3. Using Hydraulics to Actuate Engine Valves
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Executive Summary
In a typical internal combustion engine, the cam drives the motion of mechanical components,
which in turn actuate intake and exhaust valves. This report provides analysis and evaluation of an engine
where the mechanical components are replaced by hydraulic components. Mathematical and engineering
analysis was performed to determine the system’s frequency, hydraulic fluid flow rate, and hydraulic line
parameters.
While the system was initially modeled as a high-order differential equation, analysis proved that
resonance would not be achieved under normal operating conditions. Therefore, the system was modeled
using much more manageable equations. Other calculations were performed to determine minimum
hydraulic line diameter, minimum fluid velocity to purge air from the system, and minimum line wall
thickness. All calculations can be found in the appendix.
Results of data analyzed show that a 1/8 inch diameter stainless steel tube should be used as the
medium of transportation for the hydraulic fluid, which in this case will be engine oil. The baseline
pressure in the system will be supplied and maintained by an oil pump and check valves, respectively. Oil
will be pulled from the existing engine oil reservoir, removing the need for an additional fluid supply that
would take up valuable engine space. Should any air enter the system (which could cause performance
issues), the velocity of the fluid will be sufficiently high to remove it.
4. Using Hydraulics to Actuate Engine Valves
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Introduction
The year was 1886. Karl Benz had just been granted the patent for the first modern automobile
that used an internal combustion engine. It was a massive engineering feat that would make travel much
more time-efficient, but even Benz couldn’t have foreseen how his invention would change the world.
Today there are roughly 1 billion cars worldwide. Ever since the idea for the first internal combustion
engine was proposed, efforts have been made to improve its operation. Recently, with the emergence of
environmental awareness and recognition of the world’s dependence on oil and natural gas, the efficiency
of automobile engines has been at the forefront of the modern engineering landscape.
This project focuses on the feasibility of replacing mechanical components with hydraulic
components. In a typical internal combustion engine like the one shown in Figure 1, a push rod is in
contact with a cam at one end and a rocker arm at the other. As the cam rotates, it causes vertical motion
of the push rod, which in turn causes the rocker arm to rotate about a pivot point. As the rocker arm
rotates, it causes the valves that control intake and exhaust to open and close. In this project, the goal is to
replace the pushrod and rocker arm with an actuator and hydraulic tubes. A pump will supply pressure to
the incompressible hydraulic fluid, which will need to supply at least the same amount of force as the
push rod- rocker arm design in order to effectively control valve operation.
Hydraulic
Line
Hydraulic
Actuator
Cam
Figure 1.Traditional Push Rod/Rocker Arm Setup vs. Hydraulic Actuator System[1]
The primary objective of this project is to evaluate the effectiveness of such a hydraulic system.
Factors such as system pressure, hydraulic line size, and fluid flow properties will have to be analyzed in
order for the system to perform at the highest level. The system should be designed in such a way that the
performance of other engine components does not suffer. One way to accomplish this is to use an existing
engine pump to control the hydraulic pressure rather than adding a separate pump. A further objective is
to make this system easy to install and maintain. Adding a large number of separate components would
presumably overcomplicate the assembly, making it more likely that one or more of the system’s parts
will fail or have performance issues.
The idea of using hydraulics in an internal combustion engine is not a new one. Hydraulic lifters
have become a popular alternative to their mechanical counterparts. A patent filed by General Motors in
5. Using Hydraulics to Actuate Engine Valves
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1978 proposed the use of hydraulic actuators as a replacement for valve springs [2]. Another patent filed
by Richard Beaumont in 1985 suggested using a hydraulically driven distributor valve to control timing
and operation of the engine[3].However, most of these past applications revolve around a traditional inline cylinder arrangement, while this project can apply to either that traditional arrangement or the more
unorthodox rotary arrangement.
