Hans DeSmidt 

University of Tennessee, Knoxville, Tennessee

Kon-Well Wang

University of Michigan, Ann Arbor, Michigan

Edward C. Smith

Penn State University, University Park, Pennsylvania

David G. Lewicki

Glenn Research Center, Cleveland, Ohio

Variable-Speed Simulation of a Dual-Clutch 

Gearbox Tiltrotor Driveline

NASA/TM—2012-217212

January 2012

AHS2011–000178
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 7115 Standard Drive

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Hans DeSmidt 

University of Tennessee, Knoxville, Tennessee

Kon-Well Wang

University of Michigan, Ann Arbor, Michigan

Edward C. Smith

Penn State University, University Park, Pennsylvania

David G. Lewicki

Glenn Research Center, Cleveland, Ohio

Variable-Speed Simulation of a Dual-Clutch 

Gearbox Tiltrotor Driveline

NASA/TM—2012-217212

January 2012

AHS2011–000178

National Aeronautics and

Space Administration

Glenn Research Center

Cleveland, Ohio 44135

Prepared for the

67th Annual Forum and Technology Display (Forum 67)

sponsored by the American Helicopter Society (AHS)

Virginia Beach, Virginia, May 3–5, 2011
Acknowledgments

This work was supported under NASA Grant no. NNX07AC58A, with 

Technical Monitors Dr. Robert F. Handschuh and Dr. David G. Lewicki.

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Level of Review: This material has been technically reviewed by technical management. 
NASA/TM—2012-217212 1 

Variable-Speed Simulation of a Dual-Clutch Gearbox Tiltrotor Driveline 

Hans DeSmidt 

University of Tennessee 

Knoxville, Tennessee 37996 

Kon-Well Wang 

University of Michigan 

Ann Arbor, Michigan 48109-2125 

Edward C. Smith 

Penn State University 

University Park, Pennsylvania 16802 

David G. Lewicki 

National Aeronautics and Space Administration 

Glenn Research Center 

Cleveland, Ohio 44135 

Abstract 

This investigation explores the variable-speed operation and 

shift response of a prototypical two-speed dual-clutch transmission tiltrotor driveline in forward flight. Here, a Comprehensive Variable-Speed Rotorcraft Propulsion System 

Modeling (CVSRPM) tool developed under a NASA funded 

NRA program is utilized to simulate the drive system dynamics. In this study, a sequential shifting control strategy is 

analyzed under a steady forward cruise condition. This 

investigation attempts to build upon previous variable-speed 

rotorcraft propulsion studies by 1) including a fully nonlinear 

transient gas-turbine engine model, 2) including clutch 

stick-slip friction effects, 3) including shaft flexibility, 4) 

incorporating a basic flight dynamics model to account for 

interactions with the flight control system. Through exploring 

the interactions between the various subsystems, this analysis 

provides important insights into the continuing development of 

variable-speed rotorcraft propulsion systems. 

Introduction 

According to a recent NASA-Army-Industry-University 

investigation, significant benefits to rotorcraft operational 

performance, effectiveness and acoustic signature could be 

gained through the ability to adjust main rotor speed to 

accommodate various flight conditions (Ref. 1). In particular, 

variable-speed rotor technology is critical to the slowed-rotor 

compound configuration concepts and would offer significant 

benefits to future Heavy-Lift helicopter and tiltrotor configurations (Ref. 2). Current rotorcraft propulsion systems are 

designed around a fixed-ratio transmission without the 

capability to vary rotor speed except by engine speed adjustment. Since the specific fuel consumption of modern 

gas-turbine engines is optimum within a relatively narrow 

speed range, the ability to achieve a variable-speed rotor 

through engine speed adjustment is limited (Refs. 3 and 4). 

Therefore, to achieve effective variable rotor speed designs, a 

variable ratio transmission in the rotorcraft propulsion system 

becomes attractive. The NASA Heavy-Lift study (Ref. 1) 

suggests that transmission designs capable of rotor speed 

variations on the order of 50 percent will be sufficient to 

achieve significant rotorcraft performance and noise benefits. 

Recently, Stevens, Handschuh, and Lewicki (Ref. 5) presented several variable and multi-speed rotorcraft drive system 

concepts. Based on relative simplicity and necessary requirements of positive drive (i.e., no traction or fluid-based power 

transmission) for rotorcraft applications, it was concluded that 

one promising configuration would incorporate a planetary 

drive with a discrete two-speed shift capability. Therefore, a 

two-speed dual-clutch planetary gear transmission of the type 

evaluated by Lewicki, et al. (Ref. 6) is considered in the current 

investigation. In Lewicki, et al. (Ref. 6), a model and analysis 

of the dual-clutch shifting dynamics including nonlinear dry 

clutch stick-slip friction and dual-clutch control is developed 

and experimentally validated on a purpose-built test rig at the 

NASA Glenn Research Center. Furthermore, the shift dynamics of similar dual-clutch transmissions for automotive 

applications were explored (Refs. 7 and 8). One advantage 

offered by dual-clutch transmissions (DCT) is their ability to 

provide continuous power transfer when transitioning from one 

gear ratio to another. Depending on the dual-clutch shift control 

strategy, a pure neutral condition can be avoided during the 

shifting process, a seeming obvious benefit for rotorcraft 

applications. Generally speaking, to accommodate a gear ratio 

change, it is kinematically necessary for either the engine speed 

or the load speed to vary. For example, during an upshift from a 

lower gear to a higher gear, either the load speed must increase 

or the engine RPM must decrease (or some combination 

therein). In automotive systems, since the engine rotary inertia 

is typically much lower that the effective vehicle inertia, the 

engine speed primarily changes during clutch engagement. 
NASA/TM—2012-217212 2 

Once clutch lockup in the new gear ratio occurs, the engine 

speed is then adjusted to obtain the desired vehicle speed. 

In the case of rotorcraft, since it is important to maintain 

powered flight throughout the shift process and not to 

over-speed the engine power turbines, Litt, et al. (Ref. 9) 

proposed an engine-at-a-time or sequential shifting control 

(SSC) for multi engine helicopters. This approach takes 

advantage of the existing drive system free-wheeling clutch 

connections to decouple individual engines from the transmission thus allowing sequential gear ratio changes on individual 

transmission paths. In particular, Litt, et al. (Ref. 9) explored 

this concept for a 3000 hp class, two-engine single rotor 

helicopter using an integrated system model consisting of a 

linearized engine, rigid gear transmission, rotor and idealized 

clutches to capture the basic dynamics. The advantages of this 

approach are that: 1) it allows for powered flight to always be 

maintained on the reduced set of engines, and 2) it avoids full 

power transfer across engaging clutches. 

The purpose of the current investigation is to explore the 

variable-speed operation and shift response of a prototypical 

two-speed DCT tiltrotor driveline in forward flight airplane 

mode. Here, a Comprehensive Variable-Speed Rotorcraft 

Propulsion System Modeling (CVSRPM) tool developed under 

a NASA-funded NRA program is utilized to simulate the drive 

system dynamics under a two-speed shift SSC strategy for 

several mission scenarios. 

As discussed in References 1, 3, and 4, one of the potential 

beneficiaries of variable-speed rotor drive-system technologies 

is Heavy-Lift rotorcraft. Therefore, this study considers a 

two-speed version of the 30,000 hp-class NASA Large Civil 

Tiltrotor Iteration 2 (LCTR-2) (Fig. 1, (Ref. 4)) and analyzes 

the transition to and from slowed rotor cruise operation. The 

simulations in this paper include a stick-slip dry clutch friction 

DCT model based on Lewicki, et al. (Ref. 6), a new fully 

nonlinear first principles-based transient gas turbine engine 

model, cross shaft flexibility effects, and an idealized uniform 

inflow forward flight rotor/propeller model. 

Figure 1.—NASA Large Civil Tiltrotor LCTR-2 (Ref. 4). 

Two-Speed Tiltrotor Simulation 

A schematic of the two-speed dual-path tiltrotor driveline 

considered in this investigation is shown in Figure 2. Each path 

consists of an engine nacelle containing a pair of gas turbine 

engines coupled to an initial reduction gearbox, ratio ni, via 

freewheeling clutch units (FWU), see Appendix A, Nomenclature. 

1.87 :1 ,out

,in  

  FWU

DCT ni (1) 

Speed is then further reduced by a two-speed DCT gearbox 

with low and high ratios 

0.383 ,in

,low

,low  

 DCT

DCT nDCT , 0.71 ,in

,high

,high  

 DCT

DCT

nDCT (2) 

The final reduction, ratio nf, before the main rotor is represented 

as a high reduction ratio planetary gearbox 

 25 :1



  DCT

MR nf (3) 

Finally, the left and right side driveline paths are linked via a 

cross-shaft. 

The cross-shaft is located downstream from the DCT speed 

changing units (see Fig. 2) to prevent relative rotor windup 

during shift events thus ensuring that rotor indexing phase is 

maintained. 

Figure 2.—Two-speed DCT dual-path tiltrotor driveline. 

FWU

FWU

FWU

FWU

CrossShaft

Right Nacelle

(Twin Two Spool Gas 

Turbine Engines)

Left Nacelle

(Twin Two Spool Gas 

Turbine Engines) Initial 

Ratio, ni

Dual Clutch 

Transmission

nDCT,low &

nDCT,high

Final 

Ratio, nf

Forward

Speed

Blade Pitch

Control

Clutch 

Engagement

Pressure Inputs

Multi-Engine

Fuel Control

Initial 

Ratio, ni

Final 

Ratio, nf

1:1

1:1

Left Rotor

Right Rotor

Dual Clutch 

Transmission

nDCT,low &

nDCT,high

FWU

FWU

FWU

FWU

CrossShaft

Right Nacelle

(Twin Two Spool Gas 

Turbine Engines)

Left Nacelle

(Twin Two Spool Gas 

Turbine Engines) Initial 

Ratio, ni

Dual Clutch 

Transmission

nDCT,low &

nDCT,high

Final 

Ratio, nf

Forward

Speed

Blade Pitch

Control

Clutch 

Engagement

Pressure Inputs

Multi-Engine

Fuel Control

Initial 

Ratio, ni

Final 

Ratio, nf

1:1

1:1

Left Rotor

Right Rotor

Dual Clutch 

Transmission

nDCT,low &

nDCT,high
NASA/TM—2012-217212 3 

To simulate forward flight operation, the propulsion system 

model is coupled to a simplified flight dynamics block to 

account for vehicle altitude and forward speed regulation under 

variable rotor speed operation. Since all engines were considered to be identical, only one engine-per-side was simulated in 

the model, with appropriate output torque and inertia effects to 

account for both engines. Figure 3 shows the architecture of the 

overall propulsion system coupled with a vehicle flight 

dynamics block and associated closed-loop flight control 

systems and command inputs. 

