vibration_course_section1

Noise and Vibration Control

UG Course Noise and Vibration ControlCourse Objectives:To provide the student with a basic understanding of fundamental

concepts in noise and vibration control engineering. This would enable the student to analyze sound fields and to

determine the effects of different noise sources in machinery and engineered products with respect to human behavior and safety.

To be able to utilize these concepts in order to design machines or products that are quiet and functional

To be able to identify and correct potentially hazardous sound levels in the workplace or in any other noisy environment.

Credits: 4 Weekly Engagement: 3-1-2/2 Course Teacher: VHS

Indian Institute of Technology Roorkee


References:Mats Abom et al, Sound and Vibration, KTH, Stockholm, 2006.S S Rao, Mechanical Vibration, (4th Edition) Pearson Education, Delhi, 2004.J S Rao and K Gupta, Theory and Practice of Mechanical Vibration, (2nd Edition) New Age International Publishers, New Delhi, 1999.de Silva, Vibration: Fundamentals and Practice, (2nd Edition) CRC Taylor & Francis, FL USA, 2007.Fahy, F.J. and Walker, J.G.,’ Fundamentals of Noise and Vibration, E & FN, Spon, 1998

Indian Institute of Technology Roorkee 2/58


Contents:

Review of Vibration Fundamentals from a Practical Perspective.

Structural DampingExpanded Understanding of Vibration Isolation.Sounds in Ducts. Industrial Noise and Vibration Control. Fluid Machine Noise and Vibration Control.Noise and Vibration in Pipes and Ducts. Vehicle Noise and Vibration Control.Active Noise and Vibration Control.

Case Study.



1. Review of Vibration fundamentals from a practical perspective



Need for study

All mechanical systems composed of mass, stiffness and damping elements exhibit vibratory response when subject to time-varying disturbances. The prediction and control of these disturbances is fundamental to the design and operation of mechanical equipment.

Indian Institute of Technology Roorkee5/58


Terminology

Mechanical system. A mechanical system is composed of distributed elements which exhibit characteristics of mass, elasticity and damping.Degrees of freedom. The number of degrees of freedom of a system is equal to the number of independent coordinate positions required to completely describe the motionof the system.System response. All mechanical systems exhibit some form of vibratory response when excited by either internal or external forces. This motion may be irregular or mayrepeat itself at regular intervals, in which case it is called periodic motion.Period. The period T is the time taken for one complete cycle of motion.



Terminology

Harmonic motion is the simplest form of periodic motion whereby the actual or observed motion can be represented by oscillatory functions such as the sine and the cosine functions. Motion that can be described by a continuous sine or cosine function is called steady state.

Frequency is the number of cycles per second (also called hertz) of the motion and is the reciprocal of the period. Therefore frequency is specified by



Examples of SDOF



Examples of SDOF

Translatory

Rotatory

Flexural

swinging



Review of Vibration fundamentals

Single degree of freedom

The common techniques for the analysis are1. The energy method, 2. Newton's law of motion,3. The frequency response method4. The superposition theorem



Single Degree of Freedom Systems

Figure 2 shows a mechanical single degree-of-freedom (“sdof”) system consisting of a rigid mass m, a spring with spring rate k, and a viscous damper with a damping coefficient dv. The spring and the viscous damper are located between the mass and the foundation, and are considered to be massless. That implies that the forces on the opposing endpoints of each are equal and oppositely directed, for both elements.

κ dv

m

x(t)F (t)

Figure 2 Single Degree-of-Freedom System.



Newton’s Second Law gives the equation of motion of the system

),)(),(()(2

2t

dttdxtxF

dttxdm x=

Fx contains the spring force, the damper force, and external exciting force

(7)

)()()( tFdt

tdxdtxF vx +−−= κ (8)

where m is mass of the body, κ is the spring constant, dv is the viscous damping coefficient, F(t) is the external excitation, x is the displacement of the mass, dx / dt its velocity, d 2x / dt2 its acceleration. These two equations lead to a second order linear differential equation with constant coefficients

)()()(2)( 202

2tgtx

dttdx

dttxd

=++ ωδ (9)



in which the following simplifications have been incorporated:

mtFtg )()( =mκω =0 mdv 2=δ (10)

where ω0 is the eigenfrequency of the system, and δ is the damping constant. The solution to the differential equation consists of both a homogeneous part xp(t) that corresponds to the homogeneous differential equation, i.e., with the right hand side equal to zero, and a particular solution xp(t) that corresponds to the non-homogeneous differential equation, i.e., with the right hand side non-zero.

