Chapter 2 SDOF Vibration Control
2.1 Transfer Function
mxɺɺ(t ) + cxɺ (t ) + kx(t ) = F (t )
⇒ Defines the transfer function as output over input
X (s)
F (s )
= G(s) =
1
ms + cs + k
2
(1.39)
s is a complex number: s = α ± j β = −ζωn ± jωd
1
Complex s-Plane of the Poles
jω n
ωd
− ωn
Poles: the roots of
the denominator of G(s):
σ
− ζω n
ms 2 + cs + k = 0
⇒ s1,2 = −ζωn ± jωn 1 − ζ 2
2
Block Diagram Representation
X(s)
1
= 2
U(s) ms + cs + k
U(s)
Input
Plant or
Structure
Output
1
ms2 + cs + k
X(s)
SDOF
General System
3
Frequency Response Function (FRF)
4
Bode Plots: Magnitude and Phase
• Fig 1.15, phase vs
frequency for
various damping
ratios
• Fig 1.16, magnitude
vs frequency for
various damping
rations
5
2.2 Measurement and Testing
• Damping must be measured
dynamically.
• Mass and stiffness can be
measured in static
experiments.
• Free decay allows the
exponent to be measured,
and hence the damping ratio.
6
Damping Measured by Time Response
x(t )
Definition of log decrement: δ = ln
x(t + Td )
e −ζωnt sin(ωd t + φ )
⇒ δ = ln −ζωn (t +Td )
e
sin(ωd t + ωd Td + φ )
= ln(eζωnTd ) = ζωnTd
⇒ δ = ζω nTd =
⇒ζ=
2πζ
1-ζ 2
δ
4π 2 + δ 2
(1.46)
(1.47)
(1.48)
7
Frequency Domain Estimates of
Mass, Damping and Stiffness
log G ( jω )
ω →0
2
2
2

 ω   2ζω  
1 1
1
= log − log 1 − 2  + 
  = log
k 2
k
 ωn   ωn  
(1.49)
8
Damping Measured by
Frequency Response Function (FRF)
At the half-power points
1  ω2 − ω1 
ζ = 

2  ωd 
9
2.3 system Stability
Stability is defined for the solution of free response
case:
Stable:
x(t ) < M , ∀ t > 0
lim
x(t ) = 0
Asymptotically Stable:
t →∞
Unstable: if it is not stable or asymptotically stable
10
Stability
sin 2 . t
x t
y t
.
e 0.1 t . x t z t
.
e0.1 tr t
Stable
.
z t x t
Asymptotically
Stable
1
1
x t
0
5
10
y t
0
1
5
10
1
t
Fig 1.19
Fig 1.20
t
3
4
2
z t
2
r t
0
5
10
2
1
0
5
t
10
Divergent instability
4
t
Flutter instability
11
2.4 Vibration Control
• Design refers to choosing m, c and k in order to
produce a more desirable response (or ζ and ωn).
• Often this involves working with the spring constants
of various materials.
• If vibration suppression is the goal, then situations
arise when adjusting m, c and k still does not produce
the desired response, then control methods are used.
12
Shape the Response by m, c, and k
• Design a system so that its response as a desired
settling time, overshoot and time to peak.
OS == e
tp =
Fig 1.10
ts =
 −ζπ 


2
 1−ζ 


π
ωn 1 − ζ 2
3.2
ωnζ
fixes ζ
fixes ωn
is determined
13
Vibration Control
• If m, c and k cannot be adjusted to suit design goals, then control
is a possibility
• Control consists of adding hardware to affect the
response in some way
• Passive control: a fixed device or permanent change
in physical parameters (such as constrained layer
damping)
• Active control: use of some external adjustable or
active (electronic) device, called an actuator, to
provide a means of shaping or controlling the
response
14
Active Control
• Open Loop Control: the control force applied to the
system is independent of any measurement.
• Closed Loop Control: the control force depends in
some way on a measurement of the system (requires
a sensor).
15
Open Loop Control
U(s)
K
G(s)
X(s)
Choose K such that X(s) has a desired shape,
called constant gain (K) control.
X ( s)
K
= KG ( s ) = 2
⇒
U ( s)
ms + cs + k
mɺxɺ(t ) + cxɺ (t ) + kx(t ) = Ku (t )
16
Closed Loop Control
F(s)
+
-
K
G(s)
structure
X(s)
H(s)
Control law
X(s)
KG(s)
=
F(s) (1+ KG(s)H(s))
17
State Feedback
X(s)
KG(s)
=
F(s) (1+ KG(s)H(s))
State feedback:
⇒
H(s) = g1s + g2 ⇒
X(s)
K
=
F(s) ms 2 + (Kg1 + c )s + (Kg2 + k )
mxɺɺ(t ) + ( Kg1 + c) xɺ (t ) + ( Kg 2 + k ) x(t ) = KF (t )
Now we have two more parameters to adjust
in order to meet the desired performance.
18
Feedforward Control
disturbance
system
Error signal
filter
Mostly used in acoustics
and high frequency
applications.
19
An Inverted Pendulum (Example)
2


kl
2 ɺɺ
ml θ (t ) + 
− mg  θ (t ) = F (t )
 2

Need to find bounded F(t)
Choose F (t ) = −aθ (t ) − bθɺ(t ) , then
2


kl
2 ɺɺ
ɺ
ml θ (t ) + bθ (t ) + 
− mgl + a  θ (t ) = 0
 2

AS and BIOB if
kl 2
b > 0, and
− mgl + a > 0
2
AS: asymptotic stable
BIOB: bounded input and bounded output
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2.5Vibration of Nonlinear Systems
ɺxɺ(t ) + f ( x, xɺ ) = 0
k = 1000, b = 10
k
1000 b
10
g ( xg) x= k ⋅kx. x f x
f ( x) = kx − bx 3
k.x
b . x3
k.x
h x
b . x3
Hardening Spring
5000
h( x) = kx + bx 3
3750
2500
f x
Softening Spring
1250
g x
4
h x
2
0
2
4
1250
2500
3750
5000
Linear Spring
x
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State Space Formulation
 xɺ1   x2

xɺ =   = 
= F ( x)

 xɺ2  − f ( x1 , x2 )
Define the equilibrium point x e by
F (x e ) = 0
x1 and x2 are called state variables
22
A Nonlinear System of Soft Spring
Compute the equilibrium for:
ɺɺ
x + x − β 2 x3 = 0
First order form:
xɺ1 = x2
xɺ2 = x1 ( β 2 x12 − 1)
23
Equilibrium points
x2 = 0
x1 ( β x − 1) = 0 ⇒
2
2
1
0 
x ee =   ,
0 
1
 ,
β
 
 0 
 −1 
 
β
 
 0 
24
Multiple Equilibrium
• Each equilibrium has a potentially different
solution with a different stability behavior.
• One equilibrium may be stable another not.
• Depending on the initial conditions, the
solution may favor the response of one
equilibrium versus another so that the
concept of stability must be associated with
the equilibrium.
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