A description of the main elements and some key assumptions included in the simulation model are as follows: 

 Two-spool gas turbine engine transient dynamics model 

with closed-loop PID-based fuel controller to regulate 

power turbine speed pt and multi-engine load sharing. 

 DCT model with nonlinear stick-slip friction and clutch 

lockup effects and open loop clutch engagement pressure 

command inputs (Ref. 6). 

 Idealized rigid gearing (tooth meshing compliance neglected). 

 Main rotor idealized as rigid rotational inertia with 

aerodynamic torque computed based on momentum 

theory with uniform inflow (Ref. 10). 

 Torsionally flexible cross-shafting. 

 Simplified vehicle flight dynamics model with horizontal 

and vertical degrees of freedom (two DOF) accounting 

for fuselage drag, vehicle weight, rotor thrust and wing 

lift. 

 A PID-based main rotor blade pitch controller to regulate 

rotor thrust and track a desired forward speed set-point 

(controls main rotor pitch p). 

 A PID-based wing elevon pitch controller to maintain 

flight level or track a desire climb rate set-point (controls 

wing flap angle w). 

The simulation model was developed in the Matlab Simulink 

(The MathWorks, Inc.) environment. Appendix B shows the 

system and subsystem block diagrams. Further details of the 

propulsion system model are given in the subsequent subsections. 

Figure 3.—Propulsion system/vehicle dynamics/flight control block diagram assembled in the 

CVSRPM code. 

Two-Spool

Gas Turbine

Engine

PID

Fuel Control

Two Speed 

Dual Clutch/PGT

Cross-Shafting

(torsional

stiffness)

Initial 

Reduction 

Ratio, ni

Clutch Shift 

Command Inputs

(open loop)

Main Rotor

(Rigid Blades)

Forward Speed

Controller 

(Blade Pitch)

Rotor Thrust

mf  pt  p

 w

MR

set pt

Wing Pitch Control

(altitude control)

y, x

Rotorcraft Forward Speed

High 

Speed 

Clutch

Low 

Speed 

Clutch

Climb rate 

setpoint

Forward speed 

setpoint

N Parallel

Driveline Paths

Final

Reduction 

Ratio, nf

Free

Wheel

Clutch

Power Turbine 

Setpoint RPM

Vehicle Flight 

Dynamics Model

(forward & climb DOF)

Two-Spool

Gas Turbine

Engine

PID

Fuel Control

Two Speed 

Dual Clutch/PGT

Cross-Shafting

(torsional

stiffness)

Initial 

Reduction 

Ratio, ni

Clutch Shift 

Command Inputs

(open loop)

Main Rotor

(Rigid Blades)

Forward Speed

Controller 

(Blade Pitch)

Rotor Thrust

mf  pt  p

 w

MR

set pt

Wing Pitch Control

(altitude control)

y, x

Rotorcraft Forward Speed

High 

Speed 

Clutch

Low 

Speed 

Clutch

Climb rate 

setpoint

Forward speed 

setpoint

N Parallel

Driveline Paths

Final

Reduction 

Ratio, nf

Free

Wheel

Clutch

Power Turbine 

Setpoint RPM

Vehicle Flight 

Dynamics Model

(forward & climb DOF)
NASA/TM—2012-217212 4 

Figure 4.—Two-spool gas turbine engine model components and analysis stations. 

Gas Turbine Engine Model 

To model the gas turbine engines and fuel control dynamics 

during transient operation, a generic two-spool gas turbine 

engine model was developed. The components of this model, 

shown in Figure 4, include the axial compressor, combustion 

chamber, gas-generator turbine, power turbine and two mass 

accumulators (plenums). A dynamic model is formulated by 

considering the torque balances between the compressor and 

gas generator turbine as well as between the power turbine and 

load. These torques are computed based on first principles 

mechanical and thermodynamic laws following a similar 

approach as in reference (Ref. 6). The thermodynamic model is 

based on the standard Joule/Brayton cycle (Ref. 11). The main 

assumptions made in this analytical model can be summarized 

as follows: 

Engine model assumptions: 

 The air mass flows through the system are all 

one-dimensional at the mean line. 

 Air with ideal gas law behavior is assumed. 

 All expansion and compression is adiabatic. 

 The inlet stagnation temperature and pressures are constant. 

 All power transfer occurs in the rotor stages, the stators 

do no work. 

 The instantaneous compressor mass flow, pressure ratio 

and efficiency are determined via pre-calculated beta 

maps. 

 Gas generator and power turbines are in a state of choking flow (Ref. 11) 

 No air bleed occurs from the compressor. 

 The combustion is modeled as isobaric heat addition. 

 The specific heat of air was assumed constant throughout 

the engine model. 

The power required to drive the compressor is 

( ) Wc  mcc p T02 T01  (4) 

Where mc  is the compressor air mass flow rate, cp is the 

specific heat of air and T01 and T02 are the stagnation temperatures at the compressor entrance and exit respectively. Furthermore the stagnation temperature ratio is related to the 

compressor pressure ratio by 





















 













  



1 1 1

1

01

02

01

02 



 P

P

T

T

c

 (5) 

Where c is the compressor isentropic efficiency,  is the 

specific heat ratio, and P01 and P02 are the stagnation pressures 

at the compressor entrance and exit respectively. Based on (4) 

and (5), the torque required to drive the compressor is 





















 















   



1

1

01

01 02 



 P

T P TQ J m c c c

c c c c p   (6) 

Where c is the compressor rotation speed and Jc is compressor 

rotational inertia. Also, in the above equations the stagnation 

temperature and pressures at the compressor entrance are 

assumed to be the ambient atmospheric pressure, P01=Pa, and 

temperature T01 = Ta values. 

The basic engine parameters, summarized in Table 1, are 

selected based on the NASA LCTR-2 heavy lift concept engine 

prototype (Refs. 12 and 13). 

Compressor

Shaft

Gas Generator

Turbine

Power

Turbine Load

Combustion

Plenum #1 Chamber Plenum #2

1 23 4 5 6 7

1 , mc Ca 

A1

A2 A4 A5

Shaft

A6

A7

Jggt JL

mf 

T05

P05

T06

P06

T07 > Ta

P07 = Pa

T01 = Ta

P01 = Pa

4 , mggt Ca  6 , mpt Ca 

Vp1 LHV Vp2

ggt

Jpt

pt

Jc

c

T02, P02 T04, P04

Compressor

Shaft

Gas Generator

Turbine

Power

Turbine Load

Combustion

Plenum #1 Chamber Plenum #2

1 23 4 5 6 7

1 , mc Ca 

A1

A2 A4 A5

Shaft

A6

A7

Jggt JL

mf 

T05

P05

T06

P06

T07 > Ta

P07 = Pa

T01 = Ta

P01 = Pa

4 , mggt Ca  6 , mpt Ca 

Vp1 LHV Vp2

ggt

Jpt

pt

Jc

c

T02, P02 T04, P04
NASA/TM—2012-217212 5 

TABLE 1.—ENGINE NOMINAL PARAMETERS 

Parameter Value 

Ambient temperature, Ta ................................................... 288 K 

Ambient pressure, Pa ..................................................... 1.023 bar 

Nominal output (load) power, WL .................................... 7500 hp 

Nominal compressor mass flow, mcn  ............................ 13.5 kg/s 

Nominal compressor pressure ratio, prcn ................................... 28 

Nominal compressor axial velocity, Can .......................... 150 m/s 

Number of compressor stages, Ns ............................................... 8 

Compressor mean radius, Rc ............................................ 0.119 m 

Compressor inlet hub-to-tip ratio, htc ..................................... 0.35 

Nominal compressor speed, cn ............................... 26,500 RPM 

Power turbine mean radius, Rpt ..................................... 0.1811 m 

Power turbine inlet hub-to-tip ratio, htpt ................................. 0.65 

Nominal power turbine speed, ptn .......................... 12,500 RPM 

Compressor rotational inertia, Jc ............................... 0.641 kg m2

Gas generator turbine rotational inertia, Jggt .............. 0.144 kg m2

Power turbine rotational inertia, Jpt ............................. 1.04 kg m2

Plenum chamber volume, Vp .............................................. 0.1 m3

Combustion chamber loss coefficient, cc.............................. 0.85 

Combustion chamber time constant, cc .................... 5.0×10-4 sec 

Fuel lower heating value, LHV ................................... 43.1 MJ/kg 

Nominal fuel rate, mfn  ............................................. 0.225 kg/sec 

Maximum fuel rate, mf max  ...................................... 0.338 kg/sec 

Furthermore, the compressor mass flow rate, mc  , pressure 

ratio, prc, and efficiency characteristics, c, are computed by 

way of so-called beta maps (Ref. 14) with functional dependencies, 

( , , , )

( , , , )

( , , , )

c a a c c

c m a a c c

c pr a a c c

f T P

m f T P

pr f T P

 





  

 

 

 (7-a) 

Where c is the compressor beta parameter which maps the 

mass flow to pressure ratio as 

2 prc cmc    (7-b) 

In this study, the compressor pressure and efficiency maps 

are generated numerically based on the nominal compressor 

design values. Figure 5 shows the pressure-ratio versus mass 

flow characteristic curves and beta lines for the 7500 hp gas 

turbine engine considered in this study and Figure 6 shows the 

compressor adiabatic efficiencies c. 

To drive the compressor and load, separate gas generator and 

power turbines are present in the two-spool gas turbine engine 

(Fig. 4). The power generated from these turbines is 

( ) Wggt  mggtcp T04 T05  and ( ) Wpt  mptcp T06 T07  (8) 

Figure 5.—Compressor pressure ratio versus air mass flow. 

Figure 6.—Compressor adiabatic efficiency vs. mass flow. 