One might ask why the mechanical system that has been an integral component of the internal
combustion engine since its inception should be changed. As mentioned earlier, hydraulics are already
present in many automobile systems, so including them in the engine would not be a completely foreign
concept. Hydraulic components are valuable because they have a large mechanical advantage, meaning
that they can produce a large force output from a relatively low force input. They can be particularly
advantageous in automobile engines because power can be transmitted through small tubes that will not
take up valuable space in the compact area of the engine. This is especially beneficial to this project
because if multiple hydraulic lines are used, one cam should be able to drive the motion of multiple
valves. In terms of maintenance, fewer cams represent fewer possibilities of moving parts that could
malfunction.
Overall Design
For this project it was required to design, model and build a hydraulic system to actuate the
valves of an engine. For this design, we chose to use engine oil as the hydraulic fluid for the system. One
of the problems associated with this was that oil could leak out as the engine ran and would need to be
replaced.
Another problem encountered was the possibility of air getting into the system, which could
cause detrimental effects to the engine’s performance. Thus, it was necessary to devise a method to purge
the system of this air and maintain only engine oil within the line.
The final major problem encountered was the dynamics of the system and the line in particular.
These variables were to be modeled as an infinite ordered differential equation with an approximation by
a fourth ordered differential equation.
These problems and solutions will be discussed below. Also, dimensioning of the system and the
necessary flow rate from the oil pump will be explained.
Hydraulic Line Material
For this design, stainless steel tubes will be used for the hydraulic line, as rubber hoses will have
too much pliability in the radial direction at high pressure. This pliability could cause a delay or phase
shift in the system and could cause the failure or even damage to the engine.
These stainless steel tubes are made by several different companies including Matchless,
Aeroquip, and Eaton. They can be purchased at several places both in person and online. The websites
that sell these lines include Grainger, hosewarehouse.com and McMaster-Carr. Compared to the rubber
hose line, stainless steel tubes are relatively expensive, costing an average of $50 per foot compared to the
approximately $20 per foot of a rubber hose[4].
Although it would be cheaper to use this rubber hose, performance standards require the use of
steel braided tubing.
6. Using Hydraulics to Actuate Engine Valves
4
System Pressure
The pressure of the engine oil within the system will obviously change during the course of the
engine cycle. The maximum pressure will be the pressure imposed by the valve spring at the valve’s
maximum lift. The minimum pressure within the entire system will be the pressure at which the engine oil
leaves the oil pump of the engine. In most oil pumps the average pressure is around 60 psi.
The following graph shows the force that is applied by the spring throughout its displacement. In
this example, the spring constant of the valve spring is taken to be 380 pounds per inch and the lift of the
valve is taken to be 0.42 inches [5].
300
Force (lb.)
250
200
150
100
50
0
0
0.1
0.2
0.3
0.4
0.5
Valve Displacement (in.)
Figure 2. Spring Force vs. Spring Displacement
As can be seen from the graph in Figure 2, the force that is induced within the spring varies from
a minimum value of 80 pounds to a maximum of 240 pounds.
In the following figure, the pressure within the hydraulic actuating system is plotted versus the
displacement of the spring. In this example the diameter of the actuator is taken to be 0.82 inches.
7. Using Hydraulics to Actuate Engine Valves
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500
450
Pressure (psi.)
400
350
300
250
200
150
100
50
0
0
0.1
0.2
0.3
0.4
0.5
Deflection (in.)
Figure 3. Pressure of the System vs. Displacement of the Valve
As can be seen in Figure 3, the pressure within the system varies from a minimum of 60 psi at the
cam dwell to a maximum of 430 psi at the maximum displacement of the spring and valve.
With these numbers in mind we must design the system, including the hydraulic line and
actuators, to be able to withstand these pressures.
