An Atlas of Engineering Dynamic Systems, Models, and Transfer Functions

Dr. Bob Williams, [email protected]

Mechanical Engineering, Ohio University

This document presents the models and/or transfer functions of some real-world control systems.

Models are the mathematical descriptions of real-world systems, simplified by various assumptions, ignoring some nonlinear and higher effects. Models are collections of ordinary differential equations and algebraic equations. These equations must be linearized, if necessary, to work in classical controls.

Transfer functions are the mathematical vehicle of classical controls. Transfer functions are defined as the Laplace transform of the output variable divided by the Laplace transform of the input variable, with zero initial conditions. Transfer functions represent the system dynamics, as described by the simplified model – they yield the simulated system output given various inputs. Transfer functions can be derived for the open-loop, closed-loop, and/or smaller system components. Block diagrams are used for graphical representation, where the blocks have transfer functions representing the dynamics of certain system components, while the arrows represent system variables.

The models and transfer functions summarized in this document only give the bottom-line results, without derivations or much explanation. The reader is referred to the various references for more details.

2 Atlas of Engineering Dynamic Systems, Models, and Transfer Functions

Table of Contents

1. COMMON SYSTEM VARIABLES ...... 3

2. ZEROTH-ORDER SYSTEM EXAMPLES ...... 4

3. FIRST-ORDER SYSTEM EXAMPLES ...... 6

4. SECOND-ORDER SYSTEM EXAMPLES ...... 8

5. REAL-WORLD TRANSFER FUNCTIONS ...... 11

6. ADVANCED REAL-WORLD MODELS ...... 34

3 1. Common System Variables

We start with a table of like variables playing the same role in different engineering systems.

System Rate r(t) Quantity  rtdt() Effort e(t) Impulse  etdt()

translational mechanical velocity v(t) displacement x(t) force f(t) impulse

rotational mechanical angular velocity (t) angular displacement (t) torque (t) angular impulse

electrical current i(t) charge q(t) voltage v(t) flux (t)

incompressible fluid volume flow rate q(t) volume V(t) pressure p(t) none

compressible fluid mass flow rate qm(t) mass m(t) pressure p(t) none

thermal heat flow rate q(t) heat energy Q(t) temperature T(t) none

4 2. Zeroth-Order System Examples

Name Model G(s)

()tt ()  () t  ()ss () 1  ()s gear ratio n IN IN OUT OUT OUT  OUT  n ()ttt () () ()ssn ()  ()s OUT OUT IN IN IN IN

Ls() V () s Fs() 1 rack and pinion   r  lt() r () t  ()trft () ()ss () ()sr Fs() Ts() Hooke’s Law  k  k f ()tkxt ()  ()tkt  () R R Xs() ()s

11 Fs() kk Fs() series / parallel springs x()tft () f ()tkkxt ( ) ()  12  kk  12 X ()skk Xs() 12 kk12 12 Fs() Ts() viscous damping  c  c f ()tcvt ()  ()tct  () R R Vs() ()s

Newton’s Second Law As() 1 ()s 1 f ()tmat ()  ()tJt  ()   Euler’s Rotational Law Fs() m ()sJ

2 accelerometer, low-frequency 2 Xs()  ()()kmxt  x () t  (Dorf & Bishop) IN X IN ()skm

Ts() motor torque  K  ()tKit () T T Is()

VsB () back emf  K B vtBBM() K () t  ()s M

5 Zeroth-Order System Examples (continued)

Name Model G(s)

Qs() Vs() 1 capacitor (q(t) ~ charge) qt() Cvt ()  C  Vs() Qs() C

Vs() Is() 1 resistor  R  vt() Rit () Is() Vs() R

()s Is() 1 inductor ((t) ~ flux)  L  ()tLit () Is() ()sL

Vs() R potentiometer vtR() vt ()( R R ) 22 122 12 Vs() R R 112 Vs() tachometer  Kt vt() Kt () t ()s

Vs2 () DC amplifier, zero time constant  K A vt21() KvtA () Vs() 1

11 Vs() Vs() RR series / parallel resistors vt() ( R R )() it it() vt ()  R  R  12 12  Is() 12I ()sRR RR12 12

6 3. First-Order System Examples

 Name Diagram Model G(s) 

x(t) Xs() 1 c k  massless translational mechanical system c f(t) cx() t kx () t f () t Fs() csk k

x(t) Vs() 1 m c  m f(t) mv() t cv () t f () t Fs() msc c springless translational mechanical system

c ()s 1 c R  R k R ctkt() () () t RR Ts() csRR k kR inertialess rotational mechanical system (t) (t)