Where mggt  and mpt  are gas generator and power turbine mass 

flows, respectively, and T04, T05, T06 and T07 are turbine 

entrance and exit stagnation temperatures as defined in 

Figure 4. Furthermore, the turbine pressure ratios based on 

isentropic expansion thought the turbines are 

1

05

04

05

04 















  



T

T

P

P and 1

07

06

07

06 















  



T

T

P

P (9) 

and the turbine mass flows are computed based on the assumption that both turbines are operating in the condition of 

choking flow (Ref. 11). Therefore 

04

4 04

T

A P

m ggt   and 

06

6 06

T

A P m pt   (10-a) 

50

4 6 8 10 12 14 16 18

10

20

30

40

Compressor mass flow rate, mc  (kg/s)

c pr

c cn /

1.0 1.05

1.1

0.95 0.9 0.85

 0.12  c

* Surge 

line

 0.14  c

 c  0.16

 0.18  c

 c  0.2  c  0.24  0.22  c

50

4 6 8 10 12 14 16 18

10

20

30

40

Compressor mass flow rate, mc  (kg/s)

c pr

c cn /

1.0 1.05

1.1

0.95 0.9 0.85

 0.12  c

* Surge 

line

 0.14  c

 c  0.16

 0.18  c

 c  0.2  c  0.24  0.22  c

4 6 8 10 12 14 16 18

10

20

30

40

Compressor mass flow rate, mc  (kg/s)

c pr

c cn /

1.0 1.05

1.1

0.95 0.9 0.85

 0.12  c

* Surge 

line

 0.14  c

 c  0.16

 0.18  c

 c  0.2  c  0.24  0.22  c

4 6 8 10 12 14 16 18 0.75

0.8

0.85

0.9

0.95

1

Compressor mass flow rate, mc  (kg/s)

c

c cn /

1.0 1.05

1.1

0.95

0.9

0.85 1.15

0.8

0.75

0.7

4 6 8 10 12 14 16 18 0.75

0.8

0.85

0.9

0.95

1

Compressor mass flow rate, mc  (kg/s)

c

c cn /

1.0 1.05

1.1

0.95

0.9

0.85 1.15

0.8

0.75

0.7
NASA/TM—2012-217212 6 

with 





   







 



    1

1

2

1 1

R (10-b) 

Where R is the ideal gas constant. Next, since the gas generator 

turbine and compressor are on a common spool, we have 

c = ggt, Also, the torque difference between the compressor 

and gas generator turbine is 

( ) 1 ( ) ggt c

c

Jc J ggt c W W



    (11) 

and the torque difference between power turbine and load is 

 ( ) 1 ( ) pt L

pt

J pt J L pt W W



    (12) 

Where WL is the power consumed by the load and Jc , Jggt, Jpt

and JL are rotational inertias of the compressor, gas generator 

turbine, power turbine, and the load, respectively. 

During steady-state operation the air mass flow rates through 

the compressor and turbines are equal, mc mggt mpt      . 

However, during transient operation, differential mass flows 

can develop between the components due to volume packing 

(Refs. 14 and 15). To account for such transient effects, two 

plenum chambers (Plenum no. 1, situated between compressor 

and combustion chamber, and Plenum no. 2 between the two 

turbine units) are included in the model as depicted in Figure 4. 

The plenum chambers are modeled as ideal mass accumulators with no losses or heat transfer. Figure 7 shows a 

schematic of the plenum chamber element with input and 

output mass flow rates min  and mout  generated by the upstream and downstream elements. Due to the differential mass 

flows, the rate of change of density inside a plenum,  p  , is 

given by 

p

p V

min mout       (13) 

Where Vp is plenum chamber volume. Furthermore, through 

differentiation of the ideal gas law and assuming adiabatic 

compression, the following differential equation for plenum 

chamber pressure Pp is obtained 

( ) ( )

( ) ( ) in out P t

m t

m m P t p

pa

p

      (14-a) 

with 

in out 0 ( ) ( ) pa m m dt mpa m t       (14-b) 

Figure 7.—Plenum chamber element. 

Figure 8.—Combustion chamber element. 

Here mpa is the mass of air accumulating inside the plenum and 

mpa0 is the initial air mass. 

It is assumed that plenum pressure is uniform throughout its 

volume and that velocity effects and energy accumulation 

inside the plenum are negligible resulting in 

P0in  Pp and P0out  Pp (15) 

and 

T0in  T0out (16) 

Where P0in, and P0out are plenum entrance and exit stagnation 

pressures and T0in, and T0out are the plenum entrance and exit 

stagnation temperatures. 

Finally, the combustion chamber (entrance station 3, exit 

station 4 in Figs. 4 and 8) is modeled as a pure isobaric heat 

addition process. With heat addition, Qcc, based on the idealized 

combustion model 

Qcc  m f LHV (17) 

Where mf  is fuel mass flow rate and LHV is the fuel lower 

heating value. 

As a result of an energy balance, the combustion chamber 

exit stagnation temperature T04 is determined as a solution to 

m c T T m LHV

dt

dT cc cc p f   (  )   03 04

04  (18) 

P0in

min  Vp

 0 Qp Wp  0

Pp (t),  p (t)

Ain Aout

in T0

out P0

mout 

out T0

P0in

min  Vp

 0 Qp Wp  0

Pp (t),  p (t)

Ain Aout

in T0

out P0

mout 

out T0

P03

m cc

Wcc  0

A3 A4

T03

Qcc  m f LHV

cc ηcc  ,

P04

m cc

T04

P03

m cc

cc  0

A3 A4

T03

Qcc  m f LHV

cc ηcc  ,

P04

m cc

T04
NASA/TM—2012-217212 7 

Where cc is the combustion chamber time-constant and mcc  is 

the air mass flow through the combustion chamber which 

equals the mass flow drawn by the gas generator turbine 

mcc  = mggt  . Furthermore, the combustion chamber exit 

stagnation pressure, P04, is computed from the input stagnation 

pressure P03 using, 

P04 ccP03 (19) 

Where cc < 1 is the combustion chamber pressure drop ratio. 

Also, at steady-state the combustion temperature rise 

T 0cc T04 T03 is 

p

cc c

LHV T  f 0 with cc

f

m

m

f 

  (20) 

Where f is the fuel-air ratio. Based on the equations presented in 

this subsection, the transient response of the two-spool gas 

turbine engine can be simulated. Please refer to the Appendix B 

for the full engine model block diagrams. 

Dual-Clutch Gearbox Model 

As mentioned in the previous section, a two-speed DCT 

gearbox is utilized as the ratio changing element in each 

transmission path of the tiltrotor drive system (Fig. 2). A 

schematic of the DCT considered in this study is shown in 

Figure 9 along with the corresponding kinematic diagram in 

Figure 10. Here the carrier is the output. In the DCT, high ratio, 

nDCT,high is achieved by engaging the multiple clutch disks that 

couple the planet shafts (clutch 1). 

2( )( ) 1 3

2

,high

cg cg s p s

i cg r

DCT Z Z Z Z Z

Z Z Z

n

   (21-a) 

Furthermore, low ratio, nDCT,low, is achieved by fully engaging, 

a cylindrical clutch (clutch 2) which clamps the ring gear. 

2( ) ,low

p s

s DCT Z Z

Z

n

  (21-b) 

Clutch engagement is controlled by applied pressures, pcl1 and 

pcl2, normal to the clutch fiction surfaces. Due to the DCT 

kinematics, the only condition under which both sets of 

clutches can be engaged and locked simultaneously corresponds to the zero speed or fully stalled condition. When both 

clutches are applied simultaneously there is a negative 

recirculating power ratio and clutches act as a braking system. 

Thus, during steady-state constant speed operation, only one 

clutch is fully engaged and locked (pcli = pclmax) while the other 

clutch remains open (pclj = 0). 

Figure 9.—Dual-clutch transmission. 

 

Figure 10.—DCT kinematic diagram. 

To account for the DCTs in the driveline simulation, a rigid 

body dynamics model is developed which accounts for the 

DCT kinematics, the gear rotational inertias, and the nonlinear 

clutch friction torques. 

( , , , ) Mq F q q pcl1 pcl2     (22-a) 

with corresponding DOF vector 

T i r q(t)  [ ,  ] (22-b) 

Where i and r are the DCT input and ring gear rotations. Also, 

the system inertia matrix is given by 

clutch1 

(high)

clutch2 

(low)

in

out

pcl2

pcl1

pcl1

pcl2

clutch1 

(high)

clutch2 

(low)

in

out

pcl2

pcl1

pcl1

pcl2

cl1

s

cg1

cg2

cg3 cr i

in out

p

r

cl1 cl2

s

cg1

cg2

cg3 cr i

in out

p

r

cl2

clutch 1 & 2

gear mesh

cl1 & cl2

cr carrier

s sun

p planet

r ring

cg1, cg2 & cg3 control gears

i input gear

Symbol Component

clutch 1 & 2

gear mesh

cl1 & cl2

cr carrier

s sun

p planet

r ring

cg1, cg2 & cg3 control gears

i input gear

Symbol Component
NASA/TM—2012-217212 8 

 (22-c) 

and the system generalized force vector is 

















































 











2

1

out

in

2

3

1

1

2( )

0

0

2( )

1

cl

cl

cg

cg cg

p s

r

cg

cg i

p s

s

TQ

TQ

TQ

TQ

Z

N Z

Z Z

Z

Z

N Z

Z Z

Z

F (22-d) 

Where TQin and TQout are the torques on the DCT input and 

output shafts and where TQcl1 and TQcl2 are the transmitted 

clutch torques. To account for the possibility of clutch sticking 

(lock-up) or slipping, the clutch torques are computed based on 

a so-called combined Coulomb / viscous model (Refs. 6, 16, 

and 17) 











      

      

   

 

2 1 2 1

3

1 1

2 1 2 1

1

sgn( ) 3

4

( ) 

cg cg cg cg cl cl cl

cl cg cg cg cg

cl R

p

TQ (23) 









  

      

 

cl cl cl cl r r

cl r r

cl

p R L TQ 2 sgn( ) 

 

2

2

2 2

2 (24) 

Here, cg1 and cg2 are the upstream and downstream clutch 

disk rotation speeds of clutch no. 1 and r is the ring gear 

rotation speed at clutch no. 2. Here cl and cl are clutch friction 

and viscous torque coefficients (Ref. 16). Also, in clutch no. 1 

(disk-type clutch) the torque is a function of the disk radius 

Rc11, while for clutch no. 2 (cylindrical clutch) the torque is 

depends on clutch radius, Rc12, and axial length Lc12. 

