An Atlas of Engineering Dynamic Systems,
Models, and Transfer Functions
Dr. Bob Williams, williar4@ohio.edu
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
 r ( t ) dt
Effort e(t)
Impulse
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
 e ( t ) dt
4
2. Zeroth-Order System Examples
Name
gear ratio
Model
n
G(s)
 IN (t )
 (t )  OUT (t )
 IN

OUT (t ) OUT (t )  IN (t )
 OUT ( s )  OUT ( s ) 1


 IN ( s )
 IN ( s ) n
rack and pinion
l (t )  r (t )
 (t )  rf (t )
L( s) V ( s)

r
( s )  ( s )
Hooke’s Law
f (t )  kx (t )
 (t )  k R (t )
F (s)
k
X (s)
series / parallel springs
1 1 
x(t )     f (t )
 k1 k2 
f (t )  (k1  k2 ) x(t )
F ( s)
kk
 1 2
X ( s ) k1  k2
OUT ( s )
n
 IN ( s )
F (s) 1

( s ) r
T (s)
 kR
( s )
F ( s)
 k1  k 2
X ( s)
viscous damping
f (t )  cv (t )
 (t )  c R  (t )
F (s)
c
V (s)
T (s)
 cR
( s)
Newton’s Second Law
Euler’s Rotational Law
f (t )  ma (t )
 (t )  J  ( t )
A( s ) 1

F (s) m
( s) 1

( s ) J
accelerometer, low-frequency
(Dorf & Bishop)
motor torque
back emf
( k m ) x (t )   xIN (t )
X ( s)
2

X IN (s) k m
 (t )  K T i (t )
T (s)
 KT
I (s)
v B (t )  K B  M ( t )
VB ( s )
 KB
 M (s)
2
5
Zeroth-Order System Examples (continued)
Name
capacitor (q(t) ~ charge)
resistor
inductor ((t) ~ flux)
potentiometer
tachometer
DC amplifier, zero time constant
series / parallel resistors
Model
G(s)
q(t )  Cv (t )
Q(s)
C
V (s)
V ( s) 1

Q( s) C
v (t )  Ri (t )
V ( s)
R
I (s)
I (s) 1

V (s) R
 ( t )  Li ( t )
 (s)
L
I (s)
I ( s) 1

(s) L
v1 (t ) R2  v2 (t )( R1  R2 )
V2 ( s )
R2

V1 ( s ) R1  R2
v (t )  K t  (t )
V (s)
 Kt
(s)
v2 (t )  K Av1 (t )
V2 ( s )
 KA
V1 ( s )
v(t )  ( R1  R2 )i (t )
1 1
i(t )     v(t )
 R1 R2 
V ( s)
 R1  R2
I ( s)
V ( s)
RR
 1 2
I ( s ) R1  R2
6
3. First-Order System Examples
Name
Model
G(s)


cx (t )  kx (t )  f (t )
X (s)
1

F ( s ) cs  k
c
k
mv(t )  cv(t )  f (t )
V ( s)
1

F ( s ) ms  c
m
c
c R (t )  k R (t )   (t )
( s )
1

T ( s ) cR s  k R
cR
kR
J  (t )  cR (t )   (t )
(s)
1

T ( s ) Js  cR
J
cR
di (t )
 Ri (t )  v (t )
dt
I (s)
1

V ( s ) Ls  R
L
R
Diagram
x(t)
k
massless translational mechanical system
f(t)
c
x(t)
c
m
f(t)
springless translational mechanical system
cR
kR
inertialess rotational mechanical system
 (t)
 (t)
J
cR
springless rotational mechanical system
 (t)
 (t)
R
v(t)
LR series electrical circuit
+
-
i(t)
L
L
R
RC series electrical circuit
DC amplifier with time constant
+
v(t)
-
i(t)
Rq (t ) 
C
Ri ( t ) 
1
q (t )  v(t )
C
1
i ( t ) dt  v ( t )
C
 v2 (t )  v2 (t )  K Av1 (t )
Q( s)
C