Hydraulic Transmission Line Modeling
Ongoing research is being conducted regarding the improvement of the line dynamics of a hydraulic
transmission line. Fluid line transmission requires a very precise and accurate line dynamics model. In
combining the transmission line and the actuator valve dynamics, the system stability has to be studied
and analyzed before it can serve its intended purpose. In this project, the aim is to accommodate the valve
actuation system with a very unorthodox arrangement to couple a remote cam actuator with a valve
actuation follower using a hydraulic fluid line. In modeling fluid transmission lines, a lot of concerns
arisethat need to be addressed in order to develop an optimal design. Surging is a crucial problem that
creates several problems if not controlled properly [6].
Figure 4. Free Body Diagram for Line [9]
Pressure surge in a pipe occurs due to a sudden valve closing at an infinitesimal time “t”, causing a
very large pressure to build up at the valve closure point. The sudden valve closure causes the fluid
8. Using Hydraulics to Actuate Engine Valves
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flowat the valve to halt abruptly, and the fluid following behind will collide with a static fluid,creating a
shock wave which could possibly lead to the splitting of the pipe. The shock wave will
thenpropagatebackwards at the speed of sound,creating a loud banging noise and changing the
dimensions of the pipe diameter along its path. When the shock wave meets a dead end due to the closed
inlet valve, it will be reflected back as another round of waves travelling towards its origin, causing an
interference with the valve’s natural frequency. The interference will then create total system instability,
leading to uncontrolled oscillation of the pipe. The pressure surge is directly related to the pipe’s elastic
property and the compressibility of the working fluid,which is why this design utilizes a rigid steel pipe.
The properties of the steel pipe are such that instability will never be attained unless near impossible
frequencies are generated.
The main interest in this control system design process is the actuator valve. The opening and
closing of the valve is guided by the pressure difference. Due to the fact that the actuator valve gates open
and close instantaneously, controlling the pressure and flow propagation is needed. A distributed line
modeling technique is ideal for the total transmission line system design. The system is highly exposed to
unmeasured disturbances, noise, time delays and lags, making the distributed line modeling the first
choice to model the system. Using this technique is similar to finite element analysis in that each of the
segments of the capacitance and resistance of the line are considered to be connected to oneanother
throughout the length of the line.
Figure 5.Dynamic Model of Transmission Line [8]
Using the distributed line model (DLM) is different from the simplified lumped parameter model.
DLM is a nonlinear differential equation, unlike the finite order rational polynomial associated with the
lumped parameter model. The first approach was finding out the natural frequency of the pipe based on its
length.Based on the engine assembly, the initial length assumed was 9 inches. Dividing the speed of
sound propagation (Co) by the line length (L) showed that theresonant frequency was approximately 400
rad/sec. The second step taken was finding an applicable diameter for the transmission line. The most
conservative modeling technique, which is the distributed line model, was used to find the dissipation
number for the line .The dissipation number (Dn) is a unit less normalized parameter, defined as the ratio
of the viscous frequency to the characteristic frequency. This normalized value is the most accurate
modeling technique, and it is used as the reference point when comparing different frequency response
results.
9. Using Hydraulics to Actuate Engine Valves
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Normalized Impulse Response
2.5
2
8D2Z/(RLsinh )
n
1.5
1
0.5
0
-0.5
-1
0
0.5
1
1.5
2
Normalized Time,
2.5
v
3
3.5
t
Figure 6. Normalized Frequency Plot
Using the Dnhelps to find the normalized transfer function which is easily applicable and will be
good for our line properties. The transfer function generated using the normalized dissipation number of
the line is
(Eq. 1)
The applicable range of the dissipation number used for the system is 0.001<Dn< 0.5.
The diameter of the line isthen decided based on the frequency response of the system. The system
parameters frequency response, which was based on the dissipation number, indicated that the diameter
of the line has a very large range of values which led the design team to decide the diameter to be as
small as 0.125 inches. The resonance frequency, calculated after the diameter was applied on a lumped
parameter modeling technique,was approximately 61000 rad/sec, indicating that the small diameter value
decided upon is more than enough to have an acceptable pressure and flow parameters.