J

cR ()s 1 J  Jt() c () t () t R Ts() Jsc R cR springless rotational mechanical system (t) (t)

R di() t Is() 1 L + L Ri() t v () t  v(t) i(t) L - dt Vs() LsR R LR series electrical circuit

R 1 Qs() C Rqt() qt () vt ()  + v(t) i(t) C C Vs() RCs 1 RC - RC series electrical circuit 1 I()sCs Rit() itdt () vt ()  RC C  Vs() RCs 1

Vs() K  vt () vt () Kvt () 2  A DC amplifier with time constant 22A 1 Vs() s 1 1

7 First-Order System Examples (continued)

Name Model G(s)

differentiator dx() t dv() t Vs() As() vt() at()  s  s dt dt Xs() Vs()

dq() t dt() Is() Vs() it() vt()  s  s dt dt Qs() ()s

x()tvtdt () vt() atdt () Xs() 1 Vs() 1     integrator Vs() s A()ss qt() itdt ()  ()tvtdt ()   Qs() 1 ()s 1   I ()ssVs() s Is() Vs() 1  Cs  dv() t 1 it() C vt() itdt () Vs() I()sCs capacitor dt C  Qs() 1 Vs()   Rs 1 dq() t qt() vtdt () vt() R Vs() Rs Qs() resistor R  dt 1 di() t Is() 1 Vs() it() vtdt () vt() L   Ls L  dt Vs() Ls Is() inductor Ys() k Hs()SENS  ytytkytSENS() SENS () () Ys() s 1 generic sensor k gain  time constant

8 4. Second-Order System Examples

Name Diagram Model G(s)

x(t) k mx() t cx  () t kx () t f () t Xs() 1  translational mechanical system c m f(t) Fs() ms2  csk dv() t mcvtkvtdtft() ()  () dt  Vs() s  Fs() ms2  csk x(t) springless translational c   mechanical system m f(t) mx() t cx () t f () t Xs() 1  Fs() smsc ( )

x(t) damperless translational k  mechanical system m f(t) mx() t kx () t f () t Xs() 1  2 Fs() ms k

k damperless translational  mechanical system, vertical g my() t ky () t f () t Ys() 1 m  Fs() ms2  k y(t) f(t) x(t) mass-only translational Xs() 1 m  mechanical system f(t) mx() t f () t Fs() ms2

9 Second-Order System Examples (continued)

Name Diagram Model G(s)

J

c rotational mechanical system R   J ()tctktRR () () () t ()s 1  kR 2 Ts() Js csRR k

(t) (t)

springless rotational J c mechanical system R   J ()tctR () () t ()s 1  Ts() sJsc ( R )

(t) (t)

J damperless rotational  mechanical system kR J ()tktR () () t ()s 1  2 Ts() Js kR

(t)  (t)

J inertia-only rotational mechanical system Jt() () t ()s 1  Ts() Js2

(t) (t)

10 Second-Order System Examples (continued)

Name Diagram Model G(s)

torqued pendulum (linearized) (t) g g 1 ()s 1 L   ()tt ()2 () t  LmL ()s 22 g  (t) mL s  m  L

dv() t v () t 1 Vs() RLs + Cvtdtit()  ()  2 i(t) i (t) RLi (t) i (t) Cv(t)  parallel RLC circuit R L C dt R L I()sCRLsLsR  -

11 ()sRL Ct()  () t () t it ()  RL I ()sCRLsLsR2  

R di() t 1 Is() Cs L Ri() t i () t dt  v () t   2 series RLC circuit + dt C Vs() LCs RCs 1 v(t) i(t) L - 1 Qs() C Lqt() Rqt  () qt ()  vt ()  C C Vs() LCs2  RCs 1

Xs()  s2  accelerometer (Dorf & Bishop) mx() t bx  () t kx () t mx IN () t 2 X IN ()ssbmskm ( ) ( )

2 dxt() As() 2 at()  s double differentiator dt 2 Xs() Xs() 1 x()tatdt  ()  double integrator A()ss2

11 5. Real-World Transfer Functions

Simplified DC servomotor (ignoring back emf and inductor) The figure below shows a simple diagram for deriving the model of a DC servomotor, which is a rotational electromechanical system. On the circuit side, v(t) is the input armature voltage, L is the inductance constant, R is the resistance constant, and i(t) is the armature circuit current. On the rotational mechanical side, J is the lumped rotational inertia of the motor shaft and load, cR is the rotational viscous damping coefficient, and the output is angular displacement (t) (whose time derivative is angular velocity (t)).