V ( s ) RCs  1
I ( s)
Cs

V ( s ) RCs  1
V2 ( s )
KA

V1 ( s )  s  1
RC
RC

7
First-Order System Examples (continued)
Name
differentiator
integrator
capacitor
resistor
inductor
generic sensor
Model
G(s)
v (t ) 
dx (t )
dt
a (t ) 
dv(t )
dt
V (s)
s
X (s)
A( s )
s
V (s)
i (t ) 
dq ( t )
dt
v (t ) 
d (t )
dt
I (s)
s
Q(s)
V (s)
s
( s)
X (s) 1

V (s) s
V ( s) 1

A( s ) s
x (t )   v (t ) dt
v (t )   a (t ) dt
q ( t )   i ( t ) dt
 ( t )   v ( t ) dt
i (t )  C
dv (t )
dt
1
q (t )   v (t ) dt
R
1
i (t )   v(t ) dt
L
v (t ) 
1
i (t ) dt
C
dq (t )
v (t )  R
dt
di (t )
v (t )  L
dt
 y SENS (t )  ySENS (t )  ky (t )
Q (s) 1

I ( s) s
I (s)
 Cs
V (s)
 (s) 1

V ( s) s
V ( s) 1

I ( s ) Cs
Q(s) 1

V ( s ) Rs
V (s)
 Rs
Q(s)
I (s)
1

V ( s ) Ls
V (s)
 Ls
I (s)
H (s) 
k gain
YSENS ( s )
k

Y (s)
 s 1
 time constant
8
4. Second-Order System Examples
Name
Diagram
x(t)
k
translational mechanical system
c
m
Model
G(s)
mx(t )  cx (t )  kx (t )  f (t )
X (s)
1
 2
F ( s ) ms  cs  k
f(t)
m
dv (t )
 cv (t )  k  v (t )dt  f (t )
dt
V ( s)
s
 2
F ( s ) ms  cs  k
x(t)
springless translational
mechanical system
c
m
f(t)
mx(t )  cx (t )  f (t )
f(t)
mx(t )  kx(t )  f (t )
X (s)
1

F ( s ) s ( ms  c )
x(t)
damperless translational
mechanical system
damperless translational
mechanical system, vertical
k
m
X (s)
1
 2
F ( s ) ms  k
k
g
my(t )  ky (t )  f (t )
m
y(t)
Y (s)
1
 2
F ( s ) ms  k
f(t)
x(t)
mass-only translational
mechanical system
m
f(t)
mx(t )  f (t )
X (s)
1

F ( s ) ms 2
9
Second-Order System Examples (continued)
Name
Diagram
Model
G(s)
J (t )  c R (t )  k R (t )   (t )
( s )
1
 2
T ( s ) Js  cR s  k R
J (t )  c R (t )   (t )
( s )
1

T ( s ) s ( Js  cR )
J (t )  k R (t )   (t )
( s )
1
 2
T ( s ) Js  k R
J(t )   (t )
( s )
1
 2
T ( s ) Js
J
rotational mechanical system
cR
kR
 (t)
 (t)
springless rotational
mechanical system
J
cR
 (t)
 (t)
damperless rotational
mechanical system
J
kR
 (t)
inertia-only rotational
mechanical system
 (t)
J
 (t)
 (t)
10
Second-Order System Examples (continued)
Name
Diagram
torqued pendulum (linearized)
Model
(t)
g
g
L
(t )   (t ) 
L
(t)
G(s)
1
 (t )
mL2
m
parallel RLC circuit
+
i(t)
iR(t)
R
iL(t)
L
iC (t)
C
v(t)
C
-
dv (t ) v (t ) 1

  v (t ) dt  i (t )
dt
R
L
C(t ) 
R
L
series RLC circuit
v(t)
+
-
i(t)
C
accelerometer (Dorf & Bishop)
double differentiator
double integrator
L
1 
1
 (t )   (t )  i (t )
R
L
di (t )
1
 Ri (t )   i (t )dt  v(t )
dt
C
Lq(t )  Rq (t ) 
1
q(t )  v (t )
C
mx(t )  bx (t )  kx(t )  mxIN (t )
a (t ) 
x (t ) 
d 2 x (t )
dt 2
  a ( t ) dt
(s)

