below in Table 4. This best casescenario for the Skyhook Suspension Design is compared to theBaseline Suspension Design in order to show its effectiveness.The percent improvement for each input case has been calculatedand tabulated in Table 4. For ease of viewing however, it may beof more benefit to view the improvement curves in Figure 13 onthe next page.
Table 4: Comparison of Baseline and Optimized Skyhook Algorithm
V [m ph] V [ft/s] λ [ft] X Xdot Xddot X Xdot Xddot X Xdot Xddot5 7.33 10 0.294 1.466 7.311 0.060 0.480 5.186 0.80 0.67 0.295 7.33 20 0.165 0.397 1.952 0.084 0.272 8.906 0.49 0.32 -3.565 7.33 88 0.120 0.106 0.341 0.103 0.062 8.221 0.14 0.42 -23.14
13.726 20.13 10 0.093 0.901 8.564 0.036 0.550 7.942 0.61 0.39 0.0713.726 20.13 20 0.569 3.600 22.773 0.047 0.533 4.666 0.92 0.85 0.8013.726 20.13 88 0.130 0.273 0.924 0.098 0.170 8.480 0.25 0.38 -8.1840 58.67 10 0.023 0.311 8.428 0.013 0.350 13.749 0.43 -0.13 -0.6340 58.67 20 0.058 0.498 7.809 0.026 0.486 9.893 0.55 0.02 -0.2740 58.67 88 0.256 1.173 5.890 0.063 0.456 5.333 0.75 0.61 0.0960 88.00 10 0.016 0.260 10.585 0.008 0.298 17.380 0.48 -0.15 -0.6460 88.00 20 0.033 0.382 7.805 0.018 0.406 11.989 0.47 -0.06 -0.5460 88.00 88 0.572 3.593 22.576 0.047 0.532 4.687 0.92 0.85 0.79
Baseline Suspension Best Fyld = 750 lb % Im provem ent with Control
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Figure 13: % Improvement of Optimized Skyhook Algorithm Displacement andVelocity of Car Body
The above chart shows that for every case of the car bodydisplacement, the Skyhook Suspension Design significantlymitigated the baseline case responses. However, for the velocityof the car body (which is the main Critical to Quality (CTQ) fora “smooth” ride) this is not necessarily the case. While thebest case for improvement is 85% for the baseline conditions thatexhibited near resonance characteristics, the input combinationsof λ = 10 ft and v = 40 mph, λ = 10 ft and v = 6 mph, and λ = 20ft and v = 60 mph show a worse case of vehicle response. This isthe concept that will be tackled in the next section on improvingthe Skyhook Algorithm for the proposed design for the nextgeneration of CCC mid-sized automobile suspension systems.
Another interesting study that was performed is the effect of theyield force in the MR damper on the velocity of the car body for
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various input combinations. This time, the peak velocity of thecar body for each case is plotted as a function of ω/ωo, or theratio of the input frequency to the natural frequency. Thisgives an interesting look at what happens to the peak resonantresponse when the yield force is varied. This chart re-affirmsfaith that when the yield force is decreased away from themaximum value, the response in the vehicle approaches theresponse of the baseline case, as we would expect.
The Skyhook Control Algorithm is very well optimized and hasalready exhibited phenomenal improvement on the baselinesuspension system for the low cost of a simple semi-activecontrol system and additional damper with MR fluid. As statedabove, the next section will attack the regions where the Skyhookdid not improve the baseline suspension design, whilesimultaneously optimizing the rest of the vehicle suspensionresponse cases.
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Modified Skyhook Control AlgorithmThe Skyhook Control Algorithm in Appendix A.2 will now bemodified to improve upon the vehicle responses presented in thelast section. This will be accomplished using the Skyhook Matlabcode in the Appendix A.2 with a slight modification to step theyield force in the MR damper in increments of 1 from 0 to 784lbs. This modified code is presented in Appendix A.3 in order toprovide a better optimal point for each input condition than thecrude trade study performed above would. The benefit of thischange will allow a table of optimized values for each road inputcombination to be read into the Control Algorithm such that theoptimum yield force is always in use for the system. Afterrunning the optimization code in Appendix A.3, Table 5 belowshows the resultant yield forces for each input condition in theform of input frequency, ω.
Table 5: Optimized Values for each input frequency
ω [rad/s] Best F yld
4.608 7632.304 2530.524 71012.649 7766.324 7541.437 68436.861 30918.431 4164.189 74455.292 44127.646 3196.283 753
M atlab xlsread
This warrants the formation of the Modified Skyhook ControlAlgorithm presented in Appendix A.4. This algorithm has a read-in funciton that commits this reference table of values to memorysuch that the computer can have a look-up table for any range ofinput conditions. This new algorithm allows the most optimized
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value of the yield force to be used for each road inputcombination. In practice, this would have to be done for a muchfiner resolution of road input frequencies. However, for thescope of this preliminary theoretical design, this level ofoptimizaation will get the necessary point across.
