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Controlled equilibrium mounts for aircraft engine isolation
ARTICLE IN PRESS
Control Engineering Practice 14 (2006) 721–733 www.elsevier.com/locate/conengprac
Controlled equilibrium mounts for aircraft engine isolation Taehyun Shima,, Donald Margolisb a
Department of Mechanical Engineering, The University of Michigan-Dearborn, 4901 Evergreen Road, Dearborn, MI 48128, USA Department of Mechanical and Aeronautical Engineering, University of California, Davis, One Shield Avenue, Davis, CA 95616, USA
b
Received 7 October 2003; accepted 9 March 2005 Available online 4 May 2005
Abstract Softer than usual engine mounts for aircraft engine isolation can be used by adding control of the equilibrium position of the mount. The control power comes from bypass air from the engine. This isolation concept is called a controlled equilibrium mount or CEM. A model of the CEM has been developed which includes a front and rear mount with proper loading for each. These mounts possess passive stiffness and damping properties, and they can be pressurized or exhausted through control valves. The thermodynamics of the air is also modeled. A control strategy is proposed that keeps the mounts at their equilibrium positions. The simulation results show that the supply pressure and temperature of the bypass air varied with flight conditions. However, the variations are acceptable, and equilibrium can be maintained with acceptable flow rates from the bypass air for most flight conditions. r 2005 Elsevier Ltd. All rights reserved. Keywords: Aircraft engine mount; Vibration isolation; Controlled equilibrium; Load leveler
1. Introduction In any vibration isolation application, the stiffness of the isolator is chosen, in part, to handle the maximum allowable displacements based upon some assessment of the worst possible inputs. Most of the time, the worst case is not acting, and the mount is stiffer than it needs to be during this time, and the isolation is not as good as it could be if softer mounts were being used. The CEM is a concept that allows softer mounts to be used by using some active power to keep the mount near a predetermined equilibrium position. This concept is also known as a load leveling system. Application of active power in automotive suspensions for ride improvement has been extensively studied in both industrial settings and academia. Various Corresponding author. Tel.: +1 313 593 5127; fax: +1 313 593 3851. E-mail addresses: [email protected] (T. Shim), [email protected] (D. Margolis).
0967-0661/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.conengprac.2005.03.007
control algorithms utilizing different types of power actuators (electric, pneumatic, hydro-pneumatic, magnetic and commercialized magneto-rheological fluid, etc.) are documented in the literature (Margolis, 1983; Karnopp & Margolis, 1984; Karnopp, 1985; Hrovat, 1991; Elmandy & Abdujjabbar, 1990, 1992; Youn & Hac, 1995; Ogawa, Satoh, & Enomoto, 1996; Toyofuku, Yamada, Kagawa, & Fujita, 1999; White, 2000). The majority of on highway trucks (class 8) use air suspension and some type of ride height control system. When conventional steel springs are used, they are designed to provide adequate support for the fully loaded vehicle, thus they are unnecessarily stiff for smaller loads. Air suspensions coupled with some type of height control system (load leveling system) have advantages over steel springs including consistent loaded/unloaded ride heights, reduced hysteresis, soft ride, and good traction distribution (White, 2000; Sweatman, Woodrooffe, & Mcfarlane, 1997; Dudding & Wilson, 2000; Palkovics & Fries, 2001). With growing interest in vehicle safety, the role of a load leveling
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T. Shim, D. Margolis / Control Engineering Practice 14 (2006) 721–733
system is not limited to the improvement of ride comfort, but is expanded into vehicle stability and safety. There are different designs of load leveling systems and control algorithms to lessen vehicle rollover propensity in heavy trucks (Palkovics & Fries, 2001). In addition, air suspensions have the reputation of being ‘‘road friendly’’ since they reduce dynamic loading of the pavement resulting in diminished pavement wear (Sweatman et al., 1997; Sun, 2002). There has not been much research on the use of controlled equilibrium mounts for suppression of aircraft engine vibration. In this paper, a model of the CEM for an aircraft engine application is developed and simulated for different flight conditions to evaluate its effectiveness. The model includes the mount mechanics, and the fluid mechanics and thermodynamics of the valve control system.
2. Development of CEM model The CEM model proposed here consists of four major parts; conventional mount, expandable volume, control valve, and supply. There is a front mount and a rear mount with appropriate loading on each mount. These conventional mounts have elastomeric components that provide the basic isolation. In addition there is an expandable volume that can be pressurized and/or exhausted that is used to keep the mount at some equilibrium condition. Control valves are needed to either expose the mount volume to a supply pressure or to expose the mount volume to exhaust pressure. The supply for this active part of the isolation system comes from bypass air from engine. As the flight conditions change from startup and taxi, to takeoff, climb, cruise,
and descent, the supply conditions change as does the loading on the mounts due to varying thrust. Before a hardware realization is possible, it is important to know how well equilibrium can be maintained for the various flight conditions, how much bypass air is needed to accomplish the control task, and what temperatures are expected in the mounts. In order to assess this CEM concept for aircraft applications, a model was developed using bond graphs, which includes the mount mechanics, and the fluid mechanics and thermodynamics of the valve control system. Fig. 1 shows a detailed diagram of the aircraft engine and the fore and aft mounts. From this diagram, an overall system model is developed. Fig. 2 shows a reticulated schematic of the system from Fig. 1 and clearly indicates the physics of operation of the mounts in Fig. 1. This CEM includes the mount mechanics between external loading points and mount locations, as well as the fluid mechanics and thermodynamics associated with supply and exhaust of the mounts. The load mass of the front and rear mounts are modeled separately since measured loading data on each mount were provided. The mf and ma represent mass of front and rear mounts, F f and F a are external forces applied at the front and rear mounts due to engine thrust, a1 , b1 are lengths associated with the front mount lever, and a2 , b2 are lengths for the lever at the rear. There are elastomeric components, kf and bf for the front mount, and ka and ba for the rear mount, which provide basic isolation. The spring forces are set to be positive in compression. The mount volumes, V f (front) and V a (rear) are pressurized and/or exhausted based upon the respective piston positions, xf and xa . The inlet and outlet valves are opened/closed based upon control logic that is determined from the position of the mount.
Fig. 1. Schematic of aircraft engine
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Fig. 2. Schematic of controlled equilibrium mount model for aircraft engines.
It is assumed that heat transfer occurs in the tube between the supply and reservoir, and this effect is included in the CEM model. 2.1. Valve control logic The goal of CEM is to maintain the displacement of the mounts within some specified tolerance. Here, we wish to maintain the mount positions within 0:25 mm. In the front mount, for instance, when the front mount loading (external force and weight of engine) is applied a distance a1 from the pivot, the volume, V f , inside the front mount decreases and the mount position, xf , moves upward. If the mount position exceeds the positive equilibrium limit (þ0:25 mm), the inlet valve for front mount, Aif , opens and the mount will be pressurized from the reservoir until equilibrium is again attained. If the applied pressure is excessive in the mount, then the mount position will move downward and might exceed the negative equilibrium limit (0:25 mm). For this case, the exhaust (outlet) valve, Aef , will open and the high pressure inside the mount will exhaust allowing the mount to return to equilibrium. When the mount position is within the equilibrium limits, the inlet and exhaust valves will be closed. The same control logic is also applied to the rear mount. Thus the valve control strategy becomes, If xn 40:25 mm;
Ai ¼ open; Ae ¼ close,
If xn o 0:25 mm;
Ain ¼ close; Aen ¼ open,
If 0:25 mmoxn o0:25 mm;
Ain ¼ close; Aen ¼ close, (1)
where subscript ‘‘*’’ indicates front and rear.