In each case, the clutch torque transitions from a viscous 

mode to a Coulomb friction mode based on a clutch slip 

velocity threshold parameter . Here, when clutch slip speed 

exceeds , the torque transitions from a viscous mode into the 

slipping mode (Coulomb friction). By proper selection of  and 

cl, the viscous mode numerically approximates clutch lockup 

without no need for explicit model DOF reduction (Ref. 6). The 

DCT system parameters used in this investigation are summarized in Table 2. 

TABLE 2.—DCT PARAMETERS. 

Parameter Value 

Number of gear teeth Zi=40, Zcg1=42, Zcg2=29, Zcg3=52, Zr=74, 

Zp=14, Zs=46 

Gear radii, m Ri=0.1, Rcg1=0.106, Rcg2=0.074, Rcg3=0.132, 

Rr=0.19, Rp=0.036, Rs=0.12, Rcr=0.152 

Rotational inertias, kg m2 Ji=0.064, Jcg1=0.078, Jcg2=0.039, Jcg3=0.402, 

Jr=1.766, Jp=0.002, Js=0.264, Jcr=0.848 

Number of planets Np = 8 

Planet mass, kg mp = 3.576 

Planet clutch radius (clutch 1), 

m Rc11 = 0.238 

Number of planet clutches 

(clutch 1) Ncg = 2

Ring clutch dimensions 

(clutch 2), m Rc12 = 0.197, Lc12 = 0.1 

Clutch friction coefficient cl = 0.45 

Forward Flight Rotor (Propeller) Model 

To include the rotor torque and resulting thrust for the 

tiltrotor in forward flight airplane mode with vehicle speed vc, a 

rigid propeller model based on a standard blade element theory 

approach with quasi-steady aerodynamics (Ref. 10) is utilized 

(see Fig. 11). Here the differential rotor thrust dFTMR and 

torque dTQMR at blade radial location r are expressed as 

( sin cos )

( cos sin )

 

 

dTQ N r dL dD

dFT N dL dD

MR b

MR b

 

  (25) 

Where dL and dD are differential blade lift and drag force 

increments along with the number of blades Nb. 

Based on tangential blade speed, uT = MRr, together with vc

and the induced inflow velocity vind, the total inflow angle  is 

  c  ind (26-a) 

With 





 





  

r

v

MR

c c 1  tan and 2 2 2 v r

v

c MR

ind

ind

 

  (26-b) 













































































 

  















  

 















  

 





















 

   



2 2

2 2

2

2

2

2 3

3

2 2

2

2 2

2

2 2

2 2

2

1

2

1

4 ( )

4 ( )

4 ( )

4 ( )

p

p p

p s

cr out p p cr r

cg

cg cg cg

r cg

p

p p

p s

cr out p p cr r s

p

p p

p s

cr out p p cr r s

p

p p

p s

cr out p p cr s

cg

cg cg i

in i s

Z

J N

Z Z

Z J J N m R

Z

J N Z

J J

Z

J N

Z Z

Z Z J J N m R

Z

J N

Z Z

Z Z J J N m R

Z

J N

Z Z

Z J J N m R

Z

J N Z

J J J





M
NASA/TM—2012-217212 9 

Figure 11.—Propeller in forward flight. 

Where the induced inflow angle, ind, is assumed small. Now, 

assuming uniform induced inflow velocity vind together with 

uniform blade pitch, p, the following expression for total rotor 

thrust is obtained as 

v r rdr

a cN FT

c ind MR

R

c c ind p air cL b MR MR

b

( )

( ) 2 0

 

     



   

 (27) 

Similarly the corresponding rotor torque is 

 

 

  v r r dr

a cN

r v r dr c cN TQ

c MR ind

R

c c ind p air cL b MR

R

c MR c ind air D b MR MR

b

b

2

0

2

0

2

 2



   

 

 

    

    



 

 (28) 

Both with 

2 2

2

1

MR

c c

r

v



   (29) 

With air density air, rotor radius, Rb and blade chord c. Also cD

is the blade section drag coefficient and acL is blade section lift 

coefficient slope. 

Next, by equating the thrust expression in (27) with an 

expression based on momentum theory (Ref. 10), we obtain the 

following equation for the induced inflow ratio ind

0 2aind  bind  c  (30-a) 

with coefficients 

24 1 48ln 6 ln 2  



 



 

 



 



  c

c

c c cL

c

c a a 



  





 (30-b) 

       





  

 

c cL c cL p c cL

c

c

c c cL

c

a a a

b a

 24 6 tan 6 3

6 ln 1 24sin

1

3 2 1

   

 



 





 



 



 



  (30-c) 

cL c c cL c   c p c

cL c c c

c

c

c cL p

a a

a

c a

      

   







   

1 2

2 1

3

2 tan

 2 tan [ ]

1 2 ln





  























 



    



 (30-d) 

With induced and forward speed inflow ratios 

b MR

ind ind R

v



  , b MR

c c R

v



  and 2  c  1 c (31) 

and rotor solidity ratio 

b

b

R

cN



  (32) 

Based on (30)-(32), the induced velocity, vind, is determined 

from the roots of (30-a). Using this result in (27)-(28), the rotor 

thrust and torque values can be computed for a given rotor 

speed, forward speed and blade pitch angle as shown in the 

rotor thrust and torque simulation block in Figure 12. 

Using (1)-(3) and (21), the main rotor speed in high and low 

ratio are 

MR f DCT i pt

MR f DCT i pt

n n n

n n n

  

  

,low ,low

,high ,high (33) 

The rotor parameters used in this investigation are summarized 

in Table 3. 

TABLE 3.—ROTOR PARAMETERS 

Parameter Value

Rotor speed (high), MR,high ........................................... 190 RPM

Rotor speed (low), MR,low ......................................... 102.5 RPM

Number of blades, Nb ................................................................. 4

Rotor radius, Rb ................................................................. 9.91 m

Rotor solidity,  .................................................................... 0.13

Blade mass, mb ................................................................... 372 kg

Rotor rotational inertia, JMR ............................... 4.874×104

 kg m2

Section lift coefficient, cD ....................................................0.012

Section drag coefficient, aCl .................................................5.056

Figure 12.—Rotor thrust and torque 

simulation block. 

 p

MR

c v

u r T  MR

c

ind



 ind v

c v 



dL

dD

dFx

dFy

 p

MR

c v

u r T  MR

c

ind



 ind v

c v 



dL

dD

dFx

dFy

ind v

FTMR

TQMR

p 

vc

MR

Propeller

Model

ind v

FTMR

TQMR

p 

vc

MR

Propeller

Model
NASA/TM—2012-217212 10 

Figure 13.—Mission scenario; downshift and upshift in steady forward 

cruise using SSC shifting. 

Two-Speed Shift Simulation Results 

Utilizing the model summarized in the previous section, this 

portion of the paper presents a two-speed shift simulation case 

study for the tiltrotor DCT drive-system (Fig. 2) operating in 

airplane mode at an intermediate steady forward cruising speed. 

The operating conditions and parameters used in the simulation 

(see Tables 1 to 3) are based on the LCTR-2 (Refs. 12 and 18). 

Specifically, the simulated mission scenario, depicted in 

Figure 13, is a rotor speed downshift followed later by a rotor 

speed upshift both while operating at the constant forward 

speed setpoint, vc,set = 228 knots. 

To avoid shifting under full power, the SSC strategy, presented in (Ref. 9), is utilized in both downshift and upshift 

phases. Fundamentally, the SSC method utilizes the freewheel 

clutches at the power turbine output to disengage each engine 

sequentially wherein shifting along the particular driveline path 

can be performed in an unloaded condition. Since the FWU 

prevents the power turbine from being driven by the load, it will 

disengage when the load speed becomes larger that of the 

power turbine. In a multi-engine driveline equipped with a 

variable ratio gearbox there are essentially two situations which 

can cause a specific FWU to disengage; 1) by lowering the 

power turbine setpoint speed of that engine or, 2) by downshifting the gearbox in that engine’s driveline path. In the first 

situation, the remaining engines maintain the system speeds as 

the particular engine speed is lowered. In second the situation, 

the commanded downshift acts as an upshift on the gearbox 

input side thus increasing the load speed at the FWU. Since the 

engine setpoint speed remains fixed, the FWU in this driveline 

path will disengage. Both of these situations arise in the SSC 

shifting method. The downshift and upshift portions of this 

simulation are presented separately in the following two 

subsections. Here the effects of shift rate on the engine response 

will be explored in detail. 

Forward Cruise SSC Downshift 

The timeline of the SSC downshift procedure along with a 

brief description of each event is given in Table 4. 

TABLE 4.—SSC DOWNSHIFT TIMELINE* 

Time Commands Comments 

t=12 sec 

1) Disable engine torque sharing 

control loop 

2) Start left side DCT downshift 

(disengage clutch no. 1 and engage 

clutch no. 2). 4 psi/sec

30 psi/sec

2

1

 

 

cl

cl

p

p





Start of SSC downshift 

procedure 

Increases left DCT input 

shaft speed which 

disengages left engine FWU

t=32 sec 

3) Start right engine power turbine 

setpoint ramp-down (-600 

RPM/sec) 

Main rotor speed reduces 

from 190 to 102.5 RPM 

Left engine re-engages 

FWU at end of ramp-down

t=52 sec 

4) Start right side DCT downshift 

(disengage clutch no. 1 and engage 

clutch no. 2). 4 psi/sec

30 psi/sec

2

1

 

 

cl

cl

p

p





Increases right DCT input 

shaft speed which 

disengages right engine 

FWU 

t=72 sec 5) Start right engine power turbine 

setpoint ramp-up (+600 RPM/sec) 

Right engine re-engages 

FWU at end of ramp-up 

t=82 sec 6) Re-enable engine torque sharing 

control loop 

End of SSC downshift 

procedure 

*Under constant forward speed setpoint vc,set = 228 knots. 

