kuliah-3 beban harmonis

Mempelajari respon struktur

akibat beban SINUSOIDAL

P0

2T

m

K

)1.........(.................... ".

0)(C REDAMANTANPA KASUS TINJAU

'.".

:

0

0

tSinPKXXM

tSinPKXXCXM

dinamisanKeseimbang

bebansudutgetaranfrekuensi

Keterangan

:

HARMONICBEBAN

TERHADAPSTRUKTURRESPON

)4..(........................................ .

)3.....(.................... ..

kerja)beban (pengaruh partikular solusidisebut

bebas)getaran (pengaruh er komplement solusidisebut

)2........(.............................. )(

:adalah atas di dinamisgerak persamaan Solusi

tSinGX

tCosBtSinAX

X

X

XXtX

p

c

p

c

pC

)6.........(.............................. :mana di

)5.(........................................ 1

1.

:diperolehakan

(1) kemasukan dan kali dua pers.(4)Turunkan

2

0

K

PG

)8.......(..........sin1

1)(

sehingga 0

dan 1

1.

:diperolehakan 0''

dan 0

:diamkeadaan

dalam sistem pada bekerjaHarmonik beban Jika

)7....(1

1.)(

:didapat (2), pers. kemasukan

(3) pers. bersamadan (4) ke pers.(5)Masukan

2

0

2

0

0)0(

0)0(

2

0

ttSinK

PtX

B

K

PA

XX

XX

tSinK

PtBCostASintX

t

t

)9(.................... 1

1)(

)(

)()(

)(

:Respon Rasio

2

0

tSintSintR

KP

tX

statisXtX

tR

Pengaruh Getaran Bebas

Steady State Response

Pengaruh Beban

Transient Response

RESPON GABUNGAN

MPF* X(statis)

2T

2T

)(** statisXMPF

)10...(........................................ 1

1

1

'X(t)

:menjadi lengkap solusi Sehingga

1

'

:diperoleh

(7) pers. kedimasukan atas di awal kondisi Jika

''