)()()( txtxtx ph += (11)

Because the system is linear, its particular solution, when the exciting force is described by the rotating vector Eq.(12), represents an oscillation at the excitation frequency, but with a different phase and amplitude. A reasonable assumption for xp is given by Eq.(13),



tiegt ωˆ)( =gti

ptii

pp eeext ωωϕ xx ˆˆ)( ==

(12)

(13)

That assumed form, substituted into Eq.(9), provides the following result:

titip

tip

tip egeeie ωωωω ωδωω ˆˆˆ2ˆ 2

02 =++− xxx

The phase and magnitude of the complex amplitude px̂ is given by

δωωω 2)(

ˆˆ22

0 ig

p+−

=x

ϕipp exx ˆˆ =

(14)

(15)

(16)



22220 )2()(

ˆˆδωωω +−

=g

px (17)

πωω

δωϕ n+−

=20

22arctan , n = 0, 1, 2, … (18)

• From Eqs.(10) and (15), it is apparent that forω « ω0, the stiffness κ determines the displacement. Thus, the low frequency response is stiffness-controlled.

• On the other hand, for ω » ω0, the mass m determines the displacement response; the high frequency response, therefore, is mass-controlled.

• Finally, for ω ≈ ω0, the value of the viscous damping coefficient dν is decisive for the displacement; the response at frequencies around the natural frequency is therefore said to be damping-controlled.



The magnitude of the amplitude px̂ varies with circular frequency ω.

A normalized response, called the amplification factor φ, can be defined as

)0(ˆ)(ˆ

==

ω

ωφ

p

p

x

x⇒

20

20

220 )()(4))(1(

1

ωωωδωωφ

+−= (19)



Two degree-of-freedom systems• The simple single degree-of-freedom system can be coupled to another of its kind, producing a mechanical system described by two coupleddifferential equations; to each mass, there is a corresponding equation of motion. • To specify the state of the system at any instant, we need to know time tdependence of both coordinates, x1 and x2, from which follows the designation two degree-of-freedom system.

m 1 m 2

x 1 ( t) x 2 ( t)

F 1 ( t) F 2 ( t)κ 1

d v 1

κ 2

d v 2

κ 3

d v 3

Figure 3 Two degree--of-freedom system.



Newton’s second law for each mass gives

⎟⎠

⎞⎜⎝

⎛= t

dttdx

dttdx

txtxFdt

txdm x ,

)(,

)(),(),(

)( 212112

12

1 …20

⎟⎠

⎞⎜⎝

⎛= t

dttdx

dttdx

txtxFdt

txdm x ,

)(,

)(),(),(

)( 212122

22

2 …21

)()()()(

))()(()( 121

21

1212111 tFdt

tdxdt

tdxd

dttdx

dtxtxtxFx +⎟⎠

⎞⎜⎝

⎛ −−−−−−= ννκκ

…22

)()()()(

)())()(( 22

321

2232122 tFdt

tdxd

dttdx

dttdx

dtxtxtxF x +−⎟⎠

⎞⎜⎝

⎛ −+−−= ννκκ

…23



Equations (20) - (23) give

+⎟⎟⎠

⎞⎜⎜⎝

⎛−++

dttdx

dttdx

ddt

tdxd

dt

txdm

)()()()( 212

112

12

1 νν

( ) )()()()( 121211 tFtxtxtx =−++ κκ …24

−+⎟⎟⎠

⎞⎜⎜⎝

⎛−−

dttdx

ddt

tdxdt

tdxd

dt

txdm

)()()()( 23

2122

22

2 νν

)()())()(( 223212 tFtxtxtx =+−− κκ …25

Matrix and vector notation can be incorporated into Eqs. (24) and (25), which is useful for generalizing to an arbitrary number of degrees-of-freedom.