L R J  c R v(t) i(t)  (t) (t) (t)

From an earlier derivation, the RL series circuit model is: di() t L Ri() t v () t dt

where we have ignored the motor back emf voltage. Usually the time constant for the electrical system is much smaller than the time constant for the rotational mechanical system, which means that the electrical system current i(t) rises much faster than the mechanical displacement (t). Therefore, we can ignore the circuit dynamics ( L  0 ), so the electrical circuit model simplifies to Ri() t v () t , which is simply Ohm’s Law. In a DC servomotor, the generated motor torque is proportional to the circuit current, a linear proportional relationship that holds good for nearly the entire range of operation of the motor:

 ()tKit T ()

KT is the motor torque constant, which is stamped on the motor housing, available from the motor manufacturer, or determinable by experiment. The rotational mechanical system dynamic model is derived from a free-body diagram of the rotating motor shaft, using Euler’s rotational dynamics law M  Jt():   J ()tctR () () t

Substituting the electrical models into the rotational mechanical system dynamic model yields:

  K T ()s KT R Jt() cRT () t () t Kit () vt () Gs () R Vs() sJsc ( R )

This is a linear, lumped-parameter, constant-coefficient, second-order ODE. The same model written for angular velocity (t) output is a first-order model:

K T ()s KT R Jt () cR () t vt () Gs () R Vs() Jsc R

12 DC Servomotor1

L R   Jt() cRT () t () t Kit () c J  R di() t v(t) i(t) L Rit() vt () v () t vt () K () t  dt BB (t) (t) (t)

()sK Gs () 2 Vs() ( LsR )( JscR ) K

KTBKK ()sK Gs()  Vs()  2  sLsRJsc()()R K

L J If we set the armature circuit time constant to zero relative to the mechanical system time constant R cR (since the mechanical system dominates), the above transfer functions are simplified to first- and second- order, respectively (rather than the original second- and third-order systems):

()sK Gs () 2 Vs() JRs ( RcR K )

()sK Gs()  Vs()  2  JRs() RcR K s

1 R.L. Williams II and D.A. Lawrence, 2007, Linear State-Space Control Systems, John Wiley & Sons, Inc.

13 suitable for ME 3012 Term Projects

Inverted pendulum1

  2 (m12 m ) wt () mL 2 cos() t () t  mL 2 sin()  t () t  f () t

m2 Y 2  mL22() t mL cos() twt () mgL 2 sin()  t 0 g (t) non-linear L  (mmwtmLt12 ) () 2 ()  ft () X f(t) m  1 mwt22() mL () t  mg 2 () t  0 linearized

w(t)

In order to derive the overall SISO transfer function for the inverted pendulum, take the Laplace Transform of both sides of both of the linearized ODEs above. Then use algebra to eliminate W(s) between the two equations and arrive at G(s). This process yields the following Type 0, second-order, unstable open-loop transfer function:

()s 1

Gs()2 Fs() mLs112 ( m m ) g

Aircraft Pitch Control2

(t) output aircraft pitch angle ()tt  () output aircraft pitch angular velocity (t) input elevator control angle

(ss ) 1.15 0.18 Gs() (ss )2 0.74 s 0.92

2 www.engin.umich.edu/group/ctm

14 Automobile Cruise Control2

Vs() 1 Gs() u(t) road force input v(t) output speed Us() msb

Aircraft Roll Control3

()sK Gs() q(t) hydraulic fluid flow ()tt  () roll velocity Qs() s2  4 s 9 (t) roll (bank) angle

Diabetes Control3

Bs() s 2 Gs() i(t) insulin input b(t) output blood-sugar level Is() ss ( 1)

3 Dorf and Bishop, Modern Control Systems, 11th edition, Pearson Prentice Hall, 2007.

15 Elevator Control3

Vs() 1 Gs() (t) torque input vt() yt () elevator velocity ()ss2 2 s 11 y(t) elevator displacement