After running the Skymod Matlab code (Appendix A.4) for all ofthe road input wavelengths and autmobile velocities, the outputsare made available in Appendix A.7. Figure 15 below shows anexample for the defined “Reference Case” of road wavelength andvehicle velocity. These cases do not look very differentcompared to those from the original Skyhook calculation becauseonly suddle changes were made. However, it is noticable that thecases the input combinations of λ = 10 ft and v = 40 mph, λ = 10ft and v = 6 mph, and λ = 20 ft and v = 60 mph show a marginalimprovement in vehicle response.
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Figure 14: Skymod MR Suspension V = 5 mph, λ = 10 ft
Table 6 below shows a similar trade study as in Table 4 (Baselinevs. Skyhook), except this time it is the Skyhook and Skymoddesigns that are compared. It can be noticed that all of the carbody velocities show improvement from the Skyhook results, butsome of the peak displacements actually became marginally worse.
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Table 6: Comparison of Skyhook vs. Improved Skyhook (Skymod) ControlAlgorithms
V [m ph] V [ft/s] λ [ft] X Xdot Xddot X Xdot Xddot X Xdot Xddot5 7.33 10 0.060 0.480 5.186 0.060 0.480 5.316 0.00 0.00 -0.035 7.33 20 0.084 0.272 8.906 0.087 0.272 1.399 -0.03 0.00 0.845 7.33 88 0.103 0.062 8.221 0.103 0.062 7.797 0.00 0.00 0.05
13.726 20.13 10 0.036 0.550 7.942 0.035 0.545 7.925 0.02 0.01 0.0013.726 20.13 20 0.047 0.533 4.666 0.047 0.533 4.724 0.00 0.00 -0.0113.726 20.13 88 0.098 0.170 8.480 0.098 0.169 7.808 0.00 0.00 0.0840 58.67 10 0.013 0.350 13.749 0.017 0.301 11.419 -0.30 0.14 0.1740 58.67 20 0.026 0.486 9.893 0.035 0.471 9.356 -0.34 0.03 0.0540 58.67 88 0.063 0.456 5.333 0.063 0.455 5.276 0.00 0.00 0.0160 88.00 10 0.008 0.298 17.380 0.009 0.264 15.303 -0.14 0.11 0.1260 88.00 20 0.018 0.406 11.989 0.024 0.342 9.966 -0.37 0.16 0.1760 88.00 88 0.047 0.532 4.687 0.047 0.532 4.741 0.00 0.00 -0.01
Best Fyld = 750 lb Optim ized Fyld Curve % Im provem ent with Control
This data is also visually shown in comparison with the other twodesigns showing the “ride smoothness” or the magnitude of thepeak value of car body velocity in Figure 16 below. This showsthat the Baseline Suspension design case (orange) provides asignificantly rougher ride when compare with the Skyhook Semi-active MR damper (red) design as outlined in the previoussection. The Skymod MR damper design (blue) shows to bemarginally improved at the road input cases of concern from theprevious section.
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Figure 15: Comparison of Baseline, Skyhook and Skymod Suspension Systems
Therefore, this is the design that is recommended forimplementation into the next generation of mid-sized vehicles forthe CCC. This concludes the engineering analysis and trade studycalculations that were performed with the purpose of providing animproved suspension design.
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E.Conclusions
The aforementioned technologies discussed in the analysissection have been compared in terms of performance, and therecommendation of this Design Study Summary is to proceed withthe optimized and improved Skymod Control Algorithm for the newaged MR suspension system, Hover-Ride. From a theoreticalperspective, this has now been shown to be the most reliable andversatile design to provide the customer of the CCC mid-sizedvehicle with the most desirable and smooth ride possible for notjust one, but ALL road input conditions that the driver couldsee. From a cost perspective, this design is desirable because,as is shown in Appendix A.1-A.4, the amount of computing powerrequired to control the actuation system is not very large.Therefore this control law could be integrated into the existingautomotive computer system with little to no modification. Froma hardware perspective, this design incorporates the legacydesign of suspension system (used in baseline calculations)consisting of a passive spring and damper. The only additionalrequirement for this design is the additional MR fluid damper andan electromagnetic coil to create the actuation force.Therefore, this is a desirable new direction with a low cost ofimplementation relative to a complete re-design of existingtechnology. From a safety standpoint, the fact that thissuspension design minimizes the vertically dissipated kineticenergy in the automobile body (especially at the near resonancecases) gives the driver more overall control of the automobilebecause the weight of the car remains firmly pressing down infull on the road, thus improving the traction of the vehicle.The reduction in the bouncing of the vehicle may play a part inreducing the driver’s ability to avoid an accident.