Equations (24) and (25) are therefore expressed as

[ ] [ ] [ ] Fxdtxd

dtxd rr

rr

=⋅+⋅+⋅ KDM2

2

where

[ ] ⎥⎦

⎤⎢⎣

⎡=

2

10

0m

mM …27

…26

[ ] ⎥⎦

⎤⎢⎣

⎡+−

−+=

322

221

ννν

νννddd

dddD …28

[ ] ⎥⎦

⎤⎢⎣

⎡+−

−+=

322

221κκκ

κκκK …29

⎭⎬⎫

⎩⎨⎧

=)()(

)(2

1txtx

txr

…30



⎭⎬⎫

⎩⎨⎧

=)()(

)(2

1tFtF

tFr

Once again, let the excitation forces and the particular solutions be expressed by rotating vectors

tiet ω11

ˆ)( FF =

…31

…32tiet ω

22ˆ)( FF = …33

tiet ω1p1p xx ˆ)( =

tiet ωpp xx 22 ˆ)( =

Putting (32,33,34,35) into (26) gives

…34

…35

[ ] { } [ ] { } [ ] { } { }FxKxDxM ˆˆˆˆ2 =⋅+⋅+⋅− ppp iωω …36



Solving to the homogeneous equations with the force vector set equal to zero leads to the system’s eigenfrequencies. Setting, moreover, the damping matrix equal to zero, in order to obtain the undamped eigenfrequencies, the latter are found to be real. Damping, on the other hand, brings about complex-valued eigenfrequencies; the complex values contain information on both the undamped eigenfrequencies and the system damping. The eigenfrequencies ω1 and ω2 are given by the homogeneous equation

[ ] { } [ ] { } { }0ˆˆ2 =⋅+⋅− xKxMω …37

The condition for the existence of solutions to Eq. (37) is that the system determinant is identically zero, i.e.,

[ ] [ ] 0)det( 2 =+− KMω …38



For a two degree-of-freedom system, Eq. (38) has two solutions corresponding to two eigenfrequencies. A system with n degrees-of-freedom has n eigenfrequencies. The eigenfrequencies of the two degree-of-freedom system are

( ) ( )21

32312122

22

232

21

221

2

32

1

212,1 24422 mmmmmm

κκκκκκκκκκκκκκκω

−−−+

++

+±

++

+=

…39

From linear algebra, it is known that there is an eigenvector corresponding to each eigenvalue (eigenfrequency). These eigenvectors are mutually independent (orthogonal), and contain information on how the system oscillates in the vicinity of their respective eigenfrequencies. The mode shapes, x1 and x2, are obtained by substituting the eigenfrequencies, i.e., the solutions of Eq. (38), into Eq. (37), yielding



[ ] { } [ ] { } { }0ˆˆ 1121 =⋅+⋅− xKxMω …40

[ ] { } [ ] { } { }0ˆˆ 2222 =⋅+⋅− xKxMω …41



System with an arbitrary number of degrees-of-freedom

The results from the two degree-of-freedom system can be generalized to a system with an arbitrary number of masses cascaded, i.e., coupled in series, as in Figure 5-5.

m1

x1(t)

m2

x2(t)

F1(t) F2(t) κ1

dv1

κ2

dv2

κn+1

dvn+1

mn

xn(t)

• • •

Fn(t)

Figure 5 System with n cascaded masses



The equations of motion become

+⎟⎟⎠

⎞⎜⎜⎝

⎛−++

dttdx

dttdx

ddt

tdxd

dt

txdm

)()()()( 212

112

12

1 νν

( ) )()()()( 121211 tFtxtxtx =−++ κκ …42

−⎟⎟⎠

⎞⎜⎜⎝

⎛−+⎟⎟

⎠

⎞⎜⎜⎝

⎛−−

dttdx

dttdx

ddt

tdxdt

tdxd

dt

txdm

)()()()()( 323

2122

22

2 νν

…43,)())()(())()(( 2323212 tFtxtxtxtx =−+−− κκ

−⎟⎟⎠

⎞⎜⎜⎝

⎛−+⎟⎟

⎠

⎞⎜⎜⎝

⎛−− −−−

−−

− dttdx

dttdx

ddt

tdxdt

tdxd

dt

txdm nn

nnn

nn

n)()()()()( 112

121

2

1 νν

,)())()(())()(( 11121 tFtxtxtxtx nnnnnnn −−−−− =−+−− κκ …44



+⎟⎟⎠

⎞⎜⎜⎝

⎛−++ −

+ dttdx

dttdx

ddt

tdxd

dt

txdm nn

nn

nn

n)()()()( 1

12

2

νν

( ) .)()()()( 11 tFtxtxtx nnnnnn =−++ −+ κκ …45

The mass matrix, damping matrix, and stiffness matrix, respectively, become

[ ]⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

nm

mm

0000

0000

2

1

L

OM

M

L

M …46



Indian Institute of Technology Roorkee

[ ]