Ferris Wheel Control3

()ss 6 Gs() (t) torque input (t) output angular velocity ()ss ( 2)( s 4)

16 Fluid Heating System3

 T(s) temperature difference T0(s) – Te(s) Ts() 1  C thermal capacitance Gs() Hs() 1  Q constant flow rate Cs QW R  W water specific heat  R insulation thermal resistance RC  H(s) heating element heat flow rate   RQW 1   time constant

Fluid Flow Tank System3

 Q1 input flow rate

Qs2 () 1  Q2 output flow rate GsQ () Qs1 () s 1  H head   RC, time constant Hs() R  R orifice flow resistance Gs() H  C cross-sectional tank area Qs1 () RCs 1

17 Human “Paper-Pilot” Model3

()sss (2 1)( 2) Gs()  E ()t input angle error (t) output elevator angle E ()sss (0.5 1)( 2) 0.25   0.50 human pilot time constant

Hydraulic Actuator3

2 Ak A Ys() K K  x BbK   ggxP (, ) flow Gs() k k X ()ssmsB ( ) P P  A piston area g g k  k  x x P P x0 P0

18 Laser Printer Positioning3

Ys( ) 4( s 50) Gs() (t) input control torque y(t) printer head displacement (ss )2 30 s 200

Paper Processing Tensioning3

0 ()s KM Gs() et0 () input motor voltage 0 ()t windup roll velocity Es0 () s 1 L K motor constant   motor time constant M R

19 Racecar Speed Control3

Vs() 100 Gs() c(t) throttle input v(t) output racecar speed Cs() ( s 2)( s 5)

Robot Elbow Control3

()s 2 Gs() (t) torque input (t) output elbow angle ()sss ( 4)

20 Robot Position Control3

Vs() 640,000 Gs() (t) torque input vt() yt () robot velocity (ss )2 128 s 6400 y(t) robot displacement

Ship Stabilization3

()s 9 Gs()2 f ()ss 1.2 s 9

f(t) fin control input torque

(t) ship output angle

21 SkyCam Control3modified

Ys() 1 Gs() (t) torque input y(t) output displacement Ts() s (0.2 s 1)

Space Station Orientation Control3

()s 20 Gs() (t) torque input ()tt  () station velocity (ss )2 20 s 100 (t) space station angle

22 Space Telescope Pointing Control3

()s 1 Gs() (t) torque wheel input (t) output telescope angle Ts( ) ss ( 12)

Steel Rolling Thickness Control3

Ys( ) 0.25 Gs() (t) input motor torque y(t) output steel thickness ()sss ( 1)

v0  2000 ft/min nominal steel speed

Turntable Angular Speed Control (old-school records)3

()s K / R Gs()T v(t) input voltage ()t output turntable speed Vs() Jsb

KT motor torque constant R circuit resistance

23 Vehicle Steering Control3

Vs() 1 Gs() (t) steering wheel angle vt() yt () centerline velocity ()sss ( 12) y(t) centerline displacement

VTOL Aircraft Control3

Ys() 1 Gs() (t) thruster input y(t) vertical displacement Ts() ss ( 1)

24 Weld Bead Depth Control3

Ys() K Gs() i(t) input current y(t) output weld bead depth Is() (0.01 s 1)(1.5 s 1)

Welding Robot Control3

Vs() 75( s 1) Gs() (t) torque input vt() yt () robot velocity ()ss ( 5)( s 20) y(t) robot displacement

25 Antenna Azimuth Control4

()s 20.83 Gs() v(t) input voltage ()tt  () azimuth velocity Vs( ) ( s 100)( s 1.71) (t) azimuth angle

Autonomous Submersible Control5

(ss ) 0.13( 0.44) e Gs()2  (t) input elevator angle (t) output pitch angle e (ss ) 0.23 s 0.02

4 N.S. Nise, Control Systems Engineering, 2nd edition, Cummings, 1995. 5 Golnaraghi and Kuo, Automatic Control Systems, 9th edition, Wiley, 2010

26 Missile Roll Control5

Ps() l ll Gs()  p (t) aileron angle input p(t) output roll rate ()sslpa s  1 1  a   aerodynamic time constant l p

 ll p steady-state gain

Robotic Rubbertuator and Load5 (McKibben Artificial Muscle)

Xs() 10 Gs() p(t) input air pressure x(t) output displacement Ps( ) s2  10 s 29

27 Robotic Swivel5

()sK Gs() v(t) input voltage (t) output swivel velocity Vs( ) ( s2  4 s 10)