If this is indeed the selected design of choice forimplementation, there is additional forward work that will needto be performed before this design is ready to be released intothe consumer market. The optimization techniques outlined in theModified Skyhook section and in Appendix A.3 should be performedfor a finer spectrum of operating inputs, possibly from an inputfrequency of 0 up to a frequency the automobile may likely never
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see. This design should then be mounted on a shaker to test thepredictions presented in this algorithm. Some assumptions mademay severely simplify what the responses truly look like inpractice, such as the assumption that the wheel always remainsattached to the road and that the application of the yield forceis instantaneous. If these assumptions are proven incorrect,then the prediction algorithms must be modified to account forany delays in the control algorithm or for the wheel jumping offof the road at some of the more violent input cases. If theshaker test goes well, it will be time to integrate this designinto a 4 wheel application where the pitch and roll of thevehicle will have to be characterized and accounted for. All inall, this design has great potential, and if selected, should notbe overly costly or difficult to implement.
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F.References
Mark Shulz, PhD. "Integration Notes." 1983.Palm, William J. (III). "System Dynamics Second Edition." New York, NY: McGraw
Hill, 2010. 80-212.Shulz, Mark. Skyhook Control Algorithm. Draft. Cincinnati, OH: University of
Cinicinnati, 2003.Srinivasan, A. V. and D. Michael McFarland. Smart Structures. New York, NY:
Cambridge University press, 2001.Wikipedia: Wavelength. n.d. 22 October 2012
<http://en.wikipedia.org/wiki/Wavelength>.
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G.Appendix
Appendix A.1 – Baseline Suspension Model Matlab Code% Standard_Suspension.m%% This code is written as a baseline to the Skyhook algorithm model to% validate a model working with a SDOF spring mass damper system modeling% the typical automotive suspention system assuming 1 wheel and only forces% in the vertical direction.%%**********************************************************************% Author: Chad Sutton% Date: 9-OCT-2012% Introduction to Smart Structures% University of Cincinnati% Cincinnati, Ohio 45221-0072% TEL: 615-400-1658% E-MAIL: chad.h.sutton@gmail.com%*********************************************************************%%%% Clear Woskspace clear all;close all;clc %% Define all variables% Suspension System variablesW = 3220; % [lb] - Weight of car being supportedg = 32.2; % [ft/s^2] - Gravitational AccelerationM = W/g; % [slug] - Mass of car being supportedC = 134; % [lb-s/ft] - System Damping CoefficientK = 4000; % [lb/ft] - System Spring Constant % Initialize matrix for storing peak responsesPeak_Responses = zeros(5,12);jj = 1; % Forcing function inputsfor V = [5 13.726 40 60]; % [mph] Velocties of car to be consideredv = V*5280/3600; % [ft/s] Velocties of car to be considered for lambda = [10 20 88]; % [ft] Road wavelenth y0 = 0.118; % [ft] - Amplitude of the road input % Define input frequency for y(t)
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f = v/lambda; % [Hz] % Define time for system analysis T = 15; % [s] - Total time for system analysis N = 10000; % Number of points to be considered t = linspace(0,T,N); % [s] - Time vector for analysis dt = T/N; % [s] - Iteration delta time % System initial conditions displ = zeros(1,length(t)); % [ft] - Displacement of the mass (car) vel = zeros(1,length(t)); % [ft/s] - Velocity of mass (car) accel = zeros(1,length(t)); % [ft/s^2] - Acceleration of mass (car) F_s = zeros(1,length(t)); % [lb] - Spring force as f(t) F_c = zeros(1,length(t)); % [lb] - Damper force as f(t) %% Define input forcing function y(t) % Assume forcing function is sinusoidal with given Forcing function inputs w = 2*pi*f; % [rad/s] - input frequecy for y(t) y = y0*sin(w.*t); % [ft] - Displacement of Road ydot = w*y0*cos(w.*t); % [ft/s] - Velocity of Road %% Algorithm for iterating equations of motion (Newmark-Beta method) % Constants used in Newmark-Beta method (see integration nots p.6) Beta = 0.25; Lambda = 0.5; % Initialize variables at current time (t) and for n from n-1 x = 0; xdot = 0; xddot = 0; % Initialize variables to be called from previous time x0 = 0; xdot0 = 0; xddot0 = 0; for ii=1:length(t) %disp(['Loop iteration for t = ' num2str(t_iter) ' sec']) for n = 1:10 % iterate displacement, velocity, acceleration x = x0 + xdot0*dt + ((0.5-Beta)*xddot0+Beta*xddot)*dt^2; xdot = xdot0 + ((1-Lambda)*xddot0 + Lambda*xddot)*dt; xddot = -(1/M)*(K*(x-y(ii)) + C*(xdot-ydot(ii))); end % Store variables with every integration iteration displ(1,ii) = x; vel(1,ii) = xdot; accel(1,ii) = xddot; F_s(1,ii) = K*(x-y(ii)); F_c(1,ii) = C*(xdot-ydot(ii));
