##### Document Text Contents

Page 2

VIBRATION ANALYSIS FOR

ELECTRONIC EQUIPMENT

THIRD EDITION

Dave S. Steinberg

Steinberg & Associates

and University of California, Los Angeles

A WILEY-INTERSCIENCE PUBLICATION

JOHN WILEY & SONS, INC.

New York Chichester Weinheim Brisbane Singapore Toronto

Page 216

INTEGRATION METHOD FOR OBTAINING THE AREA UNDER A PSD CURVE 199

Starting with area 1 in Fig. 9.9, the b intercept on the P axis must be

determined where the frequency f equals 1 Hz. This can be obtained from

point 1 or point 2, since they are both on the same line with the same slope:

P, = 0.0133 G2/Hz

f , = 5 H z

or

P, = 0.20 G*/HZ

f 2 = 75 HZ

The slope S = 1.0 (equivalent to 3 dB/octave). Substitute into Eq. 9.16 for

the intercept b:

0.0133 0.20

b = - or - - - 0.00266 G2/Hz

(5)’ (75)’

(9.18)

Substitute into Eq. 9.17 and integrate between the limits of 5 and 75 Hz

for area 1 as shown in Fig. 9.9:

A, = ~ ~ 0 . 0 0 2 6 6 f ’ df = 0.00266 [;J -

0.00266

A, = 7 [ (75) , - ( 5 ) 2 ] = 7.46 G 2

i-

(9.19)

Comparing the results with Eq. 9.7 shows that the two different methods of

analysis agree very well.

The integration method for finding area 2 will be exactly the same as

Eq. 9.8.

The integration method for finding area 3 will be the same as the method

shown for area 1, except that a new intercept b must be obtained using

Eq. 9.16. The 200-Hz point or the 2000-Hz point can be used in Fig. 9.9,

since they are both on the same line:

P, = 0.2 G2/Hz

f l = 200 Hz

or

p2 = 0.002 G ~ / H ~

f 2 = 2000 Hz

Page 217

200 DESIGNING ELECTRONICS FOR RANDOM VIBRATION

The slope S = - 2 (equivalent to - 6 dB/octave). Substitute into Eq. 9.16 for

the intercept b:

0.20 0.002

b = = 8000 G2/Hz

(200) - 2 Or (2000) - * (9.20)

Substitute into Eq. 9.17 and integrate between the limits of 200 and 2000 Hz

for area 3, as shown in Fig. 9.9:

2000

A , = ~ ~ ~ 0 8 0 0 0 f - 2 df = 8000

8000

A , = ~ [ ( 2 0 0 0 ) - ~ - (200)-’] = 36.0 G2 /Hz (9.21)

Comparing the results with Eq. 9.9 shows the two areas and the two

different methods of analysis agree very well.

9.10 FINDING POINTS ON THE PSD CURVE

Random vibration input PSD curves are often specified in terms of the

frequency break points and the slope in dB, with only one G2/Hz point

defined as shown in Fig. 9.11. When it is necessary to find the PSD level at

these break points, then it is convenient to use the relation

(9.22)

Sample Problem-Finding PSD Values

Determine the PSD values at break points 1 and 2 as shown in Fig. 9.11.

FIGURE 9.11. Locating break points on a PSD

curve.

Slope Sloae

Frequency Hz

Page 431

This page has been reformatted by Knovel to provide easier navigation.

Index Terms Links

Vibration (Cont.)

airplanes and missiles 10 300

automobiles, trucks and trains 156

bending coupled with torsion 308

counter weights 356

die bond wires 297

fixtures 347

design summary 357

wood laminations 348

forced 30

free 17

harmonic modes 5

isolators 275 277 301

sample problem, selecting vibration isolators 278

linear systems 2

mechanical fuse 358

modes 5

nodes 5

nonlinear systems 342 351

nose cones 356

oil film slider 355 356 364

367

qualification tests 168 175 189

224 250

random 188

representation 3

servo control accelerometer 366

effects of location 367

Page 432

This page has been reformatted by Knovel to provide easier navigation.

Index Terms Links

Vibration (Cont.)

shaker machine 347

bolt patterns 361 364

mounting 347

ships and submarines 13

sinusoidal 166

sources 1

suspension systems 358

unsymmetrical fixtures 354

variable cross section 64

beams 64 66 67

electronic chassis 68

W

Wear, connectors 223 224

Wedge clamps 177 179

percent fixity 178

Work 21 22

VIBRATION ANALYSIS FOR

ELECTRONIC EQUIPMENT

THIRD EDITION

Dave S. Steinberg

Steinberg & Associates

and University of California, Los Angeles

A WILEY-INTERSCIENCE PUBLICATION

JOHN WILEY & SONS, INC.

