MECHANICAL VIBRATIONS FUNDAMENTALS
Mechanical Vibrations
Dr. Jorge A. Olórtegui Yume, Ph.D.
MECHANICAL VIBRATIONS
FUNDAMENTALS
Lecture No. 2
Mechatronics Engineering School
National University of Trujillo
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2
Mech. Vibrations Fundamentals
Dr. Jorge A. Olortegui Yume, Ph.D.
MECHANICAL VIBRATIONS FUNDAMENTALS
MECHANICAL VIBRATIONS FUNDAMENTALS
BASIC ELEMENTS OF A VIBRATING SYSTEM
BASIC ELEMENTS OF A VIBRATING SYSTEM
Spring Elements
Spring Elements
Free Body Diagrams (FBD´s)
• Linear
• Mass and Damping negligible
• Restoring Force opossed to
deformation
Lo
Assume : x1 > x2
Lo
Fext
Fext
Fint
Fint
=
Externa
(Deforming)
Fext
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Mech. Vibrations Fundamentals
Fint
Internal
(Restoring)
Dr. Jorge A. Olortegui Yume, Ph.D.
Strectching
(“Coming out”)
Shrinking
(“Coming in”)
Spring FBD
Fs
FBD of Body attached to spring
Fs
x
Lf
Fs
Stretching
Fs
Lf
W
x
Spring Force
Potential Energy stored in spring
Fs  kx
1
E pot  kx 2
2
4
(“Coming out”)
N
Shrinking
W
Fs : Spring Force (in N)
Epot : Potential Energy (in J)
x : Spring elongation (in m)
k : Spring Constant or Stiffness (in N/m)
Mech. Vibrations Fundamentals
Fs
Fs
(“Coming in”)
N
Dr. Jorge A. Olortegui Yume, Ph.D.
MECHANICAL VIBRATIONS FUNDAMENTALS
MECHANICAL VIBRATIONS FUNDAMENTALS
BASIC ELEMENTS OF A VIBRATING SYSTEM
BASIC ELEMENTS OF A VIBRATING SYSTEM
Spring Combinations
In Parallel
Spring Combinations
In Series
= k
= k
eq
eq
• Equivalent spring can replace original system
• Equivalent spring can replace original system
• All elongations are equal
• Total elongation is summation of elongations
 st  1   2
• Forces in each spring are different
F1  k11  F1  k1 st
F2  k 2 2  F2  k 2 st
• Equilibrium
W  F1  F2  k11  k 2 2
keq st  F1  F2  k1 st  k 2 st
Mech. Vibrations Fundamentals

5
keq  k1  k 2
Dr. Jorge A. Olortegui Yume, Ph.D.
 st  1   2
• Forces in each spring are equal because of equilibrium
W
 st  1   2
F1  k11  W  k 21  k1  1
F2  k 2 2  W  k 2 2  k   2
W
2
Mech. Vibrations Fundamentals
W W W
  
keq k1 k 2
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W  F1  F2
1
1 1
 
keq k1 k 2
Dr. Jorge A. Olortegui Yume, Ph.D.
MECHANICAL VIBRATIONS FUNDAMENTALS
MECHANICAL VIBRATIONS FUNDAMENTALS
BASIC ELEMENTS OF A VIBRATING SYSTEM
BASIC ELEMENTS OF A VIBRATING SYSTEM
Spring Combinations in general
Example: Find the equivalent stiffness k of the following system diameter d = 2 cm
 Fs1  Fs 2  ...Fsn  Fseq
1   2  ...   n   eq
In Parallel
keq  k1  k 2  ...  k n
Special case
k1  k 2  ...k n  k
In Series
1   2  ...   n   eq
Springs in parallel and series:
Special case
n2
k1  k 2  ...k n  k
Special case
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k eq 
k2
k1
keq 
k1+k2+k5
m
keq  nk
Fseq
F
F
F
 s1  s 2  ... sn 
k1
k2
kn
keq
1
1 1
1
   ... 
keq k1 k 2
kn
Mech. Vibrations Fundamentals
Solution:
k11  k 2 2  ...k n n  keq eq
m
k3
k3
k5
k4
k4
k1k 2
k1  k 2
k
n
Dr. Jorge A. Olortegui Yume, Ph.D.
MECHANICAL VIBRATIONS FUNDAMENTALS
BASIC ELEMENTS OF A VIBRATING SYSTEM
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Mech. Vibrations Fundamentals
Dr. Jorge A. Olortegui Yume, Ph.D.
MECHANICAL VIBRATIONS FUNDAMENTALS
BASIC ELEMENTS OF A VIBRATING SYSTEM
Solution:
Your turn !!!
Exercise: Determine the equivalent spring constant of the system shown
k1+k2+k5
m
k3
k1+k2+k5
m
m

k4
kk
1
 3 4
1 1 k3  k4

k3 k4
kk
keq  k1  k 2  k5  3 4
k3  k 4
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Mech. Vibrations Fundamentals
Dr. Jorge A. Olortegui Yume, Ph.D.
MECHANICAL VIBRATIONS FUNDAMENTALS
BASIC ELEMENTS OF A VIBRATING SYSTEM
Solution:
Mech. Vibrations Fundamentals
10
Dr. Jorge A. Olortegui Yume, Ph.D.
MECHANICAL VIBRATIONS FUNDAMENTALS
BASIC ELEMENTS OF A VIBRATING SYSTEM
Masa suspendida al final de una viga en voladizo (Flexión):
Sistema Real o Situación Física
(a) Modelo de 1GDL (asume que no hay amortiguamiento c=0)
=
keq
De Resistencia de Materiales
Deflección Estatica al final de una viga en
voladizo debido a masa “m” en el extremo.
Asumir que masa de barra << “m”
 st 
Mech. Vibrations Fundamentals
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Dr. Jorge A. Olortegui Yume, Ph.D.
Wl 3 mg l 3

