PART 2_IISC_Instructors Manual (1).pdf

7/23/2019 PART 2_IISC_Instructors Manual (1).pdf

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PART II

ADDITIONAL NOTES

FOR THE

INSTRUCTORS


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Experiment 1

Dynamics of a three storied building frame subjected to harmonic

base motion.

The performance of the experiment, as per suggested procedure, on the setupdeveloped at IISc, resulted in observations and subsequent deductions as provided in

tables 11.1-11.7 and figures11.1-11.6. The calculations related to the analysis of the

mathematical model is briefly summarized below:

Mass matrix (kg)

M = [1.8965 0 0

0 1.8965 0

0 0 1.7338]

Stiffness matrix (N/m)

K = 1.0e+003 *[5.8475 -2.9237 0.0000

-2.9237 5.8475 -2.9237

0.0000 -2.9237 2.9237]

Natural frequencies (rad/s)

{n} = [17.8939

49.7476

71.1199]

Mass normalized modal matrix

= [-0.2464 0.5401 -0.4181

-0.4416 0.2132 0.5356

-0.5451 -0.4560 -0.2679]

Damping ratios

{} = [0.024300

0.0092567

0.0067477]

Damping matrix determination

1][

t=[-0.4672 1.0244 -0.7930

-0.8374 0.4043 1.0157

-0.9450 -0.7906 -0.4644]

]2[ = [0.8696 0 0

0.0000 0.9210 0

0 0 0.9597]


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1][ = [-0.4672 -0.8374 -0.9450

1.0244 0.4043 -0.7906

-0.7930 1.0157 -0.4644]

Damping matrix (Ns/m)

C = [ ][ ] 11 2][ t = [1.7598 -0.0513 -0.0084-0.0513 1.7506 -0.0589

-0.0084 -0.0589 1.5593]

Orthogonality checks:

Mt

=[1.0000 0.0000 0.0000

0.0000 1.0000 0.0000

0.0000 0.0000 1.0000]

Kt = 1.0e+003 [0.3202 -0.0000 -0.0000-0.0000 2.4748 -0.0000

-0.0000 -0.0000 5.0580];

=

\

\2

n

Ct = [0.8696 -0.0000 0.0000-0.0000 0.9210 0.0000

-0.0000 -0.0000 0.9597];

=

\

2

\

nn

Refined methods of experimental and mathematical modeling can be brought to bear

on the structure under study. Thus the three-story shear-building frame was analyzed

mathematically using finite element (FE) analysis and experimentally by using

experimental modal analysis (EMA) procedures. The FE analysis was carried out

using NISA software and the EMA was carried out using MEscope software. The FEmodeling involved descretization of the structure using 3-D beam, 4-node shell and

concentrated mass elements resulting in a system with 1482 degrees of freedom.

Figure 11.4 shows the setup used for measurement of frequency response functions

using impulse hammer test. Tables 11.6 and 11.7 summarize the natural frequencies

of the frame using refined and approximate methods. As may be observed from this

table, these estimates show good mutual agreement. Figures 11.5 and 11.6,

respectively, show the first three mode shapes as predicted by the detailed FE analysis

and EMA.


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Figure 11.1 Plot ofX1versusf

Figure 11.2 Plot ofX2versusf

Figure11.3 Plot ofX3 versusf


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Figure 11.4 Block diagram of Shear Building Model test using impulse hammer

excitations F: Impulse hammer; A1-A3: Accelerometers.

(a)


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(b)

(c)

Figure 11.5 Mode shapes of the three-story shear building frame obtained from detailed FE model; (a)

I-mode; (b) II-mode; (c) III-mode


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(a)

(b)

(b)


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(c)

Figure 11.6 Mode shapes of the three-story shear building frame obtained from experimental modal

analysis; (a) I-mode; (b) II-mode; (c) III-mode


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Table 11.1 Physical properties of parts of the structure

Material Properties

Sl.

No.Part Material

Mass

kgYoungs

Modulus (E)

N/m2

Mass density ()

kg/m3

1 Column Aluminum Mc=0.0814 69.0E+009 2700

2 Slab Aluminum Ms=1.5430 69.0E+009 2700

3Allen screw,

M8Steel Msc=0.0035 - -

Table 11.2 Geometric data of the structure

Dimensions in mmSl. No. Part

Depth (D) Width (B) Length (L)

1 Column DA=3.00 BA=25.11 LA=400.00

2 Slab DB=12.70 BB=150.00 LB=300.00

Table 11.3 Details of the sensors used; CF: conversion factor

Sensitivity, SSl. No. Sensor

pC/ms-2

pC/gCF

Mass

kg

1 Accelerometer type 4375, B & K 0.320 3.14 0.1

mm/V

0.004


mm/V

0.011


mm/V

0.011

Table 11.4 Free vibration test data on three-story shear building frame

S.No. Quantity Notation Observations

1 Amplitude of 0th

peak A0 32.8V

2 Amplitude of nth

peak An 30.4V

3 Number of cycles n 11

4 Logarithmic decrement 0.00695 Damping ratio 0.0011


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Table 11.5.1 Base motion test data on three-story shear building frame; measurement made at first floor

S.No. Frequency,

f

(Hz)

Frequency

=2f

(rad/s)

Amplitude

1rms (mV)

Displacement Amplitude

X1= 2 (CF) 1

(mm)

1 2.2400 14.0743 0.0846 0.1196

2 2.2900 14.3885 0.0995 0.1407

3 2.3000 14.4513 0.1010 0.1428

4 2.3200 14.5770 0.1020 0.1442

5 2.4100 15.1425 0.1340 0.1895

6 2.4200 15.2053 0.1350 0.1909

7 2.5600 16.0850 0.2150 0.3041

8 2.6300 16.5248 0.3060 0.4327

9 2.7100 17.0274 0.7330 1.0366

10 2.7200 17.0903 0.7900 1.1172

11 2.7500 17.2788 1.1600 1.6405

12 2.7700 17.4044 4.1100 5.8124

13 2.7800 17.4673 0.8310 1.1752

14 2.8200 17.7186 0.3790 0.536015 2.9000 18.2212 0.2560 0.3620

16 2.9300 18.4097 0.2300 0.3253

17 3.0500 19.1637 0.1130 0.1598

18 3.3200 20.8602 0.0430 0.0608

19 3.3900 21.3000 0.0400 0.0566

20 3.5000 21.9911 0.0251 0.0355

21 4.5500 28.5885 0.0257 0.0363

22 4.9800 31.2903 0.0404 0.0571

23 5.2100 32.7354 0.0507 0.0717

24 5.4900 34.4947 0.0595 0.0841

25 6.0400 37.9504 0.0920 0.130126 6.5500 41.1549 0.1460 0.2065

27 7.1000 44.6106 0.3560 0.5035

28 7.1600 44.9876 0.4210 0.5954

29 7.3500 46.1814 1.0100 1.4284

30 7.4300 46.6841 2.9200 4.1295

31 7.4900 47.0611 5.9100 8.3580

32 7.5400 47.3752 2.0700 2.9274

33 7.6000 47.7522 1.1900 1.6829

34 7.7000 48.3805 0.6280 0.8881

35 7.7200 48.5062 0.5630 0.7962

36 7.9000 49.6372 0.3080 0.4356

37 8.0600 50.6425 0.2110 0.2984

38 8.4000 52.7788 0.1200 0.1697

39 8.7000 54.6637 0.0818 0.1157

40 9.0900 57.1142 0.0474 0.0670

41 9.2200 57.9310 0.0331 0.0468

42 9.6500 60.6327 0.0114 0.0161

43 10.2500 64.4026 0.0428 0.0605

44 10.5000 65.9734 0.0720 0.1018


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45 10.7100 67.2929 0.1300 0.1838

46 10.9400 68.7380 0.2510 0.3550

47 11.1000 69.7434 0.5280 0.7467

48 11.2600 70.7487 2.1400 3.0264

49 11.3000 71.0000 1.6700 2.3617

50 11.4700 72.0681 0.4750 0.6718

51 11.7100 73.5761 0.2460 0.347952 12.0200 75.5239 0.1580 0.2234

53 12.3800 77.7858 0.1160 0.1640

54 12.5000 78.5398 0.1070 0.1513

55 12.8000 80.4248 0.0871 0.1232

56 13.1800 82.8124 0.0742 0.1049

57 13.4800 84.6973 0.0645 0.0912

58 13.9100 87.3991 0.0570 0.0806

59 14.2500 89.5354 0.0516 0.0730

Table 11.5.2 Base motion test data on three-story shear building frame; measurement made at second

floor

S.No. Frequency,

f

(Hz)

Frequency

=2f

(rad/s)

Amplitude

2rms (mV)


X2= 2 (CF) 2

(mm)

1 2.1200 13.3204 0.0965 0.1365

2 2.2400 14.0743 0.1320 0.1867

3 2.3000 14.4513 0.1480 0.2093

4 2.3200 14.5770 0.1520 0.2150

5 2.3600 14.8283 0.1660 0.2348

6 2.4200 15.2053 0.1910 0.27017 2.5700 16.1478 0.3760 0.5317

8 2.6500 16.6504 0.6590 0.9320

9 2.7300 17.1531 1.6700 2.3617

10 2.7700 17.4044 5.0200 7.0994

11 2.9200 18.3469 0.4820 0.6817

12 2.9400 18.4726 0.4460 0.6307

13 3.0700 19.2894 0.2470 0.3493

14 3.1500 19.7920 0.1960 0.2772

15 3.3600 21.1115 0.1280 0.1810

16 3.5200 22.1168 0.1050 0.1485

17 3.5600 22.3681 0.1000 0.1414

18 3.7900 23.8133 0.0790 0.1117

19 4.0100 25.1956 0.0606 0.0857

20 4.1500 26.0752 0.0540 0.0764

21 4.3600 27.3947 0.0435 0.0615

22 4.6200 29.0283 0.0363 0.0513

23 4.8000 30.1593 0.0319 0.0451

24 5.0200 31.5416 0.0246 0.0348

25 5.3500 33.6150 0.0180 0.0255


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26 5.5500 34.8717 0.0160 0.0226

27 6.3000 39.5841 0.0072 0.0102

28 6.4000 40.2124 0.0180 0.0255

29 6.5800 41.3434 0.0347 0.0491

30 6.8300 42.9142 0.0605 0.0856

31 7.1300 44.7991 0.1600 0.2263

32 7.1500 44.9248 0.1660 0.234833 7.2000 45.2389 0.2140 0.3026

34 7.3800 46.3699 0.7600 1.0748

35 7.4200 46.6212 1.2800 1.8102

36 7.4600 46.8726 2.3900 3.3800

37 7.4700 46.9354 2.6200 3.7052

38 7.5100 47.1867 1.6000 2.2627

39 7.5200 47.2496 1.5000 2.1213

40 7.7400 48.6319 0.3110 0.4398

41 7.9200 49.7628 0.2080 0.2942

42 7.9900 50.2027 0.1900 0.2687

43 8.1300 51.0823 0.1570 0.222044 8.3500 52.4646 0.1340 0.1895

45 8.4600 53.1557 0.1220 0.1725

46 8.7000 54.6637 0.1130 0.1598

47 8.7600 55.0407 0.1120 0.1584

48 9.1700 57.6168 0.1050 0.1485

49 9.4500 59.3761 0.1090 0.1541

50 9.6500 60.6327 0.1110 0.1570

51 9.8200 61.7009 0.1150 0.1626

52 10.0600 63.2088 0.1280 0.1810

53 10.3100 64.7796 0.1490 0.2107

54 10.4800 65.8478 0.1670 0.2362

55 10.7400 67.4814 0.2330 0.3295

56 10.8000 67.8584 0.2600 0.3677

57 10.9600 68.8637 0.3920 0.5544

58 11.0600 69.4920 0.5850 0.8273

59 11.2600 70.7487 2.6000 3.6770

60 11.2700 70.8115 2.1600 3.0547

61 11.2900 70.9372 1.9500 2.7577

62 11.4400 71.8796 0.5850 0.8273

63 11.5700 72.6965 0.3040 0.4299

64 11.6600 73.2619 0.2280 0.3224

65 11.7100 73.5761 0.2020 0.2857

66 11.8500 74.4557 0.1570 0.222067 11.9800 75.2726 0.1260 0.1782

68 12.0500 75.7124 0.1150 0.1626

69 12.3500 77.5973 0.0771 0.1090

70 12.4400 78.1628 0.0714 0.1010

71 12.7700 80.2363 0.0514 0.0727

72 12.8700 80.8646 0.0482 0.0682

73 13.2300 83.1265 0.0376 0.0532


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74 13.3700 84.0062 0.0330 0.0467

