What is a single degree of freedom (SDOF) system ?
Hoe to write and solve the equations of motion?
How does damping affect the response?
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Single Degree of Freedom Systems
Mohammad Tawfik
References
• M. Bismarck-Nasr, "Structural Dynamics in Aeronautical
Engineering," AIAA Educational Series, 1999
• D. Inman and E. Austin, “Engineering Vibration,” 2nd edition,
Prentice Hall, 2001
• A. A. Shabana, "Vibration of Discrete and Continuous Systems," 2nd
edition, Springer, 1997
• D. Thorby, “Structural Dynamics and Vibration in Practice” Elsevier,
2008
• A. G. Ambekar, “Mechanical Vibrations and Noise Engineering”
Prentice Hall – India, 2006
• Leonard Meirovitch, “Fundamentals of Vibrations,” 1st edition,
McGraw Hill, 2001
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Single Degree of Freedom Systems
Mohammad Tawfik
Objectives
• Recognize a SDOF system
• Be able to solve the free vibration equation
of a SDOF system with and without
damping
• Understand the effect of damping on the
system vibration
• Apply numerical tools to obtain the time
response of a SDOF system
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Single Degree of Freedom Systems
Mohammad Tawfik
Single degree of freedom systems
• When one variable can describe the
motion of a structure or a system of
bodies, then we may call the system a 1-D
system or a single degree of freedom
(SDOF) system. e.g. x(t), q(t) Z(t), y(x).
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Single Degree of Freedom Systems
Mohammad Tawfik
Solving the ODE
• The ODE is
• The proposed
solution:
• Into the ODE you get
the characteristic
equation:
• Giving:
0)()( tkxtxm
t
aetx
)(
02
tt
ae
m
k
ae
m
k
2
m
k
j
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Single Degree of Freedom Systems
Mohammad Tawfik
Solving the ODE (cont’d)
• The proposed
solution becomes:
• For simplicity, let’s
define:
• Giving:
t
m
k
jt
m
k
j
eaeatx
21)(
m
k
tjtj
eaeatx
21)(
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Single Degree of Freedom Systems
Mohammad Tawfik
Manipulating the solution
• The solution we have:
• Rewriting:
tjtj
eaeatx
21)(
tjSintCosa
tjSintCosatx
2
1)(
tSinaajtCosaatx 2121)(
tSinAtCosAtx 21)(
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Single Degree of Freedom Systems
Mohammad Tawfik
Further manipulation
tSinAtCosAtx 21)(
2
2
2
1 AAA
A
A
Sin
A
A
Cos 12
&
tSinCostCosSinAtx )(
tASintx )(
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Single Degree of Freedom Systems
Mohammad Tawfik
Natural Frequency of Oscillation
• In the previously obtained solution:
• The frequency of oscillation is
• It depends only on the characteristics of the
oscillating system. That is why it is called the
natural frequency of oscillation
tASintx )(
m
k
n
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Single Degree of Freedom Systems
Mohammad Tawfik
Frequency
periodtheiss
2
Hz
2s2
cycles
rad/cycle2
rad/s
frequencynaturalthecalledisrad/sinis
n
nnn
n
n
T
f
We often speak of frequency in Hertz or
RPM, but we need rad/s in the arguments
of the trigonometric functions.