2.2. Heat transfer effect The elastomeric components used in the mount are not suitable for high temperatures. So the knowledge of the mount temperature is critical for designing the CEM. Since the reservoir pressure and temperature are the supply conditions for the mounts, heat transfer between the supply and the reservoir is included to determine if the reservoir temperature can be maintained near the ambient cowl temperature around the engine. The schematic and bond graph of the heat transfer model is shown in Fig. 3. The supply tube is made of stainless steel (Type 321 (UNS No.- S33100)) and has length of Lt , thickness of tt and diameter, Dt . The pseudo-bond graph for heat transfer is documented in Karnopp, Margolis, and Rosenberg (2000). As bypass air from the engine flows through the tube, conduction occurs from inside to outside of the tube due to the temperature difference between the fluid and the tube wall. This conduction is characterized by kth Asurface E_ h1 ¼ ðT 2 T h2 Þ, tt
(2)
where kth is the thermal conductivity of the tube material, Asurface is surface area of tube, tt is thickness of tube, T 2 is reservoir temperature, and T h2 is the wall temperature. This effect is characterized as a 1-port resistance in the bond graph. The tube itself stores some energy due to its thermal capacitance. This is mathematically expressed as Eq. (3) and shown as a C-element in the bond graph. T h2 ¼
1 Eh . mtube cth 2
(3)
In Eq. (3), mtube is the mass of tube, and cth is the specific heat of tube material. The tube is surrounded by the air inside the engine cowl, and it is assumed that free heat convection occurs between the outside surface of the tube and the
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724
Fig. 3. Schematic and bond graph of heat transfer between supply and reservoir in CEM.
stationary ambient air. This is characterized by, E_ h3 ¼ hAsurface ðT h2 T amb Þ,
p_ ma ¼ F a þ ma g (4)
where h is a convection coefficient. The convection coefficient, h, is updated at each time step based on the temperatures T amb and T h2 . The detailed update law for the convection coefficient can be found in Incropera and Dewitt (1990).
2.3. Equations of motion The complete bond graph model of CEM with the inclusion of heat transfer is shown in Fig. 4. Bond graphs (Karnopp et al., 2000) are a concise pictorial representation of the interactive dynamics of all types of energetic systems. They allow the model to be developed in pieces and then put together into an overall computational model. The equations of motion for this CEM can be derived from the bond graph model. For the linear momentum of front and rear mount, p_ mf ¼ F f þ mf g
b1 F1 , a1 f
(5)
a2 þ b2 F 1a , a2
(6)
where p F 1f ¼ kf qf þ Af ðP1f Patm Þ þ bf F 1a
b1 , m f a1 mf
pm a a2 þ b2 ¼ ka qa þ Aa ðP1a Patm Þ þ ba . ma a2
(7)
(8)
For the displacement of the elastomeric component in the mount, pm f b1 , (9) q_ f ¼ mf a1 q_ a ¼
pm a a2 þ b2 . ma a2
(10)
The equations of motion for the front and rear expandable volumes are b1 pm f , (11) V_ 1f ¼ Af a1 m f a2 þ b2 pm a V_ 1a ¼ Aa a2 m a
(12)
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725
Fig. 4. Bond graph model of controlled equilibrium mount for aircraft engines.
where Af is piston area of the front mount and Aa is piston area of the rear mount. The front and rear mounts are modeled as thermodynamic accumulators (Karnopp et al., 2000), and the state equations for energy and mass are E_ 1f ¼ E_ 3f þ E_ 4f P1f V_ 1f ,
(13)
E_ 1a ¼ E_ 3a þ E_ 4a P1a V_ 1a ,
(14)
_ 3f þ m _ 4f , _ 1f ¼ m m
(15)
_ 3a þ m _ 4a . _ 1a ¼ m m
(16)
For the reservoir, with the inclusion of heat transfer derived previously, the energy and mass flow rates are _ ri m _ 5f m _ 5a , _2 ¼ m m
(17)
kth Asurface E_ 2 ¼ E_ ri E_ 5f E_ 5a ðT 2 T h2 Þ, tt
(18)
kth Asurface E_ h2 ¼ ðT 2 T h2 Þ hAsurface ðT h2 T amb Þ, tt (19) T h2 ¼
1 Eh . mtube cth 2
(20)
3. Simulation and discussion The proposed CEM model was simulated using the parameters listed in Table 1. The damping ratios used for the model are appropriate for the elastomeric material being modeled. In the simulation, different flight conditions are simulated by varying applied forces, supply pressure, and supply temperature. The applied external force variations are due to the changes in engine thrust during different flight conditions. Table 2 shows the external forces, supply pressure, and temperature for the taxi, take-off, climb, cruise and descent conditions. The cruising altitude for a target airplane is 35,000 ft and the ambient air temperature variation is about 68 C over the flight envelope. As shown in Table 2, maximum engine thrust occurs during the take-off condition. At the initial condition of taxi, it is assumed that the mount pressure starts at atmospheric pressure and the mount temperature is the same as the engine cowl temperature. The loading (external force and weight) on the mount is balanced with the spring force in the mount. In order to avoid excessive mount displacement due to the loading, an imaginary stop is introduced at þ0:5 mm in the mount, and the mount sits on this stop at the beginning of simulation. The initial volumes of front and rear mounts are V 1fo ¼ V 1f Af qfo ,
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Table 1 CEM parameters used in the simulation Front mount Mass mf : a1 b1 Initial volume, V f Stiffness, kf Piston area, Af
2219.207 (lb), 1007.2 (kg) 2.721 (in), 6.9 (cm) 5.084 (in), 12.9 (cm) 4.940 (in3 ), 80.9 (cm3 ) 6500 (lbf/in), 11383.3 (N/cm) 13. ðin2 Þ, 83.8 ðcm2 Þ
rffiffiffiffiffi
Natural frequency, of ¼
kf mf
Rear mount Mass, ma : a2 b2 Initial volume, V a Stiffness, ka Piston area, Aa Natural frequency, oa ¼
Damping ratio, zf Damping constant, bf ¼ 2zf kf of
0.07
268.793 (lb), 122.0 (kg) 2.2 (inch), 5.6 (cm) 2.075 (inch), 5.3 (cm) 7.420 ðinchÞ3 , 121.6 ðcm3 Þ 6500 (lbf/in), 11383.3 (N/cm) 7.5 ðinch2 Þ, 48.4 ðcm2 Þ
qffiffiffiffiffi ka ma
Damping ratio, za Damping constant, ba ¼ 2za ka oa
0.1
Valve Heat transfer parameters Inlet ¼ outlet (reservior to mount, mount to exhaust), front ¼ rear Thermal conductivity, K th Area, Ai ¼ Ai ¼ Aef ¼ Aea 3e 5 (in2 Þ, 2e 4 ðcm2 Þ Orifice area (supply to reservoir) Specific heat, C th Ao Density of tube material, r 7:0e 4ðin2 Þ, 4:5e 3 ðcm2 Þ Mass of tube mtube Surface area, Asurf ¼ pDt Lt Tubing 3/8 (in), 0.95 (cm) Convection coefficient, h (Wm/K) is calculated Diameter, Dt Length (supply to reservoir), Lt , 30 (in), 76.2 (cm) each time step. Thickness, tt 0.020 (in), 0.05 (cm)
14.6 (W/m K) 500 (J/kg/K) 8000 (kg=m3 )
Table 2 External forces, supply temperature and pressure for different flight conditions
External forces Front mount (N) Rear mount (N) Supply Ps (gage), (KPa) T s (F) Patm (KPa)
Taxi
Take off
Climb
Cruise
Descent
351.3 351.3
4687.7 10221.2
4339.6 4339.6
2057.4 2057.6
212.5 212.5
195.8 309.8 101.4
2787.6 1034.7 101.4
1637.5 984 48.8
V 1ao ¼ V 1a Aa qao ,
(21)
where qfo and qao are 0.5 mm. The initial conditions for energy and mass in the front mount are m1fo ¼
Patm V 1fo RT e
;
E 1fo ¼
Patm V 1fo cv R
(22)
for the rear mount m1ao ¼
Patm V 1ao ; RT e
E 1ao ¼
Patm V 1ao cv R
(23)
881.2 877.7 15.98
571.6 564.1 82.7
and for the reservoir, m2 ¼
Patm V 2 ; RT e
E2 ¼
Patm V 2 cv . R
(24)
Figs. 5–18 show the simulation results for the CEM model. In the simulation each 20 s represents a different flight condition; 0–20 s for taxi, 20–40 s for take-off, 40–60 s for climb, 60–80 s for cruise, and 80–100 s for descent. The final states from the previous simulation were used as the initial states for the next simulation except for the mount temperature. A step change of the loading (external force and weight) was applied at each 20 s.
