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    Mechanical Vibrations-1 PDF

    Uploaded by

    Hrvoje Lakić
    This document provides recommended reading materials and course content for an introduction to vibration course taught by Jiraphon Srisertpol in the School of Mechanical Engineering. The recommended textbooks cover topics such as mechanical vibration, fundamentals of vibration, and free and forced vibration of single and multiple degree of freedom systems. The course content includes an introduction to vibration modeling, derivation of governing equations, solution methods, and interpretation of results. Key concepts that will be covered are free vibration, damped vibration, linear and nonlinear vibration, and deterministic and random vibration analysis. Mathematical modeling, Laplace transforms, and numerical methods will be used to obtain solutions.

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    Mechanical Vibrations-1 PDF

    Uploaded by

    Hrvoje Lakić
    269 views14 pages
    This document provides recommended reading materials and course content for an introduction to vibration course taught by Jiraphon Srisertpol in the School of Mechanical Engineering. The recommended textbooks cover topics such as mechanical vibration, fundamentals of vibration, and free and forced vibration of single and multiple degree of freedom systems. The course content includes an introduction to vibration modeling, derivation of governing equations, solution methods, and interpretation of results. Key concepts that will be covered are free vibration, damped vibration, linear and nonlinear vibration, and deterministic and random vibration analysis. Mathematical modeling, Laplace transforms, and numerical methods will be used to obtain solutions.

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    Mechanical_Vibrations-1[1].pdf

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    This document provides recommended reading materials and course content for an introduction to vibration course taught by Jiraphon Srisertpol in the School of Mechanical Engineering. The recommended textbooks cover topics such as mechanical vibration, fundamentals of vibration, and free and forced vibration of single and multiple degree of freedom systems. The course content includes an introduction to vibration modeling, derivation of governing equations, solution methods, and interpretation of results. Key concepts that will be covered are free vibration, damped vibration, linear and nonlinear vibration, and deterministic and random vibration analysis. Mathematical modeling, Laplace transforms, and numerical methods will be used to obtain solutions.

    Copyright:

    Attribution Non-Commercial (BY-NC)
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    269 views14 pages

    Mechanical Vibrations-1 PDF

    Uploaded by

    Hrvoje Lakić
    This document provides recommended reading materials and course content for an introduction to vibration course taught by Jiraphon Srisertpol in the School of Mechanical Engineering. The recommended textbooks cover topics such as mechanical vibration, fundamentals of vibration, and free and forced vibration of single and multiple degree of freedom systems. The course content includes an introduction to vibration modeling, derivation of governing equations, solution methods, and interpretation of results. Key concepts that will be covered are free vibration, damped vibration, linear and nonlinear vibration, and deterministic and random vibration analysis. Mathematical modeling, Laplace transforms, and numerical methods will be used to obtain solutions.

    Copyright:

    Attribution Non-Commercial (BY-NC)
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    of 14
    At a glance
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    The key takeaways are about modeling vibration systems using different elements like springs, masses and dampers and discussing different types of vibration systems and damping.

    Discrete and continuous systems are discussed as well as single, two and infinite degree of freedom systems.

    Viscous, dry friction, hysteretic and structural damping are discussed.

    Jiraphon Srisertpol, Ph.

    D
    School of Mechanical Engineering

    Recommended reading :
    
    ...  : !"#, Pearson Education
    Indochina 2545
    
    Singiresu S.Rao : Mechanical Vibration (Fourth Edition) ,Prentice Hall
    2004. SI Edition
    
    Leonard Meirovitch : Fundamentals of Vibrations , Mc-Graw Hill 2001.
    
    Kelly S. Graham : Fundamentals of Mechanical Vibrations,
    Mc-Graw Hill 2000.

    Introduction to Vibration and The


    Free Response
    
    
    
    

    The Spring-Mass model


    Single degree of freedom
    Simple harmonic motion
    Relationship between Displacement, Velocity
    and Acceleration
    Representations of harmonic motion

    School of Mechanical Engineering

    The Vibration of a Fixed-Fixed String

    School of Mechanical Engineering

    The main mass and dynamic absorber at three


    frequencies.