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Single Degree of Freedom Systems
Mohammad Tawfik
Amplitude & Phase from the ICs
Phase
0
01
Amplitude
2
02
0
0
0
tan,
yieldsSolving
cos)0cos(
sin)0sin(
v
xv
xA
AAv
AAx
n
n
nnn
n
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Single Degree of Freedom Systems
Mohammad Tawfik
Some useful quantities
peak valueA
T
T
dttx
T
x
0
valueaverage=)(
1
lim
valuesquaremeanroot=2
xxrms
valuesquare-mean=)(
1
lim
0
22
T
T
dttx
T
x
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Single Degree of Freedom Systems
Mohammad Tawfik
Samples of Vibrating Systems
• Deflection of continuum (beams, plates,
bars, etc) such as airplane wings, truck
chassis, disc drives, circuit boards…
• Shaft rotation
• Rolling ships
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Single Degree of Freedom Systems
Mohammad Tawfik
Bars
• Longitudinal motion
• A is the cross sectional
area (m2)
• E is the elastic modulus
(Pa=N/m2)
• l is the length (m)
• k is the stiffness (N/m)x(t)
m
EA
k
l
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Single Degree of Freedom Systems
Mohammad Tawfik
Rods
• Jp is the polar
moment of inertia of
the rod
• J is the mass
moment of inertia of
the disk
• G is the shear
modulus, l is the
length
Jp
J qt)
0
pGJ
k
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Single Degree of Freedom Systems
Mohammad Tawfik
Helical Spring
2R
x(t)
d = diameter of wire
2R= diameter of turns
n = number of turns
x(t)= end deflection
G= shear modulus of
spring material
3
4
64nR
Gd
k
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Single Degree of Freedom Systems
Mohammad Tawfik
Summary
• Write down the equation of motion using
Newton’s law
• Solve the equation of motion for a SDOF
• Use initial conditions to determine the
amplitude and phase of vibration for a
SDOF
• Evaluate the effective stiffness of
structural members
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Single Degree of Freedom Systems
Mohammad Tawfik
1. The amplitude of vibration of an undamped
system is measured to be 1 mm. the phase
shift is measured to be 2 rad and the
frequency 5 rad/sec. Calculate the initial
conditions.
2. Using the equation:
evaluate the constant A1 and A2 in terms of
the initial conditions
HW #1
tSinAtCosAtx 21)(
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Single Degree of Freedom Systems
Mohammad Tawfik
HW #1 (cont’d)
3. An automobile is modeled as 1000 kg
mass supported by a stiffness k=400000
N/m. When it oscillates, the maximum
deflection is 10 cm. when loaded with the
passengers, the mass becomes 1300 kg.
calculate the change in the frequency,
velocity amplitude, and acceleration if the
maximum deflection remain 10 cm.
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Single Degree of Freedom Systems
Mohammad Tawfik
Objectives
• Understand the damping as a force
resisting motion
• Adding viscous damping to the equation of
motion of a SDOF
• Understand the difference in the
responses of different systems depending
on the value of the damping
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Single Degree of Freedom Systems
Mohammad Tawfik
Damping
• Damping is some form of friction!
• In solids, friction between molecules result
in damping
• In fluids, viscosity is the form of damping
that is most observed
• In this course, we will use the viscous
damping model; i.e. damping proportional
to velocity
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Single Degree of Freedom Systems
Mohammad Tawfik
Solution (dates to 1743 by Euler)
0)()(2)( 2
txtxtx nn
km
c
2
=
Where the damping Ratio
is given by: (dimensionless)
Divide the equation of motion by m
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Single Degree of Freedom Systems
Mohammad Tawfik
rootstheofnaturethe
determines,1ntdiscriminatheHere
equationquadraticaofrootsthefrom
1
:inequationalgebraicannowiswhich
02
motionofeq.intosubsitute&)(Let
2
2
2,1
22
nn
t
n
t
n
t
t
aeeaea
aetx
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Single Degree of Freedom Systems
Mohammad Tawfik
Three possibilities:
00201
21
,
:conditionsinitialtheUsing
)(
221=
dampedcriticallycalled
repeated&equalareroots1)1
xvaxa
teaeatx
mkmcc
n
tt
ncr
nn
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Single Degree of Freedom Systems
Mohammad Tawfik
12
)1(
12
)1(
where
)()(
1
:rootsrealdistincttwo-damping-overcalled,1)2
2
0
2
0
2
2
0
2
0
1
1
2
1
1
2
2,1
22
n
n
n
n
ttt
nn
xv
a
xv
a
eaeaetx nnn
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Single Degree of Freedom Systems
Mohammad Tawfik
Summary
• Modeling viscous damping
• Solving the equation of motion involving
viscous damping
• Recognizing the different types of
response based on the level of damping
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Single Degree of Freedom Systems
Mohammad Tawfik
1. Use the given data to plot the response of the
SDOF system
2. Solve the equation
And plot the response
HW #1 (cont’d)
8.0,6.0,4.0,2.0,1.0,01.0
/0,1sec,/2 00
smvmmxradn
0,1
0
00
vx
xxx