(s)
1
g

mL2  s 2  
L

V (s)
RLs

2
I ( s ) CRLs  Ls  R
(s)
RL

2
I ( s ) CRLs  Ls  R
I (s)
Cs

2
V ( s ) LCs  RCs  1
Q(s)
C

2
V ( s ) LCs  RCs  1
X ( s)
s 2
 2
X IN (s) s  (b m)s  (k m)
A( s )
 s2
X ( s)
X ( s) 1

A( s ) s 2
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)).

v(t)

L
R
J
cR
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:
 (t )  KT i(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  J(t ) :
J (t )  c  (t )   (t )
R
Substituting the electrical models into the rotational mechanical system dynamic model yields:
K
(s)
KT R
J ( t )  c R ( t )   ( t )  K T i ( t )  T v ( t )
G ( s ) 

R
V ( s ) s ( Js  c 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
(s) KT R
G ( s ) 

J  ( t )  c R  ( t )  T v ( t )
R
V ( s ) Js  cR
12
DC Servomotor1
v(t)


L
R
J(t )  cR(t )   (t )  KT i(t )
J
cR
i(t)
L
 (t)
di (t )
 Ri (t )  v(t )  vB (t )  v(t )  K B(t )
dt
 (t)
(t)
G ( s ) 
( s )
K

V ( s ) ( Ls  R )( Js  cR )  K 2
G ( s ) 
( s )
K

V ( s ) s  ( Ls  R )( Js  cR )  K 2 
KT  K B  K
L
J
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 secondorder, respectively (rather than the original second- and third-order systems):
If we set the armature circuit time constant
1
G ( s ) 
( s )
K

V ( s ) JRs  ( RcR  K 2 )
G ( s ) 
( s )
K

V ( s )  JRs  ( RcR  K 2 )  s
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
(t )  m2 L cos  (t )(t )  m2 L sin  (t )(t )2  f (t )
(m1  m2 ) w
(t )  m2 gL sin  (t )  0
m2 L2(t )  m2 L cos  (t ) w
non-linear
m2
Y
 (t)
g
L
(t )  m2 L(t )  f (t )
(m1  m2 ) w
(t )  m2 L(t )  m2 g (t )  0
 m2 w
X
f(t)
m1
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:
G (s) 
( s )
1

2
F ( s ) m1 Ls  ( m1  m2 ) g
Aircraft Pitch Control2
(t) output aircraft pitch angle

 (t )   (t ) output aircraft pitch angular velocity
(t) input elevator control angle
G (s) 
2
www.engin.umich.edu/group/ctm
1.15s  0.18
( s )
 2
 ( s ) s  0.74 s  0.92
14
Automobile Cruise Control2
G (s) 
1
V ( s)

U ( s ) ms  b
u(t) road force input
v(t) output speed
q(t) hydraulic fluid flow
 (t )  (t ) roll velocity
Aircraft Roll Control3
G ( s) 
K
( s )
 2
Q(s) s  4s  9
(t) roll (bank) angle
Diabetes Control3
G (s) 
3
B(s)
s2

I ( s ) s ( s  1)
i(t) insulin input
Dorf and Bishop, Modern Control Systems, 11th edition, Pearson Prentice Hall, 2007.
b(t) output blood-sugar level
15
Elevator Control3
G (s) 
V (s)
1
 2
( s ) s  2 s  11
(t) torque input
v (t )  y (t ) elevator velocity
y(t) elevator displacement
Ferris Wheel Control3
G (s) 
( s )
s6

( s ) ( s  2)( s  4)
(t) torque input
(t) output angular velocity
16
Fluid Heating System3
T (s)
G ( s) 

H (s)








1
1

Cs  QW  
R

RC
RQW  1
T(s)
C
Q
W
R
H(s)

temperature difference T0(s) – Te(s)
thermal capacitance
constant flow rate
water specific heat
insulation thermal resistance
heating element heat flow rate
time constant
Fluid Flow Tank System3
GQ ( s ) 
GH ( s ) 
Q2 ( s )
1

