2. SECOND ORDER SYSTEM
A linear time invariant second-order system is
described by a second-order differential equation:
Other Names:
Quadratic Lag
Second Order Diff. Equations
cx
dt
dx
b
dt
x
d
a
t
f
2
2
)
( …………..(A)
3. EXAMPLE OF
SECOND ORDER SYSTEM
Classical example from mechanics
A block of mass W resting on a horizontal, frictionless
table is attached to a linear spring.
A viscous damper (dashpot) is also attached to the
block.
Assume the system is free to oscillate horizontally
under the influence of a forcing function F(t).
The origin of the coordinate system is taken as the
right edge of block when the spring is in the relaxed
or un-stretched condition.
At time zero, block is assumed to be at rest at this
origin.
Positive directions for force and displacement are
indicated by the arrows in Fig. 1.
SYSTEM EXPLANATION:
5. EXAMPLE OF
SECOND ORDER SYSTEM
Consider the block at some instant when it is to the
right of Y = 0 and when it is moving toward the right
(positive direction).
Under these conditions, the position Y and the
velocity dY/dt are both positive.
At this particular instant, the following forces are
acting on the block:
1. The force exerted by the spring (toward the left) of
-KY where K is a positive constant, called Hooke’s
constant.
2. The viscous friction force (acting to the left) of
-C dY/dt, where C is a positive constant called the
damping coefficient.
3. The external force F(t) (acting toward the right).
SYSTEM EXPLANATION:
6. EXAMPLE OF
SECOND ORDER SYSTEM
Consider the block at some instant when it is to the
right of Y = 0 and when it is moving toward the right
(positive direction).
Under these conditions, the position Y and the velocity
dY/dt are both positive.
At this particular instant, the following forces are acting
on the block:
1. The force exerted by the spring (toward the left) of -
KY where K is a positive constant, called Hooke’s
constant.
2. The viscous friction force (acting to the left) of -C
dY/dt, where C is a positive constant called the
damping coefficient.
3. The external force F(t) (acting toward the right).
FORCES ACTING ON THE SYSTEM:
7. EXAMPLE OF
SECOND ORDER SYSTEM
Newton’s law of motion, which states that
the sum of all forces acting on the mass is equal to the
rate of change of momentum (mass X acceleration),
Takes the form:
Rearrangements gives
Where
W=mass of block, lbm
Gc=32.2(lbm)(ft)/(lbf)(sec2)
C = viscous damping coefficient, lbf/(ft/sec)
K = Hooke’s constant, lbf/ft
F(t)= driving force, a function of time, lbf
MOMENTUM BALANCE:
)
(
2
2
t
F
dt
dY
C
KY
dt
y
d
g
W
c
…………………(1)
)
(
2
2
t
F
KY
dt
dY
C
dt
y
d
g
W
c
………………(2)
8. EXAMPLE OF
SECOND ORDER SYSTEM
Dividing Eq. (2) by K gives
For convenience, this is written as
Where
MOMENTUM BALANCE:
K
t
F
Y
dt
dY
K
C
dt
y
d
K
g
W
c
)
(
2
2
………………(3)
)
(
2
2
2
2
t
X
Y
dt
dY
dt
y
d
………………(4)
K
g
W
c
2
…….……………….…………….(5)
K
C
2 ……...…………….…………….(6)
K
t
F
t
X
)
(
)
( ……...…………….…………….(7)
9. EXAMPLE OF
SECOND ORDER SYSTEM
Solving for τ and ζ from Eq. (5) and (6) gives:
K
g
W
c
(sec) …………(8)
WK
C
gc
4
2
(Dimension less)…………(9)
10. TRANSFER FUNCTION
The second order equation is:
The above equation is written in a standard form that
is widely used in control theory.
If the block is motionless (dY/dt=0) and located at
its rest position (Y=0) before the forcing function is
applied, the Laplace transform of Eq(4) becomes:
TRANSFORM OF EQ(4)
)
(
2
2
2
2
t
X
Y
dt
dY
dt
y
d
………………(4)
)
(
)
(
)
(
2
)
(
2
2
s
X
s
Y
s
sY
s
Y
s
…………(10)
1
2
1
)
(
)
(
2
2
s
s
s
X
s
Y
…………(11)