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

+−⋅⋅⋅−+−⋅⋅

⋅•••⋅⋅⋅⋅••−⋅⋅−+−⋅⋅⋅−+

=

+

−−

1

11

3

3322

221

00

00

0

nnn

nnnnddd

dddd

ddddd

ddd

ννν

νννν

ν

νννν

ννν

D …47

[ ]

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

+−⋅⋅⋅−+−⋅⋅

⋅•••⋅⋅⋅⋅••−⋅⋅−+−⋅⋅⋅−+

=

+

−−

1

11

3

3322

221

00

00

0

nnn

nnnnκκκ

κκκκ

κκκκκ

κκκ

K …48

28/58


where non-zero elements not shown in the equations are marked with a •, and zero-valued elements are marked with a ⋅. One can even allow masses to be coupled in parallel, as in Figure 6.

m 1

x 1( t)

F 1( t)κ 1

d v 1

m 4

x 4( t)

F 4( t) κ 6

d v 6

κ 2

d v 2 κ 3

d v 3

κ 4

d v 4

κ 5

d v 5

m 2

m 3

F 2( t)

F 3( t)

x 3( t)

x 2( t)

Figure 6 System with parallel coupling



The equations of motion become

+⎟⎟⎠

⎞⎜⎜⎝

⎛−++

dttdx

dttdx

ddt

tdxd

dt

txdm

)()()()( 212

112

12

1 νν

( )+−++⎟⎟⎠

⎞⎜⎜⎝

⎛−+ )()()(

)()(21211

313 txtxtx

dttdx

dttdx

d κκν

( ) ,)()()( 1313 tFtxtx =−+ κ …49

−⎟⎟⎠

⎞⎜⎜⎝

⎛−+⎟⎟

⎠

⎞⎜⎜⎝

⎛−−

dttdx

dttdx

ddt

tdxdt

tdxd

dt

txdm

)()()()()( 424

2122

22

2 νν

…50,)())()(())()(( 2424212 tFtxtxtxtx =−+−− κκ



−⎟⎟⎠

⎞⎜⎜⎝

⎛−+⎟⎟

⎠

⎞⎜⎜⎝

⎛−−

dttdx

dttdx

ddt

tdxdt

tdxd

dt

txdm

)()()()()( 435

3132

32

3 νν

,)())()(())()(( 3435313 tFtxtxtxtx =−+−− κκ …51

+⎟⎟⎠

⎞⎜⎜⎝

⎛−++

dttdx

dttdx

ddt

tdxd

dt

txdm

)()()()( 244

462

42

4 νν

( )+−++⎟⎟⎠

⎞⎜⎜⎝

⎛−+ )()()(

)()(24456

345 txtxtx

dttdx

dttdx

d κκν

( ) .)()()( 4345 tFtxtx =−+ κ …52



The mass matrix, damping matrix and stiffness matrix, respectively, become

[ ]⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

4

3

2

1

000000000000

mm

mm

M …53

[ ]⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

++−−−+−−+−

−−++

=

65454

5533

4422

32321

00

00

ννννν

νννν

νννν

ννννν

ddddddddddddd

ddddd

D …54

[ ]⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

++−−−+−−+−

−−++

=

65454

5533

4422

32321

00

00

κκκκκκκκκκκκκ

κκκκκ

K …55



The general principle for generating these matrices, for systems in which the directions of forces and velocities are defined as in figures 5 and 6, can be summarized in the following way:

(i) the mass matrix is diagonal.

(ii) a diagonal element in the stiffness or damping matrix is the sum of the spring rates or damping coefficients, respectively, of all springs / dampers connected to the mass indicated by the row number of theelement.

(iii) an off-diagonal element at a specific row and column position in the stiffness or damping matrix has the opposite (negative) of the value of the spring rate or damping coefficient, respectively, for the connection between the mass indicated by the row number and that indicated by the column number.