Figures 14 to 19 give the forward cruise SSC downshift 

response of the two-speed DCT tiltrotor system. By examining 

Figures 14(a) and (b), it is apparent that the rotor speed is 

successfully lowered with very little deviation in forward 

velocity. Also note in Figure 14(b) that the left and right rotor 

speeds are nearly identical throughout the shift. Figures 14(c) 

and (d) show the open-loop DCT clutch pressure command 

inputs used to sequentially downshift the left and right side 

DCTs (step nos. 2 and 4 in Table 4). Furthermore, the engine 

power turbine speeds are shown in Figure 14(e) and (f). In 

Figure 14(e) it is observed that, even though the left side engine 

setpoint speed remains constant, the power turbine transiently 

increases due to the left side DCT downshift. This is a result of 

the rapid unloading due to FWU disengagement (see step no. 2 

in Table 4). According to step nos. 3 and 5 in Table 4, Figure 14(f) shows the ramp-down and ramp-up of the right engine 

power turbine. Similarly, when the right side DCT is downshifted, the right engine power turbine experiences a transient 

speed increase above its setpoint as its FWU disengages (see 

step no. 4 in Table 4). 

wing

pitch

PID

 PID

pitch

wing

pitch

PID

 PID

pitch

wing

pitch

PID

 PID

pitch

MR

time

Rotor 

Speed

190 RPM

102 RPM

vc

Forward 

Speed

228 knots (a)

(b)

wing

pitch

PID

 PID

pitch

wing

pitch

PID

 PID

pitch

wing

pitch

PID

 PID

pitch

MR

time

Rotor 

Speed

190 RPM

102 RPM

vc

Forward 

Speed

228 knots (a)

(b)
NASA/TM—2012-217212 11 

Figure 14.—Forward cruise SSC downshift simulation; (a) 

forward speed, (b) rotor speed, (c) left DCT shift commands, 

(d) right DCT shift commands, (e) left power turbine speed, 

(f) right power turbine speed (g) fuel rates. 

Figure 15.—Forward cruise SSC downshift—flight dynamics; 

(a) vehicle acceleration, (b) collective pitch, (c) induced 

velocity, (d) wing flap angle. 

Figure 16.—Forward cruise SSC downshift—torque response; 

(a) engine torques, (b) cross-shaft torque, (c) rotor torque. 

Figure 14(g) shows the corresponding fuel rates commanded 

by the closed-loop engine fuel control for both the left and right 

side engines. Note, the left and right side engine fuel rates are 

equivalent prior to disabling (t=12 sec) and after re-enabling 

(t=82 sec) the multi-engine torque sharing control loop (see 

step nos. 1 and 6 in Table 4). 

The flight dynamics are summarized in Figure 15. Figure 15(a) shows that both the horizontal and vertical vehicle 

accelerations are minimal. Thus, no forward speed or altitude is 

lost during the downshift. In order to maintain the constant 

forward speed setpoint under the decreasing rotor speed, the 

closed-loop blade pitch controller increases the blade collective 

pitch to keep constant thrust as shown Figure 15(b). This results 

in a corresponding increase in rotor induced velocity is seen in 

Figure 15(c). Finally, as seen in Figure 15(d), under this 

constant thrust condition, there is no appreciable change in 

wing flap angle. 

The driveline torques are summarized in Figure 16. As 

observed in the engine fuel rate response in Figure 14 (g), the 

left and right side power turbine torques are identical prior to 

t=12 sec and after t=82 sec. During the downshift, the engine 

torques become dissimilar and are due to both rotor aerodynamic torque as well as acceleration torque demands. 

As a result of the dissimilar left and right engine torques, the 

cross-shaft also becomes loaded as shown in Figure 16(b). 

Finally, Figure 16(c) shows the increasing rotor torque which is 

a result from the increasing collective pitch and constant thrust 

requirement (see Fig. 15(b)). Note that the left and right rotor 

torques are identical throughout the shift (Fig. 16(c)). 

Other important aspects of the two-spool gas turbine engine 

response are shown in Figures 17 and 18. In particular, by 

plotting compressor pressure ratio versus air mass flow, the left 

and right engine transient operating lines during the SSC 

downshift are shown in Figure 18. 

fuel rate

(kg/sec)

right side 

power 

turbine 

(kRPM)

left side 

power 

turbine 

(kRPM)

rotor

speed 

(RPM)

left side

DCT clutch 

pressure 

(psi)

right side

DCT clutch 

pressure 

(psi)

forward 

speed 

(knots)

0 10 20 30 40 50 60 70 80 90 100

228

230

0 10 20 30 40 50 60 70 80 90 100

100

200

0 10 20 30 40 50 60 70 80 90 100 0

50

0 10 20 30 40 50 60 70 80 90 100 0

50

0 10 20 30 40 50 60 70 80 90 100 5

10

15

20

0 10 20 30 40 50 60 70 80 90 100 5

10

15

20

0 10 20 30 40 50 60 70 80 90 100 0

0.2

0.4

t, sec

Actual

Set-point

Actual

Set-point

Clutch 1 (high)

Clutch 2 (low)

Clutch 1 (high)

Clutch 2 (low)

Left

Right

Left rotor

Right rotor

(a)

(b)

(c)

(d)

(e)

(f)

(g)

229

Actual

Set-point

fuel rate

(kg/sec)

right side 

power 

turbine 

(kRPM)

left side 

power 

turbine 

(kRPM)

rotor

speed 

(RPM)

left side

DCT clutch 

pressure 

(psi)

right side

DCT clutch 

pressure 

(psi)

forward 

speed 

(knots)

0 10 20 30 40 50 60 70 80 90 100

228

230

0 10 20 30 40 50 60 70 80 90 100

100

200

0 10 20 30 40 50 60 70 80 90 100 0

50

0 10 20 30 40 50 60 70 80 90 100 0

50

0 10 20 30 40 50 60 70 80 90 100 5

10

15

20

0 10 20 30 40 50 60 70 80 90 100 5

10

15

20

0 10 20 30 40 50 60 70 80 90 100 0

0.2

0.4

t, sec

Actual

Set-point

Actual

Set-point

Clutch 1 (high)

Clutch 2 (low)

Clutch 1 (high)

Clutch 2 (low)

Left

Right

Left rotor

Right rotor

(a)

(b)

(c)

(d)

(e)

(f)

(g)

229

Actual

Set-point

0 10 20 30 40 50 60 70 80 90 100 -0.1

0

0.1

0 10 20 30 40 50 60 70 80 90 100

40

60

80

0 10 20 30 40 50 60 70 80 90 100 0.5

1

1.5

0 10 20 30 40 50 60 70 80 90 100 0

5

Horizontal

vehicle Vertical

accelerations

(g)

rotor 

collective 

pitch

(degrees) 

rotor 

induced 

velocity 

(m/s)

t, sec

wing 

flap 

pitch

(degrees) 

(a)

(b)

(c)

(d)

0 10 20 30 40 50 60 70 80 90 100 -0.1

0

0.1

0 10 20 30 40 50 60 70 80 90 100

40

60

80

0 10 20 30 40 50 60 70 80 90 100 0.5

1

1.5

0 10 20 30 40 50 60 70 80 90 100 0

5

Horizontal

vehicle Vertical

accelerations

(g)

rotor 

collective 

pitch

(degrees) 

rotor 

induced 

velocity 

(m/s)

t, sec

wing 

flap 

pitch

(degrees) 

(a)

(b)

(c)

(d)

0 20 40 60 80 100

0

5

10

0 20 40 60 80 100

-20

0

20

40

0 20 40 60 80 100

300

400

500

600

Left side

engine Right side

torque 

(kNm)

cross-shaft 

torque 

(kNm)

rotor 

torque 

(kNm)

Left side

Right side

t, sec

(a)

(b)

(c)

0 20 40 60 80 100

0

5

10

0 20 40 60 80 100

-20

0

20

40

0 20 40 60 80 100

300

400

500

600

Left side

engine Right side

torque 

(kNm)

cross-shaft 

torque 

(kNm)

rotor 

torque 

(kNm)

Left side

Right side

t, sec

(a)

(b)

(c)
NASA/TM—2012-217212 12 

Figure 18 shows that both engines experience a wide range of 

operation during the SSC process and that both engines 

transiently pass over the surge line. Since Figure 18 does not 

show temporal information, the engine surge margin (SM) 

versus time is examined. Here, the working definition of SM 

used is 

100% ,surge

,surge    cc c

pr

pr pr SM (34) 

Therefore SM > 0 indicates no compressor surging. In this 

analysis, three different DCT shift rate cases (slow, medium 

and fast) as given in Table 5 are considered. Note the medium 

case corresponds to the values used thus far in Figures 14 to 18. 

TABLE 5.—DCT DOWNSHIFT RATE CASES 

Case DCT clutch pressure engagement and disengagement rates 

(psi/sec) 

Slow Clutch no. 1 disengagement rate, 

pcl1  = –15 psi/sec 

Clutch no. 2 engagement rate, 

pcl 2  = 4 psi/sec 

Medium Clutch no. 1 disengagement rate, 

pcl1  = –30 psi/sec 

Clutch no. 2 engagement rate, 

pcl 2  = 4 psi/sec 

Fast Clutch no. 1 disengagement rate, 

pcl1  = –300 psi/sec 

Clutch no. 2 engagement rate, 

pcl 2  = 4 psi/sec 

Figure 17.—Forward cruise SSC downshift—engine response; 

(a) compressor speed, (b) mass flow, (c) power turbine inlet 

temperature. 

Figure 18.—Forward cruise SSC downshift—two-spool gas 

turbine engine transient running lines. 

Figure 19.—Forward cruise SSC downshift—engine compressor surge margins for three DCT shift rates. 

The resulting SM values for the left and right engines are 

shown in Figures 19(a) and (b). In each case, the surge line is 

crossed very briefly. Here, Figure 19(a) shows that the left 

engine crosses the SM immediately after t= 52 sec as the right 

DCT begins to downshift (step no. 4, Table 4). Thus, the left 

engine surge is a result of the transient shock due to the 

unloading the right engine FWU. 

Furthermore, Figure 19(b) shows that the right engine experiences a brief surge near the start and end of the downshift 

process for similar reasons as the left engine. Specifically, the 

first surge of the right engine is due to the initial disengagement 

of the left engine FWU (step no. 1, Table 4) and the second 

surge occurs when the left engine re-engages the FWU at the 

end of its ramp-up (step no. 5, Table 4). Figure 19 shows that 

slower DCT shifting rates are more beneficial for the engine 

surge margins in the SSC downshift procedure. 