75 13.6200 85.5770 0.0272 0.0385

76 14.0100 88.0274 0.0216 0.0305

77 14.2900 89.7867 0.0190 0.0269

Table 11.5.3 Base motion test data on three-story shear building frame; measurement made at third

floor

S.No. Frequency,

f

(Hz)

Frequency

=2f

(rad/s)

Amplitude

3rms (mV)


X3= 2 (CF) 3

(mm)

1 2.1800 13.6973 0.1390 0.1966

2 2.2700 14.2628 0.1640 0.2319

3 2.3300 14.6398 0.1870 0.2645

4 2.5500 16.0221 0.3950 0.5586

5 2.6300 16.5248 0.6320 0.8938

6 2.6700 16.7761 0.9610 1.3591

7 2.7100 17.0274 1.5300 2.1637

8 2.7600 17.3416 13.7000 19.3747

9 2.7800 17.4673 7.6200 10.7763

10 2.9200 18.3469 0.6230 0.8811

11 3.0300 19.0381 0.3760 0.5317

12 3.3400 20.9858 0.1990 0.2814

13 3.4500 21.6770 0.1720 0.2432

14 3.5200 22.1168 0.1590 0.2249

15 3.6600 22.9965 0.1380 0.1952

16 3.8800 24.3788 0.1180 0.1669

17 4.1100 25.8239 0.1050 0.1485

18 4.3400 27.2690 0.0970 0.137219 4.5400 28.5257 0.0921 0.1302

20 4.6400 29.1540 0.0905 0.1280

21 4.7700 29.9708 0.0895 0.1266

22 4.9400 31.0389 0.0885 0.1252

23 5.1500 32.3584 0.0881 0.1246

24 5.3600 33.6779 0.0890 0.1259

25 5.5400 34.8088 0.0940 0.1329

26 5.6700 35.6257 0.0965 0.1365

27 5.8500 36.7566 0.1060 0.1499

28 6.0600 38.0761 0.1100 0.1556

29 6.2800 39.4584 0.1230 0.173930 6.5500 41.1549 0.1500 0.2121

31 6.7000 42.0973 0.1740 0.2461

32 6.8500 43.0398 0.2120 0.2998

33 7.1000 44.6106 0.3420 0.4837

34 7.3500 46.1814 1.0300 1.4566

35 7.3900 46.4327 1.8900 2.6729

36 7.4500 46.8097 3.9200 5.5437

37 7.5500 47.4380 1.4500 2.0506


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38 7.6700 48.1920 0.6070 0.8584

39 7.8400 49.2602 0.3300 0.4667

40 7.9100 49.7000 0.2810 0.3974

41 8.0300 50.4540 0.2170 0.3069

42 8.1800 51.3965 0.1680 0.2376

43 8.5300 53.5956 0.1150 0.1626

44 8.7400 54.9150 0.0989 0.139945 9.0400 56.8000 0.0838 0.1185

46 9.2600 58.1823 0.0766 0.1083

47 9.4500 59.3761 0.0726 0.1027

48 9.6900 60.8841 0.0684 0.0967

49 9.8400 61.8265 0.0692 0.0979

50 10.1000 63.4602 0.0726 0.1027

51 10.3500 65.0310 0.0792 0.1120

52 10.6400 66.8531 0.0984 0.1392

53 10.7800 67.7327 0.1200 0.1697

54 11.0600 69.4920 0.2680 0.3790

55 11.0900 69.6805 0.3050 0.431356 11.1100 69.8062 0.3510 0.4964

57 11.2600 70.7487 1.3300 1.8809

58 11.2900 70.9372 0.7420 1.0493

59 11.4200 71.7540 0.2280 0.3224

60 11.5200 72.3823 0.1470 0.2079

61 11.8200 74.2673 0.0620 0.0877

62 11.8600 74.5186 0.0603 0.0853

63 12.1500 76.3407 0.0370 0.0523

64 12.2000 76.6549 0.0320 0.0453

65 12.4600 78.2885 0.0226 0.0320

66 12.6900 79.7336 0.0208 0.0294

67 12.9500 81.3672 0.0155 0.0219

68 13.1200 82.4354 0.0132 0.0187

Table 11.6 Estimate of the natural frequencies of the three-story shear building frame; Experiment-I:

Impulse hammer test and modal extraction using ME scope; Experiment-II: Shake table test

Natural frequencies in Hz

Mode No. FE Model 3-DOF

ModelExperiment I

Experiment

II

1 2.84 2.8479 2.79 2.77

2 8.03 7.9176 7.91 7.47

3 11.74 11.3191 11.64 11.26


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Table 11.7 Estimate of the mode shapes of the three-story shear building frame; Experiment-I: Impulse

hammer test and modal extraction using ME scope; Experiment-II: Shake table test

Mode shapes

FE Model 3-DOF Model Experiment I Experiment II

I

mode

II

mode

III

mode

I

mode

II

mode

III

mode

I

mode

II

mode

III

mode

I

mode

II

mode

III

mode

1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000

1.7990.398

-

1.2661.792 0.394 -1.281 1.813 0.392

-

1.4152.074 0.571 -1.17

2.224 -

0.8450.638 2.212

-

0.8440.6408 2.172

-

0.8850.713 3.591

-

1.2180.480


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Experiment 2

Dynamics of a one-storied building frame with planar asymmetry

subjected to harmonic base motions.

The performance of the experiment, as per the suggested procedure, on the setup

developed at IISc, resulted in observations and subsequent analysis as provided in

tables 12.1 -12.9 and figures 12.1 and 12.2. The calculations related to the analysis of

the model is briefly summarized below:

Location of mass center (m)

bs=0.30 , ds=0.15 , ts=10.625e-3

bs1=0.286 , ds1=0.136

Mass matrix

M = [4.0444(kg) 0 0

0 4.0444 (kg) 0

0 0 0.0431 (kgm2)];

Stiffness matrix (force in N, distance in m and angle in rad).

k1=k2=k3=k4=k7=k8= 3.4240e+003 N/m

k5=k6=9.2674e+003 N/m

k*=313.1298 Nm

K = 1.0e+004 *[ 1.9539 0 0.0475

0 1.9539 -0.0824

0.0475 -0.0824 0.0805];


{n} = [66.8321

69.5064

138.0460];


=[ -0.2452 0.4308 0.0392

0.4254 0.2483 -0.0681

0.7615 0 4.7567]

Damping ratios

{} = [0.0218

0.0201


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0.0112];


1][

t =[-0.9915 1.7425 0.1587

1.7206 1.0041 -0.2755

0.0328 0 0.2050]

]2[ = [2.9139 0 0

0 2.7942 0

0 0 3.0922];

1][ = [-0.9915 1.7206 0.0328

1.7425 1.0041 -0.0000

0.1587 -0.2755 0.2050]

C = [ ][ ] 11 2][ t = [11.4260 -0.2173 0.0058-0.2173 11.6779 -0.0101

0.0058 -0.0101 0.1331];


Mt =[1.0000 0.0000 0.0000

0.0000 1.0000 0.0000

0.0000 0.0000 1.0000];

Kt

= 1.0e+004 *[ 0.4467 -0.0000 0.0000

-0.0000 0.4831 0.0000

0.0000 0.0000 1.9057];

=

\

\2

n

Ct = [2.9139 -0.0000 -0.0000

-0.0000 2.7942 -0.0000

0.0000 0.0000 3.0922];

=

\

2

\

nn

The structure under study was also analyzed using more sophisticated methods,

namely, numerical modeling using finite element method, and, frequency response

measurements using impulse hammer test. Figure 12.3 shows the set up for modal test

using impulse hammer. The frequency response functions so measured are compared


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with theoretical predictions based on equation (2.2) in figure 12.4. The mode shapes

derived from the detailed finite element model (with 630 dofs) are shown in figure

12.5. Table 12.8 summarizes the results on natural frequencies obtained using

different methods.

(a)

(b)Figure 12.1 Comparison of displacement as a function of driving frequency using theory and

experiment using shake table; =0.


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(a)

(b)

Figure 12.2 Maximum amplitude response of one-story building frame subjected to

harmonic base motion with angle of incidence varying from 0 to 90 degrees;

frequency of excitation = 7.42 Hz; maximum support displacement = 0.291mm. (a)

response alongxdirection; (b) response alongy direction;


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Figure 12.3 Block diagram of building modal test using impulse hammer excitation F:

Impulse hammer; A1-A2: Accelerometers


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(a) (b)

(c)

Figure 12.4 Plots of frequency response function of building frame model using

impulse hammer test; impulse applied in the x-direction (a) displacement along x-

direction; (b) displacement along y-direction; (c) rotation about z-axis.


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(a)

(b)


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(c)

Figure 12.5 Mode shapes of the three-story shear building frame obtained from

detailed FE model; (a) I-mode; (b) II-mode; (c) III-mode


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Table 12.1 Equipment used in free vibration and forced vibration test of one-story building frame

S.No. Equipment Quantity

1 Oscilloscope 1

2 Accelerometers 2

3 Conditioning amplifiers 2 channels

4 Shake table 1


Material Properties

Sl.

No.

Part Qty.

Nos.Material

Mass

Kg

Mass

density

()

kg/m3

Modulus

of

elasticity

(E) N/m2

Poissons

ratio ()

1 Columns 3 Aluminum(m1+m2+m3)

= 0.32642700 69.0E+009 0.3

2 Column 1 Steel m4=0.3037 7800 2.00E+011 0.3

3 Slab 1 Steel Ms=3.7294 7800 2.00E+011 0.3


Sl. No. Part Dimensions in mm

Depth (t) 10.625

Length (bs ) 3001 Slab

Width (ds) 150

Diameter (Dal) 10.132Aluminum

Column Length (L) 500

Diameter (Ds) 9.958

3 Steel Column Length (L) 500


Sensitivity, SSl.