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The initial temperatures of the front and rear mounts were set to equal the cowl (ambient air) temperature of 250 F throughout the simulations. This is done since after equilibrium is reached and the valves are all shut, heat transfer would take place and bring the mounts back to the cowl temperature. If the mount cools and the pressure remains constant (to keep the load in force equilibrium) then the volume has to change, and the piston position would also change. When this happens the inlet valve would open to get the load back to equilibrium. Since we are not simulating this entire sequence, we simply initialize the next flight condition with the appropriate values to have the mount at equilibrium at the cowl temperature (250 F). This requires that mass has to be added to the mounts. The front and rear mounts are lifted from their stops when the mount pressures exceed their respective load forces shown in Eqs. (25) and (26), P1f Patm X
1 Af
600
a1 ðF f þ mf gÞ kf qf b1
(25)
727
for the front, and, 1 a2 ðF a þ ma gÞ ka qa P1a Patm X Aa a2 þ b2
for the rear. The response of gage pressure for the front and rear mounts is shown in Figs. 5 and 6 along with magnified responses for the transition time due to flight condition changes. The gage pressure of the reservoir is in Fig. 7. The ‘‘HTR’’ indicates that the model includes the heat transfer between the supply and reservoir, and ‘‘NHTR’’ represents the case of no included heat transfer. At taxi condition (0–20 s), the supply pressure is not sufficient to lift the front mount from its stop location, so the pressure inside the front mount increased and saturated. However, the load on the rear mount is much less than the load on the front mount. Thus the supply pressure at taxi condition can lift the rear mount off its stop and move the piston to the desired mount position. The front and rear mount (piston) positions are illustrated in Figs. 8 and 9. As mentioned before, the
Front-HTR Front-NHTR
1400 Rear-HTR Rear-NHTR
Gage pressure,mount, [KPa]
400 300 200 100 0
1200 1000 800 600 400 200
-100 20
30
40 50 60 Time (sec)
200
0
100
19.5
20
20.5
340 320 300 39
0
39.5
40
400 350
59.5
60
Time (sec)
60.5
Gage pressure, mount, [KPa]
550 Gage pressure mount, [KPa]
450
500 450 400 350 79
79.5
80
80.5
Time (sec)
Fig. 5. Front mount gage pressure for different flight conditions with and without heat transfer in the CEM.
10
20
30
40 50 60 Time (sec)
250 200 150 100 19
40.5
Time (sec)
Time (sec)
300 59
90
360
250
150 19
80
Gage pressure, mount, [KPa]
300
70
Gage pressure, mount, [KPa]
10
Gage pressure mount, [KPa]
Gage pressure, mount, [KPa]
0
19.5 20 Time (sec)
700
80
90
100
39.5 40 Time (sec)
40.5
79.5 80 Time (sec)
80.5
1200 1100 1000
450
680 660 640 620 59
70
1300
900 39
20.5
Gage pressure, mount, [KPa]
Gage pressure,mount, [KPa]
500
Gage pressure, mount, [KPa]
(26)
59.5 60 Time (sec)
60.5
400 350 300 79
Fig. 6. Rear mount gage pressure for different flight conditions with and without heat transfer in the CEM.
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728
8
Reservoir-HTR Reservoir-NHTR
2500
x 10-3 Upper limit Lower limit Rear-HTR Rear-NHTR
6 Rear piston position (m)
Gage pressure,reservoir,[Kpa]
3000
2000 1500 1000 500
4 2 0 -2
0
0
10
20
30
40 50 60 Time(sec)
70
80
90
100 -4
0
10
20
30
Fig. 7. Reservoir gage pressure for different flight conditions with and without heat transfer in the CEM.
40 50 60 Time (sec)
70
80
90
100
Fig. 9. Rear mount position for different flight conditions with and without heat transfer in CEM.
20
x 10-4
15
Front-HTR Rear-HTR Front-NHTR Rear-NHTR
450 400 Mount Temp. F
Front piston position (m)
500
Upper limit Lower limit Front-HTR Front-NHTR
10
5
350 300 250 200
0 150 -5
100 0
10
20
30
40 50 60 Time(sec)
70
80
90
100
0
10
20
30
40 50 60 Time (sec)
70
80
90
100
Fig. 8. Front mount position for different flight conditions with and without heat transfer in the CEM.
Fig. 10. Front and rear mount temperature for different flight conditions with and without heat transfer in the CEM.
front mount sits on its stop location during the taxi condition due to lack of sufficient supply pressure. During the take-off condition (20–40 s), a higher supply pressure is available and the front mount piston is lifted from the stop and moves to the desired mount location as shown in Fig. 8. The front mount position can be maintained within the desired equilibrium position for most of the flight conditions except the descent case (80–100 s). The rear mount position can be maintained between the desired equilibrium positions for all flight conditions. From Table 2, the biggest positive external force is applied during the take-off condition. As this external
force is applied at t ¼ 20 s, the rear mount (piston) position goes downward as shown in Fig. 2 (positive direction in Fig. 9) and it decreases the rear mount volume. With the opening of the inlet valve, the mount is pressurized and the piston position moves upward (to the desired equilibrium position). As flight conditions change from take-off to climb, the applied external force reduces significantly. The pressure inside the rear mount, which was balanced with the external force for the take-off condition, is too high for the climb case and it moves the mount upward, i.e. the mount position moves in the negative direction at t ¼ 40 s as shown in Fig. 9. At this instant, the exhaust valve opens and the
ARTICLE IN PRESS T. Shim, D. Margolis / Control Engineering Practice 14 (2006) 721–733
290 Reservoir-HTR Reservoir-NHTR
285
275 270 265 260 255 250 245 0
10
20
30
40 50 60 Time (sec)
70
80
90
100
Fig. 11. Reservior temperature for different flight conditions with and without heat transfer in the CEM.
pressure inside the mount is exhausted in order to achieve the desired mount position. As flight conditions change from climb to cruise and from cruise to descent, the applied external force reduces and the exhaust valve is opened again to exhaust pressure. The response of the front and rear mount temperatures is shown in Fig. 10. The mount temperature starts initially from 250 F for all flight conditions. The knowledge of a maximum mount temperature is very important for the design of CEM because of the temperature limitations of the elastomeric components. An elastomeric component, a stock rated to 212 F, was designed to be used for the front mount. For the rear mount, an elastomeric component, a stock rated to 350 F, was designed to be used. Even with inclusion of heat transfer in the model, the temperature of the front mount reaches 420 F in the taxi condition and the rear mount temperature goes to 450 F during the take-off
x 10-3 7 HTR NHTR
Front mount,total mass flow [kg/min]
6
5
4
3
2
1
Front mount,totalmass flow[kg/min]
0
0
10
20
30
40
x 10-4 HTR NHTR
7 6.5 6 5.5 5 1
2
3 4 Time(sec)
5
6
50 60 Time(sec) Front mount,totalmass flow[kg/min]
Reservoir Temp. F
280
729
70
80
90
100
x10-3 HTR NHTR
1.3 1.2 1.1 1 0.9 0.8 80
80.2 80.4 80.6 80.8 Time(sec)
81
Fig. 12. Mass flow rate of front mount for different flight conditions with and without heat transfer in the CEM.
ARTICLE IN PRESS T. Shim, D. Margolis / Control Engineering Practice 14 (2006) 721–733
condition. Without heat transfer in the model, the front mount temperature is 4 F higher during taxi condition and the temperature in the rear mount 17 F higher during take off condition. Typical elastomers for vibration isolation should be used in environments not exceeding 350 F. This model predicts that temperatures exceeding 400 F can be expected under certain flight conditions. This information is extremely useful for the design of CEM mounts. The front mount temperature rises to 420 F during taxi condition when the supply temperature is 309 F. This can be explained from energy considerations. Any energy that lows into the front mount from the reservoir shows up as internal energy in the mount. Temperature is the only indicator of this internal energy. The energy that flows into the mount has both internal energy and potential energy due to the compression of the fluid as indicated by its pressure. During the time interval when the mount temperature rises quite a bit, the mass of fluid in the mount more than doubles. Significant energy is brought into the mount, and the temperature rises appropriately. The rear mount temperature drops in cruise condition (t ¼ 40–60 s) as energy in the rear mount is exhausted. The response of the reservoir temperature for different flight conditions with and without heat transfer is shown in Fig. 11. Without heat transfer in the CEM model, the reservoir temperature increased about 40 F from the cowl temperature during the take-off condition while it increased only 5 F with heat transfer. Figs. 12 and 13 show the mass flow rate and total mass in the front mount for different flight conditions. Heat transfer does not affect this response. Maximum mass flow rate during the take-off conditions through the inlet valve is 0.0068 (kg/min) and is an acceptable flow rate.