0 10 20 30 40 50 60 70 80 90 100

10

15

20

25

30

35

0 10 20 30 40 50 60 70 80 90 100

0

5

10

15

20

0 10 20 30 40 50 60 70 80 90 100

0

500

1000

1500

2000

Left side

Right side

compressor

speed 

(kRPM)

compressor 

mass flow 

(kg/sec)

power 

turbine inlet 

temperature 

(K)

t, sec

(a)

(b)

(c)

0 10 20 30 40 50 60 70 80 90 100

10

15

20

25

30

35

0 10 20 30 40 50 60 70 80 90 100

0

5

10

15

20

0 10 20 30 40 50 60 70 80 90 100

0

500

1000

1500

2000

Left side

Right side

compressor

speed 

(kRPM)

compressor 

mass flow 

(kg/sec)

power 

turbine inlet 

temperature 

(K)

t, sec

0 10 20 30 40 50 60 70 80 90 100

10

15

20

25

30

35

0 10 20 30 40 50 60 70 80 90 100

0

5

10

15

20

0 10 20 30 40 50 60 70 80 90 100

0

500

1000

1500

2000

Left side

Right side

compressor

speed 

(kRPM)

compressor 

mass flow 

(kg/sec)

power 

turbine inlet 

temperature 

(K)

t, sec

(a)

(b)

(c)

mass flow, kg/sec

pressure 

ratio

2 4 6 8 10 12 14 16 18 0

5

10

15

20

25

30

35

40

Left side engine

Right side engine

surge line

mass flow, kg/sec

pressure 

ratio

2 4 6 8 10 12 14 16 18 0

5

10

15

20

25

30

35

40

Left side engine

Right side engine

surge line

left engine 

compressor 

SM (%)

right engine 

compressor 

SM (%)

t, sec

0 10 20 30 40 50 60 70 80 90 100

-20

0

20

40

60

0 10 20 30 40 50 60 70 80 90 100

-20

0

20

40

60

no surge 

surge

no surge 

surge

(a)

(b)

Slow

Medium

Fast

Shift Rate Case

left engine 

compressor 

SM (%)

right engine 

compressor 

SM (%)

t, sec

0 10 20 30 40 50 60 70 80 90 100

-20

0

20

40

60

0 10 20 30 40 50 60 70 80 90 100

-20

0

20

40

60

no surge 

surge

no surge 

surge

(a)

(b)

Slow

Medium

Fast

Shift Rate Case
NASA/TM—2012-217212 13 

Forward Cruise SSC Upshift 

Next, the simulation results of the SSC upshift procedure, 

which is essentially the process reverse of the SSC downshift, 

are described. The SSC upshift timeline is given in Table 6. 

TABLE 6.—SSC UPSHIFT TIMELINE* 

Time Commands Comments 

t=7 sec 

1) Disable engine torque sharing 

control loop 

2) Start right engine power 

turbine setpoint ramp-down 

(–600 RPM/sec) 

Start of SSC upshift procedure 

Disengages right engine FWU 

t=27 

sec 

3) Start right side DCT upshift 

(disengage clutch no. 2 and 

engage clutch no. 1). 

30 psi/sec

4 psi/sec

2

1

 

 

cl

cl

p

p





Decreases right DCT input shaft 

speed. 

Right engine FWU becomes 

re-engaged

t=47 

sec 

4) Start right engine power 

turbine setpoint ramp-up 

(+230 RPM/sec) 

Main rotor speed increases from 

102.5 to 190 RPM 

Speed increase disengages left 

engine FWU 

t=82 

sec 

5) Start left side DCT upshift 

(disengage clutch no. 2 and 

engage clutch no. 1). 

30 psi/sec

4 psi/sec

2

1

 

 

cl

cl

p

p



 

Decreases left DCT input shaft 

speed which re-engages left 

engine FWU 

t=92 

sec 

6) Re-enable engine torque 

sharing control loop 

End of SSC upshift procedure 

*Under constant forward speed setpoint vc,set = 228 knots. 

Figures 20 to 25 show the forward cruise SSC upshift response of the two-speed DCT tiltrotor. By examining the 

forward speed and the rotor speed in Figures 20(a) and(b), it is 

apparent that rotor speed is successfully increased with little 

deviation in forward velocity. Figures 20(c) and (d) show the 

open-loop DCT clutch pressure command inputs used to 

sequentially upshift the right and left side DCTs (step nos. 3 

and 5 in Table 6). Furthermore, the engine power turbine 

speeds are shown in Figure 20(e) and (f). In Figure 20(e) it is 

observed that the left power turbine speed transiently increases 

due to the right engine ramp-up. This is due to the unloading of 

the left engine FWU. (See step no. 4 in Table 6). According to 

step nos. 2 and 4 in Table 6, Figure 20(f) shows the ramp-down 

and ramp-up of the right engine power turbine. Figure 20(g) 

shows the corresponding fuel rates commanded by the 

closed-loop engine fuel control for both the left and right 

engines. Note, the left and right side engine fuel rates are 

equivalent prior to disabling (t=7 sec) and after re-enabling 

(t=92 sec) the multi-engine torque sharing control loop (see 

step nos. 1 and 6 in Table 6). 

Also note that the fuel rate is essentially flat during the rotor 

speed acceleration phase (step no. 4, Table 6). This is due to the 

fact that as rotor speed increases, the collective pitch is 

automatically decreased by the forward speed pitch controller 

to maintain constant thrust as further shown in Figure 21. 

Figure 20.—Forward cruise SSC upshift simulation; (a) forward 

speed, (b) rotor speed, (c) left DCT shift commands, (d) right 

DCT shift commands, (e) left power turbine speed, (f) right 

power turbine speed (g) fuel rates. 

Figure 21.—Forward cruise SSC upshift—light dynamics; 

(a) vehicle acceleration, (b) collective pitch, (c) induced 

velocity, (d) wing flap angle. 

t, sec

0 10 20 30 40 50 60 70 80 90 100 228

229

230

0 10 20 30 40 50 60 70 80 90 100

100

200

0 10 20 30 40 50 60 70 80 90 100 0

50

0 10 20 30 40 50 60 70 80 90 100 0

50

0 10 20 30 40 50 60 70 80 90 100 5

10

15

20

0 10 20 30 40 50 60 70 80 90 100 5

10

15

20

0 10 20 30 40 50 60 70 80 90 100 0

0.2

0.4

fuel rate

(kg/sec)

right side 

power 

turbine 

(kRPM)

left side 

power 

turbine 

(kRPM)

rotor

speed 

(RPM)

left side

DCT clutch 

pressure 

(psi)

right side

DCT clutch 

pressure 

(psi)

forward 

speed 

(knots)

Actual

Set-point

Actual

Set-point

Clutch 1 (high)

Clutch 2 (low)

Clutch 1 (high)

Clutch 2 (low)

Left

Right

Left rotor

Right rotor

Actual

Set-point (a)

(b)

(c)

(d)

(e)

(f)

(g)

t, sec

0 10 20 30 40 50 60 70 80 90 100 228

229

230

0 10 20 30 40 50 60 70 80 90 100

100

200

0 10 20 30 40 50 60 70 80 90 100 0

50

0 10 20 30 40 50 60 70 80 90 100 0

50

0 10 20 30 40 50 60 70 80 90 100 5

10

15

20

0 10 20 30 40 50 60 70 80 90 100 5

10

15

20

0 10 20 30 40 50 60 70 80 90 100 0

0.2

0.4

fuel rate

(kg/sec)

right side 

power 

turbine 

(kRPM)

left side 

power 

turbine 

(kRPM)

rotor

speed 

(RPM)

left side

DCT clutch 

pressure 

(psi)

right side

DCT clutch 

pressure 

(psi)

forward 

speed 

(knots)

Actual

Set-point

Actual

Set-point

Clutch 1 (high)

Clutch 2 (low)

Clutch 1 (high)

Clutch 2 (low)

Left

Right

Left rotor

Right rotor

Actual

Set-point

t, sec

0 10 20 30 40 50 60 70 80 90 100 228

229

230

0 10 20 30 40 50 60 70 80 90 100

100

200

0 10 20 30 40 50 60 70 80 90 100 0

50

0 10 20 30 40 50 60 70 80 90 100 0

50

0 10 20 30 40 50 60 70 80 90 100 5

10

15

20

0 10 20 30 40 50 60 70 80 90 100 5

10

15

20

0 10 20 30 40 50 60 70 80 90 100 0

0.2

0.4

fuel rate

(kg/sec)

right side 

power 

turbine 

(kRPM)

left side 

power 

turbine 

(kRPM)

rotor

speed 

(RPM)

left side

DCT clutch 

pressure 

(psi)

right side

DCT clutch 

pressure 

(psi)

forward 

speed 

(knots)

Actual

Set-point

Actual

Set-point

Clutch 1 (high)

Clutch 2 (low)

Clutch 1 (high)

Clutch 2 (low)

Left

Right

Left rotor

Right rotor

Actual

Set-point (a)

(b)

(c)

(d)

(e)

(f)

(g)

0 10 20 30 40 50 60 70 80 90 100 -0.05

0

0.05

0 10 20 30 40 50 60 70 80 90 100

40

60

80

0 10 20 30 40 50 60 70 80 90 100 0.5

1

1.5

0 10 20 30 40 50 60 70 80 90 100 0

5

Horizontal

vehicle Vertical

accelerations

(g)

rotor 

collective 

pitch

(degrees) 

rotor 

induced 

velocity 

(m/s)

t, sec

wing 

flap 

pitch

(degrees) 

(a)

(b)

(c)

(d)

0 10 20 30 40 50 60 70 80 90 100 -0.05

0

0.05

0 10 20 30 40 50 60 70 80 90 100

40

60

80

0 10 20 30 40 50 60 70 80 90 100 0.5

1

1.5

0 10 20 30 40 50 60 70 80 90 100 0

5

Horizontal

vehicle Vertical

accelerations

(g)

rotor 

collective 

pitch

(degrees) 

rotor 

induced 

velocity 

(m/s)

t, sec

wing 

flap 

pitch

(degrees) 

0 10 20 30 40 50 60 70 80 90 100 -0.05

0

0.05

0 10 20 30 40 50 60 70 80 90 100

40

60

80

0 10 20 30 40 50 60 70 80 90 100 0.5

1

1.5

0 10 20 30 40 50 60 70 80 90 100 0

5

Horizontal

vehicle Vertical

accelerations

(g)

rotor 

collective 

pitch

(degrees) 

rotor 

induced 

velocity 

(m/s)

t, sec

wing 

flap 

pitch

(degrees) 

(a)

(b)

(c)

(d)
NASA/TM—2012-217212 14 

Figure 22.—Forward cruise SSC upshift—torque response; (a) 

engine torques, (b) cross-shaft torque, (c) rotor torque. 