Free motion of SDOF systems

Based upon the observation that mechanical systems respond harmonically in freemotion, the solution of equation can be assumed to be of the form given b equation Therefore we assume that the actual motion can be described as

where Ar and A are real amplitudes of motion.Substitution of equation differentiating with respect totime and eliminating common terms, provides a relation for the frequency ton at which the system will naturally vibrate.



Free motion of SDOF systems

This motion can also be written as

the phase angle is specified from equation

amplitude of motion that results from equation



Damped motion of SDOF systems

All vibrations in realistic systems occur with some form of damping mechanism, where the energy of vibration is dissipated during a cycle of motion. The simplest form of damping is when the resisting force associated with the damping is proportional to, and acts in an opposite direction to, the velocity of the element. Thus the damping force isspecified by

where C is the damping coefficient. SDOF system with this form of damping which is called viscous damping.



Damped motion of SDOF systems

response will oscillate at a damped natural frequency

The actual displacement is then given by

where the phase angle is now given by

and the real amplitude by



Inclusion of damping into the relationship for the forced response modifies the resulting amplitude to



The corresponding phase response is plotted in Fig. For very light damping the phase of the displacement response relative to the excitation force of the system flips through nearly a 180 ° phase change as the excitation frequency is increased through the resonance frequency.



Multi-degree-of-freedom (MDOF) systems

When N independent coordinates are required to completely specify the system response, the system is said to have N degrees of freedom. Such a system is also said to have multi-degrees of freedom (MDOF). When a mechanical system has many degrees of freedom it is more convenient to use a matrix representation to describe and analysesthe motion. In this section we formulate the equations of motion in matrix form.



Adding Mass, Stiffness and DampingAn increase in the systems stiffness reduces the displacement amplitude proportionately. An increase in k of course implies an increased value of but this makes the approximation even more valid. At such a low frequency, any increase in the damping will have a negligible effect.



Adding Mass Stiffness and Damping

Increase in the system mass (inertia) reduces the displacement amplitude proportionately. It may again be noted that an increase in m makes the approximationeven more valid. An increment in the Damping will have a negligible effect in the displacement amplitude in this range of frequency as well. This high-frequency range is termed as the inertia-controlled region.



Adding Mass Stiffness and Damping



Energy storage elements

Inertia (m)Consider an inertia element of lumped mass excited by force as shown in Figure. The resulting velocity is v.

Newton's second law gives

Kinetic energy stored is



Spring (K)

Consider a massless spring element of lumped stiffness k , as shown in Figure One end of the spring is fixed and the other end is free. A force ‘f’ is applied at the free end

By hookes law

Elastic potential energy



Gravitational potential energy

The work done in raising an object against the gravitationalenergy of the object. Consider a lumped mass as shownfrom some reference level.



The Energy Method

A vibrating system is said to be conservative if, damper and external excitations are absent from the system interval's when a conservative s systems set into oscillatory motion, the mechanical energy is partly kinetic and partly potential. the kinetic energy is stored in mass due to its velocity



The Energy Method

kinetic energy in Newtonian notations is



The Energy Method

As is clear from the figure, the total deflection produced in the unstrained spring , work done is

the total work done in deforming the spring from

potential energy of the system is



The Energy Method

the total energy in the conservative system

Differentiating with respect to time

Homogeneous second-order ordinary differential equation of motion



Method Based on Newton's Second Law of Motion

The rate of change of momentum is proportional to the impressed force and takes place in the direction in which the force acts



Rayleigh's MethodSince the sum of kinetic and potential energies always constant it follows from the energy method that the potential energy is maximum when the kinetic energy is zero and conversely when the kinetic energy is maximum, the potential energy must be zero.



Checklist of vibration control means



LAGRANGE'S EQUATIONSNo vibrations discussion is complete without a discussion of Lagrange's equations. All of you have already taken a dynamics class or two, which in the vast majority of cases has involved Newtonian mechanics.



LAGRANGE'S EQUATIONS

LAGRANGE'S EQUATIONS is giving by



LAGRANGE'S EQUATIONS

where KE and PE are the system kinetic and potential energies, respectively, and L is called the Lagrangian



Thank you


vibration_course_section1

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