Ch01_ Introduction 2015

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Mechanical

Vibrations

Singiresu S.

Rao

SI Edition

Chapter 1 Fundamentals of Vibration

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1. Fundamentals of Vibration

2. Free Vibration of Single DOF Systems

3. Harmonically Excited Vibration

4. Vibration under General Forcing

Conditions

5. Two DOF Systems

6. Multidegree of Freedom Systems

7. Determination of Natural Frequencies

and Mode Shapes

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8. Continuous Systems

9. Vibration Control

10. Vibration Measurement and Applications

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1.1 Preliminary Remarks

1.2 Brief History of Vibration

1.3 Importance of the Study of Vibration

1.4 Basic Concepts of Vibration

1.5 Classification of Vibration

1.6 Vibration Analysis Procedure

1.7 Spring Elements

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1.8 Mass or Inertia Elements

1.9 Damping Elements

1.10 Harmonic Motion

1.11 Harmonic Analysis

Chapter Outline

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1.1 Preliminary Remarks

• Examination of vibration’s important role

• Vibration analysis of an engineering system • Definitions and concepts of vibration

• Concept of harmonic analysis for general periodic motions

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1.3 Importance of the Study of Vibration

• Why study vibration?

 Vibrations can lead to excessive deflections

and failure on the machines and structures

 To reduce vibration through proper design of

machines and their mountings

 To utilize profitably in several consumer and

industrial applications

 To improve the efficiency of certain machining,

casting, forging & welding processes

 To stimulate earthquakes for geological

research and conduct studies in design of nuclear reactors

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1.4 Basic Concepts of Vibration

 _Vibration_{= any motion that repeats itself after}

an interval of time

 _{Vibratory System}_{consists of:}

1) spring or elasticity 2) mass or inertia

3) damper

_Involves_transfer_{of potential energy to kinetic}

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1.4 Basic Concepts of Vibration

 _{Degree of Freedom (d.o.f.)}₌

min. no. of independent coordinates required to determine completely the positions of all

parts of a system at any instant of time

_{Examples of}_single_{degree-of-freedom}

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1.4 Basic Concepts of Vibration

_{Examples of}_single_{degree-of-freedom}

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1.4 Basic Concepts of Vibration

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1.4 Basic Concepts of Vibration

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1.4 Basic Concepts of Vibration

_{Example of}_Infinite-

number-of-degrees-of-freedom system:

_{Infinite number of degrees of freedom system}

are termed continuous or distributed systems

_{Finite number of degrees of freedom are}

termed discrete or lumped parameter systems

_{More accurate results obtained by}_increasing

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1.5 Classification of Vibration

_Free_Vibration:

A system is left to vibrate on its own after an

initial disturbance and no external force acts on the system. E.g. simple pendulum

_Forced_Vibration:

A system that is subjected to a repeating

external force. E.g. oscillation arises from diesel engines

_Resonance_{occurs when the frequency of the}

external force coincides with one of the natural frequencies of the system

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1.5 Classification of Vibration

_Undamped_Vibration:

When no energy is lost or dissipated in friction or other resistance during oscillations

_Damped_Vibration:

When any energy is lost or dissipated in

friction or other resistance during oscillations

_Linear_Vibration:

When all basic components of a vibratory system, i.e. the spring, the mass and the damper behave linearly

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1.6 Vibration Analysis Procedure

Step 1: Mathematical Modeling

Step 2: Derivation of Governing Equations Step 3: Solution of the Governing Equations Step 4: Interpretation of the Results

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1.6 Vibration Analysis Procedure

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Example 1.1

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_{Using mathematical model to represent the}

actual vibrating system

E.g. In figure below, the mass and damping

of the beam can be disregarded; the system can thus be modeled as a spring-mass

system as shown.

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1.9 Damping Elements

_Viscous_Damping:

Damping force is proportional to the velocity of the vibrating body in a fluid medium such as air, water, gas, and oil.

_Coulomb_or_{Dry Friction}_Damping:

Damping force is constant in magnitude but

opposite in direction to that of the motion of the vibrating body between dry surfaces

_Material_or_Solid_or_Hysteretic_Damping:

Energy is absorbed or dissipated by material during deformation due to friction between internal planes

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Example 1.10 Equivalent Spring and Damping Constants of a Machine Tool Support

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Let the total forces acting on all the springs and all the dampers be F_sand F_d, respectively (see Fig. 1.37d). The force equilibrium equations can thus be expressed as

Example 1.10 Solution

E.1)

(

4 ,

3 ,

2 ,

1 ;

4 ,

3 ,

2 ,

1 ;



i

x

c

F

i

x

k

F

i di i si



E.2)

(

4 3 2 1 4 3 2 1 d d d d d s s s s s

F









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Example 1.10 Solution

E.3)

(

x

c

F

x

k

F

eq d eq s





where F_s + F_d = W, with W denoting the total

vertical force (including the inertia force) acting on the milling machine. From Fig. 1.37(d), we have

Equation (E.2) along with Eqs. (E.1) and (E.3), yield

E.4)

(

4

4 3 2 1 4 3 2 1

c

k

eq eq











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 

1.1

kx

F 

F = spring force,

k = spring stiffness or spring constant, and x = deformation (displacement of one end

with respect to the other)

1.7 Spring Elements

_Linear_{spring is a type of mechanical link that is}

generally assumed to have negligible mass and damping

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_{Work done (U)}_{in deforming a spring or the}

strain (potential) energy is given by:

 

1.2 2 1 ₂ kx U 

1.7 Spring Elements

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1.7 Spring Elements

_{Static deflection}_{of a beam at the free end is}

given by:

_{Spring Constant}_{is given by:}

 

1.6 3 3 EI Wl st  

W = mg is the weight of the mass m, E = Young’s Modulus, and

I = moment of inertia of cross-section of beam

 

1.7 3 3 l EI W k st   

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1.7 Spring Elements

_{Combination of Springs:}

1) Springs in parallel – if we have n spring

constants k1, k2, …, kn in parallel, then the equivalent spring constant keq is:

1.11

2

1 ... n

eq k k k

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1.7 Spring Elements

_{Combination of Springs:}

2) Springs in series – if we

have n spring constants k1,

k2, …, kn in series, then the equivalent spring constant keq is:



1.17



1 ... 1 1 1 2 1 n eq k k k k    

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

1.30



sin sin A t A x    



1.31



cos t A dt dx   



1.32



sin 2 2 2 2 x t A dt x d        1.10 Harmonic Motion

_Periodic_{Motion: motion repeated after equal}

intervals of time

_Harmonic_{Motion: simplest type of periodic}

motion

_{Displacement (x): (on horizontal axis)} _Velocity:

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_{Complex number}_{representation of harmonic}

motion:

where i = √(–1) and a and b denote the real and imaginary x and y components of X,

respectively.



1.35



ib a X   1.10 Harmonic Motion

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_Definitions_{of Terminology:}

_{Amplitude (A) is the maximum displacement}

of a vibrating body from its equilibrium position

_{Period of oscillation (T) is time taken to}

complete one cycle of motion

_{Frequency of oscillation (f) is the no. of}

cycles per unit time



1.59



2    T



1.60



2 1     T f 1.10 Harmonic Motion

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1.10 Harmonic Motion

_Definitions_{of Terminology:}

_{Natural frequency is the frequency which a}

system oscillates without external forces

_{Phase angle (}_{) is the angular difference}

between two synchronous harmonic motions











1.62



sin 61 . 1 sin 2 2 1 1       t A x t A x