No.Sensor mV/ms

-

2

mV/g

Mass

gm

1 B & K Deltatron Accelerometer

Type 4507 002, Sl. No. 10308

93.8 920 4.8


Type 4507 002, Sl. No. 10309

94.6 928 4.8


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Table 12.5 Base motion test data on one-story building frame; Measurement made at first floor along

X direction; Amplitude of base motion, =0.055 mm

No. Frequency,

f

(Hz)

Frequency

=2f

(rad/s)

Amplitude

x

rms (mV)

Conversion

Factor

CF(V/m)

Displacement

Amplitude

X = 2 (CF) x

(mm)1 2.68 16.84 29.45 1000 0.041648190

2 2.86 17.97 33.30 1000 0.047092860

3 3.19 20.05 32.85 1000 0.046456470

4 3.50 22.00 31.70 1000 0.044830140

5 3.55 22.31 25.22 1000 0.035666124

6 3.88 24.38 26.20 1000 0.037052040

7 4.25 26.71 32.32 1000 0.045706944

8 4.62 29.04 33.76 1000 0.047743392

9 4.91 30.86 34.50 1000 0.048789900

10 4.95 31.11 35.72 1000 0.050522295

11 5.19 32.62 35.23 1000 0.049822266

12 5.41 34.00 30.42 1000 0.043027035

13 5.49 34.50 35.36 1000 0.050006112

14 5.56 34.94 33.60 1000 0.047517120

15 5.85 36.77 33.55 1000 0.047446410

16 6.17 38.78 36.16 1000 0.051137472

17 6.51 40.92 39.71 1000 0.056157882

18 6.80 42.74 40.06 1000 0.056652852

19 7.17 45.06 45.88 1000 0.064883496

20 7.25 45.57 48.50 1000 0.068588700

21 7.31 45.94 41.32 1000 0.058434744

22 7.77 48.84 54.42 1000 0.076967835

23 8.10 50.91 55.50 1000 0.078488100

24 8.15 51.22 59.35 1000 0.083932770

25 8.30 52.17 83.03 1000 0.117421026

26 8.32 52.29 82.50 1000 0.116671500

27 8.55 53.74 95.67 1000 0.13530358528 8.62 54.18 98.50 1000 0.139298700

29 8.87 55.75 117.67 1000 0.166408914

30 9.09 57.13 136.00 1000 0.192331200

31 9.30 58.45 155.67 1000 0.220148514

32 9.48 59.58 182.50 1000 0.258091500

33 9.68 60.84 265.00 1000 0.374763000

34 9.83 61.78 349.25 1000 0.493909350

35 10.15 63.80 177.50 1000 0.251020500

36 10.2 64.11 384.00 1000 0.543052800

37 10.36 65.12 758.33 1000 1.072430286

38 10.36 65.12 766.50 1000 1.083984300

39 10.60 66.62 863.40 1000 1.221020280

40 10.66 67.00 670.00 1000 0.947514000

41 10.68 67.13 659.50 1000 0.93266490042 10.84 68.13 381.75 1000 0.539870850

43 11.13 69.96 222.00 1000 0.313952400

44 11.14 70.02 225.50 1000 0.318902100

45 11.67 73.35 127.50 1000 0.180310500

46 11.83 74.36 111.50 1000 0.157683300

47 11.85 74.48 107.67 1000 0.152266914

48 12.20 76.68 85.83 1000 0.121380786

49 12.25 77.00 80.45 1000 0.113772390

50 12.59 79.13 67.50 1000 0.095458500


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51 12.67 79.64 64.00 1000 0.090508800

52 12.80 80.45 61.43 1000 0.086874306

53 13.08 82.21 52.12 1000 0.073715175

54 13.19 82.90 48.20 1000 0.06816444

55 13.25 83.28 49.53 1000 0.070049569

56 13.35 83.91 46.94 1000 0.066382548

57 13.46 84.60 45.44 1000 0.064261248

58 13.53 85.04 43.30 1000 0.06123486059 13.82 86.86 38.88 1000 0.054984096

60 13.96 87.74 37.92 1000 0.053626464

61 14.15 88.94 35.97 1000 0.050875845

62 14.36 90.26 30.98 1000 0.043811916

63 14.37 90.32 32.14 1000 0.045452388

64 14.76 92.77 29.46 1000 0.041662332

65 14.81 93.09 29.27 1000 0.041400705

66 15.10 94.91 26.67 1000 0.037723785

67 15.16 95.29 23.57 1000 0.033344008

68 15.40 96.80 23.36 1000 0.033035712

69 20.75 130.42 8.86 1000 0.012529812

70 22.05 138.60 24.40 1000 0.034506480

Table 12.6 Base motion test data on one-story building frame; measurement made at first floor along Y

direction; Amplitude of base motion, =0.055 mm

Sl.

No.

Frequency,

f

(Hz)

Frequency

=2f

(rad/s)

Amplitude

x1rms (mV)

Amplitude

x2rms (mV)

Conversion

Factor

CF1(V/m)

Displacement

Amplitude

X1= 2

(CF1) x1(mm)

ConversionFactor

CF2(V/m)

DisplacementAmplitude

X2= 2

(CF2) x2(mm)

Rotation**

z

(rad)

1 3.61 22.72 34.50 36.20 1000 0.04878990 1000 0.051194040 0.017677500

2 3.62 22.77 34.10 35.60 1000 0.04822422 1000 0.050345520 0.015597794

3 5.56 34.94 27.00 26.00 1000 0.03818340 1000 0.036769200 0.010398529

4 5.64 35.47 17.90 18.20 1000 0.02531418 1000 0.025738440 0.003119559

5 5.66 35.63 18.60 19.40 1000 0.02630412 1000 0.027435480 0.008318824

6 5.88 36.96 35.40 34.40 1000 0.05006268 1000 0.048648480 0.010398529

7 6.32 39.73 53.50 51.60 1000 0.07565970 1000 0.072972720 0.019757206

8 6.34 39.88 53.60 51.70 1000 0.07580112 1000 0.073114140 0.019757206

9 6.46 40.60 33.10 35.80 1000 0.04681002 1000 0.050628360 0.028076029

10 6.48 40.78 53.40 51.70 1000 0.07551828 1000 0.073114140 0.017677500

11 6.85 43.11 61.10 59.10 1000 0.08640762 1000 0.083579220 0.020797059

12 6.87 43.23 59.20 58.00 1000 0.08372064 1000 0.082023600 0.012478235

13 7.1 44.62 63.80 60.70 1000 0.09022596 1000 0.085841940 0.032235441

14 7.50 47.19 75.20 72.10 1000 0.10634784 1000 0.101963820 0.032235441

15 7.56 47.52 73.90 71.90 1000 0.10450938 1000 0.101680980 0.020797059

16 7.75 48.71 79.20 76.10 1000 0.11200464 1000 0.107620620 0.032235441

17 7.77 48.84 78.50 74.50 1000 0.11101470 1000 0.105357900 0.041594118

18 7.81 49.11 81.20 79.10 1000 0.11483304 1000 0.111863220 0.02183691219 7.92 49.80 84.40 82.40 1000 0.11935848 1000 0.116530080 0.020797059

20 8.02 50.44 86.10 81.50 1000 0.12176262 1000 0.115257300 0.047833235

21 8.14 51.16 91.30 85.90 1000 0.12911646 1000 0.121479780 0.056152059

22 8.22 51.69 97.70 94.40 1000 0.13816734 1000 0.133500480 0.034315147

23 8.54 53.68 103.0 94.30 1000 0.14566260 1000 0.133359060 0.090467206

24 8.57 53.90 114.0 105.0 1000 0.16121880 1000 0.148491000 0.093586764

25 8.80 55.33 127.0 118.0 1000 0.1796034 1000 0.166875600 0.093586764

26 9.11 57.26 161.0 147.0 1000 0.2276862 1000 0.207887400 0.145579411

27 9.29 58.41 188.0 172.0 1000 0.2658696 1000 0.243242400 0.166376469


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28 9.45 59.41 216.0 197.0 1000 0.3054672 1000 0.278597400 0.197572056

29 9.66 60.73 276.0 250.0 1000 0.3903192 1000 0.353550000 0.270361758

30 9.78 61.50 328.0 295.0 1000 0.4638576 1000 0.417189000 0.343151457

31 9.82 61.74 338.0 306.0 1000 0.4779996 1000 0.432745200 0.332752929

32 9.96 62.60 451.0 402.0 1000 0.6378042 1000 0.568508400 0.509527897

33 10.14 63.73 890.0 754.0 1000 1.2586380 1000 1.066306800 1.414199057

34 10.31 64.80 188.0 130.0 1000 0.2658696 1000 0.183846000 0.603114633

35 10.34 64.99 140.0 157.0 1000 0.1979880 1000 0.222029400 0.17677499836 10.56 66.37 1100.0 1050.0 316 4.922848101 316 4.699082278 1.645335448

37 10.64 66.88 465.0 670.0 316 2.081022152 316 2.998462025 6.745779097

38 10.78 67.76 210.0 199.0 316 0.939816456 316 0.890587975 0.361974109

39 11.09 69.70 108.0 99.10 316 0.483334177 316 0.443503861 0.292869966

40 11.25 70.71 85.00 78.20 316 0.380401899 316 0.349969747 0.223765819

41 11.45 71.97 67.40 61.10 316 0.301636329 316 0.273441835 0.207312450

42 11.61 72.97 57.20 51.70 316 0.255988101 316 0.231373861 0.180987061

43 11.74 73.79 50.70 45.80 316 0.226898544 316 0.204969494 0.161243018

44 11.82 74.29 48.50 43.80 316 0.217052848 316 0.196018861 0.154661670

45 12.10 76.05 40.00 35.10 316 0.179012658 316 0.157083608 0.161243018

46 12.20 76.68 36.80 32.30 316 0.164691646 316 0.144552722 0.148080323

47 12.45 78.25 28.10 25.00 316 0.125756392 316 0.111882911 0.102010889

48 12.52 78.69 89.80 79.60 1000 0.126995160 1000 0.112570320 0.106065000

49 12.92 81.21 77.60 67.80 1000 0.109741920 1000 0.095882760 0.101905588

50 13.20 82.97 69.60 59.60 1000 0.098428320 1000 0.084286320 0.103985294

51 13.60 85.48 59.50 51.60 1000 0.084144900 1000 0.072972720 0.082148382

52 14.00 88.00 51.10 42.40 1000 0.072265620 1000 0.059962080 0.090467206

53 14.20 89.25 47.10 39.50 1000 0.066608820 1000 0.055860900 0.079028823

54 21.50 135.14 555.0 489.0 3160 0.248380063 3160 0.218842975 0.217184472

55 21.80 137.02 116.0 134.0 316 0.519136709 316 0.599692405 0.592321226

56 22.26 139.92 249.0 362.0 3160 0.111435380 3160 0.162006456 0.371846130

57 23.21 145.89 88.80 154.0 3160 0.039740810 3160 0.068919873 0.214551933

Table 12.7 Base motion test data on one-story building frame; Frequency of excitation = 7.42 Hz;

Amplitude of base motion, = 0.291mm

No. Angle of

Incidence(degrees)

Amplitude

xrms (V)

Amplitude

yrms (V)

Conversion

FactorCF

(V/m)

Displacement

AmplitudeX = 2 (CF) x

(mm)

Displacement

AmplitudeY = 2 (CF)

y

(mm)

1 0 0.4156 0.0117 1000 0.58689 0.01655

2 5 0.4136 0.0172 1000 0.58407 0.02432

3 10 0.4106 0.0221 1000 0.57982 0.03125

4 15 0.3991 0.0278 1000 0.56356 0.03932

5 20 0.3876 0.1312 1000 0.54730 0.18526

6 25 0.3706 0.1653 1000 0.52325 0.23334

7 30 0.3536 0.1943 1000 0.49921 0.27436

8 35 0.3375 0.2274 1000 0.47659 0.32102

9 40 0.3155 0.2604 1000 0.44547 0.36769

10 45 0.2915 0.2884 1000 0.41153 0.4072911 50 0.2634 0.3055 1000 0.37194 0.43133

12 55 0.2274 0.3155 1000 0.32102 0.44547

13 60 0.2013 0.3355 1000 0.28425 0.47376

14 65 0.1758 0.3515 1000 0.24819 0.49638

15 70 0.1432 0.3596 1000 0.20223 0.50770

16 75 0.1077 0.3636 1000 0.15203 0.51336

17 80 0.0775 0.3616 1000 0.10939 0.51053

18 85 0.0378 0.2864 1000 0.05332 0.40446

19 90 0.0178 0.1422 1000 0.02517 0.20082


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Table 12.8 Estimate of the natural frequencies of the one-story building frame; Experiment-I: Impulsehammer test and modal extraction using ME scope; Experiment-II: shake table test.


Mode No. FE Model 3-DOFModel ExperimentI ExperimentII

1 10.1559 10.6367 10.1380 10.1400

2 10.8203 11.0623 10.3820 10.6400

3 18.0018 21.9707 21.7250 21.8000

Table 12.9 Estimate of the damping of one-story building frame using the data from Experiment-I:

Impulse hammer test

Mode No.Experiment

I

1 0.0218

2 0.02013 0.0112


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Experiment 3

Dynamics of a three storied building frame subjected to periodic

(non-harmonic) base motion.