    Vibration Characteristics of a
    Spring

    Fundamental Torsional Mode of a


    Valve Support Stand

    Deflected Elastomer Shock


    Isolation

    Structural Vibration

    10

    !"#$%&')*+,!)-.&'%&')/0.$1'2*))
    
    
    
    

    11

    
     
     ! "" #$% &'()*+#+
    ,)-.."""  .
    /" 
    .)01# */2 "% #$""

    12

    
      
    

    
      

    #BCDEFGHI (Degree of Freedom, DOF) - LF


    M(Coordinate) !OPQ!OR!OSGTCUDTVU"LHIP"WP
    BQXYT"T"Z C"!RTFWBB!OC[HF#YU"

    Single degree of freedom system

    13

    14

    Three degree of freedom system

    Two degree of freedom system

    15

    16

    Infinite number of degree of freedom


    system

    
      
    

    Discrete System (Lumped System)-""*'#/


    
    $(#(//+
    '/ " 2*' /#
    Continuous System (Distributed System)- ""* *
    '/ " 2)-
    (' /

    17

      (Free Vibration)

     
    
    
    
    
    
    
    
    

    18

     ."" (Free Vibration)


     ."""  " (Forced Vibration)
     .""(*# (Undamped Vibration)
     .""*# (Damped Vibration)
     ."", (Linear Vibration)
     .""(, (Nonlinear Vibration)
     .""'#/(/ (Deterministic Vibration)
     ."" (Random Vibration)

    19

    $ ""
    *#
    *"""*# +/
    +*//
     2. $'#/ 2.
       2/'
    ()%/+(*.3+' """*
    +
    """)-'#/ / #$'#/
    4#$ 2 ."" 

    20

    XBBSGTGOEFGYTF"
    (Undamped Vibration)

    XBBB"EB (Forced Vibration)


    

    $ ""3+.'3+ &.'3+*2


    )-.
    &2'#$(&2' 4(/
     
    *2 ,  $(/
    $ */
    #
    *
    .""*2()   "*5,""  *2*

    **, (amplitude)  *  *+ 


    *2
       (Resonance)

    
    

    #+ *(*1*+
    # "./
    "" (+
    ).*+/ #$.$/
    $""
    $*.""(*#'#
    ""#
    
    $**2**
     *(*#""/2(/ 2
     .""(*#.
     .""*""*+
    *5, (Natural Frequency)

    21

    22

        (Linear Vibration)

    XBBGOEFGYTF" (Damped Vibration)


    

    #+  ** 1*+


    #/
    $*
    "" (/+#/4
    %/+ ().
     3 )- 2 )- .""*
    #." 2 2

    23

    "" &)"/+ 
    ) .
     # *
    !)
    *+.)
    .'()+, "
    +#$4  /*,
     ."",, #
    '.#
    (Principle of Superposition)

    24

    XBBSGTHIH"HP (Nonlinear Vibration)


    

    XBBYSP (Deterministic Vibration)

    "" &)"/+ 
    ) .
     # *
    !)
    *+.)
    .'()+()-,
    "+#$4  /*,

    ""* / +3+.'3+


    
    /4 .*'+ 2'#//.(/
    #$"  56. & )-706, 
    *.
     2'

    25

    CDFHEY!"FFG
    (Vibration Analysis Procedure)

    XBBRTG (Random Vibration)


    

    26

    ""* / +3+.'3+


    
    /4 .*'""('#//
    .(/

    
    
    
    

    27

    .""'
     6 (Mathematical Modeling)
    #
    $* (Derivation of Governing Equations)
    #8
    9
    +
    $* (Solution of Governing Equations)
    #68
    *(/ (Interpretation of the Results)

    28

    P"XBBL#"!"E[
    (Mathematical Modeling)
    

    
    
    

    YGHE#`!O
    (Derivation of Governing Equations)

    :"$2""*/  %/+.**
    /+.""'
    +3 ,
    ..#
    
     +6 .""""/+)(spring )
    .*'#/1*+
    /+ #(damped)
    .*)-.#
    
    
    6 /+ 
    (mass) , 9$+

    
    
    

    . 56
    )
    *+)
    )
    & (Laplace)s transform)
    5*, 
     (Numerical method)

    Dynamic System Modeling and Analysis, Hung V Vu and Ramin S. Esfandiari,


    McGraw-Hill 1998

    29

    30

    FHEYb#!OSP
    (Interpretation of the Results)

    Yb#Hc#QGHE#`!O
    (Solution of Governing Equations)

    .
    $* "" ".""'
    *
    2&(/#
    +5* ,
     ;
    $* 
     ; 6
    
    

    *
    *
    *

    31

    5"+.
    )8

    56*(/
    /.*'
    +
    $(#
    4, 

    32

    Mathematical Model of
    Motorcycle

    Vibration Analysis Procedure

    ."'
    *+*/

    Single-degree of freedom model *./) b.