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% Assign current ii solutions to ii-1 for next loop iteration x0 = x; xdot0 = xdot; xddot0 = xddot; end %% Create plots of results for viewing and consideration figure subplot(4,1,1) plot(t,y), hold on, plot(t,displ,'r-') grid on title('Road Displacement & Car Displacement vs. Time') xlabel('Time [s]') ylabel('Displacement [ft]') legend('Road Input','Car Output') xlim([0 min([5/f T])]) subplot(4,1,2) plot(t,vel) grid on title('Baseline Velocity vs. Time') xlabel('Time [s]') ylabel('Velocity [ft/s]') xlim([0 min([5/f T])]) subplot(4,1,3) plot(t,accel) grid on title('Baseline Acceleration vs. Time') xlabel('Time [s]') ylabel('Acceleration [ft/s^2]') xlim([0 min([5/f T])]) subplot(4,1,4) plot(t,F_s), hold on, plot(t,F_c,'r-') grid on title('Suspension Forces vs. Time') xlabel('Time [s]') ylabel('Force [lb]') legend('Spring Force','Damper Force') xlim([0 min([5/f T])]) disp(['Viewing plots for V = ' num2str(V) ' [mph] and lambda = '
num2str(lambda) ' [ft]']) % Save Disp, Vel, Accel figure for current velocity and lambda combination set(gcf, 'Position', get(0,'Screensize')); % Maximize figure. saveas(gcf, ['V = ' num2str(v) ' fps_lam = ' num2str(lambda) '
ft_Outputs.png']);
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%pause close all %% Grab peaks for each combination of lambda and velocity Peak_Responses(1,jj) = v; Peak_Responses(2,jj) = lambda; Peak_Responses(3,jj) = max(abs(displ)); Peak_Responses(4,jj) = max(abs(vel)); Peak_Responses(5,jj) = max(abs(accel)); jj = jj+1; endend % Transpose Response maatrixPeak_Responses = Peak_Responses';
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Appendix A.2 – Standard Skyhook Suspension System (Matlab Code)% MR_Suspension_Skyhook.m%% This code is written as the Skyhook algorithm model to show improvement% from the SDOF spring mass damper system modeling the typical automotive% suspention system assuming 1 wheel and only forces in the vertical% direction.%%**********************************************************************% Author: Chad Sutton% Date: 14-OCT-2012% Introduction to Smart Structures% University of Cincinnati% Cincinnati, Ohio 45221-0072% TEL: 615-400-1658% E-MAIL: chad.h.sutton@gmail.com%*********************************************************************%%%% Clear Woskspace clear all;close all;clc %% Define all variables% Suspension System variablesW = 3220; % [lb] - Weight of car being supportedg = 32.2; % [ft/s^2] - Gravitational AccelerationM = W/g; % [slug] - Mass of car being supportedC = 134; % [lb-s/ft] - System Damping CoefficientK = 4000; % [lb/ft] - System Spring Constant % Magnetorheleological Damping CharacteristicsCmr = 21; % [lb-s/ft] - MR Damping CoefficientFyld = 750; % [lb] - Yield force for the MR Damper (Max 784) ADJUST THIS!!! % Initialize matrix for storing peak responsesPeak_Responses = zeros(5,12);jj = 1; % Forcing function inputsfor V = [5 13.726 40 60]; % [mph] Velocties of car to be considered v = V*5280/3600; % [ft/s] Velocties of car to be considered for lambda = [10 20 88]; % [ft] Road wavelenth y0 = 0.118; % [ft] - Amplitude of the road input
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% Define input frequency for y(t) f = v/lambda; % [Hz] % Define time for system analysis T = 15; % [s] - Total time for system analysis N = 10000; % Number of points to be considered t = linspace(0,T,N); % [s] - Time vector for analysis dt = T/N; % [s] - Iteration delta time % System initial conditions displ = zeros(1,length(t)); % [ft] - Displacement of the mass (car) vel = zeros(1,length(t)); % [ft/s] - Velocity of mass (car) accel = zeros(1,length(t)); % [ft/s^2] - Acceleration of mass (car) F_MR = zeros(1,length(t)); % [lb] - Force in the MR damper as f(t) F_s = zeros(1,length(t)); % [lb] - Spring force as f(t) F_c = zeros(1,length(t)); % [lb] - Damper force as f(t) %% Define input forcing function y(t) % Assume forcing function is sinusoidal with given Forcing function inputs w = 2*pi*f; % [rad/s] - input frequecy for y(t) y = y0*sin(w.*t); % [ft] - Displacement of Road ydot = w*y0*cos(w.*t); % [ft/s] - Velocity of Road %% Algorithm for iterating equations of motion (Newmark-Beta method) % Constants used in Newmark-Beta method (see integration nots p.6) Beta = 0.25; Lambda = 0.5; % Initialize variables at current time (t) and for n from n-1 x = 0; xdot = 0; xddot = 0; % Initialize variables to be called from previous time x0 = 0; xdot0 = 0; xddot0 = 0; for ii = 1:length(t) %disp(['Loop iteration for t = ' num2str(t_iter) ' sec']) for n = 1:10 % iterate displacement, velocity, acceleration x = x0 + xdot0*dt + ((0.5-Beta)*xddot0+Beta*xddot)*dt^2; xdot = xdot0 + ((1-Lambda)*xddot0 + Lambda*xddot)*dt; %% Apply Skyhook Control Laws for xdot > 0 % Car is moving faster in the upward direction than road
(could be % positive or negative), turn ON MR to reduce motion of car.