The performance of the experiment, as per suggested procedure, on the setup

developed at IISc, resulted in observations and subsequent deductions as provided in

figures13.1-13.3. Refer to the notes provided in Experiment 1 for details of the mass,

stiffness and damping matrices. Figure 13.1 shows the base motions for different

RPM of the motor resulting in base motion periods of 0.9870,1.0328, 1.1845 and

1.2490 s respectively. The base motions re-constructed from the derived Fourier

coefficients are also shown in this figure. Figure 13.2 shows the computed floor

displacement from the mathematical model. Here the solution is obtained using two

alternative strategies: the first method involves modal decomposition of the system

and Fourier decomposition of excitation (equation 3.2) and, the second, employs

direct integration technique using the base motion without Fourier decomposition.

The two solutions show the expected mutual agreement. The steady oscillatoryresponse of the system obtained using experiments and theory are compared in figure

13.3. It may be noted that the phase difference that is observed between the two

results in these figures is because of the arbitrary choice of time axis for the two

results. The instructor may note that significant amount of computer work is needed

for obtaining the analytical results of this study.


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Period=0.9870 s Period=1.0328 s

Period=1.1845 s Period=1.2490 s

Figure 13.1 Typical base motion profiles with differing periods; the reconstructed

signals using Fourier series are also shown in these figures.


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Figure 13.2 Displacement of floors (a) Floor I (b) Floor II (c) Floor III. Period=1.0328

s


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(a)

(b)

(c)Figure 13.3 Comparison of experimental and analytical results on the total floor

displacement in the steady state. Note that only the oscillatory motion is measured in

the experiment; Period=1.0328 s; (a) First floor; (b) Second floor; (c) third floor.


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Experiment 4

Vibration isolation of a secondary system

The performance of the experiment, as per suggested procedure here in, on the setup

developed at IISc, resulted in observations and subsequent analysis as shown in tables

14.1-14.8 and figures 14.1-14.5. Refined methods of experimental and mathematical

modeling can be brought to bear on the structure under study. Thus the frame was

analyzed mathematically using finite element (FE) analysis and experimentally by

using experimental modal analysis (EMA) procedures. The FE analysis was carried

out using NISA software and the EMA was carried out using MEscopeVES software.

The FE modeling involved discretization of the structure using 3-D beam, 4-node

shell and concentrated mass elements resulting in a system with 1482 degrees of

freedom. Figure 14.2 shows the setup used for measurement of frequency response

functions using impulse hammer test. Table 14.8 summarizes the natural frequencies

of the frame using refined and approximate methods. As may be observed from this

table, these estimates show good mutual agreement Figures 14.3 and 14.4 respectively

show the first three mode shapes as predicted by detailed FE analysis and EMA.

In the development of mathematical model, as described in section 4.3, the structural

matrices were obtained as

Mass matrix (kg)

M1= MS+ 4* MC+ Msc1

M2= MS+ 4* MC+ Msc1+ Mbb+0.5* Msp+ Macc

M3= MS+ 4*0.5* MC+ Msc1

M4= Miso+ 0.5* Msp+ Macc+ Msc2

Amplitude of the Base motion= 0.1015mm

M = [1.8965 0 0 0

0 1.9543 0 0

0 0 1.7338 0

0 0 0 0.0915]


K = 1.0e+003 *[ 5.8475 -2.9237 0 0

-2.9237 5.9379 -2.9237 -0.0904

0 -2.9237 2.9237 00 -0.0904 0 0.0904]

The solution of the eigenvalue problem MK 2= lead to the natural frequencies

(rad/s)

{n} = [17.5682

31.6945

49.7510


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70.7656]

and mass normalized mode shapes

= [-0.2406 -0.0400 0.5392 -0.4208

-0.4331 -0.0540 0.2127 0.5253

-0.5301 -0.1336 -0.4547 -0.2667-0.6299 3.2391 -0.1413 -0.1291]

The damping ratios derived from the measured frequency response functions were

found to be

{} = [0.0100

0.0080

0.0060

0.0040];

The C matrix associated with the above damping ratios was obtained as follows:

1][

t = [-0.4564 -0.0759 1.0225 -0.7981

-0.8464 -0.1055 0.4157 1.0266

-0.9191 -0.2316 -0.7884 -0.4624

-0.0577 0.2965 -0.0129 -0.0118]

]2[ = [0.3514 0 0 0

0 0.5071 0 0

0 0 0.5970 0

0 0 0 0.5661]

1][ =[-0.4564 -0.8464 -0.9191 -0.0577

-0.0759 -0.1055 -0.2316 0.2965

1.0225 0.4157 -0.7884 -0.0129

-0.7981 1.0266 -0.4624 -0.0118]

C = [ ][ ] 11 2][ t = [1.0609 -0.0703 -0.1160 -0.0047-0.0703 0.9572 -0.1786 -0.0088

-0.1160 -0.1786 0.8161 -0.0070

-0.0047 -0.0088 -0.0070 0.0459]

Furthermore, the following orthogonality checks were also made:

Mt

=[1.0000 0.0000 -0.0000 -0.0000

0.0000 1.0000 -0.0000 -0.0000

-0.0000 -0.0000 1.0000 -0.0000

-0.0000 -0.0000 -0.0000 1.0000];


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Kt

= 1.0e+003 [0.3086 0.0000 0.0000 0.0000

0.0000 1.0045 0.0000 0.0000

0.0000 0.0000 2.4752 -0.0000

0.0000 0.0000 -0.0000 5.0078]

Ct = [0.3514 -0.0000 0.0000 0.0000

-0.0000 0.5071 0.0000 0.0000-0.0000 -0.0000 0.5970 -0.0000

0.0000 0.0000 -0.0000 0.5661]

The following calculations were involved in making a sdof model:

M4= Miso+ 0.5* Msp+ Macc+ Msc2

= 0.0675+ 0.5*0.0073+0.012+ 0.00836 (from tables 14.2 and 14.3)

= 0.0915 kg

12

3

2

bdI =

= 0.02*0.0083/12

= 8.5333e-13 m4

3

2

224

3

L

IEk =

= 3*69e9*8.5333e-13/0.1253k

= 90.4397 N-m

Hence, in this approximation, 0.0915/4397.90/ 44 == mkn = 31.43 rad/s (5.0036

Hz)


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(a)

(b)

(c ) (d)

Figure 14.1 Comparison of experimental results and theoretical predictions.

Experimental results were obtained for the frame under harmonic base motions

supplied by the shake table and theoretical results from the 4-dof model of equation

4.6. (a) Amplitude of the base motion as the speed of the motor is varied; note that

this amplitude is expected to be independent of the motor speed; (b) Displacement of

mass M2; (c) Displacement of mass M4; (d) Displacement transmissibility ratio.


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Figure 14.2 Test set-up for the impulse hammer test on the frame in figures 4.1c and

d; impulse hammer excitations; F: Impulse hammer; A1-A3: Accelerometers.

(a)


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(b)

(c)


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197

(d)

Figure 14.3 Mode shapes of the the frame in figures 4.1c and d obtained from detailed

FE model; (a) I-mode; (b) II-mode; (c) III-mode; (d) IV-mode


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I-modeII-mode

III-mode IV-mode

Figure 14.4 Mode shapes of the frame in figures 4.1c and d obtained from

experimental modal analysis using impulse hammer tests and modal extractionsoftware.


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Figure 14.5 Displacement transmissibility ratios obtained using different alternativeapproaches. (Experiment-I: Impulse hammer test and modal extraction using ME scope;Experiment-II: Shake table test)


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Table 14.1 Equipments used in free vibration and forced vibration test of three-story shear building

frame

No. Equipments Quantity

1 Oscilloscope/Data acquisition system 1

2 Accelerometer 3

3 Charge amplifier/Transducer conditioner 14 Shake table 1


Material Properties

No. Part Material

Mass kg

Youngs

Modulus (E)

N/m2

Mass density ()kg/m

3

1 Column Aluminum Mc= 0.0814 69.0E+009 2700

2 Slab Aluminum Ms=1.5430 69.0E+009 2700

3Allen screw,

M8Steel Msc1=0.0280 2.0E+011 7800

4 Allen screw,M3 Steel Msc2=0.00836

5 Base block Aluminum Mbb=0.0422 69.0E+009 2700

6Spring (Al

strip)Aluminum Msp

*=0.0073 69.0E+009 2700

7Mass to be

isolatedAluminum Miso=0.0675 69.0E+009 2700

8

Mass of

accelerometer - Macc=0.012

*Msp = *d*b*L2


Dimensions in mm

No. PartDepth (D) Width (B)

Length (L) Effective

length

1 Column DC=3.00 BC=25.11 LC=400.00 -

2 Slab DS=12.70 BS=150.00 LS=300.00 -

3 Base block Dbb=25.00 Bbb=25.00 Lbb=25.00 -

4 Spring (Al strip) d=0.80 b=20.00 L2=170.00 LE2=125.00

5Mass to be

isolatedDiso=10.00 Biso =50.00 Liso=50.00 -


No. SensorSensitivity, S

mV/g

CF

m/s

2

/V

Mass

kg1 Flat Pack, Model JTF, Sensotec 129.3298 75.8526 0.012

2 Flat Pack, Model JTF, Sensotec 326.6722 30.0301 0.012

3 Flat Pack, Model JTF, Sensotec 336.5686 29.1471 0.012


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Table 14.5 Free vibration test data on three-story shear building frame with a mass, on second floor, to

be isolated

No. Quantity Notation Observations

1 Amplitude of 0th

peak A0 0.5448 m/s2

2 Amplitude of nth

peak An 0.4489 m/s2


4 Logarithmic decrement 0.0108

5 Damping ratio 0.0017

Table 14.6 Base motion test data on frame in figures 4.1c and d

No. Frequency,

f

(Hz)

Frequency

=2f

(rad/s)

Amplitude

1rms (mm)

Amplitude

2rms (mm)

Amplitude

3rms (mm)

Displacement

Amplitude

Y =

2 (CF) 1

(mm)

Displacement

Amplitude

X2=

2 (CF) 2(mm)

Displacement

Amplitude

X4=

2 (CF) 3(mm)

1 2.1362 13.4221 0.0770 0.2083 0.2838 0.1089 0.2946 0.4014

2 2.3499 14.7649 0.0761 0.3063 0.4464 0.1076 0.4332 0.6313

3 2.5024 15.7230 0.0750 0.4797 0.7347 0.1061 0.6784 1.0390

4 2.5635 16.1069 0.0757 0.6222 0.9846 0.1070 0.8799 1.3924

5 2.7161 17.0658 0.0770 2.1028 3.3068 0.1090 2.9738 4.6765

6 2.8381 17.8323 0.0761 1.0113 1.8413 0.1076 1.4301 2.6040

7 2.9907 18.7911 0.0728 0.4373 0.8376 0.1030 0.6185 1.1845

8 3.0823 19.3667 0.0744 0.2813 0.5800 0.1053 0.3978 0.8202

9 3.4180 21.4759 0.0766 0.1454 0.3268 0.1084 0.2056 0.4621

10 3.6926 23.2013 0.0738 0.1003 0.1998 0.1043 0.1418 0.2826

11 4.1199 25.8861 0.0723 0.0617 0.1477 0.1023 0.0873 0.2088

12 4.3030 27.0365 0.0723 0.0520 0.1612 0.1022 0.0735 0.2279

13 4.5471 28.5703 0.0712 0.0403 0.1872 0.1007 0.0570 0.2648

14 4.7913 30.1046 0.0718 0.0341 0.2137 0.1015 0.0482 0.3022

15 4.9744 31.2551 0.0719 0.0297 0.2891 0.1016 0.0421 0.4089

16 5.0354 31.6384 0.0718 0.0271 0.4273 0.1016 0.0384 0.6043

17 5.0659 31.8300 0.0712 0.0263 0.4430 0.1007 0.0372 0.626518 5.0964 32.0216 0.0710 0.0256 0.6779 0.1004 0.0362 0.9587