:diamtidak

yang sistem pada bekerjaHarmonik beban Jika

2

0

02

00

2

00

0

0)0(

0)0(

tSinK

P

tCosXtSinK

PX

K

PXA

XB

XX

XX

t

t

t (det) X-A X-B X-TOT t(det) X-A X-B X-TOT

0.000 3.774 -3.774 0.000 0.000 0.000 0.000 0.000

0.020 3.698 -3.247 0.451 0.020 0.750 -0.657 0.093

0.040 3.476 -1.815 1.661 0.040 1.470 -1.130 0.339

0.060 3.114 0.124 3.238 0.060 2.131 -1.288 0.842

0.080 2.629 2.028 4.657 0.080 2.707 -1.087 1.620

0.100 2.039 3.366 5.405 0.100 3.175 -0.583 2.593

0.120 1.367 3.765 5.133 0.120 3.517 0.084 3.602

0.140 0.641 3.114 3.756 0.140 3.719 0.728 4.447

0.160 -0.110 1.594 1.484 0.160 3.772 1.168 4.940

0.180 -0.857 -0.370 -1.228 0.180 3.675 1.283 4.958

0.200 -1.570 -2.232 -3.802 0.200 3.431 1.039 4.471

0.220 -2.221 -3.471 -5.691 0.220 3.051 0.506 3.557

0.240 -2.783 -3.741 -6.524 0.240 2.549 -0.168 2.380

0.260 -3.234 -2.968 -6.202 0.260 1.945 -0.796 1.149

0.280 -3.556 -1.367 -4.923 0.280 1.264 -1.201 0.063

0.300 -3.736 0.615 -3.120 0.300 0.533 -1.272 -0.739

0.320 -3.767 2.426 -1.341 0.320 -0.220 -0.987 -1.207

0.340 -3.648 3.560 -0.088 0.340 -0.964 -0.427 -1.392

Beban Fungsi SINUS Beban Fungsi COSINUS

0.360 -3.384 3.701 0.317 0.360 -1.670 0.252 -1.418

0.380 -2.985 2.809 -0.176 0.380 -2.309 0.861 -1.448

0.400 -2.467 1.134 -1.333 0.400 -2.856 1.229 -1.626

0.420 -1.850 -0.858 -2.708 0.420 -3.289 1.255 -2.034

0.440 -1.160 -2.610 -3.770 0.440 -3.591 0.931 -2.660

0.460 -0.423 -3.634 -4.058 0.460 -3.750 0.347 -3.403

0.480 0.330 -3.645 -3.315 0.480 -3.759 -0.334 -4.093

0.500 1.070 -2.638 -1.568 0.500 -3.619 -0.922 -4.540

0.520 1.768 -0.896 0.872 0.520 -3.334 -1.252 -4.586

0.540 2.395 1.097 3.492 0.540 -2.916 -1.233 -4.149

0.560 2.927 2.783 5.710 0.560 -2.382 -0.870 -3.253

0.580 3.342 3.693 7.035 0.580 -1.753 -0.265 -2.018

0.600 3.623 3.573 7.196 0.600 -1.054 0.415 -0.640

0.620 3.761 2.456 6.216 0.620 -0.314 0.979 0.665

0.640 3.748 0.654 4.402 0.640 0.440 1.269 1.709

0.660 3.586 -1.331 2.255 0.660 1.176 1.206 2.382

0.680 3.281 -2.944 0.337 0.680 1.865 0.806 2.671

0.700 2.845 -3.736 -0.891 0.700 2.479 0.182 2.661

0.720 2.296 -3.486 -1.190 0.720 2.995 -0.494 2.501

0.740 1.655 -2.263 -0.608 0.740 3.391 -1.031 2.360

0.760 0.948 -0.409 0.539 0.760 3.653 -1.281 2.371

0.780 0.204 1.559 1.763 0.780 3.768 -1.174 2.594

0.800 -0.549 3.092 2.543 0.800 3.733 -0.739 2.995

0.820 -1.280 3.763 2.483 0.820 3.550 -0.098 3.452

-8.000

-6.000

-4.000

-2.000

0.000

2.000

4.000

6.000

8.000

0.000 0.100 0.200 0.300 0.400 0.500 0.600 0.700 0.800 0.900 1.000

Res

po

n P

erp

ind

ah

an

(c

m)

Waktu (detik)

Respon Beban Harmonis (Beban Fungsi COSINUS, C=0)

pengaruh beban

pengaruh getaran bebas

Respon Total

-6.000

-4.000

-2.000

0.000

2.000

4.000

6.000

0.000 0.200 0.400 0.600 0.800 1.000 1.200 1.400 1.600

Pe

rpin

da

ha

n (

cm

)

Waktu(detik)

Respon Beban Harmonis (Beban Fungsi SINUS, C=0)

pengaruh beban

pengaruh getaran bebas

RESPON SISTEM

t(det)

0.1 0.3 0.5 0.7 0.9 0.99

A 3.367003 3.663004 4.444444 6.535948 17.54386 167.5042

` 2.9277 8.783101 14.6385 20.4939 26.3493 28.98423

0.1 0.3 0.5 0.7 0.9 0.99

A 3.367003 3.663004 4.444444 6.535948 17.54386 167.5042

` 2.9277 8.783101 14.6385 20.4939 26.3493 28.98423

0.000 0.000 0.000 0.000 0.000 0.000 0.000

0.020 0.011 0.033 0.055 0.076 0.097 0.107

0.040 0.083 0.248 0.409 0.563 0.708 0.769

0.060 0.258 0.762 1.237 1.663 2.025 2.164

0.080 0.540 1.579 2.501 3.240 3.753 3.903

0.100 0.900 2.586 3.947 4.830 5.162 5.132

0.120 1.281 3.584 5.175 5.783 5.377 4.911

0.140 1.617 4.350 5.762 5.499 3.779 2.698

0.160 1.857 4.711 5.408 3.678 0.358 -1.271

0.180 1.979 4.595 4.042 0.481 -4.137 -5.892

0.200 2.000 4.056 1.865 -3.446 -8.337 -9.495

0.220 1.969 3.254 -0.699 -7.120 -10.680 -10.439

0.240 1.948 2.404 -3.118 -9.494 -9.972 -7.804

0.260 1.996 1.704 -4.898 -9.776 -5.897 -1.897

0.280 2.144 1.275 -5.726 -7.696 0.726 5.668

0.300 2.390 1.120 -5.548 -3.618 8.071 12.387

0.320 2.694 1.129 -4.567 1.517 13.810 15.698

0.340 2.995 1.122 -3.167 6.444 15.857 13.940

t(det)