Heat Transfer System5

Tso () 1 Gs() ti(t) input temperature to(t) output temperature Tsi () RCs 1 RC thermal time constant

Pneumatic System5

Psb () 1 Gs() p1(t) input pressure pb(t) output pressure Ps1() RCs 1 q mass flow rate RC pneumatic time constant

28 Magnetically-Levitated Ball6

Xs() 1 Gs() I ()sAsB2 

x(t) output ball height i(t) input current

i i A  0 B  0 2g x0

i0 and x0 are the nominal current and ball height at which the linearization was performed

Autonomous Automobile BackupDr. Bob

Xs() K Gs() f(t) road force input x(t) backup distance Fs() smsc ( )

K open-loop gain m car mass c effective viscous damping coefficient

What is Subaru Reverse Automatic Braking? (goldsteinsubaru.com)

6 H. Huang, H. Du & W. Li, 2015, "Stability enhancement of magnetic levitation ball system with two controlled electromagnets," Power Engineering Conference (AUPEC), 2015 Australasian Universities: 1-6.

29 Third-Order and Fourth-Order Systems

suitable for ME 3012 Term Projects using an internal pre-filter GPi(s)

Flexible Robot Control3

()ss 500 Gs() (t) torque input (t) angle output (sss ) ( 0.03)( s2 2.57 s 6667)

Helicopter Pitch Control3

()ss 25( 0.03) Gs() (t) torque input (t) output pitch angle (ss ) ( 0.4)( s2 0.36 s 0.16)

30 Robot Force Control3

Fs() Ks ( 2.5) Gs() (t) torque input f(t) force output ()sss (22 2 2)( ss 4 5)

31 Third-Order and Fourth-Order Systems

less suitable for ME 3012 Term Projects

Ball-and-Beam System1

J b mptmgtmptt() sin()  () ()2  0 2 r

2   mpt() J Jb  () t  2 mpt () pt () () t  mgpt ()cos() t  () t

t) p( non-linear

Jb (t) mpt() mgt ()  0 2 r 2  (t) mL JJ() t  mgpt ()  () t 4 b  linearized

L is the constant half-length of the beam, m and r are the mass and radius of the ball, respectively, Jb and J are the mass moment of inertia of the ball and beam, respectively.

In order to derive the overall SISO transfer function for the ball-and-beam system, take the Laplace Transform of both sides of both of the linearized ODEs above. Then use algebra to eliminate (s) between the two equations and arrive at G1(s). This process yields the following Type 0, fourth-order, unstable open-loop transfer function: Ps()  mg

Gs1 () 422 Ts() JEE12 J s mg

2 J mL  where: Jmb J JJ E1 2 E2  b  r  4 

This process could alternatively eliminate P(s) between the two equations and arrive at G2(s), the following Type 0, fourth-order, unstable open-loop transfer function:

2 ()s JsE1 Gs2 () 422 Ts() JEE12 J s mg

32 Missile Yaw Control3 (cannot use internal pre-filter for positive poles)

(ss ) 0.5(2 2500) Gs() (t) torque input (t) yaw acceleration (sss ) ( 3)(2 50 s 1000)

33 Printer Belt Drive3 (closed-loop) r s Xs() J Ts()1  2 Tsd () 32brk12Kmkkr12 ssksb 2  JmJmJR

d(t) disturbance torque input x1()trtyt  () () output displacement error

m mass 0.2 kg k1 light sensor 1 V/m k2 velocity feedback gain 0.1 Vs/m r radius 0.15 m L inductance 0 (ignore) b rotational viscous damping 0.25 Nms/rad R resistance 2  Km torque constant 2 Nm/A 2 JJmotor J pulley inertia 0.01 kg-m

34 6. Advanced Real-World Models

less suitable for ME 3012 Term Projects

(use state-space controller design techniques instead of classical methods)

Three-dof translational mechanical system1

y (t) y (t) y (t) 1 2 3

u1 (t) u2 (t) u3 (t) k1 k2 k3 k4

m1 m2 m3 c1 c2 c3 c4

my11() t ( c 1 c 2 )y  1 () t ( k 1 k 2 ) y 1 () t cy 22  () t k 22 y () t  u 1 () t

myt22() ( c 2 cyt 3 )  2 ()  ( k 2 kyt 3 ) 2 ()  cyt 21  ()  kyt 21 ()  cyt 33  ()  kyt 33 ()  ut 2 ()

my33() t ( c 3 c 4 )y  3 () t  ( k 3 k 4 ) y 3 () t  cy 32  () t  ky 32 () t  u 3 () t