19 5.2490 32.9804 0.0706 0.0321 0.5760 0.0998 0.0454 0.8146

20 5.3101 33.3643 0.0718 0.0295 0.3615 0.1015 0.0418 0.5112

21 5.4321 34.1309 0.0705 0.0258 0.2207 0.0997 0.0364 0.3121

22 5.4626 34.3225 0.0701 0.0249 0.2043 0.0992 0.0353 0.2889

23 5.7068 35.8569 0.0697 0.0207 0.0975 0.0986 0.0292 0.1379

24 5.8289 36.6241 0.0698 0.0261 0.0684 0.0987 0.0370 0.0967

25 6.1340 38.5411 0.0702 0.0116 0.0275 0.0993 0.0165 0.0388

26 6.2561 39.3082 0.0704 0.0101 0.0194 0.0996 0.0142 0.0274

27 6.4697 40.6503 0.0707 0.0068 0.0087 0.1000 0.0097 0.0124

28 7.0190 44.1017 0.0723 0.0146 0.0171 0.1022 0.0207 0.0242

29 7.0801 44.4856 0.0723 0.0182 0.0204 0.1023 0.0258 0.0289

30 7.2632 45.6360 0.0719 0.0291 0.0305 0.1017 0.0412 0.0431

31 7.3853 46.4032 0.0720 0.0432 0.0408 0.1018 0.0611 0.0577

32 7.6294 47.9369 0.0714 0.0927 0.0771 0.1009 0.1311 0.1090

33 7.7820 48.8957 0.0714 0.2502 0.1848 0.1009 0.3539 0.2614

34 7.8430 49.2790 0.0709 0.4139 0.2953 0.1003 0.5854 0.4177

35 7.9651 50.0462 0.0701 0.9613 0.6324 0.0992 1.3595 0.8943

36 7.9956 50.2378 0.0705 0.4943 0.3257 0.0997 0.6991 0.4606

37 8.1787 51.3883 0.0701 0.1706 0.1061 0.0992 0.2412 0.1500

38 8.3008 52.1555 0.0700 0.1267 0.0752 0.0990 0.1792 0.1064

39 8.4229 52.9226 0.0700 0.1010 0.0574 0.0990 0.1428 0.0812

40 8.6060 54.0731 0.0698 0.0821 0.0435 0.0987 0.1161 0.0615


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41 8.9111 55.9901 0.0693 0.0669 0.0318 0.0980 0.0947 0.0450

42 8.9416 56.1817 0.0695 0.0658 0.0308 0.0983 0.0931 0.0436

43 9.0942 57.1405 0.0693 0.0622 0.0272 0.0980 0.0880 0.0385

44 9.3079 58.4833 0.0690 0.0590 0.0237 0.0976 0.0835 0.0335

45 9.4910 59.6337 0.0689 0.0576 0.0219 0.0975 0.0815 0.0310

46 9.7351 61.1674 0.0690 0.0573 0.0204 0.0976 0.0810 0.0288

47 10.0098 62.8934 0.0693 0.0597 0.0194 0.0980 0.0845 0.0275

48 10.2539 64.4272 0.0696 0.0646 0.0205 0.0985 0.0914 0.029049 10.4370 65.5776 0.0695 0.0690 0.0200 0.0983 0.0976 0.0283

50 10.6812 67.1120 0.0701 0.0816 0.0227 0.0992 0.1154 0.0321

51 11.1389 69.9878 0.0712 0.1460 0.0369 0.1006 0.2064 0.0522

52 11.3220 71.1382 0.0712 0.2168 0.0523 0.1007 0.3066 0.0739

53 11.5356 72.4803 0.0719 2.7661 0.6252 0.1017 3.9118 0.8842

54 11.5967 72.8642 0.0716 1.8836 0.4200 0.1013 2.6639 0.5940

55 11.9019 74.7818 0.0728 0.1822 0.0391 0.1030 0.2576 0.0553

56 12.2375 76.8905 0.0733 0.0812 0.0166 0.1036 0.1148 0.0235

57 12.2986 77.2744 0.0737 0.0712 0.0144 0.1042 0.1007 0.0204

58 12.6343 79.3836 0.0738 0.0449 0.0088 0.1044 0.0635 0.0124

59 13.0005 81.6846 0.0732 0.0306 0.0060 0.1035 0.0432 0.0085

60 13.5193 84.9443 0.0745 0.0206 0.0038 0.1054 0.0291 0.0054


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Table 14.7 Frequency ratio and the displacement transmissibility ratio

No.

Frequency ratio

f / fn

Amplitude ratio

X4/ X2

1 0.4192 1.3652

2 0.4611 1.4574

3 0.4910 1.53154 0.5030 1.5825

5 0.5329 1.5726

6 0.5569 1.8208

7 0.5868 1.9152

8 0.6048 2.0619

9 0.6707 2.2475

10 0.7246 1.9927

11 0.8084 2.3916

12 0.8443 3.1017

13 0.8922 4.6431

14 0.9401 6.2660

15 0.9761 9.7197

16 0.9880 15.7434

17 0.9940 16.8614

18 1.0000 26.5040

19 1.0299 17.9559

20 1.0419 12.2436

21 1.0659 8.5645

22 1.0719 8.1901

23 1.1198 4.7185

24 1.1437 2.6171

25 1.2036 2.3581

26 1.2276 1.9267

27 1.2695 1.2771

28 1.3772 1.1698

29 1.3892 1.1200

30 1.4252 1.045531 1.4491 0.9437

32 1.4970 0.8314

33 1.5270 0.7387

34 1.5389 0.7134

35 1.5629 0.6579

36 1.5689 0.6588

37 1.6048 0.6217

38 1.6288 0.5937

39 1.6527 0.5688

40 1.6886 0.5296

41 1.7485 0.4750

42 1.7545 0.4685

43 1.7844 0.4379

44 1.8264 0.4010

45 1.8623 0.3797

46 1.9102 0.3560

47 1.9641 0.3255

48 2.0120 0.3171

49 2.0479 0.2895

50 2.0958 0.2779

51 2.1856 0.2516

52 2.2216 0.2411

53 2.2635 0.2260


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54 2.2755 0.2230

55 2.3354 0.2147

56 2.4012 0.2097

57 2.4132 0.2027

58 2.4791 0.1949

59 2.5509 0.1910

60 2.6527 0.1846

Table 14.8 Estimate of the natural frequencies of theframe in figures 4.1c and d; Experiment-I:

Impulse hammer test and modal extraction using ME scope; Experiment-II: Shake table test.


Mode No. FE Model 4-DOF ModelExperiment I Experiment II

1 2.79916 2.7914 2.74658 2.7161

2 5.16252 5.0438 5.06592 5.0964

3 8.05047 7.9083 7.93457 7.9651

4 11.68590 11.2579 11.59668 11.5356


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Experiment 5

Dynamics of a vibration absorber

The performance of the experiment, as per suggested procedure here in, on the setup

developed at IISc, resulted in observations and subsequent analysis as provided in

tables 15.1-15.9 and figures15.1-15.3. Refined methods of experimental and

mathematical modeling can be brought to bear on the structure under study. Thus the

beam vibration absorber system was analyzed mathematically using finite element

(FE) analysis and experimentally by using experimental modal analysis (EMA)

procedures. The FE analysis was carried out using NISA software and the EMA was

carried out using MEscope software. The FE modeling involved descretization of the

structure using 2-D beam and concentrated mass elements resulting in a system with

40 degrees of freedom. Figure 15.4 shows the setup used for measurement of

frequency response functions using impulse hammer test. Figures 15.5 and 15.6

respectively show the first three mode shapes as predicted by detailed FE analysis and

EMA. Both the FE model and the EMA model predict the presence of an intermediate

mode that gets sandwiched between the two modes predicted by the approximate 2-dof model for the beam-absorber system.Figure 15.7 shows the frequency response

functions obtained using refined and approximate methods.

Calculations for main beam(1 dof system)

Mass of main beam, M0 = (0.3714*MA)+MB+MC+2*MD+ME+ Mac1 = 3.9999 Kg

Stiffness of the main beam, K1=2.5898e+004 N/m

Frequency of the main beam, f =12.8065 Hz

Damping ratio = 0.202 % obtained from table15.5

Calculations for main beam and absorber beam (2 dof system)

M1=(0.3714*MA)+MB+MC+2*MD+ME+MF + Mac1= 4.1952 kg

K1=192*E*I/L3

e 1 = 2.5898e+004 N/m

M2=0.2357*2*MG+2*MH + Mac2 = 0.4516 kg

K2=2*3*E*I/L3

e 2 = 0.30229 e+004 N/m

Mass matrix (kg)

M=[4.1952 0

0 0.4516]



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K= 1.0e+004 * [2.89209 -0.30229

-0.30229 0.30229]


{n} = =[67.7897

94.8288]


= [-0.3373 -0.3530

-1.0759 1.0281]

Damping ratios obtained from half power bandwidth method

{} = [0.026508

0.021382]


1][

t =[-1.4150 -1.4808

-0.4858 0.4643];

]2[ = [3.5939 0

0 4.0554];

1][ = [-1.4150 -0.4858

-1.4808 0.4643];


C = [ ][ ] 11 2][ t = [16.0892 -0.3172-0.3172 1.7224];


Mt =[1.0000 0.0000

0.0000 1.0000]

Kt

= 1.0e+003 *[4.5954 -0.0000

0.0000 8.9925];

=

\

\2

n


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Ct = [3.5939 0.0000-0.0000 4.0554];

=

\

2

\

nn


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Figure 15.1 Plot ofX/Fversusf

Figure 15.2 Plot of Y/Fversusf

Figure 15.3 Plot ofZ/Fversusf


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F

A1

A2A3

Figure 15.4 Setup for impulse hammer test; F: point of application of impulse; A1, A2,

A3: accelerometers.

Charge Amplifier

Signal

Conditioning

Amplifiers

Anti-aliasing

Filters

Simultaneous

Sampling and

hold Board

Analyzer

&

Display


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(b)

(c)

Figure 15.5 Mode shapes of the combined system obtained from detailed FE model; (a) I-mode; (b) II-

mode; (c) III-mode


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(a)

(b)


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(c)

Figure 15.6 Mode shapes of the combined system obtained from experimental modal analysis; (a) I-

mode; (b) II-mode; (c) III-mode

(a)


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(b)

(c)

Figure 15.7 Frequency response functions using refined and approximate methods; (a)F

X&versusf;(b)

F

Y&versus f ; (c)

F

Z&& versus f; Experiment-I: Impulse hammer test and modal extraction using ME

scope; Experiment-II: sine test using motor with eccentric mass.


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Table 15.1 Equipments used in free vibration and forced vibration test of DVA

S.No. Equipments Quantity

1 Oscilloscope 1

2 Accelerometers 2

3 Transducer conditioning amplifiers 2

4 Regulated DC power supplier 1


Mass kg Material PropertiesNotat

ionPart Material Formula Value Youngs Modulus

(E) N/m2

Mass density

() kg/m3

A Main Beam Mild Steel*D1*B1*

Le1MA= 3.0153 2.10E+11

7800

B D.C. Motor -

C Fly wheel Aluminum

- MB+MC=

2.2550.69E+11 2700

D Eccentric mass Mild Steel- MD=0.03 2.10E+11 7800

E Connectingplates

Mild Steel - ME=0.510 2.10E+11 7800

F Connecting rod Mild Steel*Dc*Bc*

LcMF =0.1953 2.10E+11 7800

GAbsorber beam

Aluminum*D2*B2*

L2MG = 0.0762 0.69E+11 2700

HMass on

absorber beamMild Steel

*(pi*DG2

/4)*TGMH = 0.2058 2.10E+11 7800

ac1 Accelrometer1 - - Mac1= 0.055 - -

ac2 Accelrometer2 - - Mac2= 0.004 - -


Dimensions in mmPart

Depth (D) Width (B) Length (L) Effective Length (Le)

Main beam D1=6.43 B1=50.10 L1=1500 Le 1= 1200

Absorber beam D2=3.15 B2=29.85 L2=300* Le 2= 220

Connecting rod Dc=11.92 Bc=11.92 Lc=176.26 -

Diameter (DG ) 40Mass on absorber

beam Thickness (TG ) 21

Eccentricity of eccentric mass on the flywheel, e=50 mm

* The absorber is modeled as a cantilever beam. The total length of the absorber beam is 2*L2