Q1 ( s )  s  1
H ( s )
R

Q1 ( s ) RCs  1






Q1
Q2
H

R
C
input flow rate
output flow rate
head
RC, time constant
orifice flow resistance
cross-sectional tank area
17
Human “Paper-Pilot” Model3
G (s) 
 (s)
 (2 s  1)( s  2)

 E ( s ) (0.5 s  1)( s  2)
 E ( t ) input angle error
(t) output elevator angle
0.25    0.50
human pilot time constant
Hydraulic Actuator3
G ( s) 
Y (s)
K

X ( s ) s ( ms  B )
Ak x
K
kP
kx 
g
x
B b K 
kP 
x0
g
P
P0
A2
kP


g  g ( x, P ) flow
A piston area
18
Laser Printer Positioning3
G(s) 
Y (s)
4( s  50)
 2
( s ) s  30 s  200
(t) input control torque
y(t) printer head displacement
e0 (t ) input motor voltage
0 (t ) windup roll velocity
K M motor constant

Paper Processing Tensioning3
G (s) 
0 (s)
K
 M
E0 ( s )  s  1
L
motor time constant
R
19
Racecar Speed Control3
G(s) 
V (s)
100

C ( s ) ( s  2)( s  5)
c(t) throttle input
v(t) output racecar speed
(t) torque input
(t) output elbow angle
Robot Elbow Control3
G (s) 
( s )
2

( s ) s ( s  4)
20
Robot Position Control3
G ( s) 
V (s)
640, 000
 2
( s ) s  128s  6400
(t) torque input
v (t )  y (t ) robot velocity
y(t) robot displacement
Ship Stabilization3
G(s) 
(s)
9
 2
 f (s) s 1.2s  9
f(t) fin control input torque
(t) ship output angle
21
SkyCam Control3modified
G (s) 
Y (s)
1

T ( s ) s (0.2 s  1)
(t) torque input
y(t) output displacement
(t) torque input
 (t )  (t ) station velocity
Space Station Orientation Control3
G (s) 
( s )
20
 2
( s ) s  20 s  100
(t) space station angle
22
Space Telescope Pointing Control3
G (s) 
(s)
1

T ( s ) s ( s  12)
(t) torque wheel input
(t) output telescope angle
(t) input motor torque
y(t) output steel thickness
v0  2000 ft/min
nominal steel speed
Steel Rolling Thickness Control3
G (s) 
Y (s)
0.25

( s ) s ( s  1)
Turntable Angular Speed Control (old-school records)3
G( s) 
( s) KT / R

V ( s) Js  b
v(t) input voltage
 (t ) output turntable speed
K T motor torque constant
R circuit resistance
23
Vehicle Steering Control3
G (s) 
V (s)
1

( s ) s ( s  12)
(t) steering wheel angle
v (t )  y (t ) centerline velocity
y(t) centerline displacement
VTOL Aircraft Control3
G ( s) 
Y (s)
1

T ( s ) s ( s  1)
(t) thruster input
y(t) vertical displacement
24
Weld Bead Depth Control3
G (s) 
Y (s)
K

I ( s ) (0.01s  1)(1.5s  1)
i(t) input current
y(t) output weld bead depth
(t) torque input
v (t )  y (t ) robot velocity
Welding Robot Control3
G (s) 
V (s)
75( s  1)

( s ) ( s  5)( s  20)
y(t) robot displacement
25
Antenna Azimuth Control4
G (s) 
( s )
20.83

V ( s ) ( s  100)( s  1.71)
v(t) input voltage
 (t )  (t ) azimuth velocity
(t) azimuth angle
Autonomous Submersible Control5
G (s) 
4
5
( s )
0.13( s  0.44)
 2
 e ( s ) s  0.23s  0.02
e(t) input elevator angle
N.S. Nise, Control Systems Engineering, 2nd edition, Cummings, 1995.
Golnaraghi and Kuo, Automatic Control Systems, 9th edition, Wiley, 2010
(t) output pitch angle
26
Missile Roll Control5
G (s) 
l l
l
P(s)
    p
( s ) s  l p  a s  1
(t) aileron angle input
a  
1
lp
p(t) output roll rate
aerodynamic time constant
 l l p
steady-state gain
Robotic Rubbertuator and Load5 (McKibben Artificial Muscle)
G(s) 
X (s)
10
 2
P ( s ) s  10 s  29
p(t) input air pressure
x(t) output displacement
27
Robotic Swivel5
G(s) 
( s )
K
 2
V ( s ) ( s  4 s  10)
v(t) input voltage
(t) output swivel velocity
ti(t) input temperature
to(t) output temperature
Heat Transfer System5
G ( s) 
To ( s )
1