4.5
x 10-3
6
HTR NHTR
5 4 3 2 1 0 -1 -2 -3
0
10
20
30
40 50 60 Time(sec)
70
80
1.4
HTR NHTR
3 2.5 2 1.5
100
x 10-3 HTR NHTR
1.2
3.5
90
Fig. 14. Mass flow rate of rear mount for different flight conditions with and without heat transfer in the CEM.
Rear mount, total mass [kg]
Front mount, total mass [kg]
7
x 10-4
4
1 0.8 0.6 0.4 0.2
1 0.5
The mass flow rate and total mass of the rear mount are illustrated in Figs. 14 and 15. Maximum mass flow rate occurs during the take-off condition through inlet valve. For climb, cruise, and descent conditions, the mass flow goes out through exhaust valve. The inlet and outlet valve positions for the front mount valve are shown in Fig. 16. The outlet valve of the front mount is not opened during the different flight conditions. Fig. 17 shows the inlet and outlet valve positions for the rear mount valve for different flight conditions. During the taxi and take-off conditions, the inlet valve is opened while for the rest of the flight conditions the outlet valve is opened.
Rear mount,total mass flow [kg/min]
730
0 0
10
20
30
40 50 60 Time(sec)
70
80
90
100
Fig. 13. Total mass of front mount for different flight conditions with and without heat transfer in the CEM.
0
10
20
30
40 50 60 Time(sec)
70
80
90
100
Fig. 15. Total mass of rear mount for different flight conditions with and without heat transfer in the CEM.
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With heat Transfer
Front,Inlet valve area [m2]
x 10-8
Front,outlet valve area [m2]
731
2 1.5 1 0.5 0
0
10
20
30
40
50 60 Time(sec)
70
80
90
100
0
10
20
30
40
50 60 Time(sec)
70
80
90
100
1 0.5 0 -0.5 -1
Fig. 16. Inlet and outlet valve position of the front mount for different flight conditions with heat transfer in the CEM.
x 10-8
With heat Transfer
Rear,Inlet valve area [m2]
2 1.5 1 0.5 0 0
10
20
30
40
50 60 Time(sec)
70
80
90
100
10
20
30
40
50 60 Time(sec)
70
80
90
100
Rear,outlet valve area [m2]
x 10-8 2 1.5 1 0.5 0 0
Fig. 17. Inlet and outlet valve position of the rear mount for different flight conditions with heat transfer in the CEM.
The response of the convection coefficient used in the different flight conditions is shown in Fig. 18. With heat transfer, the variation of convection coefficient is
minimal. However, the variation of convection coefficient without heat transfer is significant and should be included in models of CEM systems.
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732
bf
Convection coefficient, [Watt/m]
825 With heat transfer Without heat transfer
820
Vf xf Af
815 810
Rear mount 805
ma Fa
800 795 0
10
20
30
40 50 60 Time(sec)
70
80
90
100
Fig. 18. Convection coefficients at the tube for different flight conditions with and without heat transfer in the CEM.
a2 b2 ka ba Va
4. Conclusions A model of the CEM for an aircraft engine application has been developed and evaluated for different flight conditions. This model includes the mount mechanics, and the fluid mechanics and thermodynamics of the valve control system. The proposed CEM model demonstrates that the front and rear mount displacements can be maintained within specified tolerance with acceptable flow rates from the bypass air for most flight conditions (take-off, climb, cruise). This conclusion would be applicable to virtually any engine subject to realistic flight conditions. However, design constraints on the mounts might result in excessive bleed air requirements to accomplish the task efficiently. This would have to be evaluated on a case by case basis. However, the results show that, even with heat transfer between the inlet conditions and the mounts, the temperature of the mount may become excessive for the elastomeric parts of the mount. This must be further considered for the design of the CEM.
Appendix A Front mount mf Ff a1 b1 kf
damping constant of elastomeric component at the front mount front mount volume for pressurization and exhaustion piston position at front mount piston area of front mount
mass of front mount external force applied at the front mount due to engine thrust length associated with the front mount lever length associated with the front mount lever stiffness of elastomeric component at the front mount
xa Aa
mass of rear mount external force applied at the rear mount due to engine thrust length associated with the rear mount lever length associated with the rear mount lever stiffness of elastomeric component at the rear mount damping constant of elastomeric component at the rear mount rear mount volume for pressurization and exhaustion piston position at rear mount piston area of rear mount
Valves Ai f A ef Ai a A ea Ao
inlet valve area for front mount outlet valve area for front mount inlet valve area for rear mount outlet valve area for rear mount valve orifice area
Tubing Lt tt Dt
length of tube thickness of tube diameter of tube
Parameters associated with thermodynamics kth Asurface T h2 mtube cth h T amb E h1 E h2 E h3 pm f qf qf qa
thermal conductivity of tube material surface area of tube absolute temperature at tube wall mass of tube specific heat of tube material convection coefficient absolute ambient temperature thermal energy loss due to conduction thermal energy stored in the tube thermal energy loss due to convection linear momentum of front mount linear momentum of rear mount displacement of elastomeric component at front mount ¼ xf displacement of elastomeric component at rear mount ¼ xa
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P1f P1a Patm Ps Ts P2 T2 E 1f E 3f E 4f E 1f E 3a E 4a m1f m3f m4f m1a m3a m4a E ri mri
pressure at front mount pressure at rear mount ambient pressure pressure at supply chamber absolute temperature at supply chamber pressure at reservoir absolute temperature at reservoir thermal energy in the front mount thermal energy from exhaust valve in the front mount thermal energy from inlet valve in the front mount thermal energy in the rear mount thermal energy from exhaust valve in the rear mount thermal energy from inlet valve in the rear mount containing mass in the front mount containing mass from exhaust valve in the front mount containing mass from inlet valve in the front mount containing mass in the rear mount containing mass from exhaust valve in the rear mount containing mass from inlet valve in the rear mount thermal energy from orifice to reservoir containing mass from orifice to reservoir
References Dudding, A. T., & Wilson, W. (2000). Development of a new front air suspension and steer axle system for on highway commercial vehicles. SAE Technical Paper no. 2000-01-3449.
733
Elmandy, M. M., & Abdujjabbar, Z. (1990). Design evaluation of advanced suspension systems for truck ride comfort. Computers & Structures, 36(2), 321–331. Elmandy, M. M., & Abdujjabbar, Z. (1992). On the ride and attitude control of road vehicles. Computers & Structures, 42(2), 245–254. Hrovat, D. (1991). Optimal suspension performance for 2-D vehicle models. Journal of Sound and Vibration, 146, 93–110. Incropera, F. P., & Dewitt, D. P. (1990). Introduction to heat transfer. (2nd ed.), New York: Wiley. (pp. 510–511). Karnopp, D. (1985). Active suspensions based on fast load levelers. Vehicle System Dynamics, 16, 355–380. Karnopp, D., & Margolis, D. (1984). Adaptive suspension concepts for road vehicles. Vehicle System Dynamics, 13, 145–160. Karnopp, D. C., Margolis, D. L., & Rosenberg, R. C. (2000). System dynamics: Modeling and simulation of mechatronic systems. New York: Wiley. Margolis, D. (1983). Semi-active control of wheel hop in ground vehicles. Vehicle System Dynamics, 12, 317–330. Ogawa, K., Satoh, K., & Enomoto, T. (1996). Development of damping control system for air suspension. JSAE Review, 17, 319–324. Palkovics, L., & Fries, A. (2001). Intelligent electronic systems in commercial vehicles for enhanced traffic safety. Vehicle System Dynamics, 35(4–5), 227–289. Sun, L. (2002). Optimum design of ‘‘road-friendly’’ vehicle suspension systems subjected to rough pavement surfaces. Applied Mathematical Modeling, 26, 635–652. Sweatman, P. F., Woodrooffe, J. H. , & Mcfarlane, S. (1997). Improving the interaction between heavy trucks, roads, and bridges. SAE Technical Paper, 973265. Toyofuku, K., Yamada, C., Kagawa, T., & Fujita, T. (1999). Study on dynamics characteristics analysis of air spring with auxiliary chamber. JSAE Review, 20, 349–355. White, D. L. (2000). Parametric study of leveling system characteristics on roll stability of trailing arm air suspension for heavy trucks. SAE Technical Paper, 2000-01-3480. Youn, I., & Hac, A. (1995). Semi-active suspensions with adaptive capability. Journal of Sound and Vibration, 180(3), 475–492.