Sensitivity, SSl. No. Sensor

pC/ms-2

pC/g

CFx10-6

Mass

kg1 Accelerometer type 4370, B & K 10.05 98.7 1 m/mV 0.055

2 Accelerometer type 4375, B & K 0.320 3.14 1 m/mV 0.004


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Table 15.5 Free vibration test data on primary beam (without the absorber)


1 Amplitude of 0th

peak A0 2600 mV

2 Amplitude of nth

peak An 2440 mV


4 Logarithmic decrement 0.0135 Damping ratio 0.00202

Table 15.6 Forced vibration test data on primary beam (without the absorber)

No. Frequency,

f

(Hz)

Frequency

=2f

(rad/s)

Amplitude

rms (mV)

Amplitude

X = 2 (CF)

(m) x10-3

Force,

F=2MD e2

(N)

Receptance

X/F

(m/N) x10-3

1 5.1000 32.0442 60 0.0849 3.0805 0.0275

2 5.6000 35.1858 80 0.1131 3.7141 0.0305

3 6.2000 38.9557 114 0.1612 4.5527 0.0354

4 6.6000 41.4690 140 0.1980 5.1590 0.0384

5 7.2000 45.2389 184 0.2602 6.1397 0.0424

6 7.7000 48.3805 233 0.3295 7.0220 0.0469

7 8.2000 51.5221 290 0.4101 7.9636 0.05158 8.7000 54.6637 373 0.5275 8.9644 0.0588

9 9.2000 57.8053 464 0.6562 10.0244 0.0655

10 9.8000 61.5752 630 0.8910 11.3745 0.0783

11 10.4000 65.3451 880 1.2445 12.8100 0.0972

12 10.8000 67.8584 1130 1.5981 13.8143 0.1157

13 11.3000 71.0000 1660 2.3476 15.1230 0.1552

14 11.6000 72.8849 2000 2.8284 15.9366 0.1775

15 11.9000 74.7699 2830 4.0022 16.7716 0.2386

16 12.0000 75.3982 3140 4.4406 17.0547 0.2604

17 12.2500 76.9690 4730 6.6892 17.7727 0.3764

18 12.3000 77.2832 5130 7.2549 17.9181 0.4049

19 12.6000 79.1681 11000 15.5563 18.8028 0.8273

20 12.7000 79.7965 20000 28.2843 19.1024 1.4807

21 12.8000 80.4248 35000 49.4975 19.4044 2.5508

22 12.8500 80.7389 40000 56.5685 19.5563 2.8926

23 12.9000 81.0531 50000 70.7107 19.7088 3.5878

24 13.0000 81.6814 15000 21.2132 20.0156 1.0598

25 13.6000 85.4513 4050 5.7276 21.9058 0.2615

26 13.8500 87.0221 3370 4.7659 22.7185 0.2098

27 14.0500 88.2788 2910 4.1154 23.3794 0.1760

28 14.3500 90.1637 2470 3.4931 24.3885 0.1432

29 14.6500 92.0487 2130 3.0123 25.4189 0.1185

30 15.0200 94.3734 1870 2.6446 26.7190 0.0990

31 15.3500 96.4469 1700 2.4042 27.9060 0.0862

32 16.0000 100.5310 1450 2.0506 30.3194 0.0676

33 16.7000 104.9292 1270 1.7961 33.0304 0.0544

34 17.1000 107.4425 1190 1.6829 34.6317 0.048635 17.6000 110.5841 1130 1.5981 36.6865 0.0436

36 18.2000 114.3540 1050 1.4849 39.2305 0.0379

37 19.1000 120.0088 978 1.3831 43.2064 0.0320

38 19.9000 125.0354 925 1.3081 46.9015 0.0279


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Table 15.7 Free vibration test data combined system.

S.No. Quantity Notation Observations on

primary beam

Observations on

absorber beam

1 Amplitude of 0th

peak A0 20.0 V 2.12 V

2 Amplitude of n

th

peak An 18.4 V 1.92 V3 Number of cycles n 8 7

4 Logarithmic decrement 0.0104 0.0142

5 Damping ratio 0.00165 0.00225

Average value of damping ratio, =0.00195

Table 15.8 Forced vibration test data on combined system; measurement made on main beam

No. Frequency,

f

(Hz)

Frequency

=2f

(rad/s)

Amplitude

rms (mV)

Amplitude

Y = 2 (CF)

(m) x10-3

Force,

F=2MD e2

(N)

Receptance

Y/F

(m/N) x10-3

1 5.0000 31.4159 60 0.0849 2.9609 0.0287

2 5.7000 35.8142 91 0.1287 3.8480 0.03343 6.3600 39.9611 131 0.1853 4.7907 0.0387

4 6.9500 43.6681 182 0.2574 5.7207 0.0450

5 7.7000 48.3805 269 0.3804 7.0220 0.0542

6 8.3300 52.3389 383 0.5416 8.2181 0.0659

7 8.9600 56.2973 567 0.8019 9.5082 0.0843

8 9.5600 60.0673 880 1.2445 10.8242 0.1150

9 10.0800 63.3345 1530 2.1637 12.0338 0.1798

10 10.5500 66.2876 3500 4.9497 13.1821 0.3755

11 10.8700 68.2982 28000 39.5980 13.9939 2.8297

12 11.4400 71.8796 2700 3.8184 15.5000 0.2463

13 12.0200 75.5239 1300 1.8385 17.1116 0.1074

14 12.7400 80.0478 703 0.9942 19.2229 0.0517

15 13.2500 83.2522 405 0.5728 20.7928 0.0275

16 13.6400 85.7026 240 0.3394 22.0348 0.015417 14.1000 88.5929 36 0.0509 23.5461 0.0022

18 14.6000 91.7345 340 0.4808 25.2457 0.0190

19 15.1300 95.0646 1040 1.4708 27.1118 0.0542

20 15.5800 97.8920 3200 4.5255 28.7485 0.1574

21 15.7700 99.0858 13300 18.8090 29.4540 0.6386

22 16.3700 102.8557 5240 7.4105 31.7379 0.2335

23 16.7200 105.0549 3090 4.3699 33.1096 0.1320

24 17.1200 107.5681 2230 3.1537 34.7127 0.0909

25 17.8300 112.0292 1610 2.2769 37.6516 0.0605

26 18.2500 114.6681 1420 2.0082 39.4463 0.0509

27 19.0000 119.3805 1200 1.6971 42.7551 0.0397

28 19.7600 124.1557 1060 1.4991 46.2439 0.0324


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Table 15.9 Forced vibration test data on combined system; measurement made on absorber beam

No. Frequency,

f

(Hz)

Frequency

=2f

(rad/s)

Amplitude

rms (mV)

Amplitude

Z = 2 (CF)

(m) x10-3

Force,

F=2MD e2

(N)

Receptance

Z /F

(m/N) x10-3

1 5.2000 32.6726 72 0.1018 3.2025 0.0032

2 6.0000 37.6991 256 0.3620 4.2637 0.0085

3 6.5000 40.8407 436 0.6166 5.0039 0.01234 7.2000 45.2389 940 1.3294 6.1397 0.0217

5 7.7200 48.5062 1680 2.3759 7.0586 0.0337

6 8.3600 52.5274 3550 5.0205 8.2774 0.0607

7 8.8800 55.7947 6630 9.3762 9.3391 0.1004

8 9.5000 59.6903 13800 19.5161 10.6888 0.1826

9 10.0000 62.8319 27500 38.8909 11.8435 0.3284

10 10.4800 65.8478 67000 94.7523 13.0078 0.7284

11 10.8500 68.1726 70000 989.9495 13.9425 7.1002

12 11.3000 71.0000 14700 207.8894 15.1230 1.3747

13 11.5000 72.2566 8610 121.7638 15.6631 0.7774

14 11.9000 74.7699 6620 93.6209 16.7716 0.5582

15 12.2000 76.6549 4470 63.2153 17.6279 0.3586

16 12.8500 80.7389 3720 52.6087 19.5563 0.269017 13.5000 84.8230 3310 46.8105 21.5848 0.2169

18 14.0000 87.9646 3510 49.6389 23.2133 0.2138

19 14.5000 91.1062 4060 57.4171 24.9010 0.2306

20 15.0000 94.2478 5270 74.5291 26.6479 0.2797

21 15.5500 97.7035 11000 155.5635 28.6379 0.5432

22 15.7500 98.9602 20700 292.7422 29.3793 0.9964

23 15.8000 99.2743 32800 463.8620 29.5662 1.5689

24 16.2000 101.7876 12600 178.1909 31.0821 0.5733

25 16.6000 104.3009 7230 102.2476 32.6360 0.3133

26 17.0000 106.8142 4070 57.5585 34.2278 0.1682

27 17.5000 109.9557 2500 35.3553 36.2708 0.0975

28 18.1000 113.7257 1390 19.6576 38.8006 0.0507

29 18.7300 117.6841 904 12.7845 41.5486 0.0308

30 19.2000 120.6372 720 10.1823 43.6600 0.023331 19.8000 124.4071 550 7.7782 46.4314 0.0168

Note: Here in this case CF=1 m/mV from 1-10 points and 0.1 m/mV from 11-31 points


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EXPERIMENT 6

Dynamics of a four storied building model with and without an open

ground floor

The frequency response functions for displacement of different floors were obtained

using impulse hammer test. Figure 16.1 shows the experimental setup and figures

16.2 and 16.3 show the comparison between analytical and experimental results.

Tables 16.1-16.8 document the results of experiment as per the suggested procedure.

Figure 16.4 show the plot of absolute floor displacements for the two frames obtained

using theory and experiment. Results on spring forces from theory and experiment are

shown in figure 16.5 for the two frames. Tables 16.9 and 16.10 compare the results on

fundamental natural frequency for the two frames using theory and experiment. From

Table 16.9 it may be observed that the match between theoretical and experiment is

not good. This could be due to the approximate nature of determination of inter-storey

stiffness.

Brief details of analysis of the 4-dof model is given below.

Calculations for building frame without soft first story

Mass matrix (kg)

M = [3.0390 0 0 0

0 3.0390 0 0

0 0 3.0390 0

0 0 0 2.2910];


K = [558260 -279130 0 0

-279130 558260 -279130 0

0 -279130 558260 -279130

0 0 -279130 279130];


{ }n = [111.2212317.4533

478.3553

575.2993];


= [-0.1418 0.3483 0.3719 -0.2221

-0.2645 0.3145 -0.1827 0.3561

-0.3516 -0.0644 -0.2821 -0.3489

-0.3913 -0.3726 0.3213 0.2033] ;


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Damping ratios

{} = [0.0190

0.0074

0.0076

0.0056];


[ ] 1 t = [-0.4309 1.0586 1.1301 -0.6750-0.8037 0.9557 -0.5552 1.0823

-1.0684 -0.1957 -0.8573 -1.0603

-0.8964 -0.8537 0.7360 0.4657];

[ ]2 = [4.23534.6922

7.2446

6.4940];

[ ] 1 = [-0.4309 -0.8037 -1.0684 -0.89641.0586 0.9557 -0.1957 -0.8537

1.1301 -0.5552 -0.8573 0.7360

-0.6750 1.0823 -1.0603 0.4657];


[ ] [ ][ ] == 11 2tC [18.2551 -3.0760 -1.3936 1.3802-3.0760 16.8615 -1.2451 -0.4640

-1.3936 -1.2451 17.6396 -2.93791.3802 -0.4640 -2.9379 12.1562];


[ ][ ] Mt = [1.0000 0.0000 0.0000 -0.00000.0000 1.0000 -0.0000 -0.0000

0.0000 -0.0000 1.0000 0.0000

-0.0000 -0.0000 0.0000 1.0000];

[ ][ ] Kt = 1.0e+005 *[0.1237 0.0000 0.0000 0.00000.0000 1.0078 0.0000 0.0000

0.0000 0.0000 2.2882 0.0000-0.0000 0.0000 0.0000 3.3097];

[ ][ ] Ct = [4.2353 -0.0000 -0.0000 -0.00000.0000 4.6922 -0.0000 -0.0000

0.0000 0.0000 7.2446 -0.0000

-0.0000 -0.0000 0.0000 6.4940];