Figure 23.—Forward cruise SSC upshift—engine response; 

(a) compressor speed, (b) mass flow, (c) power turbine inlet 

temperature. 

Figure 24.—Forward cruise SSC upshift—two-spool gas 

turbine engine transient running lines. 

Similar to the downshift case, Figure 21(a) shows that both 

the horizontal and vertical vehicle accelerations are minimal 

thus with no forward speed or altitude lost during the upshift. 

To maintain constant forward speed under the increasing rotor 

speed, the closed-loop blade pitch controller decreases the 

blade collective pitch to keep constant thrust as shown Figure 

21(b). This causes the drop in induced velocity as seen in 

Figure 21(c). Finally, as seen in Figure 21(d), under the 

constant thrust condition, there is no appreciable change in 

wing flap angle. 

The driveline torques are summarized in Figure 22. As 

observed in the engine fuel rate response in Figure 20(g), the 

left and right side power turbine torques are identical prior to 

t=7 sec and after t=92 sec. During the downshift, the engine 

torques become dissimilar and are due to both rotor aerodynamic torque as well as acceleration torque demand. 

As a result of the dissimilar left and right engine torques, the 

cross-shaft also becomes loaded as shown in Figure 22(b). 

Here, the cross-shaft is subjected to a fully reversing torque 

load. This is also true in the downshift case. Finally, Figure 22(c) shows the deceasing rotor torque which is a result 

from the decreasing collective pitch. 

Other important aspects of the two-spool gas turbine engine 

response are shown in Figures 23 and 24. As seen in Figure 24, 

similar to the downshift case, the transient running lines of each 

engine undergo small excursions past the compressor surge line 

during the SSC upshift process. 

Next, as in the downshift case, three different DCT shift rates 

(slow, medium and fast), given by Table 7, are examined. Note the 

medium case corresponds to the values used in Figures 20 to 24. 

TABLE 7.—UPSHIFT RATE CASES 

Case DCT clutch pressure engagement and disengagement rates 

(psi/sec) 

Slow Clutch no. 1 engagement rate, 

pcl1  = 4 psi/sec 

Clutch no. 2 disengagement rate, 

pcl 2  = –15 psi/sec 

Medium Clutch no. 1 engagement rate, 

pcl1  = 4 psi/sec 

Clutch no. 2 disengagement rate, 

pcl 2  = –30 psi/sec 

Fast Clutch no. 1 engagement rate, 

pcl1  = 4 psi/sec 

Clutch no. 2 disengagement rate, 

pcl 2  = –300 psi/sec 

The resulting SM values for each shift rate are shown in 

Figure 25. In each case, the surge line is crossed briefly. 

Figure 25(a) shows that the left engine crosses the surge line 

when the FWU re-engages due to the left side DCT upshift 

(step no. 5, Table 6). Furthermore the right engine experiences 

a surge near t = 30 sec and t = 48 sec. The first surge of the right 

engine is due to re-engagement of the right engine FWU as a 

result of the right side DCT upshift (step no. 3, Table 6). The 

second surge, near t = 48 sec, is due to the right engine power 

turbine set-point ramp-up (step no. 4, Table 6) which causes the 

left engine to disengage from its FWU. 

Finally, Figure 25 shows that faster DCT shifting rates are 

more beneficial for the engine surge margins in the SSC upshift 

procedure. 

0 10 20 30 40 50 60 70 80 90 100

0

5

10

0 10 20 30 40 50 60 70 80 90 100

-20

0

20

40

0 10 20 30 40 50 60 70 80 90 100

300

400

500

600

engine 

torque 

(kNm)

cross-shaft 

torque 

(kNm)

rotor 

torque 

(kNm)

t, sec

Left side

Right side

Left side

Right side

(a)

(b)

(c)

0 10 20 30 40 50 60 70 80 90 100

0

5

10

0 10 20 30 40 50 60 70 80 90 100

-20

0

20

40

0 10 20 30 40 50 60 70 80 90 100

300

400

500

600

engine 

torque 

(kNm)

cross-shaft 

torque 

(kNm)

rotor 

torque 

(kNm)

t, sec

Left side

Right side

Left side

Right side

0 10 20 30 40 50 60 70 80 90 100

0

5

10

0 10 20 30 40 50 60 70 80 90 100

-20

0

20

40

0 10 20 30 40 50 60 70 80 90 100

300

400

500

600

0 10 20 30 40 50 60 70 80 90 100

0

5

10

0 10 20 30 40 50 60 70 80 90 100

-20

0

20

40

0 10 20 30 40 50 60 70 80 90 100

300

400

500

600

engine 

torque 

(kNm)

cross-shaft 

torque 

(kNm)

rotor 

torque 

(kNm)

t, sec

Left side

Right side

Left side

Right side

(a)

(b)

(c)

0 10 20 30 40 50 60 70 80 90 100 10

20

30

0 10 20 30 40 50 60 70 80 90 100 0

10

20

0 10 20 30 40 50 60 70 80 90 100 0

1000

2000

Left side

Right side

compressor

speed 

(kRPM)

compressor 

mass flow 

(kg/sec)

power 

turbine inlet 

temperature 

(K)

t, sec

(a)

(b)

(c)

0 10 20 30 40 50 60 70 80 90 100 10

20

30

0 10 20 30 40 50 60 70 80 90 100 0

10

20

0 10 20 30 40 50 60 70 80 90 100 0

1000

2000

Left side

Right side

compressor

speed 

(kRPM)

compressor 

mass flow 

(kg/sec)

power 

turbine inlet 

temperature 

(K)

t, sec

0 10 20 30 40 50 60 70 80 90 100 10

20

30

0 10 20 30 40 50 60 70 80 90 100 0

10

20

0 10 20 30 40 50 60 70 80 90 100 0

1000

2000

Left side

Right side

compressor

speed 

(kRPM)

compressor 

mass flow 

(kg/sec)

power 

turbine inlet 

temperature 

(K)

t, sec

(a)

(b)

(c)

2 4 6 8 10 12 14 16 18

0

5

10

15

20

25

30

35

40

mass flow, kg/sec

pressure 

ratio

Left side engine

Right side engine

surge line

2 4 6 8 10 12 14 16 18

0

5

10

15

20

25

30

35

40

mass flow, kg/sec

pressure 

ratio

Left side engine

Right side engine

surge line
NASA/TM—2012-217212 15 

Figure 25.—Forward cruise SSC upshift—engine compressor 

surge margins for three DCT shift rates. 

Conclusions 

This investigation explores the variable-speed operation and 

transient shift response of a prototypical two-speed dual-clutch 

transmission tiltrotor driveline based on the NASA LCTR-2 in 

forward flight airplane mode. A comprehensive variable-speed 

rotorcraft propulsion system modeling tool developed under a 

NASA-funded NRA program is utilized to simulate the drive 

system dynamics. In this study, a sequential shifting control 

strategy is analyzed under a steady cruise condition. This 

investigation attempts to build upon previous variable-speed 

rotorcraft propulsion studies by 1) including a fully nonlinear 

transient gas-turbine engine model, 2) including clutch stick-slip 

friction effects, 3) including shaft flexibility, and 4), incorporating a basic flight dynamics model to account for interactions with 

the flight control system. Sequential shifting provided smooth 

rotor speed changes while maintaining a constant forward speed 

setpoint and a constant altitude for the tiltrotor studied. However, 

it was also found that transient shocks produced by engine 

engagements and disengagements from their freewheeling 

clutches tended to produce brief excursions of the engine 

compressor surge limit. It was found that proper choice of 

dual-clutch transmission shift rate could minimize these 

excursions. In the sequential downshift case, it was found that 

slower shift rates tended to reduce the engine surge effects while 

the opposite trend was found for the upshift case. Through 

exploring these interactions between the various subsystems, this 

analysis provides important insights into the continuing development of variable-speed rotorcraft propulsion technology. 

References 

1. Johnson, W., Yamauchi G.K., and Watts M.E., “NASA Heavy Lift 

Rotorcraft Systems Investigation.” NASA 213467, December 

2005. 

2. Goi, T., Kawakami, K., Yamakawa, E., and Tanaka, H., “Variable 

Rotor Speed Transmission with High Speed Traction Drive,” 

American Helicopter Society 55th Annual Forum, Montreal, 

Quebec, Canada, May 25-27, 1999. 

3. Kish, J., “Vertical Lift Drive System Concept Studies Variable 

Speed/Two-Speed Transmissions,” NASA/CR—2002-211564, 

2002. 

4. Acree, C.W., Yeo, H., and Sinsay, J.D., “Performance Optimization of the NASA Large Civil Tiltrotor,” International Powered 

Lift Conference, London, UK, Jul. 22-24, 2008. 

5. Stevens, M.A, Handschuh, R.F., and Lewicki, D.G., “Concepts for 

Variable/Multi-Speed Rotorcraft Drive System,” AHS 64th Annual Forum and Technology Display, Montreal Canada, April 

29–May 1, 2008 

6. Lewicki, D.G., DeSmidt H., Smith, E.C., and Bauman, S.W., 

“Dynamics of a Dual Clutch Gearbox System: Analysis and Experimental Validation,” AHS 66th Annual Forum and Technology 

Display, Phoenix, AZ, May 11-13, 2010. 

7. Kulkarni, M., Shim, T., and Zhang, Y., “Shift Dynamics and 

Control of Dual-Clutch Transmissions”, Mechanisms and Machine Theory, 42, pp. 168-182, (2007). 

8. Lui, C.H., “Simulation Study of Dual Clutch Transmission for 

Medium-Duty Truck Applications,” SAE, 2005-01-3590 (2005). 

9. Litt, J.S., Edwards, J.M., and DeCastro, J.A., “A Sequential 

Shifting Algorithm for Variable Rotor Speed Control,” 

NASA/TM—2007-214842 (2007). 

10. Johnson, W., Helicopter Theory, University Press, Princeton, NJ, 

1980. 

11. Walsh, P. P., and Fletcher, P., Gas Turbine Performance, Wiley-Blackwell; 2nd Edition, Oxford, 2004. 

12. Veres, J. P., 2009, “Compressor Study to Meet Large Civil Tilt 

Rotor Engine Requirements,” AHS International, 65th Annual 

Forum & Technology Display, Grapevine, TX, May 27–29, 2009. 