The transfer function based on lumped parameter model indicates that the system is stable for the set
variables.
(Eq.2)
10. Using Hydraulics to Actuate Engine Valves
-150
8
System: G
Frequency (rad/sec): 6.09e+004
Bode Diagram
Magnitude (dB): -157
Magnitude (dB)
-200
-250
-300
-350
-400
360
Phase (deg)
270
180
90
0
2
10
3
4
10
10
5
10
6
10
7
10
8
10
Frequency (rad/sec)
Figure 7. Frequency Response for Transmission Line
Determination of theDiameter of the Tube
The minimum diameter of the hydraulic tube connecting two lifters is determined considering
fluid dynamics and the strength of the tube. Several approaches can be used to determine the diameterof
the hydraulic tube. For the purposes of this the design, the damping ratio was used to calculate the
diameter.The damping ratio of the system is a function of thedimension of the tube and properties of
fluid. The fluid considered in this analysis is standard engine oil. The equation used for diameter
determination is
(Eq. 3)
Where:
= Damping factor
= Kinematic viscosity
= speed of sound in fluid
l = length of the tube
d = diameter of the tube
11. Using Hydraulics to Actuate Engine Valves
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The range of thedamping ratio for the system to be stable is 0.1 < < 0.5. The design length of the
tube is 9 in. The minimum diameter of the tube is less than two hundredths of an inch. Working with such
small dimensions would bevery difficult and extremely expensive to manufacture. As such,a diameter
of0.125 (1/8)inchwas used.
Determination of Thickness of the Tube
Even though the tube can be modeled as a thin walled tube, the required thickness to withstand
the pressure is calculated. Tube thickness is calculated using the equivalent bulk modulus of the system.
The bulk modulus of the fluid is the property that indicates the “springiness” of the fluid and is defined as
the pressure needed to cause a given decrease in volume. It is a measure of the fluid’s resistance to
compressibility. Typical oil will decrease about 0.5% in volume for every 1000 psi increase in pressure.
The bulk modulus of the tube should also be considered [10]. If thereis any entrapped air in the tube,
thebulk modulus of the gas should be considered, but sincethe system will be purged,air can be neglected.
The equivalent bulk modulus of the system is given by
(Eq. 4)
Where,
βe = equivalent bulk modulus
βl =bulk modulus of fluid
βc = bulk modulus of the tube
The bulk modulus of the tube gives the measure of stiffness of the tube. The hydraulic tube is
considered to be a thin-walled cylinder and thebulk modulus of the tube is givenas
(Eq. 5)
t = thickness
E = Young’s modulus of the tube
d = outer diameter of the tube
Substituting
in equation 2, the following equation is developed.
(Eq. 6)
The obtained minimum thickness required is
in. As expected, minimum thickness is
very small, so using a commercially available standard stainless tube serves the purpose.
12. Using Hydraulics to Actuate Engine Valves
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Purging the System
As in any hydraulic system, keeping the air out becomes an issue. In most designs, the air is bled
out manually, effectively eliminating the problem. For example, in a brake system for automobiles, the air
is bled out of the system by manually pumping the brakes and then relieving the pressure at the brake
caliper. While this system works effectively for brakes, it will not be applicable for the hydraulic
actuating system used for lifting an engine valve. This is because there is a constant possibility for air to
get into the hydraulic line,whereas brake systems have a very small chance of developing air once bled
correctly [11].
The air could be brought into the system in two main ways. The first possibility for air to get in is
at the initial start-up. For example, if the engine has not been started for a long period of time, it is likely
that the oil will leak out of the line and drain to the bottom of the oil pan. The oil leakage could occur
from the clearance between the piston and the cylinder wall. Therefore air will be in the system as soon as
the engine is fired. This will cause the engine to not start due to the compressibility of the air in the line
causing the valves to not open. The possibility is illustrated in Figure 10.
