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Structure

Objectives

17.1 INTRODUCTION

In the last unit, you were explained forced vibration of mechanical systems. Systemshave forced vibration due to an exciting force. The force of excitation may acting due toeither external agency or may be developed in the system due to unbalance.

The system may also be excited due vibration of the base. The base may be having

vibrations due to neighbouring machine which is transmitting large amount of force tothe ground. This results in disturbance of other machines. To measure vibrations, suitableinstruments are required. The design principle can be evolved by analyzing this type ofvibration.

It is always desirable that least force is transmitted to the ground so that neighbourhoodmachines are not disturbed. Therefore, there is necessity to control and isolate vibration.

Objectives

After studying this unit, you should be able to

analyse the vibration due to base excitation, understand principles of design of vibration instruments, control vibrations transmitted to the ground, and isolate vibration.

17.2 VIBRATION DUE TO THE BASE EXCITATION

A spring mass system is shown in Figure 17.1 in which the base of suspension isvibrating with the function.

. . . (17.1)0 siny y t=

0

The equation of forces can be written as

. . . (17.2)( ) ( )mx c x y k x y+ + =

or mx cx k x c y k y+ + = +

SARDAR PATEL INSTITUTE OF TECHNOLOGY,PILUDARA

SUB: D.O.M BY: H.A.PATELBRANCH: 6 thMECH.VIBRATION MEASURING INSTRUMENTS

1 Introduction2 Vibration due to the Base Excitation3 Vibration Measuring Instruments

3.1 Principle of Design of Seismometer3.2 Principle of Design of Accelerometer3.3 Domain of Operation of Seismometer and Accelerometer

4 Force Transmitted to the Ground5 Transmissibility and Vibration Isolation

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or 0 0sin coscwmx cx kx k y t y t+ + = +

or 2 20{ ( ) }sin (mx cx kx y k c t+ + = + + ) . . . (17.3)

where 1tanc

k

=

. . . (17.4)

is phase angle between base excitation and force.

Let 2 20 0( )y k c F+ =

. . . (17.5)0 sin ( )mx cx kx F t+ + = +

y=yo sint

k

Figure 17.1

As explained in the last unit, steady state the solution of the above equation is given by

sin ( )x X t= +

is phase angle between forceF0 and displacement

where

20

0 1F c

X DM DM yk k

= = +

or

2

0

21

n

X DM y

= +

. . . (17.6)

The angle () is the phase difference between the base excitation and vibration ofmass m.

Figure 17.2

The response of mass m and phase angle have been plotted in Figure 17.2. Thefollowing points may be noted.

x

c

m

= 0

increasing

1.0

x/y

o

2 /n1.0

/2

/n

= 0.1

0.2

0.3

1.0

increasing

= 0

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(a) The response is either equal to the excitation or greater than it for 2n

.

(b) At 1n

=

, the response is very large for small amount of damping.

(c) For 2n

>

, the response is smaller than the excitation and damping has

very little effect. Damping increases response.

SAQ 1

When the damping is most effective and when it is disadvantageous?

17.3 VIBRATION MEASURING INSTRUMENTS

The vibration response can be expressed in terms of various parameters, e.g.displacement, velocity, acceleration and induced stress. The selection of the parameterdepends on the objective and the field of application. For a vehicle designer the comfortof the passenger is important, therefore, vibration in a vehicle is normally expressed interms of acceleration. To measure vibration a transducer is used which generates voltageproportional to the magnitude of parameter.

A vibration equipment can be thought of a spring mass system enclosed in a box type ofa thing. It is shown in Figure 17.3. The box is fitted on the vibrating base. The dynamical

differential equation is represented bymx ( ) 0k x y+ =

Let z x y=

x z y or x z y= + = +

( )m z y kz+ + = 0

or . . . (17.7)20 sinmz kz my z t+ =

Scale

Pointer

k

xm

y=yo sint

Figure 17.3

The solution of this differential equation is given by

sin ( )z Z t=

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where

2

0

2

1

n

n

y

Z

=

. . . (17.8)

If damping is also considered

2

0

22 22

1 2

n

n n

y

Z

=

+

. . . (17.9)

17.3.1 Principle of Design of Seismometer

This instrument is used to measure the displacement. Considering expression of Z givenin Eq. (17.8), if

1n

> >

0Z y

This means the seismometer has very low natural frequency so that this condition is truly

followed for operating lower value of. The value of is generally greater than2.5n. It is evident from the Figure 17.4 that the range of frequency for displacementy can be approximated by z is maximum forj =0.707. Therefore, in actual transduceralso damping factor is kept near this value. To satisfy the necessary condition the naturalfrequency should be very low, hence mass should be fairly heavy and springs should beof very low stiffness. Therefore, seismometer is a heavy transducer and may causeloading of the system whose displacement is to be measured. Due to this reason

seismometer is not used mechanical applications.

Figure 17.4

17.3.2 Principle of Design of Accelerometer

When the frequency is very large than 1n