Control Engineering Practice 14 (2006) 721–733 www.elsevier.com/locate/conengprac
Controlled equilibrium mounts for aircraft engine isolation Taehyun Shima,, Donald Margolisb a
Department of Mechanical Engineering, The University of Michigan-Dearborn, 4901 Evergreen Road, Dearborn, MI 48128, USA Department of Mechanical and Aeronautical Engineering, University of California, Davis, One Shield Avenue, Davis, CA 95616, USA
b
Received 7 October 2003; accepted 9 March 2005 Available online 4 May 2005
Abstract Softer than usual engine mounts for aircraft engine isolation can be used by adding control of the equilibrium position of the mount. The control power comes from bypass air from the engine. This isolation concept is called a controlled equilibrium mount or CEM. A model of the CEM has been developed which includes a front and rear mount with proper loading for each. These mounts possess passive stiffness and damping properties, and they can be pressurized or exhausted through control valves. The thermodynamics of the air is also modeled. A control strategy is proposed that keeps the mounts at their equilibrium positions. The simulation results show that the supply pressure and temperature of the bypass air varied with flight conditions. However, the variations are acceptable, and equilibrium can be maintained with acceptable flow rates from the bypass air for most flight conditions. r 2005 Elsevier Ltd. All rights reserved. Keywords: Aircraft engine mount; Vibration isolation; Controlled equilibrium; Load leveler
1. Introduction In any vibration isolation application, the stiffness of the isolator is chosen, in part, to handle the maximum allowable displacements based upon some assessment of the worst possible inputs. Most of the time, the worst case is not acting, and the mount is stiffer than it needs to be during this time, and the isolation is not as good as it could be if softer mounts were being used. The CEM is a concept that allows softer mounts to be used by using some active power to keep the mount near a predetermined equilibrium position. This concept is also known as a load leveling system. Application of active power in automotive suspensions for ride improvement has been extensively studied in both industrial settings and academia. Various Corresponding author. Tel.: +1 313 593 5127; fax: +1 313 593 3851. E-mail addresses: [email protected] (T. Shim), [email protected] (D. Margolis).
0967-0661/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.conengprac.2005.03.007
control algorithms utilizing different types of power actuators (electric, pneumatic, hydro-pneumatic, magnetic and commercialized magneto-rheological fluid, etc.) are documented in the literature (Margolis, 1983; Karnopp & Margolis, 1984; Karnopp, 1985; Hrovat, 1991; Elmandy & Abdujjabbar, 1990, 1992; Youn & Hac, 1995; Ogawa, Satoh, & Enomoto, 1996; Toyofuku, Yamada, Kagawa, & Fujita, 1999; White, 2000). The majority of on highway trucks (class 8) use air suspension and some type of ride height control system. When conventional steel springs are used, they are designed to provide adequate support for the fully loaded vehicle, thus they are unnecessarily stiff for smaller loads. Air suspensions coupled with some type of height control system (load leveling system) have advantages over steel springs including consistent loaded/unloaded ride heights, reduced hysteresis, soft ride, and good traction distribution (White, 2000; Sweatman, Woodrooffe, & Mcfarlane, 1997; Dudding & Wilson, 2000; Palkovics & Fries, 2001). With growing interest in vehicle safety, the role of a load leveling
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T. Shim, D. Margolis / Control Engineering Practice 14 (2006) 721–733
system is not limited to the improvement of ride comfort, but is expanded into vehicle stability and safety. There are different designs of load leveling systems and control algorithms to lessen vehicle rollover propensity in heavy trucks (Palkovics & Fries, 2001). In addition, air suspensions have the reputation of being ‘‘road friendly’’ since they reduce dynamic loading of the pavement resulting in diminished pavement wear (Sweatman et al., 1997; Sun, 2002). There has not been much research on the use of controlled equilibrium mounts for suppression of aircraft engine vibration. In this paper, a model of the CEM for an aircraft engine application is developed and simulated for different flight conditions to evaluate its effectiveness. The model includes the mount mechanics, and the fluid mechanics and thermodynamics of the valve control system.
2. Development of CEM model The CEM model proposed here consists of four major parts; conventional mount, expandable volume, control valve, and supply. There is a front mount and a rear mount with appropriate loading on each mount. These conventional mounts have elastomeric components that provide the basic isolation. In addition there is an expandable volume that can be pressurized and/or exhausted that is used to keep the mount at some equilibrium condition. Control valves are needed to either expose the mount volume to a supply pressure or to expose the mount volume to exhaust pressure. The supply for this active part of the isolation system comes from bypass air from engine. As the flight conditions change from startup and taxi, to takeoff, climb, cruise,
and descent, the supply conditions change as does the loading on the mounts due to varying thrust. Before a hardware realization is possible, it is important to know how well equilibrium can be maintained for the various flight conditions, how much bypass air is needed to accomplish the control task, and what temperatures are expected in the mounts. In order to assess this CEM concept for aircraft applications, a model was developed using bond graphs, which includes the mount mechanics, and the fluid mechanics and thermodynamics of the valve control system. Fig. 1 shows a detailed diagram of the aircraft engine and the fore and aft mounts. From this diagram, an overall system model is developed. Fig. 2 shows a reticulated schematic of the system from Fig. 1 and clearly indicates the physics of operation of the mounts in Fig. 1. This CEM includes the mount mechanics between external loading points and mount locations, as well as the fluid mechanics and thermodynamics associated with supply and exhaust of the mounts. The load mass of the front and rear mounts are modeled separately since measured loading data on each mount were provided. The mf and ma represent mass of front and rear mounts, F f and F a are external forces applied at the front and rear mounts due to engine thrust, a1 , b1 are lengths associated with the front mount lever, and a2 , b2 are lengths for the lever at the rear. There are elastomeric components, kf and bf for the front mount, and ka and ba for the rear mount, which provide basic isolation. The spring forces are set to be positive in compression. The mount volumes, V f (front) and V a (rear) are pressurized and/or exhausted based upon the respective piston positions, xf and xa . The inlet and outlet valves are opened/closed based upon control logic that is determined from the position of the mount.
Fig. 1. Schematic of aircraft engine
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723
Fig. 2. Schematic of controlled equilibrium mount model for aircraft engines.
It is assumed that heat transfer occurs in the tube between the supply and reservoir, and this effect is included in the CEM model. 2.1. Valve control logic The goal of CEM is to maintain the displacement of the mounts within some specified tolerance. Here, we wish to maintain the mount positions within 0:25 mm. In the front mount, for instance, when the front mount loading (external force and weight of engine) is applied a distance a1 from the pivot, the volume, V f , inside the front mount decreases and the mount position, xf , moves upward. If the mount position exceeds the positive equilibrium limit (þ0:25 mm), the inlet valve for front mount, Aif , opens and the mount will be pressurized from the reservoir until equilibrium is again attained. If the applied pressure is excessive in the mount, then the mount position will move downward and might exceed the negative equilibrium limit (0:25 mm). For this case, the exhaust (outlet) valve, Aef , will open and the high pressure inside the mount will exhaust allowing the mount to return to equilibrium. When the mount position is within the equilibrium limits, the inlet and exhaust valves will be closed. The same control logic is also applied to the rear mount. Thus the valve control strategy becomes, If xn 40:25 mm;
Ai ¼ open; Ae ¼ close,
If xn o 0:25 mm;
Ain ¼ close; Aen ¼ open,
If 0:25 mmoxn o0:25 mm;
Ain ¼ close; Aen ¼ close, (1)
where subscript ‘‘*’’ indicates front and rear.