3EI
3EI
Mech. Vibrations Fundamentals
Ley de Hooke
F  k
k
12
F

Analogía
k
W
 st

3EI
l3
Dr. Jorge A. Olortegui Yume, Ph.D.
MECHANICAL VIBRATIONS FUNDAMENTALS
BASIC ELEMENTS OF A VIBRATING SYSTEM
Masa suspendida al final de una barra (Torsión):
MECHANICAL VIBRATIONS FUNDAMENTALS
BASIC ELEMENTS OF A VIBRATING SYSTEM
Constantes de Rigidez para otros Tipos Elementos Simples (Ejemplo)
(a) Sistema Real o Situación Física
(b) Modelo de 1GDL (asume que no hay amortiguamiento )
d
G
(a)
Gd 4
32 L
kt 
(b)
L
M
t 
De Resistencia de Materiales
Desplazamiento angular quasi-estatico al
final de una barra redonda debido a torque
“M” en el extremo.
 st 
M  kt
ML
32 ML

 d 4  G d 4

G 
 32 
ML

GI p
kt 
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Mech. Vibrations Fundamentals
Analogía
Ley de Hooke
kt 
M

M
 st

M
Gd 4

32ML
32 L
Gd 4
Dr. Jorge A. Olortegui Yume, Ph.D.
MECHANICAL VIBRATIONS FUNDAMENTALS
BASIC ELEMENTS OF A VIBRATING SYSTEM
Example: The figure shows the suspension system of a freight truck with a parallel
spring arrangement . Find the equivalent spring constant of the suspension if
each of the three helical springs is made of steel (G=80x109 N/m2) and has five
effective turns, mean coil diameter D =20 cm, and wire diameter d = 2 cm
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Mech. Vibrations Fundamentals
Dr. Jorge A. Olortegui Yume, Ph.D.
MECHANICAL VIBRATIONS FUNDAMENTALS
BASIC ELEMENTS OF A VIBRATING SYSTEM
Ejemplo: Determine the
torsional spring constant
of the steel propeller shaft
shown
Solution:
Solution:
The stiffness of each helical spring is:
k


Gd 4
80 109 0.02 

 40,000 N / m
3
8D 3n
80.20  5
4
Parallel spring arrangement:
keq  3k  340,000 N / m   120,000 N / m
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Mech. Vibrations Fundamentals
Dr. Jorge A. Olortegui Yume, Ph.D.
• Consider shaft by parts: 12 y 23
• Induced torque in any cross section of the shaft equal to the applied torque
“T” (draw imaginary sections at A-A y B-B)
• Segments 12 y 23 regarded as series springs
kt 12  GJ12  G D12  d12  
4
l12
4
32l12
Mech. Vibrations Fundamentals
MECHANICAL VIBRATIONS FUNDAMENTALS
80 10  0.3  0.2   25.53 10 N  m / rad
4
9
4
6
322 
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Dr. Jorge A. Olortegui Yume, Ph.D.
BIBLIOGRAPHY
BASIC ELEMENTS OF A VIBRATING SYSTEM
BASIC:
Solution: (cont´d)
•Thomson, W.T., Dahleh, M.D., 1997, “Teoria de Vibraciones con Aplicaciones”, Prentice Hall
Iberoamericana, 5ta Edición, México.
•Inman, D., 2007, “Engineering Vibration”, Prentice Hall, 3rd Edition, USA.
•Moore, H., 2008, “Matlab for Engineers”, Prentice Hall, 2nd Edition, USA.
ADDITIONAL:
•Balachandran, B., Magrab, E., 2006, “Vibraciones”, Thomson, 5ta Edición, México
•Rao, S.S., 2004, “Mechanical Vibrations”, Ed. Prentice Hall, 4th Edition, USA.
SPECIALIZED:
 



4
4
80 10  0.25  0.15
kt 23  GJ 23  G D23  d 23 
 8.9 10 6 N  m / rad
l23
32l23
323
Series spring
1

1

1
kt eq kt 12 kt 23
Mech. Vibrations Fundamentals
kt eq 
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4
4
•Hartog, D., 1974, “Mecánica de las Vibraciones”, Cecsa, Mexico.
•Harris, C., Piersol, A., 2001, “Harri´s Shock and Vibration Handbook”, McGraw Hill Professional,
5th Edition. USA.
kt 12 kt 23
 6.6  10 6 N  m / rad
kt 12  kt 23
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Dr. Jorge A. Olortegui Yume, Ph.D.
Mech. Vibrations Fundamentals
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Dr. Jorge A. Olortegui Yume, Ph.D.