New York Chichester Weinheim Brisbane Singapore Toronto

Page 216

INTEGRATION METHOD FOR OBTAINING THE AREA UNDER A PSD CURVE 199

Starting with area 1 in Fig. 9.9, the b intercept on the P axis must be

determined where the frequency f equals 1 Hz. This can be obtained from

point 1 or point 2, since they are both on the same line with the same slope:

P, = 0.0133 G2/Hz

f , = 5 H z

or

P, = 0.20 G*/HZ

f 2 = 75 HZ

The slope S = 1.0 (equivalent to 3 dB/octave). Substitute into Eq. 9.16 for

the intercept b:

0.0133 0.20

b = - or - - - 0.00266 G2/Hz

(5)’ (75)’

(9.18)

Substitute into Eq. 9.17 and integrate between the limits of 5 and 75 Hz

for area 1 as shown in Fig. 9.9:

A, = ~ ~ 0 . 0 0 2 6 6 f ’ df = 0.00266 [;J -

0.00266

A, = 7 [ (75) , - ( 5 ) 2 ] = 7.46 G 2

i-

(9.19)

Comparing the results with Eq. 9.7 shows that the two different methods of

analysis agree very well.

The integration method for finding area 2 will be exactly the same as

Eq. 9.8.

The integration method for finding area 3 will be the same as the method

shown for area 1, except that a new intercept b must be obtained using

Eq. 9.16. The 200-Hz point or the 2000-Hz point can be used in Fig. 9.9,

since they are both on the same line:

P, = 0.2 G2/Hz

f l = 200 Hz

or

p2 = 0.002 G ~ / H ~

f 2 = 2000 Hz

Page 217

200 DESIGNING ELECTRONICS FOR RANDOM VIBRATION

The slope S = - 2 (equivalent to - 6 dB/octave). Substitute into Eq. 9.16 for

the intercept b:

0.20 0.002

b = = 8000 G2/Hz

(200) - 2 Or (2000) - * (9.20)

Substitute into Eq. 9.17 and integrate between the limits of 200 and 2000 Hz

for area 3, as shown in Fig. 9.9:

2000

A , = ~ ~ ~ 0 8 0 0 0 f - 2 df = 8000

8000

A , = ~ [ ( 2 0 0 0 ) - ~ - (200)-’] = 36.0 G2 /Hz (9.21)

Comparing the results with Eq. 9.9 shows the two areas and the two

different methods of analysis agree very well.

9.10 FINDING POINTS ON THE PSD CURVE

Random vibration input PSD curves are often specified in terms of the

frequency break points and the slope in dB, with only one G2/Hz point

defined as shown in Fig. 9.11. When it is necessary to find the PSD level at

these break points, then it is convenient to use the relation

(9.22)

Sample Problem-Finding PSD Values

Determine the PSD values at break points 1 and 2 as shown in Fig. 9.11.

FIGURE 9.11. Locating break points on a PSD

curve.

Slope Sloae

Frequency Hz

Page 431

This page has been reformatted by Knovel to provide easier navigation.

Index Terms Links

Vibration (Cont.)

airplanes and missiles 10 300

automobiles, trucks and trains 156

bending coupled with torsion 308

counter weights 356

die bond wires 297

fixtures 347

design summary 357

wood laminations 348

forced 30

free 17

harmonic modes 5

isolators 275 277 301

sample problem, selecting vibration isolators 278

linear systems 2

mechanical fuse 358

modes 5

nodes 5

nonlinear systems 342 351

nose cones 356

oil film slider 355 356 364

367

qualification tests 168 175 189

224 250

random 188

representation 3

servo control accelerometer 366

effects of location 367

Page 432

This page has been reformatted by Knovel to provide easier navigation.

Index Terms Links

Vibration (Cont.)

shaker machine 347

bolt patterns 361 364

mounting 347

ships and submarines 13

sinusoidal 166

sources 1

suspension systems 358

unsymmetrical fixtures 354

variable cross section 64

beams 64 66 67

electronic chassis 68

W

Wear, connectors 223 224

Wedge clamps 177 179

percent fixity 178

Work 21 22