Ti ( s ) RCs  1
RC
thermal time constant
Pneumatic System5
G (s) 
Pb ( s )
1

P1 ( s ) RCs  1
p1(t) input pressure
pb(t) output pressure
q mass flow rate
RC pneumatic time constant
28
Magnetically-Levitated Ball6
G (s) 
X (s)
1
 2
I ( s ) As  B
x(t) output ball height
i(t) input current
A
i0
2g
B
i0
x0
i0 and x0 are the nominal current and ball height
at which the linearization was performed
Autonomous Automobile BackupDr. Bob
G ( s) 
X ( s)
K

F ( s ) s ( ms  c )
f(t) road force input
K
m
c
x(t) backup distance
open-loop gain
car mass
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
G (s) 
( s )
s  500

( s ) s ( s  0.03)( s 2  2.57 s  6667)
(t) torque input
(t) angle output
(t) torque input
(t) output pitch angle
Helicopter Pitch Control3
G(s) 
( s )
25( s  0.03)

( s ) ( s  0.4)( s 2  0.36 s  0.16)
30
Robot Force Control3
G ( s) 
F ( s)
K ( s  2.5)
 2
( s ) ( s  2 s  2)( s 2  4 s  5)
(t) torque input
f(t) force output
31
Third-Order and Fourth-Order Systems
less suitable for ME 3012 Term Projects
Ball-and-Beam System1
p(t
 Jb

p(t )  m g sin  (t )  m p(t ) (t )2  0
 r 2  m  


 m p(t )2  J  J b  (t )  2 m p(t ) p (t ) (t )  m g p(t ) cos  (t )   (t )
non-linear
)
 Jb

p(t )  m g (t )
 r 2  m  


 (t)
 m L2
 
 4  J  J b   (t )  m g p(t )


 (t)
 0
  (t )
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:
P(s)
 mg
G1 ( s ) 

T ( s ) J E1 J E 2 s 4  m 2 g 2
where:
J

J E1   2b  m 
r

JE2
 m L2


 J  Jb 
 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:
(s)
J E1s2
G2 (s) 

T (s) J E1J E 2 s4  m2 g 2
32
Missile Yaw Control3 (cannot use internal pre-filter for positive poles)
G (s) 
(s)
 0.5( s 2  2500)

 ( s ) ( s  3)( s 2  50 s  1000)
(t) torque input
(t) yaw acceleration
33
Printer Belt Drive3 (closed-loop)
X (s)