Figure 10. Oil Leakage from Piston Clearance
Another possibility for air to get into the system is during maneuvering of the vehicle. If the car is
being steered in a direction that causes the oil to slosh to one side, it is possible for air to be pumped in
from the oil pump. Figure 11 shown below illustrates how this can happen.
13. Using Hydraulics to Actuate Engine Valves
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Figure 11. Engine Oil Sloshes
Due to the various possibilities of air being developed in the system, a self-purging design must
be developed to ensure proper performance. A good way to keep the system purged is to allow a constant
flow of oil through the hydraulic line while in use. However, it is necessary for the constant flow to shut
off while actuating the valve with the camshaft. Without proper shutoff, the system will release all of the
displaced oil out of the outlet hole shown in Figure 12.
Figure 12. Purging Design
The design of the purging system is shown in Figure 12. The figure was developed in SolidWorks
by modifying a 16 valve engine head downloaded from GrabCad[12]. The idea is to pump oil into the
valve actuator and have it flow towards the camshaft. A hole is then placed appropriately so that it shuts
off when the cam is actuated and opens when the cam is in the dwell portion of rotation. To prevent
backflow towards the oil pump, a check valve should be placed at the inlet of the oil. The outlet hole will
allow air to be pushed out and removed from the system before the cam is used to lift the valve. For
convenience, the design is also shown below in Figure 13 with a larger scale.
14. Using Hydraulics to Actuate Engine Valves
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Figure 13. Detailed Purging Design
To ensure that the air is removed on every cycle the velocity of the oil must be high enough to
push the bubble out in one cycle. The bubble will have approximately 50% of the time it takes for one full
rotation of the cam to be removed. During the other 50% of the cam rotation, the outlet hole will be
closed off. This idea is illustrated in Figure 14.
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Figure 14. Outlet Hole Shutoff
To make the best use of this time, the line should be designed in a way shown below in Figure
15. This particular design allows the buoyancy force to move the bubble to the top of the line. Therefore,
even when the outlet hole is closed off, the bubble will still be moving towards the outlet for the fall
period of the cam. This places the bubble at portion B of the hydraulic line when the dwell time is
initiated. At this time the outlet hole is opened and the flow of oil is developed. The bubble will have to
travel a vertical distance of “h” to be removed from the system.
Figure 15. Design of the Hydraulic Line
16. Using Hydraulics to Actuate Engine Valves
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To minimize the velocity needed, the value of h should be kept to a minimum. This will reduce
the flow rate needed to keep the system running at its optimum level. The velocity needed to initiate
movement of the bubble can be found from equation7[13].
(Eq. 7)
This equation shows that the buoyancy force is the opposing force whereas the drag force is the
force pushing the bubble out. A free body diagram of the air bubble is shown in Figure 16.
Figure 16. Free Body Diagram of Air Bubble
The minimum velocity needed to initiate movement was found to be approximately 1 ft/s, and the
full derivation is shown in the Appendix in section (C). In addition, the average velocity needed to push
the bubble completely out can be found from the kinematic equation8[14], assuming that the cam is
rotating at 2000 rpm and that the overall height is 2 inches. The average velocity was found to be 10 ft/s.
The complete derivation is shown in the Appendix in section (C).
(Eq. 8)
There are also other advantages of this design that increase the reliability of the engine. For
example, since the oil is being removed at the camshaft end of the system, the oil can be used to lubricate
the cam and actuator. In addition, there is no need to worry about the oil leaking out after delayed usage
of the engine since the oil will be replaced upon startup.
Calculating Diameter Using Secondary Criterion
As a secondary criterion for transmission line size calculations, hydraulic line manufacturers
recommend that the maximum velocity of fluid flowing through the line be less than 20 feet per
second[15].Thus, to determine what size the line must be to achieve an acceptable value, calculations
regarding volumetric flow rate and velocity must be made.