2.2. Heat transfer effect The elastomeric components used in the mount are not suitable for high temperatures. So the knowledge of the mount temperature is critical for designing the CEM. Since the reservoir pressure and temperature are the supply conditions for the mounts, heat transfer between the supply and the reservoir is included to determine if the reservoir temperature can be maintained near the ambient cowl temperature around the engine. The schematic and bond graph of the heat transfer model is shown in Fig. 3. The supply tube is made of stainless steel (Type 321 (UNS No.- S33100)) and has length of Lt , thickness of tt and diameter, Dt . The pseudo-bond graph for heat transfer is documented in Karnopp, Margolis, and Rosenberg (2000). As bypass air from the engine flows through the tube, conduction occurs from inside to outside of the tube due to the temperature difference between the fluid and the tube wall. This conduction is characterized by kth Asurface E_ h1 ¼ ðT 2 T h2 Þ, tt
(2)
where kth is the thermal conductivity of the tube material, Asurface is surface area of tube, tt is thickness of tube, T 2 is reservoir temperature, and T h2 is the wall temperature. This effect is characterized as a 1-port resistance in the bond graph. The tube itself stores some energy due to its thermal capacitance. This is mathematically expressed as Eq. (3) and shown as a C-element in the bond graph. T h2 ¼
1 Eh . mtube cth 2
(3)
In Eq. (3), mtube is the mass of tube, and cth is the specific heat of tube material. The tube is surrounded by the air inside the engine cowl, and it is assumed that free heat convection occurs between the outside surface of the tube and the
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724
Fig. 3. Schematic and bond graph of heat transfer between supply and reservoir in CEM.
stationary ambient air. This is characterized by, E_ h3 ¼ hAsurface ðT h2 T amb Þ,
p_ ma ¼ F a þ ma g (4)
where h is a convection coefficient. The convection coefficient, h, is updated at each time step based on the temperatures T amb and T h2 . The detailed update law for the convection coefficient can be found in Incropera and Dewitt (1990).
2.3. Equations of motion The complete bond graph model of CEM with the inclusion of heat transfer is shown in Fig. 4. Bond graphs (Karnopp et al., 2000) are a concise pictorial representation of the interactive dynamics of all types of energetic systems. They allow the model to be developed in pieces and then put together into an overall computational model. The equations of motion for this CEM can be derived from the bond graph model. For the linear momentum of front and rear mount, p_ mf ¼ F f þ mf g
b1 F1 , a1 f
(5)
a2 þ b2 F 1a , a2
(6)
where p F 1f ¼ kf qf þ Af ðP1f Patm Þ þ bf F 1a
b1 , m f a1 mf
pm a a2 þ b2 ¼ ka qa þ Aa ðP1a Patm Þ þ ba . ma a2
(7)
(8)
For the displacement of the elastomeric component in the mount, pm f b1 , (9) q_ f ¼ mf a1 q_ a ¼
pm a a2 þ b2 . ma a2
(10)
The equations of motion for the front and rear expandable volumes are b1 pm f , (11) V_ 1f ¼ Af a1 m f a2 þ b2 pm a V_ 1a ¼ Aa a2 m a
(12)
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725
Fig. 4. Bond graph model of controlled equilibrium mount for aircraft engines.
where Af is piston area of the front mount and Aa is piston area of the rear mount. The front and rear mounts are modeled as thermodynamic accumulators (Karnopp et al., 2000), and the state equations for energy and mass are E_ 1f ¼ E_ 3f þ E_ 4f P1f V_ 1f ,
(13)
E_ 1a ¼ E_ 3a þ E_ 4a P1a V_ 1a ,
(14)
_ 3f þ m _ 4f , _ 1f ¼ m m
(15)
_ 3a þ m _ 4a . _ 1a ¼ m m
(16)
For the reservoir, with the inclusion of heat transfer derived previously, the energy and mass flow rates are _ ri m _ 5f m _ 5a , _2 ¼ m m
(17)
kth Asurface E_ 2 ¼ E_ ri E_ 5f E_ 5a ðT 2 T h2 Þ, tt
(18)
kth Asurface E_ h2 ¼ ðT 2 T h2 Þ hAsurface ðT h2 T amb Þ, tt (19) T h2 ¼
1 Eh . mtube cth 2
(20)
3. Simulation and discussion The proposed CEM model was simulated using the parameters listed in Table 1. The damping ratios used for the model are appropriate for the elastomeric material being modeled. In the simulation, different flight conditions are simulated by varying applied forces, supply pressure, and supply temperature. The applied external force variations are due to the changes in engine thrust during different flight conditions. Table 2 shows the external forces, supply pressure, and temperature for the taxi, take-off, climb, cruise and descent conditions. The cruising altitude for a target airplane is 35,000 ft and the ambient air temperature variation is about 68 C over the flight envelope. As shown in Table 2, maximum engine thrust occurs during the take-off condition. At the initial condition of taxi, it is assumed that the mount pressure starts at atmospheric pressure and the mount temperature is the same as the engine cowl temperature. The loading (external force and weight) on the mount is balanced with the spring force in the mount. In order to avoid excessive mount displacement due to the loading, an imaginary stop is introduced at þ0:5 mm in the mount, and the mount sits on this stop at the beginning of simulation. The initial volumes of front and rear mounts are V 1fo ¼ V 1f Af qfo ,
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Table 1 CEM parameters used in the simulation Front mount Mass mf : a1 b1 Initial volume, V f Stiffness, kf Piston area, Af
2219.207 (lb), 1007.2 (kg) 2.721 (in), 6.9 (cm) 5.084 (in), 12.9 (cm) 4.940 (in3 ), 80.9 (cm3 ) 6500 (lbf/in), 11383.3 (N/cm) 13. ðin2 Þ, 83.8 ðcm2 Þ
rffiffiffiffiffi
Natural frequency, of ¼
kf mf
Rear mount Mass, ma : a2 b2 Initial volume, V a Stiffness, ka Piston area, Aa Natural frequency, oa ¼
Damping ratio, zf Damping constant, bf ¼ 2zf kf of
0.07
268.793 (lb), 122.0 (kg) 2.2 (inch), 5.6 (cm) 2.075 (inch), 5.3 (cm) 7.420 ðinchÞ3 , 121.6 ðcm3 Þ 6500 (lbf/in), 11383.3 (N/cm) 7.5 ðinch2 Þ, 48.4 ðcm2 Þ
qffiffiffiffiffi ka ma
Damping ratio, za Damping constant, ba ¼ 2za ka oa
0.1
Valve Heat transfer parameters Inlet ¼ outlet (reservior to mount, mount to exhaust), front ¼ rear Thermal conductivity, K th Area, Ai ¼ Ai ¼ Aef ¼ Aea 3e 5 (in2 Þ, 2e 4 ðcm2 Þ Orifice area (supply to reservoir) Specific heat, C th Ao Density of tube material, r 7:0e 4ðin2 Þ, 4:5e 3 ðcm2 Þ Mass of tube mtube Surface area, Asurf ¼ pDt Lt Tubing 3/8 (in), 0.95 (cm) Convection coefficient, h (Wm/K) is calculated Diameter, Dt Length (supply to reservoir), Lt , 30 (in), 76.2 (cm) each time step. Thickness, tt 0.020 (in), 0.05 (cm)
14.6 (W/m K) 500 (J/kg/K) 8000 (kg=m3 )
Table 2 External forces, supply temperature and pressure for different flight conditions
External forces Front mount (N) Rear mount (N) Supply Ps (gage), (KPa) T s (F) Patm (KPa)
Taxi
Take off
Climb
Cruise
Descent
351.3 351.3
4687.7 10221.2
4339.6 4339.6
2057.4 2057.6
212.5 212.5
195.8 309.8 101.4
2787.6 1034.7 101.4
1637.5 984 48.8
V 1ao ¼ V 1a Aa qao ,
(21)
where qfo and qao are 0.5 mm. The initial conditions for energy and mass in the front mount are m1fo ¼
Patm V 1fo RT e
;
E 1fo ¼
Patm V 1fo cv R
(22)
for the rear mount m1ao ¼
Patm V 1ao ; RT e
E 1ao ¼
Patm V 1ao cv R
(23)
881.2 877.7 15.98
571.6 564.1 82.7
and for the reservoir, m2 ¼
Patm V 2 ; RT e
E2 ¼
Patm V 2 cv . R
(24)
Figs. 5–18 show the simulation results for the CEM model. In the simulation each 20 s represents a different flight condition; 0–20 s for taxi, 20–40 s for take-off, 40–60 s for climb, 60–80 s for cruise, and 80–100 s for descent. The final states from the previous simulation were used as the initial states for the next simulation except for the mount temperature. A step change of the loading (external force and weight) was applied at each 20 s.