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if xdot > 0 && xdot > ydot(ii) u = 1; % Car is moving upward slower than the road (positive), turn
OFF MR % to reduce force transmitted to car. elseif xdot > 0 && xdot < ydot(ii) u = 0; end %% Apply Skyhook Control Laws for xdot < 0 % Car is moving downward slower than the road (negative), turn
OFF % MR damper so that tire motion downward will not force car
down % with it. if xdot < 0 && xdot > ydot(ii) u = 0; % Since car is moving faster in the negative direction than
the % road (could be positive or negative), damper should be on to % reduce motion of the car. elseif xdot < 0 && xdot < ydot(ii) u = 1; end %% Catch xdot = 0 situation if xdot == 0 u = 0; end %% Apply Newton's 2nd Law for equilibrium xddot = -(1/M) * (K*(x-y(ii)) + (C+Cmr)*(xdot-ydot(ii)) +
Fyld*u*sign(xdot-ydot(ii))); end % Store variables with every integration iteration displ(1,ii) = x; vel(1,ii) = xdot; accel(1,ii) = xddot; F_MR(1,ii) = Fyld*u*sign(xdot-ydot(ii)) + Cmr*(xdot-ydot(ii)); F_s(1,ii) = K*(x-y(ii)); F_c(1,ii) = (C+Cmr)*(xdot-ydot(ii)); % Assign current ii solutions to ii-1 for next loop iteration x0 = x; xdot0 = xdot; xddot0 = xddot; end %% Create plots of results for viewing and consideration figure
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subplot(4,1,1) plot(t,y), hold on, plot(t,displ,'r-') grid on title('Road Displacement & Car Displacement vs. Time') xlabel('Time [s]') ylabel('Displacement [ft]') legend('Road Input','Car Output') xlim([0 min([5/f T])]) subplot(4,1,2) plot(t,vel) grid on title('Skyhook Velocity vs. Time') xlabel('Time [s]') ylabel('Velocity [ft/s]') xlim([0 min([5/f T])]) subplot(4,1,3) plot(t,accel) grid on title('Skyhook Acceleration vs. Time') xlabel('Time [s]') ylabel('Acceleration [ft/s^2]') xlim([0 min([5/f T])]) ylim([-max(abs(accel))-1 max(abs(accel))+1]) subplot(4,1,4) plot(t,F_s), hold on, plot(t,F_c,'r-'), plot(t,F_MR,'k-') grid on title('Suspension Forces vs. Time') xlabel('Time [s]') ylabel('Force [lb]') legend('Spring Force','Damper Force','MR Force') xlim([0 min([5/f T])]) disp(['Viewing plots for V = ' num2str(V) ' [mph] and lambda = ' num2str(lambda) ' [ft]']) % Save Disp, Vel, Accel figure for current velocity and lambda combination set(gcf, 'Position', get(0,'Screensize')); % Maximize figure. saveas(gcf, ['V = ' num2str(v) ' fps_lam = ' num2str(lambda) '
ft_Outputs.png']); pause close all %% Grab peaks for each combination of lambda and velocity Peak_Responses(1,jj) = v;
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Peak_Responses(2,jj) = lambda; Peak_Responses(3,jj) = max(abs(displ)); Peak_Responses(4,jj) = max(abs(vel)); Peak_Responses(5,jj) = max(abs(accel)); jj = jj+1; endend % Transpose Response maatrixPeak_Responses = Peak_Responses';
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Appendix A.3 – Optimization Code for Modifying Skyhook Suspension (MatlabCode) % MR_Suspension_FyldCalc.m%% This code is written as a modification to the Skyhook algorithm model to% minimize xdot with Fyld. This will be compared with a SDOF spring mass% damper system modeling the typical automotive suspention system assuming% 1 wheel and only forces in the vertical direction.%%**********************************************************************% Author: Chad Sutton% Date: 14-OCT-2012% Introduction to Smart Structures% University of Cincinnati% Cincinnati, Ohio 45221-0072% TEL: 615-400-1658% E-MAIL: chad.h.sutton@gmail.com%*********************************************************************%%%% Clear Woskspace clear all;close all;clc %% Define all variables% Suspension System variablesW = 3220; % [lb] - Weight of car being supportedg = 32.2; % [ft/s^2] - Gravitational AccelerationM = W/g; % [slug] - Mass of car being supportedC = 134; % [lb-s/ft] - System Damping CoefficientK = 4000; % [lb/ft] - System Spring Constant % Magnetorheleological Damping CharacteristicsCmr = 21; % [lb-s/ft] - MR Damping CoefficientFyld_max = 784; % [lb] - Yield force for the MR Damper (Max 784)Fyld_Opt = zeros(2,Fyld_max); % Initialize matrix for storing peak responsesPeak_Responses = zeros(4,12);jj = 1; % Forcing function inputsfor V = [5 13.726 40 60]; % [mph] Velocties of car to be considered v = V*5280/3600; % [ft/s] Velocties of car to be considered