Using standard equations governing cam velocity, it was found that the maximum velocity of the
cam is approximately 55 inches per second. The cam pushes the actuator which then transfers the energy
of the cam into the hydraulic fluid, through the system and then to the valve actuator, effectively opening
17. Using Hydraulics to Actuate Engine Valves
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the valve. The area of the actuator that the cam pushes through is about a quarter inch. Using the
equation:
(Eq. 9)
the volumetric flow rate of the fluid flowing through the actuator is then 14 cubic inches per second.
Now that the flow rate in the actuator is known, the flow rate throughout the system can be
assumed to be the same (with some small amount of error). To solve for the diameter of the line necessary
to maintain a maximum velocity of twenty feet per second, equation 9 can be solved for area and area can
then be solved for the diameter.
The diameter decided upon in the previous section would be the preferred diameter of the line
using the secondary criterion. Thus, an eighth of an inch is the desired diameter for the system. Using the
maximum 20 feet per second as the velocity flowing through the line, the diameter is found to be
approximately 0.15 inches. Noting that the flow rate through the line will be nominally smaller than the
flow rate through the actuator, the diameter that was solved for is acceptably close to the eighth of an inch
diameter that dynamics calls for.
Byusing this secondary criterion, the diameter of an eighth of an inch was verified and is the best
choice for use within the system.
Conclusion
In conclusion, this report presents a design for a hydraulic actuating system for a valve train that
functions proficiently provided the person implementing stays within an acceptable range of such
constraints as line length, line diameter and speed of engine. All of these constraints provide ample room
for implementation and are applicable for many different engines and applications from automotive to
agriculture to even aeronautical.
The preliminary test engine parameters, such as pressure within the system and the forces
associated with this pressure, were analyzed. Accordingly, the entire design of the system revolved
around these pressures and forces, and the system was designed to sustain them and function properly
with them in mind.
During the design of this system, careful considerations were given to the dynamic effects
associated with the transmission line of the system. These concerns, in the end, were unfounded as the
dynamic effects such as surge and other timing issues can be neglected due to the very high natural
frequencies associated with a system like this. Thus, a much simpler model was introduced and a suitable
response diagram was attained.
Using principles of the bulk modulus, the necessary thickness of the line was calculated. This is
the minimum thickness that will prevent timing problems due to the flex from the transmission line
pressures. Also, purging the system of any air will be accomplished by maintaining a constant fluid flow
throughout the system. This requires implementation of minimal additional components foreign to the
standard automobile seen in practice. It does possibly require some upgrades to parts such as the oil pump
of the automobile and the oil reservoir already mounted to the engine.
After careful consideration of all of the characteristics and properties associated with a hydraulic
actuating system, an acceptable design has been developed. This design includes the use of an eighth of
an inch line that is approximately nine inches long and an actuator with a piston that is approximately six
tenths of an inch in diameter. This system should be run at a rate not exceeding three thousand rotations
per minute and the hydraulic fluid used within the system should be standard SAE grade oil. Much
beyond these constraints, it cannot be guaranteed that the dynamics of the system can be neglected and
that this simple model can be used.
18. Using Hydraulics to Actuate Engine Valves
16
References
[1]
DOE-HDBK-1018/1-93, “Diesel Engine Fundamentals,” Diesel Engines,
nuclearpowertraining.com, Web 25, September 2013, from
http://nuclearpowertraining.tpub.com/h1018v1/css/h1018v1_31.htm
[2]
Trenne, Myron (To General Motors Corporation), “Hydraulic Valve Actuator System,”
U.S. Patent 05/901,452, May 1, 1978.
[3]
Beaumont, Richard. “Internal Combustion Hydraulic Engine,” U.S. Patent 06/733,074. May 13,
1985.
[4]
“Hose Master- Flexible Metal Hose.” Grainger.com. Accessed December 2013.
[http://m.grainger.com/mobile/product/HOSE-MASTER-Flexible-Metal-Hose-2ZV57]
[5]
“Edelbrock Valve Springs and Retainers.” Edelbrock.com. Accessed September 2013.