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The initial temperatures of the front and rear mounts were set to equal the cowl (ambient air) temperature of 250 F throughout the simulations. This is done since after equilibrium is reached and the valves are all shut, heat transfer would take place and bring the mounts back to the cowl temperature. If the mount cools and the pressure remains constant (to keep the load in force equilibrium) then the volume has to change, and the piston position would also change. When this happens the inlet valve would open to get the load back to equilibrium. Since we are not simulating this entire sequence, we simply initialize the next flight condition with the appropriate values to have the mount at equilibrium at the cowl temperature (250 F). This requires that mass has to be added to the mounts. The front and rear mounts are lifted from their stops when the mount pressures exceed their respective load forces shown in Eqs. (25) and (26), P1f Patm X
1 Af
600
a1 ðF f þ mf gÞ kf qf b1
(25)
727
for the front, and, 1 a2 ðF a þ ma gÞ ka qa P1a Patm X Aa a2 þ b2
for the rear. The response of gage pressure for the front and rear mounts is shown in Figs. 5 and 6 along with magnified responses for the transition time due to flight condition changes. The gage pressure of the reservoir is in Fig. 7. The ‘‘HTR’’ indicates that the model includes the heat transfer between the supply and reservoir, and ‘‘NHTR’’ represents the case of no included heat transfer. At taxi condition (0–20 s), the supply pressure is not sufficient to lift the front mount from its stop location, so the pressure inside the front mount increased and saturated. However, the load on the rear mount is much less than the load on the front mount. Thus the supply pressure at taxi condition can lift the rear mount off its stop and move the piston to the desired mount position. The front and rear mount (piston) positions are illustrated in Figs. 8 and 9. As mentioned before, the
Front-HTR Front-NHTR
1400 Rear-HTR Rear-NHTR
Gage pressure,mount, [KPa]
400 300 200 100 0
1200 1000 800 600 400 200
-100 20
30
40 50 60 Time (sec)
200
0
100
19.5
20
20.5
340 320 300 39
0
39.5
40
400 350
59.5
60
Time (sec)
60.5
Gage pressure, mount, [KPa]
550 Gage pressure mount, [KPa]
450
500 450 400 350 79
79.5
80
80.5
Time (sec)
Fig. 5. Front mount gage pressure for different flight conditions with and without heat transfer in the CEM.
10
20
30
40 50 60 Time (sec)
250 200 150 100 19
40.5
Time (sec)
Time (sec)
300 59
90
360
250
150 19
80
Gage pressure, mount, [KPa]
300
70
Gage pressure, mount, [KPa]
10
Gage pressure mount, [KPa]
Gage pressure, mount, [KPa]
0
19.5 20 Time (sec)
700
80
90
100
39.5 40 Time (sec)
40.5
79.5 80 Time (sec)
80.5
1200 1100 1000
450
680 660 640 620 59
70
1300
900 39
20.5
Gage pressure, mount, [KPa]
Gage pressure,mount, [KPa]
500
Gage pressure, mount, [KPa]
(26)
59.5 60 Time (sec)
60.5
400 350 300 79
Fig. 6. Rear mount gage pressure for different flight conditions with and without heat transfer in the CEM.
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728
8
Reservoir-HTR Reservoir-NHTR
2500
x 10-3 Upper limit Lower limit Rear-HTR Rear-NHTR
6 Rear piston position (m)
Gage pressure,reservoir,[Kpa]
3000
2000 1500 1000 500
4 2 0 -2
0
0
10
20
30
40 50 60 Time(sec)
70
80
90
100 -4
0
10
20
30
Fig. 7. Reservoir gage pressure for different flight conditions with and without heat transfer in the CEM.
40 50 60 Time (sec)
70
80
90
100
Fig. 9. Rear mount position for different flight conditions with and without heat transfer in CEM.
20
x 10-4
15
Front-HTR Rear-HTR Front-NHTR Rear-NHTR
450 400 Mount Temp. F
Front piston position (m)
500
Upper limit Lower limit Front-HTR Front-NHTR
10
5
350 300 250 200
0 150 -5
100 0
10
20
30
40 50 60 Time(sec)
70
80
90
100
0
10
20
30
40 50 60 Time (sec)
70
80
90
100
Fig. 8. Front mount position for different flight conditions with and without heat transfer in the CEM.
Fig. 10. Front and rear mount temperature for different flight conditions with and without heat transfer in the CEM.
front mount sits on its stop location during the taxi condition due to lack of sufficient supply pressure. During the take-off condition (20–40 s), a higher supply pressure is available and the front mount piston is lifted from the stop and moves to the desired mount location as shown in Fig. 8. The front mount position can be maintained within the desired equilibrium position for most of the flight conditions except the descent case (80–100 s). The rear mount position can be maintained between the desired equilibrium positions for all flight conditions. From Table 2, the biggest positive external force is applied during the take-off condition. As this external
force is applied at t ¼ 20 s, the rear mount (piston) position goes downward as shown in Fig. 2 (positive direction in Fig. 9) and it decreases the rear mount volume. With the opening of the inlet valve, the mount is pressurized and the piston position moves upward (to the desired equilibrium position). As flight conditions change from take-off to climb, the applied external force reduces significantly. The pressure inside the rear mount, which was balanced with the external force for the take-off condition, is too high for the climb case and it moves the mount upward, i.e. the mount position moves in the negative direction at t ¼ 40 s as shown in Fig. 9. At this instant, the exhaust valve opens and the
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290 Reservoir-HTR Reservoir-NHTR
285
275 270 265 260 255 250 245 0
10
20
30
40 50 60 Time (sec)
70
80
90
100
Fig. 11. Reservior temperature for different flight conditions with and without heat transfer in the CEM.
pressure inside the mount is exhausted in order to achieve the desired mount position. As flight conditions change from climb to cruise and from cruise to descent, the applied external force reduces and the exhaust valve is opened again to exhaust pressure. The response of the front and rear mount temperatures is shown in Fig. 10. The mount temperature starts initially from 250 F for all flight conditions. The knowledge of a maximum mount temperature is very important for the design of CEM because of the temperature limitations of the elastomeric components. An elastomeric component, a stock rated to 212 F, was designed to be used for the front mount. For the rear mount, an elastomeric component, a stock rated to 350 F, was designed to be used. Even with inclusion of heat transfer in the model, the temperature of the front mount reaches 420 F in the taxi condition and the rear mount temperature goes to 450 F during the take-off
x 10-3 7 HTR NHTR
Front mount,total mass flow [kg/min]
6
5
4
3
2
1
Front mount,totalmass flow[kg/min]
0
0
10
20
30
40
x 10-4 HTR NHTR
7 6.5 6 5.5 5 1
2
3 4 Time(sec)
5
6
50 60 Time(sec) Front mount,totalmass flow[kg/min]
Reservoir Temp. F
280
729
70
80
90
100
x10-3 HTR NHTR
1.3 1.2 1.1 1 0.9 0.8 80
80.2 80.4 80.6 80.8 Time(sec)
81
Fig. 12. Mass flow rate of front mount for different flight conditions with and without heat transfer in the CEM.
ARTICLE IN PRESS T. Shim, D. Margolis / Control Engineering Practice 14 (2006) 721–733
condition. Without heat transfer in the model, the front mount temperature is 4 F higher during taxi condition and the temperature in the rear mount 17 F higher during take off condition. Typical elastomers for vibration isolation should be used in environments not exceeding 350 F. This model predicts that temperatures exceeding 400 F can be expected under certain flight conditions. This information is extremely useful for the design of CEM mounts. The front mount temperature rises to 420 F during taxi condition when the supply temperature is 309 F. This can be explained from energy considerations. Any energy that lows into the front mount from the reservoir shows up as internal energy in the mount. Temperature is the only indicator of this internal energy. The energy that flows into the mount has both internal energy and potential energy due to the compression of the fluid as indicated by its pressure. During the time interval when the mount temperature rises quite a bit, the mass of fluid in the mount more than doubles. Significant energy is brought into the mount, and the temperature rises appropriately. The rear mount temperature drops in cruise condition (t ¼ 40–60 s) as energy in the rear mount is exhausted. The response of the reservoir temperature for different flight conditions with and without heat transfer is shown in Fig. 11. Without heat transfer in the CEM model, the reservoir temperature increased about 40 F from the cowl temperature during the take-off condition while it increased only 5 F with heat transfer. Figs. 12 and 13 show the mass flow rate and total mass in the front mount for different flight conditions. Heat transfer does not affect this response. Maximum mass flow rate during the take-off conditions through the inlet valve is 0.0068 (kg/min) and is an acceptable flow rate.
4.5
x 10-3
6
HTR NHTR
5 4 3 2 1 0 -1 -2 -3
0
10
20
30
40 50 60 Time(sec)
70
80
1.4
HTR NHTR
3 2.5 2 1.5
100
x 10-3 HTR NHTR
1.2
3.5
90
Fig. 14. Mass flow rate of rear mount for different flight conditions with and without heat transfer in the CEM.