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Calculations for building frame with soft first story

Mass matrix (kg)

M = [2.6398 0 0 0

0 3.0390 0 0

0 0 3.0390 00 0 0 2.2910];


K = [306801 -279130 0 0

-279130 558260 -279130 0

0 -279130 558260 -279130

0 0 -279130 279130];


{ }n = [ 48.0745261.8895

456.5069

571.1084];


= [-0.2768 0.4047 0.3263 -0.1788

-0.2981 0.1823 -0.2845 0.3550

-0.3120 -0.1762 -0.2498 -0.3718

-0.3181 -0.4031 0.3516 0.2217] ;

Damping ratios

{} = [0.0050

0.0050

0.0050

0.0050];


[ ] 1 t = [-0.7306 1.0683 0.8614 -0.4720-0.9061 0.5540 -0.8645 1.0788

-0.9483 -0.5355 -0.7592 -1.1299

-0.7287 -0.9236 0.8056 0.5079];

[ ]2 = [0.48072.6189

4.5651

5.7111];


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221

[ ] 1 = [-0.7306 -0.9061 -0.9483 -0.72871.0683 0.5540 -0.5355 -0.9236

0.8614 -0.8645 -0.7592 0.8056

-0.4720 1.0788 -1.1299 0.5079];


[ ] [ ][ ] == 11 2tC [7.9050 -4.4390 -1.1052 -0.5289-4.4390 11.2563 -4.3286 -1.0729

-1.1052 -4.3286 11.1055 -4.4424

-0.5289 -1.0729 -4.4424 6.9254];


[ ][ ] Mt = [1.0000 -0.0000 -0.0000 0.0000-0.0000 1.0000 -0.0000 -0.0000

-0.0000 -0.0000 1.0000 0.0000

0.0000 -0.0000 0.0000 1.0000];

[ ][ ] Kt = 1.0e+005 *[0.0231 0.0000 -0.0000 0.00000.0000 0.6859 -0.0000 0.0000

-0.0000 -0.0000 2.0840 0.0000

0.0000 0.0000 0.0000 3.2616];

[ ][ ] Ct = [0.4807 -0.0000 -0.0000 -0.00000.0000 2.6189 -0.0000 0.0000

-0.0000 -0.0000 4.5651 -0.0000

0.0000 -0.0000 0.0000 5.7111];


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Figure 16.1 Setup for impact hammer tests on the two frames.


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223

(a)

(b)

(c)

Figure 16.2 Frequency response function for the structure with soft first storey;

response measured at (a) first floor (drive point); (b) second floor; (c) third floor.


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(a)

(b)

(c)

Figure 16.3 Frequency response function for the structure without soft first storey;

response measured at (a) first floor (drive point); (b) second floor; (c) third floor.


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5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 100

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

Frequency, Hz

AbsoluteDisplacement,m

I Floor

II Floor

III Floor

IV Floor

(a)

5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 100

0.002

0.004

0.006

0.008

0.01

0.012

Frequency, Hz


I Floor

II Floor

III Floor

(b)

10 11 12 13 14 15 16 17 18 19 200

1

2

3

4

5

6

7 x 10-3

Frequency, Hz


I Floor

II Floor

III Floor

IV Floor

(c)

10 11 12 13 14 15 16 17 18 19 200

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5 x 10-3

Frequency, Hz


I Floor

II Floor

III Floor

(d)

Figure 16.4 Absolute displacement response of the four storied building model (a)

analytical (with soft first storey); (b) experimental (with soft first storey); (c)

analytical (without soft first storey); (d) experimental (without soft first storey).


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5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 100

50

100

150

200

250

300

350

400

450

Frequency, Hz

Springforce,

N

I Floor

II Floor

III Floor

IV Floor

(a)

5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 100

50

100

150

200

250

300

Frequency, Hz

Springforce,

N

I Floor

II Floor

III Floor

(b)

10 11 12 13 14 15 16 17 18 19 200

100

200

300

400

500

600

Frequency, Hz

Springforce,

N

I FloorII Floor

III Floor

IV Floor

(c)

10 11 12 13 14 15 16 17 18 19 20-50

0

50

100

150

200

250

300

350

400

450

Frequency, Hz

Springforce,

N

I FloorII Floor

III Floor

(d)

Figure 16.5 Spring forces in four storied building model (a) analytical (with soft first

storey); (b) experimental (with soft first storey); (c) analytical (without soft first

storey); (d) experimental (without soft first storey).


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Table 16.1 Details of sensors and equipment used.

Sl.No. Equipments Quantity

1 Accelerometers 4

2 Nexus conditioning amplifier 1

3 Shake Table 1

4 Data Acquisition System 1


Material PropertiesSl.

No.Part Quantit

y

MaterialMass

kgYoungs Modulus

(E) N/m2

Mass density ()

kg/m3

1 Column 4 nos. Aluminum Mc= 0.6976 69.0E+009

2700

2 Slab 1 no. Aluminum Ms= 1.5430 69.0E+009

2700

3Stiffener

A

2 nos.Aluminum Msa= 0.2592 69.0E+009

2700

4Stiffener

B

2 nos.Aluminum Msb= 0.5391 69.0E+009

2700


Dimensions in mmSl.

No.

Part

Depth Width Length

1 Slab 150 300 12.7

2 Column 25.1391 6.4233 400

3 Stiffener A 150 2 160

4 Stiffener B 2 312 160

Table 16.4 Details of the sensors usedSensitivity, SSl.

No.Sensor

mV/ms-2

mV/g

Mass

gm

1 B & K Deltatron Accelerometer Type 4507

002, Sl. No. 10308

93.8 920 4.8

2 B & K Deltatron Accelerometer Type 4507

002, Sl. No. 10309

94.6 928 4.8

3 B & K Deltatron Accelerometer Type 4507,

Sl. No. 11574

9.80 96.1 4.8

4 B & K Deltatron Accelerometer Type 4507,

Sl. No. 11575

10.04 98.4 4.8


70/114

Table 16.5 Base motion test data on four storey building frame with soft first storey

Sl.no Frequency

Hz

Base motion

Amplitudex*10

-4

rms (V)

First floor

Amplitudex1*10

-4

rms (V)

Second floor

Amplitudex2*10

-4

rms (V)

Third floor

Amplitudex3*10

-4

rms (V)

Base motionDisplacement

Amplitude

Xg= 2 x

(m) *10-4

First floorDisplacement

Amplitude

X1= 2 x1

(m) *10-4

D

1 2.29 1.0930 3.6610 3.9680 4.2210 1.545721 5.177386

2 2.40 1.0810 1.4110 1.3380 1.3410 1.528750 1.995436

3 2.62 1.0840 1.4590 1.4890 1.3970 1.532993 2.063318

4 2.90 1.0870 1.4360 1.5300 1.4170 1.537235 2.030791

5 3.00 1.1360 1.5350 1.5050 1.4600 1.606531 2.170797

6 3.30 1.1220 2.4460 2.6490 2.6870 1.586732 3.459133

7 3.70 1.1520 1.7940 1.6850 1.7230 1.629158 2.537075

8 4.00 1.1510 1.7600 1.8170 1.8830 1.627744 2.488992

9 4.30 1.1750 2.0680 2.1060 2.1000 1.661685 2.924566

10 4.40 1.2160 2.1920 2.1920 2.2350 1.719667 3.099926

11 4.50 1.2180 2.1390 2.2150 2.2520 1.722496 3.024974

12 4.75 1.2100 2.4190 2.4340 2.4980 1.711182 3.420950

13 4.90 1.2280 2.5780 2.7920 2.8530 1.736638 3.645808

14 4.95 1.2640 2.6830 2.7610 2.8690 1.787549 3.794299

15 5.10 1.2580 3.0420 3.2030 3.2120 1.779064 4.301996

16 5.25 1.2550 3.4880 3.6870 3.7870 1.774821 4.932730

17 5.40 1.3360 4.0300 4.2500 4.4080 1.889371 5.699226

18 5.68 1.3240 4.5040 4.7730 5.0380 1.872401 6.369557

19 5.81 1.3620 5.3090 5.6370 5.9360 1.926140 7.507988

20 5.95 1.3950 6.1890 6.6130 7.0100 1.972809 8.752484

21 6.30 1.5220 11.6120 12.5850 13.4630 2.152412 16.421690


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22 6.40 1.5510 13.3400 14.3090 15.3550 2.193424 18.865428

23 6.70 1.0330 69.5040 75.5870 82.3930 1.460869 98.292557

24 6.80 0.9240 25.3700 27.4880 29.7840 1.306721 35.878254

25 6.90 0.9900 17.8570 19.4060 20.9650 1.400058 25.253369

26 7.20 1.1690 6.1100 6.6630 7.2090 1.653200 8.640762

27 7.60 1.2040 4.3970 4.8490 5.1800 1.702697 6.218237

28 7.80 1.2390 3.1800 3.5220 3.7620 1.752194 4.497156

29 8.20 1.2110 2.0140 2.2400 2.4220 1.712596 2.848199

30 8.70 1.2170 1.7510 1.8360 1.8850 1.721081 2.476264

31 9.26 1.2260 1.5690 1.5460 1.6080 1.733809 2.218880

32 9.95 1.2010 1.5100 1.5700 1.5890 1.698454 2.135442

33 10.70 1.1620 1.1220 1.2040 1.2060 1.643300 1.586732

34 12.00 1.1440 0.7930 0.7420 0.7780 1.617845 1.121461

35 12.95 1.1370 0.7830 0.8350 0.7270 1.607945 1.107319

36 13.89 1.1830 0.8330 0.8700 0.6130 1.672999 1.178029

37 15.30 1.2610 0.7100 0.7990 0.5210 1.783306 1.004082

38 17.20 1.4310 0.8040 0.8490 0.5070 2.023720 1.137017

39 17.60 1.4730 0.8990 0.8760 0.6740 2.083117 1.271366

40 19.70 1.2740 0.7550 0.6440 0.4430 1.801691 1.067721


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Table 16.6 Base motion test data on four storey building frame without soft first storey

Sl.no Frequency

(Hz)

Base motion

Amplitude

x*10-4

rms(V)

First floor

Amplitude

x1*10-4

rms(V)

Second

floor

Amplitude

x2*10-4

rms(V)

Third floor

Amplitude

x3*10-4

rms(V)

Base motion


Xg= 2 x

(m) *10-4

First floor


X1= 2 x1

(m) *10-4

Second fl

DisplacemAmplitu

X2= 2

(m) *10

1 2.44 1.0110 1.2370 1.3210 1.1800 1.4298 1.7494 1.8682

2 2.44 1.0400 1.2380 1.2620 1.1700 1.4708 1.7508 1.7847

3 4.88 1.1200 1.3560 1.4070 1.3130 1.5839 1.9177 1.9898

4 5.11 1.1300 1.3340 1.4720 1.3780 1.5980 1.8865 2.0817

5 5.49 1.1440 1.3340 1.4700 1.4000 1.6178 1.8865 2.0789

6 5.68 1.1540 1.3720 1.5690 1.4110 1.6320 1.9403 2.2189

7 5.62 1.1880 1.4320 1.4650 1.4530 1.6801 2.0251 2.0718

8 6.17 1.1800 1.4060 1.4810 1.4910 1.6688 1.9884 2.0944

9 6.58 1.1960 1.3990 1.6550 1.5410 1.6914 1.9785 2.3405

10 6.90 1.2240 1.4910 1.6680 1.6570 1.7310 2.1086 2.3589

11 7.30 1.2410 1.6280 1.8720 1.9060 1.7550 2.3023 2.6474

12 7.70 1.2780 1.7600 2.0330 2.1470 1.8073 2.4890 2.8751

13 7.95 1.3220 1.7060 1.9510 2.0110 1.8696 2.4126 2.7591

14 8.20 1.3290 1.6510 1.8390 1.9830 1.8795 2.3348 2.6007

15 8.70 1.3600 1.6810 1.8920 2.0330 1.9233 2.3773 2.6757

16 8.93 1.3780 1.9470 2.0190 2.1350 1.9488 2.7534 2.8553

17 9.30 1.4230 1.7830 2.0960 2.2430 2.0124 2.5215 2.9642

18 9.45 1.4350 1.8240 2.1480 2.3490 2.0294 2.5795 3.0377


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19 9.54 1.4550 1.8470 2.2170 2.4170 2.0577 2.6120 3.1353