[http://edelbrock.com/automotive_new/mc/valvetrain/springs_retainers.shtml]
[6]
Hullender, D.A., 1990. “Effects of Fluid Transmission Lines in Pressure Measurement”
Instrumentation and Control – Fundamentals and Applications, Section 10.5, pp. 434-438, John
Wiley & Sons, Inc., Somerset, NJ
[7]
Craig, Kevin. "Hydraulic Transmission Lines."EDN.N.p., n.d. Web. 18 Nov. 2013.
<http://www.edn.com/electronics-blogs/mechatronics-indesign/4423778/Hydraulictransmission-lines>.
[8]
Shinners, Stanley M.Advanced modern control system theory and design. New York: Wiley,
1998. Print.
[9]
"Keep an Eye on Hydraulic Transmission Lines." N4SAcom. N.p., n.d. Accessed 18 Nov. 2013.
[http://n4sa.com/arkiv/107629.]
[10]
Woods, R.L. and Lawrence, K., 1997, Modeling and Simulation of Dynamic Systems, Prentice
Hall, Upper Saddle River, NJ.
[11]
“Service Manual Procedure – Brake Bleeding Procedure.” Dodge Rem Service Manual. Accessed
October 2013.[dodgeram.org http://dodgeram.org/tech/repair/Brakes/beeding.htm. ]
[12]
Murarik, Peter. “Valve Train dohc.” GRABCAD.com. Accessed December 2013.
[https://grabcad.com/library/valve-train-dohc-1/files.]
[13]
Munson, Bruce R., “Buoyancy, Flotation, and Stability,” Fundamentals of Fluid Mechanics, 2009
19. Using Hydraulics to Actuate Engine Valves
17
[14]
Giancoli, Douglas C., “Kinematics in One Dimension,” Physics for Scientist and Engineers with
Modern Physics, 2009
[15]
“Line Sizing and Fluid Velocity,” RHM Fluid Power Inc, Westland, MI
[http://www.rhmfp.com/tech-tips/161-line-sizing-and-fluid-velocity. Accessed 10/3/2013.]
20. Using Hydraulics to Actuate Engine Valves
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Appendix A
Normalized Frequency Program
function [NGZoverSinh,GZoverSinh] = ZoverSinh(Den,Beta,Mu,d,L )
formatshortg
r=0.00625;
Mu=4.6e-5;
Nu=.015;
Den= 870;
L=0.25;
d=2*r;
Bf=1.152e9;
Bl=1.73e7;
Beta=(Bf * Bl)/(Bf + Bl);
Dn=Nu*L*sqrt(Den/Beta)/r^2;% Dissipation number
RL=128*Mu*L/(pi*d^4);% Steady state flow resistance
Wv=Nu/r^2;% Normalizing frequency, rad/sec
ifDn>0.5
Dn
'Dn is greater than 0.5 which is outside the range for the computed coefficients'
elseifDn<0.001
Dn
'Dn is less than 0.001 which is outside the range for the computed coefficients'
else
ifDn<=0.2
a3=0.0454*Dn^3.5523;
a2=-0.1012*Dn^2.0585;
else
a3=-0.2167*Dn^4+0.2569*Dn^3-0.1077*Dn^2+0.0225*Dn;
a2=-0.8047*Dn^4+1.7023*Dn^3-1.005*Dn^2+0.2079*Dn-0.0173;
end
ifDn<=0.1
a1=-5436.6*Dn^5+1213.3*Dn^4-91.209*Dn^3+2.9331*Dn^2-.0036*Dn+3e-5;