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for lambda = [10 20 88]; % [ft] Road wavelenth y0 = 0.118; % [ft] - Amplitude of the road input % Define input frequency for y(t f = v/lambda; % [Hz] % Define time for system analysis T = 15; % [s] - Total time for system analysis N = 10000; % Number of points to be considered t = linspace(0,T,N); % [s] - Time vector for analysis dt = T/N; % [s] - Iteration delta time % System output variable storage matricies displ = zeros(1,length(t)); % [ft] - Displacement of the mass (car) vel = zeros(1,length(t)); % [ft/s] - Velocity of mass (car) accel = zeros(1,length(t)); % [ft/s^2] - Acceleration of mass (car) F_MR = zeros(1,length(t)); % [lb] - Force in the MR damper as f(t) % Assume forcing function is sinusoidal with given Forcing function inputs w = 2*pi*f; % [rad/s] - input frequecy for y(t) y = y0*sin(w.*t); % [ft] - Displacement of Road ydot = w*y0*cos(w.*t); % [ft/s] - Velocity of Road yddot = -w^2*y0*sin(w.*t); % [ft/s^2] - Accleration of Road %% Vary Fyld to determine which gives best response for minimizing xdot for Fyld = 1:Fyld_max %% Algorithm for iterating equations of motion (Newmark-Beta method) % Constants used in Newmark-Beta method (see integration nots p.6) Beta = 0.25; Lambda = 0.5; % Initialize variables at current time (t) and for n from n-1 x = 0; xdot = 0; xddot = 0; % Initialize variables to be called from previous time x0 = 0; xdot0 = 0; xddot0 = 0; for ii = 1:length(t) for n = 1:10 % iterate displacement, velocity, acceleration x = x0 + xdot0*dt + ((0.5-Beta)*xddot0+Beta*xddot)*dt^2; xdot = xdot0 + ((1-Lambda)*xddot0 + Lambda*xddot)*dt; %% Apply Skyhook Control Laws for xdot > 0
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% Car is moving faster in the upward direction than road (could be
% positive or negative), turn ON MR to reduce motion of car.
if xdot > 0 && xdot > ydot(ii) u = 1; % Car is moving upward slower than the road (positive),
turn OFF MR % to reduce force transmitted to car. elseif xdot > 0 && xdot < ydot(ii) u = 0; %% Apply Skyhook Control Laws for xdot < 0 % Car is moving downward slower than the road (negative),
turn OFF % MR damper so that tire motion downward will not force
car down % with it. elseif xdot < 0 && xdot > ydot(ii) u = 0; % Since car is moving faster in the negative direction
than the % road (could be positive or negative), damper should be
on to % reduce motion of the car. elseif xdot < 0 && xdot < ydot(ii) u = 1; %% Catch xdot = 0 situation elseif xdot == 0 u = 0; end %% Apply Newton's 2nd Law for equilibrium xddot = -(1/M) * (K*(x-y(ii)) + (C+Cmr)*(xdot-ydot(ii)) +
Fyld*u*sign(xdot-ydot(ii))); end % Store variables with every integration iteration displ(1,ii) = x; vel(1,ii) = xdot; accel(1,ii) = xddot; F_MR(1,ii) = Fyld*u*sign(xdot-ydot(ii)) + Cmr*(xdot-ydot(ii)); % Assign current ii solutions to ii-1 for next loop iteration x0 = x; xdot0 = xdot; xddot0 = xddot; end
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%% Store Fyld for each iteration with max xdot value Fyld_Opt(1,Fyld) = Fyld; Fyld_Opt(2,Fyld) = max(abs(vel)); disp(['Calculating for V = ' num2str(V) ' [mph] and lambda = '
num2str(lambda) ' [ft] and Fyld = ' num2str(Fyld) ' [lb]']) end % Find minimum xdot response and corresponding Fyld [Min,I] = min(Fyld_Opt,[],2); %% Grab peaks for each combination of lambda and velocity Peak_Responses(1,jj) = v; Peak_Responses(2,jj) = lambda; Peak_Responses(3,jj) = Fyld_Opt(1,I(2,1)); Peak_Responses(4,jj) = Fyld_Opt(2,I(2,1)); jj = jj+1; endend