Non-linear Proof-Mass Actuator System1

q(t)

(Mmqtkqtmet ) () () ( ()cos() t   2 ()sin()) t t  0 2  M (Jmet ) () meqt ()cos() t  nt () n(t) k f(t)

e J (t) m

Two-dof translational mechanical system1

y (t) y (t) 1 2    u (t) u (t) my11() t ( c 1 c 2 ) y 1 () t ( k 1 k 2 ) y 1 () t cy 22 () t k 22 y () t  u 1 () t k 1 k 2 1 2 my22() t cy 22  () t ky 22 () t cy 21  () t ky 21 () t  u 2 () t m m c 1 c 2 1 2

35 Automobile Suspension System2

mx11()(()())(()())() t b 1 x  1 t x  2 t k 1 x 1 t x 2 t ut

mxtbxtwtkxtwtcytbxtxtkxtxt22( ) 22 (  ( )  ( )) 22 ( ( ) ( )) 22  ( ) 11 (  ( )  2 ( )) 11 ( ( ) 2 ( )) ut ( )

2 Xs12() Xs () ( m 1222 ms ) bsK Gsu ()22 2 U() s ( ms1112121211 bs K )( m s  ( b b ) s  ( K  K ))(  bs  K )

2 Xs12() Xs () mbsKs 122 ( ) Gsw ()22 2 W() s ( ms1112121211 bs K )( m s  ( b b ) s  ( K  K ))(  bs  K )

In this model, the Body Mass is m1, generally one-fourth of the vehicle mass (excluding tires). m2 is the Suspension Mass, the mass of one tire. As seen in the above transfer functions, this model is 4th- order.

To make this model fit the second-order system controller designs we focus on in ME 3012:

 Eliminate the tire mass (m2, the Suspension Mass) and include half of this into m1.  Then combine the two springs and two dampers in series to obtain one equivalent spring stiffness and one equivalent damper coefficient.

36 Human Skeletal Muscle Model7

kkkkk() Ft () k yt()21212 yt ()yt  ()  yt () m  2 ( F () t  F () t mg kL ) bmmbmmbA mR11

where:

y(t) absolute displacement of the muscle end m lumped muscle mass k1 linear spring stiffness representing the muscle fascia, parallel elastic component k2 linear spring stiffness representing the connecting tendons, series elastic component b viscous damping coefficient representing the muscle energy loss FA(t) muscle actuation force (the integrated effects of all contracting sarcomeres) Fm(t) external load applied to the muscle g acceleration due to gravity L1R neutral length of the muscle (the unstretched length of spring k1).

7 Dr. Bob’s ME 4670 / 5670 Biomechanics NotesBook Supplement, derived by Elvedin Kljuno

37 Armature circuit / DC servomotor / gear box / robot joint1 – ME 3012 Term Example

LR n cL

J L (t) cM + +  L (t) v A (t) v B (t) J M - -  L (t)  i A (t) M (t) L (t)  M (t) M (t)

K LJ() t ( Lc RJ )  () t  ( Rc K K )   () t T v () t LLTBLAn

K LJ() t ( Lc RJ )  () t  ( Rc K K )  () t T v () t LLTBLAn

J c where JJL and ccL are the effective polar inertia and viscous damping coefficient M n 2 M n2 reflected to the motor shaft. Numerical Parameters

Parameter Value Units Name L 0.0006 H armature inductance R 1.40  armature resistance kB 0.00867 V/deg/s motor back emf constant 2 JM 0.00844 lbf-in-s motor shaft polar inertia bM 0.00013 lbf-in/deg/s motor shaft damping constant kT 4.375 lbf-in/A torque constant n 200 unitless gear ratio 2 JL 1 lbf-in-s load shaft polar inertia bL 0.5 lbf-in/deg/s load shaft damping constant

 ()sKn / 5 Gs() LT   Vs( ) LJsLcRJsRcKK22 ( )  ( ) s  11 s 1010 ATB LT()sKn / 5 Gs () 22  VsATB( ) sLJsLcRJsRcKK ( ( )  ( )) ss (  11 s 1010)