else
a1=1.1585*Dn^4-1.8786*Dn^3+0.0199*Dn^2+0.1218*Dn-0.0067;
end
b4=0.0047*Dn^4.0812;
b3=0.035*Dn^3.6798;
b2=0.1912*Dn^2.0661;
b1=0.6685*Dn^1.6288;
fprintf('Normalized Transfer Function, 8Dn^2Z/[RLSinh]')
NGZoverSinh=tf([a3 a2 a1 1],[b4 b3 b2 b1 1 0]);
NGZoverSinhzpk=zpk(NGZoverSinh);
[Normalized_poles]=roots([b4 b3 b2 b1 1 0]);
figure
[y,t]=impulse(NGZoverSinh);
plot(t,y,'r','LineWidth',2)
xlabel('Normalized Time, omega_v t')
21. Using Hydraulics to Actuate Engine Valves
ylabel('8D_n^2Z/(R_LsinhGamma)')
title('Normalized Impulse Response')
figure
fprintf('Un-normalized transfer function, Z/Sinh')
GZoverSinh=tf([a3/Wv^3 a2/Wv^2 a1/Wv 1]*RL/(8*Dn^2),[b4/Wv^5 b3/Wv^4 b2/Wv^3 b1/Wv^2
1/Wv 0]);
GZoverSinhzpk=zpk(GZoverSinh)
[Poles]=roots([b4/Wv^5 b3/Wv^4 b2/Wv^3 b1/Wv^2 1/Wv 0])
[Y,T]=impulse(GZoverSinh);
plot(T,Y,'k','Linewidth',2)
xlabel(' Time, seconds')
ylabel('Z/sinhGamma, (N/m^2)/(m^3/s)')
title('Impulse Response, Delta P_b/(Delta Q_a) or -Delta P_a/(Delta Q_b)')
end
end
19
22. Using Hydraulics to Actuate Engine Valves
20
Appendix B
Lumped Parameter Program to Find Out the Resonant Frequency Based on the Set Parameters
p=870; %Kg/m^3
M=0.05;%kg
b=115; % N.s/m DAMPING
Vis=4.6e-9;
Bf=1.21e9;
Bl=1.73e7;
L=.25;%m
Cd=1;
K=66548.2;%N/m
d=0.003175;%m
r=d/2;
Ps=0.687e6;%pa
Pi=3.14;
V=L*pi*r^2;
Av=5.29e-4;%m^2
Ac=5.29e-4;%m^2
Be = (Bf * Bl)/(Bf + Bl);
C1 = 32*Vis/(p*d^2);
C2 = Pi*d^2 / 8*L(Cd)^2*(Av)^2;
C3 = pi*d^2 /4*p*L;
B1 = b / M;
B2 = K / M;
B3 = Ac / M;
D1 = 4*Be/ (Pi*d^2);
D2 = 4*Be*Ac /(Pi*d^2);
A=[0 1 0 0;-B2 -B1 0 B3;0 0 -C1 -C3;0 -D2 D1 0];
B=[0;0;C3;0];
C=[1,0,0,0];
D=[0];
23. Using Hydraulics to Actuate Engine Valves
21
Appendix C
Diameter of the Tube
Diameter
Figure 8. Diameter of Transmission Line
Using,
Here,
= 0.0045*
l = 9 in
= 0.1
=
= (16)(0.0045*
)*(
)
d = 0.0017 in = 0.42 mm
Therefore, 0.42 mm is the minimum diameter of the tube.
24. Using Hydraulics to Actuate Engine Valves
22
Appendix D
Determination of Thickness of the Tube
Thickness
Figure 9. Thickness of Transmission Line
Assuming,
E=
psi
psi
Equivalentbulkmodulus
in.
The obtained thickness is the minimum thickness of the tube.
25. Using Hydraulics to Actuate Engine Valves
23
Appendix E
Buoyant Forces and Drag Force
Where:
26. Using Hydraulics to Actuate Engine Valves
24
Appendix F
Average Velocity Needed to Push Bubble Out
(dwell portion of cam)
Assuming 2000 rpm
Therefore each cam rotation takes .03 sec.
(dwell portion of cam)