13. Snyder, C.A., and Thurman, D.R., “Gas turbine characteristics for 

a Large Civil Tilt-Rotor (LCTR),” AHS International, 65th Annual 

Forum & Technology Display, Grapevine, TX, May 27–29, 2009. 

14. Camporeale, S.M., Fortunato, B., and Mastrovito, M. “A Modular 

Code for Real Time Dynamics Simulation of Gas Turbines in 

Simulink,” Trans. ASME: J. Eng. Gas Turbines Power, 128, pp. 

506-517, (2006). 

15. Sekhon, R., Bassily, H., Wagner, J., and Gaddis, J.. 2006, “Stationary Gas Turbines – A Real Time Dynamic Model with Experimental Validation,” Proc. of the 2006 American Control 

Conference, Minneapolis, MN, June 14-16, 2006. 

16. Crowther, A., Zhang, N., Liu, D.K., Jeyakumaran, J.K., “Analysis 

and Simulation of Clutch Engagement Judder and Stick-Slip in 

Automotive Powertrain Systems, Proc. Instn Mech. Engrs, Part D: 

J. Automobile Engineering, 218, pp. 1427-1446, (2004). 

17. Liu, Y., Qin, D., Jiang, H. and Zhang, Y., “A Systematic Model for 

Dynamics and Control of Dual Clutch Transmissions,” Journal of 

Mechanical Design, ASME Trans. 131 (2009) 061012. 

18. Acree, C.W. and Johnson, W., “Aeroelastic Stability of the 

LCTR-2 Civil Tiltrotor,” Presented at the AHS Technical Specialists’ Meeting, Dallas, TM, October 15-17, 2008. 

left engine 

compressor 

SM (%)

right engine 

compressor 

SM (%)

t, sec

0 10 20 30 40 50 60 70 80 90 100

-20

0

20

40

60

0 10 20 30 40 50 60 70 80 90 100

-20

0

20

40

60

no surge 

surge

no surge 

surge

(a)

(b)

Slow

Medium

Fast

Shift Rate Case

left engine 

compressor 

SM (%)

right engine 

compressor 

SM (%)

t, sec

0 10 20 30 40 50 60 70 80 90 100

-20

0

20

40

60

0 10 20 30 40 50 60 70 80 90 100

-20

0

20

40

60

no surge 

surge

no surge 

surge

(a)

(b)

Slow

Medium

Fast

Shift Rate Case

NASA/TM—2012-217212 17 

Appendix A.—Nomenclature

Ai [i=1-7] engine stage cross-section areas, m2

acL blade lift coefficient slope 

a, b , c inflow ratio solution coefficients

Ca compressor axial flow velocity, m/s 

Can nominal compressor axial flow velocity, m/s 

CVSRPM comprehensive variable-speed rotorcraft propulsion system modeling 

c blade chord, m 

cD blade drag coefficient 

cp specific heat of air, kJ/(kg K) 

DCT dual-clutch transmission 

DOF degrees of freedom 

dD differential blade element drag, N 

dFx, dFy differential blade element force components, N 

dL differential blade element lift, N 

f fuel-air ratio 

F DCT generalized force vector 

FTMR main rotor thrust, Nm 

FWU freewheeling clutch units 

htc compressor inlet hub-to-tip ratio 

htpt power turbine inlet hub-to-tip ratio 

Jc compressor rotational inertia, kg m2 

Jcgi [i=1-3] DCT control gear rotational inertias, kg m2

Jcr DCT carrier rotational inertia, kg m2

Jggt gas generator turbine rotational inertia, kg m2

Ji DCT input gear rotational inertia, kg m2

Jin DCT input-side rotational inertia, kg m2

Jout, DCT output-side rotational inertia, kg m2

Jp DCT planet gear rotational inertia, kg m2

Jpt power turbine rotational inertia, kg m2

Jr DCT ring gear rotational inertia, kg m2

Js DCT sun gear rotational inertia, kg m2

Lcl2 ring gear clutch length, m 

LHV fuel lower heating value, J/kg 

M DCT inertia matrix 

mb blade mass, kg 

mp DCT planet mass, kg 

mpa plenum chamber air mass, kg 

mc  compressor mass flow rate, kg/s 

mcc  combustion chamber mass flow rate, kg/s 

mcn  nominal compressor mass flow rate, kg/s 

mf  fuel flow rate, kg/s 

mf ,max  maximum fuel flow rate, kg/s 

mfn  nominal fuel flow rate, kg/s 

mggt  gas generator mass flow rate, kg/s 

min  plenum chamber input mass flow rate, kg/s 

mout  plenum chamber output mass flow rate, kg/s

mpt  power turbine mass flow rate, kg/s 

Nb number of main rotor blades 

Ncg number of DCT planet clutches

Np number of DCT planets 

Ns number of compressor stages 

nDCT,high DCT high speed ratio 

nDCT,low DCT low speed ratio 

nf final gearbox ratio 

ni initial reduction gearbox ratio 

Pa ambient pressure, bar 

PGT planetary gear transmission 

PID proportional-integral-derivative 

pcli [i=1-2] clutch engagement pressures, N/m2 

pclmax maximumclutch engagement pressure, N/m2

Pp plenum chamber pressure, N/m2

P0i[i=1-7] engine stagnation pressures, N/m2

prc compressor pressure ratio 

prcn nominal compressor pressure ratio 

q DCT degree-of-freedom vector

Qcc combustion chamber heat addition, W 

r blade element radial location, m 

R ideal gas constant, J/(mol K) 

Rb blade radius, m 

Rc compressor mean radius, m 

Rcgi [i=1-3] DCT control gear radii, m 

Rcli [i=1-2] clutch radii, m 

Rcr DCT carrier radius, m 

Ri DCT input gear radius, m 

Rp DCT planet gear radius, m 

Rpt power turbine mean radius, m 

Rr DCT ring gear radius, m 

Rs DCT sun gear radius, m 

SM compressor surge margin, percent 

SSC sequential shifting control 

t time, s 

T0i[i=1-7] engine stage stagnation temperatures, K 
NASA/TM—2012-217212 18 

Ta ambient temperature, K

TQc compressor torque, Nm 

TQcl,i [i=1-2] clutch torques, Nm 

TQin DCT input shaft torque, Nm 

TQout DCT output shaft torque, Nm 

uT blade element tangential speed, m/s 

vc forward speed, knots 

vc,set forward speed setpoint, knots 

vind induced velocity, knots 

Vp plenum chamber volume, m3 

Wc compressor power, W 

WL load power, W

Wggt gas generator turbine power, W 

Wpt power turbine power, W 

x vehicle horizontal position coordinate, m 

y vehicle vertical position coordinate, m 

Zcgi [i=1-3] number of DCT control gear teeth 

Zi number of DCT input gear teeth 

Zp number of DCT planet gear teeth 

Zr number of DCT ring gear teeth 

Zs number of DCT sun gear teeth 

 angle of attack, rad

c compressor beta parameter, s2/kg2 

T0cc combustion chamber temperature rise, K 

 clutch slip velocity threshold, RPM

 total inflow angle, rad 

c forward speed inflow angle, rad 

i DCT input gear rotation, rad 

ind induced inflow angle, rad 

r DCT ring gear rotation, rad

 specific heat ratio 

c forward speed inflow parameter 

c compressor isentropic efficiency 

cc combustion chamber loss coefficient 

 choking mass flow parameter, s/(m K )

c forward speed inflow ratio 

ind induced inflow ratio

cl clutch dry friction coefficient 

p main rotor pitch angle, rad 

w wing elevon flap angle, rad 

air ambient air density, kg/m3 

p plenum chamber gas density, kg/m3 

 rotor solidity ratio

cc combustion chamber time constant, s 

cl clutch viscous torque coefficient, kg m2/s 

c total speed ratio 

c compressor speed, RPM 

cn compressor nominal speed, RPM 

DCT two-speed transmission output speed, RPM 

DCT,high two-speed transmission high output speed, RPM 

DCT,in two-speed transmission input speed, RPM 

DCT,low two-speed transmission low output speed, RPM 

FWU,out freewheeling clutch output speed, RPM 

cg1,2 DCT planet clutch speeds, RPM 

MR main rotor speed, RPM 

pt power turbine speed, RPM 

ptn nominal power turbine speed, RPM 

pt,set power turbine setpoint speed, RPM 

r DCT ring gear speed, RPM 

 Note, if not explicitly stated, a dot above a 

variable represents change with respect to time. 
NASA/TM—2012-217212 19 

Appendix B.—Comprehensive Variable-Speed 

Rotorcraft Propulsion System Model 

Figure B-1.—Two-speed rotorcraft driveline comprehensive simulation—main block. 
NASA/TM—2012-217212 20 

Figure B-2.—Two-speed tiltrotor driveline/engine/clutch/rotor/fuel control/flight dynamics block. 
NASA/TM—2012-217212 21 

Figure B-3.—Two-DOF (altitude and forward speed) vehicle flight dynamics block. 

Figure B-4.—Rigid rotor and blade pitch forward speed controller block. 
NASA/TM—2012-217212 22 

Figure B-5.—Two-spool gas turbine engine simulation block. 

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Variable-Speed Simulation of a Dual-Clutch Gearbox Tiltrotor Driveline 

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13. SUPPLEMENTARY NOTES

14. ABSTRACT

This investigation explores the variable-speed operation and shift response of a prototypical two-speed dual-clutch transmission tiltrotor 

driveline in forward flight. Here, a Comprehensive Variable-Speed Rotorcraft Propulsion System Modeling (CVSRPM) tool developed

under a NASA funded NRA program is utilized to simulate the drive system dynamics. In this study, a sequential shifting control strategy is 

analyzed under a steady forward cruise condition. This investigation attempts to build upon previous variable-speed rotorcraft propulsion 

studies by 1) including a fully nonlinear transient gas-turbine engine model, 2) including clutch stick-slip friction effects, 3) including shaft 

flexibility, 4) incorporating a basic flight dynamics model to account for interactions with the flight control system. Through exploring the 

interactions between the various subsystems, this analysis provides important insights into the continuing development of variable-speed 

rotorcraft propulsion systems. 

15. SUBJECT TERMS

Tiltrotor aircraft; Transmission (machine elements); Compressors; Turbines; Clutches 

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