% Transpose Response maatrixPeak_Responses = Peak_Responses'; Appendix A.4 – Modified Skyhook Matlab Code% MR_Suspension_Modified.m%% This code is written as a modification to the Skyhook algorithm model to% be compared with a SDOF spring mass damper system modeling the typical% automotive suspention system assuming 1 wheel and only forces in the% vertical direction.%%**********************************************************************% Author: Chad Sutton% Date: 14-OCT-2012% Introduction to Smart Structures% University of Cincinnati% Cincinnati, Ohio 45221-0072% TEL: 615-400-1658% E-MAIL: chad.h.sutton@gmail.com%*********************************************************************%%%% Clear Woskspace clear all;close all;clc
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%% Define all variables% Suspension System variablesW = 3220; % [lb] - Weight of car being supportedg = 32.2; % [ft/s^2] - Gravitational AccelerationM = W/g; % [slug] - Mass of car being supportedC = 134; % [lb-s/ft] - System Damping CoefficientK = 4000; % [lb/ft] - System Spring Constant % Magnetorheleological Damping CharacteristicsCmr = 21; % [lb-s/ft] - MR Damping Coefficient% Yield force for the MR Damper Fyld_range = xlsread('Responses.xlsx','Improve Algo','P4:Q15'); % [lb] % Initialize matrix for storing peak responsesPeak_Responses = zeros(5,12);jj = 1; % Forcing function inputsfor V = [5 13.726 40 60]; % [mph] Velocties of car to be consideredv = V*5280/3600; % [ft/s] Velocties of car to be considered for lambda = [10 20 88]; % [ft] Road wavelenth y0 = 0.118; % [ft] - Amplitude of the road input % Define input frequency for y(t) f = v/lambda; % [Hz] % Define time for system analysis T = 15; % [s] - Total time for system analysis N = 10000; % Number of points to be considered t = linspace(0,T,N); % [s] - Time vector for analysis dt = T/N; % [s] - Iteration delta time % System output variable storage matricies displ = zeros(1,length(t)); % [ft] - Displacement of the mass (car) vel = zeros(1,length(t)); % [ft/s] - Velocity of mass (car) accel = zeros(1,length(t)); % [ft/s^2] - Acceleration of mass (car) F_MR = zeros(1,length(t)); % [lb] - Force in the MR damper as f(t) F_s = zeros(1,length(t)); % [lb] - Force in the Spring as f(t) F_c = zeros(1,length(t)); % [lb] - Force in the passive damper as
f(t) %% Define input forcing function y(t)
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% Assume forcing function is sinusoidal with given Forcing function inputs
w = 2*pi*f; % [rad/s] - input frequecy for y(t) y = y0*sin(w.*t); % [ft] - Displacement of Road ydot = w*y0*cos(w.*t); % [ft/s] - Velocity of Road yddot = -w^2*y0*sin(w.*t); % [ft/s^2] - Accleration of Road %% Algorithm for iterating equations of motion (Newmark-Beta method) % Constants used in Newmark-Beta method (see integration nots p.6) Beta = 0.25; Lambda = 0.5; % Initialize variables at current time (t) and for n from n-1 x = 0; xdot = 0; xddot = 0; % Initialize variables to be called from previous time x0 = 0; xdot0 = 0; xddot0 = 0; for ii = 1:length(t) %disp(['Loop iteration for t = ' num2str(t_iter) ' sec']) for n = 1:10 % iterate displacement, velocity, acceleration x = x0 + xdot0*dt + ((0.5-Beta)*xddot0+Beta*xddot)*dt^2; xdot = xdot0 + ((1-Lambda)*xddot0 + Lambda*xddot)*dt; %% Apply Skyhook Control Laws for xdot > 0 % Car is moving faster in the upward direction than road
(could be % positive or negative), turn ON MR to reduce motion of
car. if xdot > 0 && xdot > ydot(ii) u = 1; % Car is moving upward slower than the road (positive),
turn OFF MR % to reduce force transmitted to car. elseif xdot > 0 && xdot < ydot(ii) u = 0; %% Apply Skyhook Control Laws for xdot < 0 % Car is moving downward slower than the road (negative),
turn OFF % MR damper so that tire motion downward will not force
car down % with it.
