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Dynamics IIDr. Jorge A. Olórtegui Yume, Ph.D.
MECHANICAL VIBRATIONS
FUNDAMENTALS
Mechanical Engineering School
National University of Trujillo
Lecture No. 2
Mech. Vibrations Fundamentals Dr. Jorge A. Olortegui Yume, Ph.D.2
MECHANICAL VIBRATIONS FUNDAMENTALS
Mech. Vibrations Fundamentals Dr. Jorge A. Olortegui Yume, Ph.D.3
MECHANICAL VIBRATIONS FUNDAMENTALSBASIC ELEMENTS OF A VIBRATING SYSTEM
Spring Elements
• Linear • Mass and Damping negligible• Restoring Force opossed todeformation
Assume : x1 > x2
Fext FextFint Fint
Fext = Fint
External Internal(Deforming) (Restoring)
Mech. Vibrations Fundamentals Dr. Jorge A. Olortegui Yume, Ph.D.4
MECHANICAL VIBRATIONS FUNDAMENTALSBASIC ELEMENTS OF A VIBRATING SYSTEMSpring Elements
Free Body Diagrams (FBD´s)
Strectching(“Coming out”)
Shrinking(“Coming in”)
Lo
Lo x
Fs
Lf
Fs
xLf
Fs Fs
Spring FBD
Shrinking
Fs
W
N
(“Coming in”)
Stretching
Fs
W
N
(“Coming out”)
FBD of Body attached to spring
Fs : Spring Force (in N)Epot : Potential Energy (in J)x : Spring elongation (in m)k : Spring Constant or Stiffness (in N/m)
Spring Force Potential Energy stored in spring
kxFs 2
2
1kxEpot
Mech. Vibrations Fundamentals Dr. Jorge A. Olortegui Yume, Ph.D.5
MECHANICAL VIBRATIONS FUNDAMENTALSBASIC ELEMENTS OF A VIBRATING SYSTEMSpring Combinations
In Parallel
• Equivalent spring can replace original system
• All elongations are equal
• Forces in each spring are different
• Equilibrium
keq=
21 st
111 kF stkF 22 stkF 11 222 kF
221121 kkFFW
stststeq kkFFk 2121 21 kkkeq
Mech. Vibrations Fundamentals Dr. Jorge A. Olortegui Yume, Ph.D.6
MECHANICAL VIBRATIONS FUNDAMENTALSBASIC ELEMENTS OF A VIBRATING SYSTEMSpring Combinations
In Series
• Equivalent spring can replace original system
• Total elongation is summation of elongations
• Forces in each spring are equal because of equilibrium
=
21 st
21 FFW
111 kF 12kW
22kW 222 kF
1
1
k
W
2
2
k
W
21 st
21 k
W
k
W
k
W
eq 21
111
kkkeq
keq
Mech. Vibrations Fundamentals Dr. Jorge A. Olortegui Yume, Ph.D.7
MECHANICAL VIBRATIONS FUNDAMENTALSBASIC ELEMENTS OF A VIBRATING SYSTEMSpring Combinations in general
neq kkkk ...21
eqn ...21
eqeqnn
seqsnss
kkkk
FFFF
...
...
2211
21In Parallel
nkkeq Special case
kkkk n ...21
neq kkkk
1...
111
21
kkkk n ...21n
kkeq
2n21
21
kk
kkkeq
eqn ...21
eq
seq
n
snss
k
F
k
F
k
F
k
F ...
2
2
1
1In Series
Special case
Special case
Mech. Vibrations Fundamentals Dr. Jorge A. Olortegui Yume, Ph.D.8
MECHANICAL VIBRATIONS FUNDAMENTALSBASIC ELEMENTS OF A VIBRATING SYSTEM
Solution:
Example: Find the equivalent stiffness k of the following system diameter d = 2 cm
Springs in parallel and series:
k1k2
k3
k4
m
k5
k3
k4
m
k1+k2+k5
Mech. Vibrations Fundamentals Dr. Jorge A. Olortegui Yume, Ph.D.9
MECHANICAL VIBRATIONS FUNDAMENTALSBASIC ELEMENTS OF A VIBRATING SYSTEM
Solution:
m
43
43521
kk
kkkkkkeq
m
k1+k2+k5
1
1
k31
k4
k3k4k3 k4
k3
k4
m
k1+k2+k5
Mech. Vibrations Fundamentals Dr. Jorge A. Olortegui Yume, Ph.D.10
MECHANICAL VIBRATIONS FUNDAMENTALSBASIC ELEMENTS OF A VIBRATING SYSTEM
Exercise: Determine the equivalent spring constant of the system shown
Your turn !!!
Mech. Vibrations Fundamentals Dr. Jorge A. Olortegui Yume, Ph.D.11
MECHANICAL VIBRATIONS FUNDAMENTALSBASIC ELEMENTS OF A VIBRATING SYSTEM
Solution:
Mech. Vibrations Fundamentals Dr. Jorge A. Olortegui Yume, Ph.D.12
MECHANICAL VIBRATIONS FUNDAMENTALSBASIC ELEMENTS OF A VIBRATING SYSTEMMasa suspendida al final de una viga en voladizo (Flexión):
Mech. Vibrations Fundamentals Dr. Jorge A. Olortegui Yume, Ph.D.13
MECHANICAL VIBRATIONS FUNDAMENTALSBASIC ELEMENTS OF A VIBRATING SYSTEMMasa suspendida al final de una barra (Torsión):
Mech. Vibrations Fundamentals Dr. Jorge A. Olortegui Yume, Ph.D.14
MECHANICAL VIBRATIONS FUNDAMENTALSBASIC ELEMENTS OF A VIBRATING SYSTEMConstantes de Rigidez para otros Tipos Elementos Simples (Ejemplo)
Mech. Vibrations Fundamentals Dr. Jorge A. Olortegui Yume, Ph.D.15
MECHANICAL VIBRATIONS FUNDAMENTALSBASIC 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
Mech. Vibrations Fundamentals Dr. Jorge A. Olortegui Yume, Ph.D.16
MECHANICAL VIBRATIONS FUNDAMENTALSBASIC ELEMENTS OF A VIBRATING SYSTEM
Solution:
Ejemplo: Determine the torsional spring constant of the steel propeller shaft shown
Mech. Vibrations Fundamentals Dr. Jorge A. Olortegui Yume, Ph.D.17
MECHANICAL VIBRATIONS FUNDAMENTALSBASIC ELEMENTS OF A VIBRATING SYSTEM
Solution: (cont´d)
Mech. Vibrations Fundamentals Dr. Jorge A. Olortegui Yume, Ph.D.18
BIBLIOGRAPHY
BASIC:•Thomson, W.T., Dahleh, M.D., 1997, “Teoria de Vibraciones con Aplicaciones”, Prentice HallIberoamericana, 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:•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.