    keq {kt , kr , ks } equivalentstiffness.


    ceq {cs , cr } equivalentdamping constant.
    meq {mr , mv , mw } equivalent mass
    r rider ,t tires, s struts , v vehicle body, w wheels,

    33

    Mathematical Model of
    Motorcycle

    34

    Mathematical Model of
    Motorcycle

    r rider ,t tires, s struts ,v vehicle body, w wheels,

    r rider ,t tires, s struts ,v vehicle body, w wheels,

    35

    36

    Mathematical Model of
    Motorcycle

        !"# $


    Springs Elements
     Mass or Inertia Elements
     Damping Elements
    

    r rider ,t tires, s struts ,v vehicle body, w wheels,

    37

    38

    Spring Elements
    
    
    
    
    
    

    Stiffness (N/m)
    Youngs modulus (N/m)
    Density (kg/m)
    Shear modulus G(N/m)
    Springs in series
    Springs in parallel

    ).) 6/ ""**/#+*, $


    
     +6""
    Spring force

    F = kx

    k spring contant or spring stiffness


    x displacement(deformation)

    Potential energy in the spring :

    39

    U=

    1 2
    kx
    2

    40

    Springs in Parallel

    Springs in Series
    1. Static of the system ( st )

    Equilibrium equation

    W = k1 st + k 2 st

    st = 1 + 2

    2. Equilibrium equation W = k11

    W = keq st

    W = k2 2

    where keq = k1 + k 2

    3. keq for the same static deflection

    W = keq st

    k11 = k 2 2 = keq st
    or

    Equivalent spring constant (keq ) in parallel

    Equivalent spring constant (keq ) in series

    keq = k1 + k 2 + k3 + + k n

    1
    1 1 1
    1
    = + + ++
    keq k1 k 2 k3
    kn

    1 =

    that is,

    keq st
    k1

    , 2 =

    keq st
    k2

    1
    1 1
    = +
    keq k1 k 2

    41

    42

    Damping Elements

    EX: Springs in Parallel

    shear modulus G = 80 109 N m 2

    mean coil diameter D = 20 cm


    wire diameter d = 2 cm

    
    The stiffness of helical spring is given by

     #$#$/ (Viscous Damping)


     #$.*+/#.4 ".4 (Dry
    Friction or Coulomb Damping)
     #$(+/$ #+ / (Hysteretic Damping or
    Structural Damping)

    d 4G (0.02 ) 80 109
    =
    40,000 N m
    k=
    3
    8D 3n
    8(0.2 ) 5
    4

    The equivalent spring constant of the suspension system is given by

    keq = 3k = 3 40,000 = 120,000 N m


    43

    44

    Viscous Damping

    Damping

    All real systems dissipate energy when they vibrate. To


    account for this we must consider damping. The most simple
    form of damping (from a mathematical point of view) is called
    viscous damping. A viscous damper (or dashpot) produces a
    force that is proportional to velocity.
    Mostly a mathematically motivated form, allowing
    a solution to the resulting equations of motion that predicts
    reasonable (observed) amounts of energy dissipation.

    Damper (c)
    x

    f c = cv(t ) = cx (t )

    fc
    45

    46

    Viscous Damping
    

    Equivalentdampingconstant (ceq ) in series

    1 1 1
    1
    = + +
    ceq c1 c2
    cn

    
    

    Damper
    Damping coefficient
    Critical damping coefficient
    Damping ratio

    Equivalentdampingconstant (ceq ) in parallel


    ceq = c1 + c2 + + c3

    47

    48

    Ex: Horizontal milling machine


    
    
    

    Underdamped Motion
    Overdamped Motion
    Critically Damped Motion

    49

    Ex: Horizontal milling machine

    50

    Ex: Horizontal milling machine


    Fs = keq x

    Fsi = ki x ; i = 1,2,3,4
    Fdi = ci x ; i = 1,2,3,4

    Fd = ceq x

    Fs = Fs1 + Fs 2 + Fs 3 + Fs 4
    Fd = Fd1 + Fd 2 + Fd 3 + Fd 4

    where

    keq = k1 + k2 + k3 + k4
    ceq = c1 + c2 + c3 + c4

    Fs + Fd = W total vertical force

    G center of mass , Fsi forces acting on the springs , Fdi forces acting on the dampers
    Fs forces acting on all the springs , Fd forces acting on all the dampers

    51

    52

    
    
    
    
    

    Newtons second law


    Conservation of Energy
    Potential Energy
    Kinetic Energy
    Natural frequency

    53

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