Rear mount, total mass [kg]
Front mount, total mass [kg]
7
x 10-4
4
1 0.8 0.6 0.4 0.2
1 0.5
The mass flow rate and total mass of the rear mount are illustrated in Figs. 14 and 15. Maximum mass flow rate occurs during the take-off condition through inlet valve. For climb, cruise, and descent conditions, the mass flow goes out through exhaust valve. The inlet and outlet valve positions for the front mount valve are shown in Fig. 16. The outlet valve of the front mount is not opened during the different flight conditions. Fig. 17 shows the inlet and outlet valve positions for the rear mount valve for different flight conditions. During the taxi and take-off conditions, the inlet valve is opened while for the rest of the flight conditions the outlet valve is opened.
Rear mount,total mass flow [kg/min]
730
0 0
10
20
30
40 50 60 Time(sec)
70
80
90
100
Fig. 13. Total mass of front mount for different flight conditions with and without heat transfer in the CEM.
0
10
20
30
40 50 60 Time(sec)
70
80
90
100
Fig. 15. Total mass of rear mount for different flight conditions with and without heat transfer in the CEM.
ARTICLE IN PRESS T. Shim, D. Margolis / Control Engineering Practice 14 (2006) 721–733
With heat Transfer
Front,Inlet valve area [m2]
x 10-8
Front,outlet valve area [m2]
731
2 1.5 1 0.5 0
0
10
20
30
40
50 60 Time(sec)
70
80
90
100
0
10
20
30
40
50 60 Time(sec)
70
80
90
100
1 0.5 0 -0.5 -1
Fig. 16. Inlet and outlet valve position of the front mount for different flight conditions with heat transfer in the CEM.
x 10-8
With heat Transfer
Rear,Inlet valve area [m2]
2 1.5 1 0.5 0 0
10
20
30
40
50 60 Time(sec)
70
80
90
100
10
20
30
40
50 60 Time(sec)
70
80
90
100
Rear,outlet valve area [m2]
x 10-8 2 1.5 1 0.5 0 0
Fig. 17. Inlet and outlet valve position of the rear mount for different flight conditions with heat transfer in the CEM.
The response of the convection coefficient used in the different flight conditions is shown in Fig. 18. With heat transfer, the variation of convection coefficient is
minimal. However, the variation of convection coefficient without heat transfer is significant and should be included in models of CEM systems.
ARTICLE IN PRESS T. Shim, D. Margolis / Control Engineering Practice 14 (2006) 721–733
732
bf
Convection coefficient, [Watt/m]
825 With heat transfer Without heat transfer
820
Vf xf Af
815 810
Rear mount 805
ma Fa
800 795 0
10
20
30
40 50 60 Time(sec)
70
80
90
100
Fig. 18. Convection coefficients at the tube for different flight conditions with and without heat transfer in the CEM.
a2 b2 ka ba Va
4. Conclusions A model of the CEM for an aircraft engine application has been developed and evaluated for different flight conditions. This model includes the mount mechanics, and the fluid mechanics and thermodynamics of the valve control system. The proposed CEM model demonstrates that the front and rear mount displacements can be maintained within specified tolerance with acceptable flow rates from the bypass air for most flight conditions (take-off, climb, cruise). This conclusion would be applicable to virtually any engine subject to realistic flight conditions. However, design constraints on the mounts might result in excessive bleed air requirements to accomplish the task efficiently. This would have to be evaluated on a case by case basis. However, the results show that, even with heat transfer between the inlet conditions and the mounts, the temperature of the mount may become excessive for the elastomeric parts of the mount. This must be further considered for the design of the CEM.
Appendix A Front mount mf Ff a1 b1 kf
damping constant of elastomeric component at the front mount front mount volume for pressurization and exhaustion piston position at front mount piston area of front mount
mass of front mount external force applied at the front mount due to engine thrust length associated with the front mount lever length associated with the front mount lever stiffness of elastomeric component at the front mount
xa Aa
mass of rear mount external force applied at the rear mount due to engine thrust length associated with the rear mount lever length associated with the rear mount lever stiffness of elastomeric component at the rear mount damping constant of elastomeric component at the rear mount rear mount volume for pressurization and exhaustion piston position at rear mount piston area of rear mount
Valves Ai f A ef Ai a A ea Ao
inlet valve area for front mount outlet valve area for front mount inlet valve area for rear mount outlet valve area for rear mount valve orifice area
Tubing Lt tt Dt
length of tube thickness of tube diameter of tube
Parameters associated with thermodynamics kth Asurface T h2 mtube cth h T amb E h1 E h2 E h3 pm f qf qf qa
thermal conductivity of tube material surface area of tube absolute temperature at tube wall mass of tube specific heat of tube material convection coefficient absolute ambient temperature thermal energy loss due to conduction thermal energy stored in the tube thermal energy loss due to convection linear momentum of front mount linear momentum of rear mount displacement of elastomeric component at front mount ¼ xf displacement of elastomeric component at rear mount ¼ xa
ARTICLE IN PRESS T. Shim, D. Margolis / Control Engineering Practice 14 (2006) 721–733
P1f P1a Patm Ps Ts P2 T2 E 1f E 3f E 4f E 1f E 3a E 4a m1f m3f m4f m1a m3a m4a E ri mri
pressure at front mount pressure at rear mount ambient pressure pressure at supply chamber absolute temperature at supply chamber pressure at reservoir absolute temperature at reservoir thermal energy in the front mount thermal energy from exhaust valve in the front mount thermal energy from inlet valve in the front mount thermal energy in the rear mount thermal energy from exhaust valve in the rear mount thermal energy from inlet valve in the rear mount containing mass in the front mount containing mass from exhaust valve in the front mount containing mass from inlet valve in the front mount containing mass in the rear mount containing mass from exhaust valve in the rear mount containing mass from inlet valve in the rear mount thermal energy from orifice to reservoir containing mass from orifice to reservoir
References Dudding, A. T., & Wilson, W. (2000). Development of a new front air suspension and steer axle system for on highway commercial vehicles. SAE Technical Paper no. 2000-01-3449.
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Elmandy, M. M., & Abdujjabbar, Z. (1990). Design evaluation of advanced suspension systems for truck ride comfort. Computers & Structures, 36(2), 321–331. Elmandy, M. M., & Abdujjabbar, Z. (1992). On the ride and attitude control of road vehicles. Computers & Structures, 42(2), 245–254. Hrovat, D. (1991). Optimal suspension performance for 2-D vehicle models. Journal of Sound and Vibration, 146, 93–110. Incropera, F. P., & Dewitt, D. P. (1990). Introduction to heat transfer. (2nd ed.), New York: Wiley. (pp. 510–511). Karnopp, D. (1985). Active suspensions based on fast load levelers. Vehicle System Dynamics, 16, 355–380. Karnopp, D., & Margolis, D. (1984). Adaptive suspension concepts for road vehicles. Vehicle System Dynamics, 13, 145–160. Karnopp, D. C., Margolis, D. L., & Rosenberg, R. C. (2000). System dynamics: Modeling and simulation of mechatronic systems. New York: Wiley. Margolis, D. (1983). Semi-active control of wheel hop in ground vehicles. Vehicle System Dynamics, 12, 317–330. Ogawa, K., Satoh, K., & Enomoto, T. (1996). Development of damping control system for air suspension. JSAE Review, 17, 319–324. Palkovics, L., & Fries, A. (2001). Intelligent electronic systems in commercial vehicles for enhanced traffic safety. Vehicle System Dynamics, 35(4–5), 227–289. Sun, L. (2002). Optimum design of ‘‘road-friendly’’ vehicle suspension systems subjected to rough pavement surfaces. Applied Mathematical Modeling, 26, 635–652. Sweatman, P. F., Woodrooffe, J. H. , & Mcfarlane, S. (1997). Improving the interaction between heavy trucks, roads, and bridges. SAE Technical Paper, 973265. Toyofuku, K., Yamada, C., Kagawa, T., & Fujita, T. (1999). Study on dynamics characteristics analysis of air spring with auxiliary chamber. JSAE Review, 20, 349–355. White, D. L. (2000). Parametric study of leveling system characteristics on roll stability of trailing arm air suspension for heavy trucks. SAE Technical Paper, 2000-01-3480. Youn, I., & Hac, A. (1995). Semi-active suspensions with adaptive capability. Journal of Sound and Vibration, 180(3), 475–492.