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elseif xdot < 0 && xdot > ydot(ii) u = 0; % Since car is moving faster in the negative direction
than the % road (could be positive or negative), damper should be
on to % reduce motion of the car. elseif xdot < 0 && xdot < ydot(ii) u = 1; %% Catch xdot = 0 situation elseif xdot == 0 u = 0; end %% Modify Algorithm based on Fyld Calculations Fyld = Fyld_range(jj,2); %% Apply Newton's 2nd Law for equilibrium xddot = -(1/M) * (K*(x-y(ii)) + (C+Cmr)*(xdot-ydot(ii)) +
Fyld*u*sign(xdot-ydot(ii))); end % Store variables with every integration iteration displ(1,ii) = x; vel(1,ii) = xdot; accel(1,ii) = xddot; F_MR(1,ii) = Fyld*u*sign(xdot-ydot(ii)) + Cmr*(xdot-ydot(ii)); F_s(1,ii) = K*(x-y(ii)); F_c(1,ii) = (C+Cmr)*(xdot-ydot(ii)); % Assign current ii solutions to ii-1 for next loop iteration x0 = x; xdot0 = xdot; xddot0 = xddot; end %% Create plots of results for viewing and consideration figure subplot(4,1,1) plot(t,y), hold on, plot(t,displ,'r-') grid on title('Road Displacement & Car Displacement vs. Time') xlabel('Time [s]') ylabel('Displacement [ft]') legend('Road Input','Car Output') xlim([0 min([5/f T])]) subplot(4,1,2)
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plot(t,vel) grid on title('Skymod Velocity vs. Time') xlabel('Time [s]') ylabel('Velocity [ft/s]') xlim([0 min([5/f T])]) subplot(4,1,3) plot(t,accel) grid on title('Skymod Acceleration vs. Time') xlabel('Time [s]') ylabel('Acceleration [ft/s^2]') xlim([0 min([5/f T])]) ylim([-max(abs(accel))-1 max(abs(accel))+1]) subplot(4,1,4) plot(t,F_s), hold on, plot(t,F_c,'r-'), plot(t,F_MR,'k-') grid on title('Suspension Forces vs. Time') xlabel('Time [s]') ylabel('Force [lb]') legend('Spring Force','Damper Force','MR Force') xlim([0 min([5/f T])]) disp(['Calculating for V = ' num2str(V) ' [mph] and lambda = '
num2str(lambda) ' [ft] and Fyld = ' num2str(Fyld) ' [lb]']) % Save Disp, Vel, Accel figure for current velocity and lambda
combination set(gcf, 'Position', get(0,'Screensize')); % Maximize figure. saveas(gcf, ['V = ' num2str(v) ' fps_lam = ' num2str(lambda) '
ft_Outputs.png']); %pause close all %% Grab peaks for each combination of lambda and velocity Peak_Responses(1,jj) = v; Peak_Responses(2,jj) = lambda; Peak_Responses(3,jj) = max(abs(displ)); Peak_Responses(4,jj) = max(abs(vel)); Peak_Responses(5,jj) = max(abs(accel)); jj = jj+1; endend % Transpose Response maatrix
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Appendix A.5 – Baseline Suspension Output Plots
Figure 16: Standard Suspension V = 5 mph, λ = 20 ft
Figure 17: Standard Suspension V = 5 mph, λ = 20 ft
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Figure 18: Standard Suspension V = 5 mph, λ = 88 ft
Figure 19: Standard Suspension V = 13.726 mph, λ = 10 ft
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Figure 20: Standard Suspension V = 13.726 mph, λ = 20 ft
Figure 21: Standard Suspension V = 13.726 mph, λ = 88 ft
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Figure 22: Standard Suspension V = 40 mph, λ = 10 ft
Figure 23: Standard Suspension V = 40 mph, λ = 20 ft
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Figure 24: Standard Suspension V = 40 mph, λ = 88 ft
Figure 25: Standard Suspension V = 60 mph, λ = 10 ft
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Figure 26: Standard Suspension V = 60 mph, λ = 20 ft
Figure 27: Standard Suspension V = 60 mph, λ = 88 ft
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Appendix A.6 – Skyhook Suspension Output Plots
Figure 28: Skyhook Suspension V = 5 mph, λ = 10 ft
Figure 29: Skyhook Suspension V = 5 mph, λ = 20 ft
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Figure 30: Skyhook Suspension V = 5 mph, λ = 88 ft
Figure 31: Skyhook Suspension V = 13.726 mph, λ = 10 ft
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Figure 32: Skyhook Suspension V = 13.726 mph, λ = 20 ft
Figure 33: Skyhook Suspension V = 13.726 mph, λ = 88 ft
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Figure 34: Skyhook Suspension V = 40 mph, λ = 10 ft
Figure 35: Skyhook Suspension V = 40 mph, λ = 20 ft
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Figure 37: Skyhook Suspension V = 60 mph, λ = 10 ft
Figure 38: Skyhook Suspension V = 60 mph, λ = 20 ft
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Figure 39: Skyhook Suspension V = 60 mph, λ = 88 ft
Appendix A.6 – Modified Skyhook Suspension Output Plots
Figure 40: Modified Skyhook Suspension V = 5 mph, λ = 10 ft
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Figure 42: Modified Skyhook Suspension V = 5 mph, λ = 88 ft
Figure 43: Modified Skyhook Suspension V = 13.726 mph, λ = 10 ft
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Figure 45: Modified Skyhook Suspension V = 13.726 mph, λ = 88 ft
Figure 46: Modified Skyhook Suspension V = 40 mph, λ = 10 ft
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Figure 47: Modified Skyhook Suspension V = 40 mph, λ = 20 ft
Figure 48: Modified Skyhook Suspension V = 40 mph, λ = 88 ft
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Figure 49: Modified Skyhook Suspension V = 60 mph, λ = 10 ft
Figure 50: Modified Skyhook Suspension V = 60 mph, λ = 20 ft