20 9.80 1.4860 2.1270 2.3180 2.5570 2.1015 3.0080 3.2781

21 10.42 1.4970 2.0960 2.4660 2.7290 2.1171 2.9642 3.4874

22 10.64 1.5050 2.2070 2.5430 2.8600 2.1284 3.1211 3.5963

23 10.70 1.5120 2.3360 2.7350 2.9990 2.1383 3.3036 3.8678

24 11.00 1.5370 2.3280 2.8420 3.1330 2.1736 3.2923 4.0192

25 11.30 1.5310 2.2940 2.9070 3.3970 2.1651 3.2442 4.1111

26 11.50 1.5630 2.4860 3.1400 3.7450 2.2104 3.5157 4.4406

27 11.90 1.5630 2.7140 3.4230 4.1670 2.2104 3.8381 4.8408

28 12.20 1.5790 2.9110 3.8420 4.6580 2.2330 4.1167 5.4334

29 12.40 1.6100 2.9760 3.9290 4.8510 2.2769 4.2087 5.5564

30 12.50 1.6030 2.9710 4.0240 4.8710 2.2670 4.2016 5.6907

31 12.82 1.6220 3.2440 4.5840 5.7250 2.2938 4.5877 6.4827

32 13.00 1.6450 3.3390 4.6220 5.8200 2.3264 4.7220 6.5364

33 13.20 1.6670 3.7240 5.2750 6.7020 2.3575 5.2665 7.4599

34 13.35 1.7050 3.8260 5.5730 7.1550 2.4112 5.4107 7.8813

35 13.55 1.7070 4.2920 6.1980 8.1130 2.4140 6.0697 8.7652

36 13.65 1.7330 4.3690 6.2910 8.2480 2.4508 6.1786 8.8967

37 13.71 1.7360 4.4560 6.3670 8.3410 2.4551 6.3017 9.0042

38 13.80 1.8250 5.6400 8.6810 11.7310 2.5809 7.9761 12.2767

39 13.90 1.8720 6.7390 10.7160 14.6900 2.6474 9.5303 15.1546

40 14.29 1.3430 11.4610 20.1390 29.9940 1.8993 16.2081 28.4806

41 14.29 1.3280 11.0010 19.2880 28.8310 1.8781 15.5576 27.277

42 14.50 0.9850 7.7460 14.1780 21.4670 1.3930 10.9544 20.0505

43 15.40 0.8750 4.5600 8.7890 13.3670 1.2374 6.4488 12.4294

44 15.15 0.9050 4.7640 9.0380 13.6860 1.2799 6.7372 12.7815

45 15.62 0.9100 3.6440 7.1750 10.8740 1.2869 5.1533 10.1469

46 15.80 0.9040 3.0770 6.2160 9.4210 1.2784 4.3515 8.7907


74/114

47 16.40 0.9560 2.4460 5.0660 7.7640 1.3520 3.4591 7.1643

48 16.50 1.0170 1.8250 4.1010 6.3160 1.4382 2.5809 5.7996

49 16.80 1.0090 1.7150 3.3770 5.2740 1.4269 2.4254 4.7758

50 17.10 1.0540 1.3010 2.9370 4.6540 1.4906 1.8399 4.1535

51 17.50 1.0800 1.0120 2.6460 4.0600 1.5273 1.4312 3.7420

52 17.70 1.0970 1.1360 2.1580 3.5570 1.5514 1.6065 3.0518

53 18.31 1.1400 1.2010 2.1120 3.2940 1.6122 1.6985 2.9868

54 18.50 1.1490 0.8010 1.8190 2.9320 1.6249 1.1328 2.5724

55 18.50 1.1370 1.2130 1.6990 2.9220 1.6079 1.7154 2.4027

56 18.52 1.1940 0.8350 1.5770 2.7990 1.6886 1.1809 2.2302

57 19.23 1.2140 1.3720 1.3570 2.5530 1.7168 1.9403 1.9191

58 19.70 1.2400 0.7690 1.1550 2.1430 1.7536 1.0875 1.6334


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233

Table 16.7: Free vibration test data for the one storey structure without walls


1 Amplitude of 0th

peak A0 16.0 mV

2 Amplitude of nth

peak An 10.0 mV


4 Logarithmic decrement 0.0294

5 Damping ratio 0.00476 Natural Frequency

(experiment)

f 19.23 Hz

7 Total mass (SDOF

approximation)

M1 1.8954 kg

8 Stiffness (from experiment) Kopen 2.7671E+04 N/m

Table 16.8: Free vibration test data for the one storey structure with walls


1 Amplitude of 0th

peak A0 18.4 mV

2 Amplitude of nth

peak An 6.4 mV

3 Number of cycles n 24 Logarithmic decrement 0.5280

5 Damping ratio from modal

analysis

0.019

6 Natural Frequency

(experiment)

fs 55.51 Hz

7 Total mass (SDOF

approximation)

M2 2.2946 kg

8 Stiffness (from experiment) Kclose 2.7913E+05 N/m

Table 16.9 Estimate of the first natural frequency in Hz of the building frame with and without soft first

story

Frame with soft first story Frame without soft first story

Analytical Experimental Analytical Experimental

7.65 6.70 17.70 14.29

Table 16.10 Estimate of the fundamental mode shape of the building frame with and without soft first

story

Frame with soft first storyFrame without soft first

story

Analytical Experimental Analytical Experimental1.0000 1.0000 1.0000 1.0000

1.0759 1.0887 1.8562 1.7572

1.1266 1.1905 2.4653 2.6171

1.1456 * 2.7433 *

*: this coordinate was not measured.


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234

Experiment 7

Dynamics of one-span and two-span beams

Tables 17.1-17.11 show the results of experiment conducted on the setup developed at

IISc. The two beam structures were also studied experimentally using impulse

hammer test and, computationally, using commercial finite element software. The

results of these studies are superposed on the experimental results in figure 17.1. The

results show reasonable mutual agreement. In the FE model developed, and also in the

two-dof analytical model developed, the stiffening effect of due to the placement of

the motor and the dummy mass has not been included. Similarly the support

conditions have been assumed to correspond to perfect simple support situations. This

may not be correctly realized in the experimental setup. These features possibly

explain the difference between the analytical and experimental predictions. Some of

the calculations involved in the development of the two-dof model for the two beams

systems are summarized below.

Calculations for one-span beam

Mass matrix (kg)

M1 = 0.5* MA + ME + MF

M2 = 0.5*MA + MB + MC+2* MD1+ MF

M = [4.0826 0

0 4.0891]

Flexibility matrix (m/N)

F = 1.0e-004 *[0.9122 0.7095

0.7095 0.9122]


K = 1.0e+004 *[2.7748 -2.1582

-2.1582 2.7748]


{ }n = [38.8479109.8784]


= [-0.3498 -0.3501

-0.3499 0.3495]

Damping ratios

{} = [0.0372


77/114

235

0.0148]


[ ] 1 t = [-1.4280 -1.4295-1.4306 1.4291]

[ ]2 = [2.8903 00 3.2524]

[ ] 1 = [-1.4280 -1.4306-1.4295 1.4291]


[ ] [ ][ ] == 11 2tC [12.5398 -0.7398-0.7398 12.5582]


[ ][ ] Mt = [1.0000 0.00000.0000 1.0000]

[ ][ ] Kt = 1.0e+004 [0.1509 0.00000.0000 1.2073]

[ ][ ] Ct = [2.8903 0.00000.0000 3.2524];

Calculations for two-span beam

Mass matrix (kg)

M1 = 0.25*MA + ME + MF

M2 = 0.25*MA + MB+MC+2* MD2 + MF

M = [3.3338 0

0 3.3386]

Flexibility matrix (m/N)

F = 1.0e-004 *[0.1457 -0.0570

-0.0570 0.1457]


78/114

236


K = 1.0e+004 *[8.1041 3.1712

3.1712 8.1041]


{ }n = [121.5982183.8392];


= [-0.3869 0.3876

0.3873 0.3866];

Damping ratios

{} = [0.0143

0.0113];


[ ] 1 t =[1.2899 -1.2923-1.2932 - 1.2908]

[ ]2 = [3.4777 00 4.1548]

[ ] 1 = [1.2899 -1.2932-1.2923 -1.2908]


[ ] [ ][ ] == 11 2tC [12.7246 1.12941.1294 12.7388];


[ ][ ] Mt = [1.0000 0.00000.0000 1.0000];

[ ][ ] Kt = 1.0e+004 [1.4786 00 3.3797]

[ ][ ] Ct = [3.4777 -0.0000-0.0000 4.1548]


79/114

237

(a) (b)

(c) (d)

Figure 17.1 Comparison of experimentally measured frequency response function, normalized with respect to the driving forceamplitude, with the corresponding results from analysis; (a) one-span beam; response under the dummy mass; (b) one-span beam

response under drive point; (c) two-span beam; response under the dummy mass; (d) two-span beam; response under drive point.

Experiment I: Impulse hammer test; Experiment II: harmonic excitation test using the excitation induced by the electric motor.


80/114

238

(a)

(b)

Figure 17.2 Setup for impact hammer tests on (a) one span beam (b) two span beam


81/114

239

Table 17.1 Equipments used in free vibration and forced vibration test of simply supported beam &

continuous beam

S.No. Equipments Quantity

1 Oscilloscope 1

2 Accelerometers 2

3 Signal conditioning amplifier 14 Regulated DC power supply 1

5 D.C Motor 1


Material Properties

Part MaterialMass

KgYoungs Modulus

(E) N/m2

Mass density

() kg/m3

Main Beam Mild Steel MA= 2.9952 2.00E+011

7800

D.C. Motor -

Fly wheel AluminumMB+MC= 2.000 69.0E+009 2700

Eccentric mass

(Simply supported beam)

Mild Steel MD1= 0.00324 2.00E+011 7800

Eccentric mass

(Continuous beam)

Mild Steel MD2= 0.00240 2.00E+011 7800

Lumped Mass Mild Steel ME= 2.000 2.00E+011 7800

Base plate Mild Steel MF = 0.5850 2.00E+011 7800


Dimensions in mmPart

Depth (DA) Width (BA) Length (LA) Effective Length (Le)

Main beam 6.4 50 1350 1200

Eccentricity of eccentric mass on the flywheel, e=25 mm

Formula for calculating, MA =DA*BA*Le*

Table 17.4 Details of the sensors used

Sensitivity, SSl.

No.Sensor

mV/ms-2

mV/g

Mass

gm


Type 4507, Sl. No. 11574

9.80 96.1 4.8


Type 4507, Sl. No. 11575

10.04 98.4 4.8

Table 17.5 Free vibration test data on simply supported beam (observations I) and continuous beam(observations II)

S.No. Quantity Notation Observations I Observations II

1 Amplitude of 0th

peak A0 50mV 220mV

2 Amplitude of nth

peak An 6mV 80mV

3 Number of cycles n 19 18

4 Logarithmic decrement 0.11159 0.0562

5 Damping ratio 0.01776 0.0089


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240

Table 17.6 Flexibility coefficients obtained from the finite element analysis

One span beam (m/N) Continuous beam (m/N)

f11

= 9.12233E-05 f12

= 7.09514E-05 f11

= 1.45704E-05 f12

= -5.70146E-06

f21= 7.09514E-05 f22= 9.12233E-05 f21= -5.70146E-06 f22= 1.45704E-05


83/114

Table 17.7 Forced vibration test data on simply supported beam

S.No. Frequency,

f

(Hz)

Frequency

=2f

(rad/s)

Amplitude

x1rms (mV)

Amplitude

x2rms (mV)

Conversion

Factor

CF

(V/m)

Displacement

Amplitude

X1=

2 (CF) x1(mm)

Displacement

Amplitude

X2=

2 (

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