T (s)  1
Td ( s )
r
  s
J
 1 r2 
K m k1k2 r 
2k 
b 2
3
s    s  2k    s 
b 

mJ 
R 
J
m J 
d(t) disturbance torque input
m
k1
k2
r
L
b
R
Km
J  J motor  J pulley
x1 (t )  r (t )  y (t ) output displacement error
mass
light sensor
velocity feedback gain
radius
inductance
rotational viscous damping
resistance
torque constant
inertia
0.2 kg
1 V/m
0.1 Vs/m
0.15 m
0 (ignore)
0.25 Nms/rad
2
2 Nm/A
0.01 kg-m2
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
y1(t)
u1(t)
k1
m1
c1
y3(t)
y2(t)
u2(t)
k2
m2
c2
u3(t)
k3
c3
m3
k4
c4
m1 
y1 (t )  (c1  c2 ) y1 (t )  (k1  k2 ) y1 (t )  c2 y 2 (t )  k2 y2 (t )  u1 (t )
m2 
y2 (t )  (c2  c3 ) y 2 (t )  (k2  k3 ) y2 (t )  c2 y1 (t )  k2 y1 (t )  c3 y3 (t )  k3 y3 (t )  u2 (t )
m3 
y3 (t )  (c3  c4 ) y3 (t )  (k3  k4 ) y3 (t )  c3 y 2 (t )  k3 y2 (t )  u3 (t )
Non-linear Proof-Mass Actuator System1
q(t)
( M  m)q(t )  kq(t )  me((t ) cos  (t )   2 (t ) sin  (t ))  0
( J  me2 )(t )  meq(t ) cos  (t )  n(t )
M
n(t)
k
f(t)
e
J
 (t) m
Two-dof translational mechanical system1
y1(t)
u1(t)
k1
c1
m1
y2(t)
u2(t)
k2
c2
m2
m1 
y1 (t )  (c1  c2 ) y1 (t )  (k1  k2 ) y1 (t )  c2 y2 (t )  k2 y2 (t )  u1 (t )
m2 
y2 (t )  c2 y2 (t )  k2 y2 (t )  c2 y1 (t )  k2 y1 (t )  u2 (t )
35
Automobile Suspension System2
m1
x1 (t )  b1 ( x1 (t )  x2 (t ))  k1 ( x1 (t )  x2 (t ))  u(t )
m2 
x2 (t )  b2 ( x2 (t )  w (t ))  k2 ( x2 (t )  w(t ))  c2 y2 (t )  b1 ( x1 (t )  x2 (t ))  k1 ( x1 (t )  x2 (t ))  u(t )
Gu ( s ) 
X1 (s)  X 2 (s)
(m1  m2 ) s 2  b2 s  K 2

U ( s)
(m1s 2  b1s  K1 )(m2 s 2  (b1  b2 ) s  ( K1  K 2 ))  (b1s  K1 ) 2
Gw ( s ) 
X1 (s)  X 2 (s)
 m1 (b2 s  K 2 ) s 2

W ( s)
(m1s 2  b1s  K1 )(m2 s 2  (b1  b2 ) s  ( K1  K 2 ))  (b1s  K1 ) 2
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 4thorder.
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

y (t ) 
(k  k )
k2
kk

y (t )  1 2 y (t )  1 2 y (t ) 
b
m
mb
Fm (t ) k2
( FA (t )  Fm (t )  mg  k1 L1R )

m
mb
where:
y(t)
m
k1
k2
b
FA(t)
Fm(t)
g
L1R
7
absolute displacement of the muscle end
lumped muscle mass
linear spring stiffness representing the muscle fascia, parallel elastic component
linear spring stiffness representing the connecting tendons, series elastic component
viscous damping coefficient representing the muscle energy loss
muscle actuation force (the integrated effects of all contracting sarcomeres)
external load applied to the muscle
acceleration due to gravity
neutral length of the muscle (the unstretched length of spring k1).
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
L
n
R
cL
v A (t)
+
+
v B (t)
-
i A (t)
J L (t)
cM
JM
 M (t)
 M (t)
 M (t)
LJL (t )  ( Lc  RJ )L (t )  ( Rc  K T K B )L (t ) 
LJ L (t )  ( Lc  RJ ) L (t )  ( Rc  K T K B )L (t ) 
 L (t)
 L (t)
 L (t)
KT
vA (t )
n
KT
vA (t )
n
c
where J  J M  J L2 and c  cM  L2 are the effective polar inertia and viscous damping coefficient
n
n
reflected to the motor shaft.
Numerical Parameters
Parameter
L
R
kB
JM
bM
kT
n
JL
bL
Value
0.0006
1.40
0.00867
0.00844
0.00013
4.375
200
1
0.5
Units
H

V/deg/s
lbf-in-s2
lbf-in/deg/s
lbf-in/A
unitless
lbf-in-s2
lbf-in/deg/s
Name
armature inductance
armature resistance
motor back emf constant
motor shaft polar inertia
motor shaft damping constant
torque constant
gear ratio
load shaft polar inertia
load shaft damping constant
G ( s ) 
L (s)
VA ( s )

KT / n
5
 2
LJs  ( Lc  RJ ) s  ( Rc  K T K B ) s  11s  1010
G ( s ) 
L ( s)
VA ( s )

KT / n
5

2
s ( LJs  ( Lc  RJ ) s  ( Rc  K T K B )) s ( s  11s  1010)
2
2