0.360 3.230 0.912 -1.776 9.917 13.108 7.088

0.380 3.353 0.376 -0.729 11.058 5.926 -3.039

0.400 3.356 -0.501 -0.166 9.608 -3.832 -13.233

0.420 3.263 -1.613 -0.005 5.987 -13.287 -19.910

0.440 3.129 -2.764 0.009 1.170 -19.398 -20.367

0.460 3.019 -3.728 0.193 -3.603 -19.941 -13.838

0.480 2.985 -4.316 0.793 -7.178 -14.309 -1.950

0.500 3.052 -4.440 1.875 -8.801 -3.821 11.640

0.520 3.206 -4.136 3.281 -8.293 8.560 22.364

0.540 3.401 -3.550 4.664 -6.055 19.072 26.277

0.560 3.573 -2.888 5.594 -2.892 24.310 21.497

0.580 3.662 -2.349 5.699 0.264 22.333 9.016

0.600 3.632 -2.055 4.791 2.644 13.362 -7.433

0.620 3.483 -2.016 2.947 3.851 -0.171 -22.478

0.640 3.251 -2.129 0.498 3.927 -14.284 -30.865

0.660 2.996 -2.213 -2.055 3.267 -24.631 -29.287

0.680 2.781 -2.078 -4.182 2.415 -27.850 -17.654

0.700 2.653 -1.589 -5.472 1.832 -22.660 0.680

0 HARMONIS BEBAN C

diabaikandapat jadi kecil,

makin struktur respon pengaruh

panjang yang waktu Dalam

..

erComplement Solusi

)11(.......... '2"

.'.".

:ankeseimbangpersamaan

)(

02

0

tCosBtSinAeX

tSinM

PXXX

tSinPXKXCXM

t

tC

bekerja efektip belumbeban saat pada

awaldetik -deti pada maks.respon

kecilmakin lamamakin

PARTIKULAR SOLUSIluarbeban dengan sama yang

fasa mempunyaiselalu tidak

maka (damping) ada Karena

)(tPX

2220

222

2

0

)(

21

22

21

11

(11) pers. kean subsitusikdan turunkan

)12....(.................... .2.1

KP

G

KP

G

tCosGtSinGX tp

TOTAL RESPON :sehingga diabaikan, )( tCX

)13.........( 21

21

1)( 2

222

0 tCostSinK

PtX

ARGAND DIAGRAM

)15...(.......... 21:factor ion magnificat Dynamic

.)(

:ditulisdapat n PerpindahaRespon

1

2tan

)14.....(.......... 21

222

0

2

1

2220

KP

D

D

tSintX

K

P

tt

222

0

21

2

K

P 222

2

0

21

1

K

P

D

0

%2

1 2

1

awal kondisi

kanmemperhatidengan ditentukan dan Sedangkan

21

2

21

1

:dimana

)15.....( )(

:menjadistruktur respon maka

gkandiperhitun bebasgetaran pengaruh Jika

222

222

2

BA

D

C

tDCostCSintBCostASinetX t

Po = 500000 N

K = 59709 N/mm

m = 285.88 N-det2/mm

0.1 0.5 0.9 1.3

5%

20%

40%

60%

80%

90%

99%

RESPON

SYSTEM ?

0.05 0.2 0.4 0.6 0.8 0.9 0.99

0.010101 0.040382 0.080633 0.120624 0.160231 0.179853 0.197396

8.458172 8.451707 8.431121 8.397141 8.350253 8.322165 8.294343

0 -0.08543 -0.3412 -0.67909 -1.01044 -1.33225 -1.48871 -1.62665

0.04 0.403363 0.147271 -0.19242 -0.52713 -0.85376 -1.01316 -1.15403

0.08 0.890811 0.635255 0.294886 -0.04205 -0.37242 -0.53423 -0.67755

0.12 1.375283 1.121116 0.78121 0.443165 0.110164 -0.05351 -0.19881

0.16 1.85516 1.603233 1.264924 0.9269 0.59238 0.427391 0.280596

0.2 2.328839 2.079992 1.744412 1.407538 1.072617 0.906862 0.759066

0.24 2.794738 2.549803 2.218073 1.883474 1.549271 1.383304 1.234999

0.28 3.2513 3.011096 2.684323 2.353118 2.020749 1.855124 1.706806

0.32 3.697 3.462329 3.141605 2.8149 2.485476 2.320746 2.172911

0.36 4.13035 3.901995 3.588392 3.267278 2.941899 2.778616 2.631757

0.4 4.5499 4.328625 4.02319 3.708741 3.388495 3.227202 3.08181

0.44 4.95425 4.740794 4.444548 4.137814 3.823769 3.665007 3.521568

0.48 5.342048 5.137125 4.851057 4.553062 4.246269 4.090567 3.949561

0.52 5.712 5.516293 5.24136 4.9531 4.654583 4.502462 4.364359

0.56 6.062869 5.877032 5.614152 5.33659 5.047347 4.899315 4.764576

0.6 6.393483 6.218138 5.968188 5.702252 5.423249 5.2798 5.148876

0.64 6.702737 6.538469 6.302286 6.048863 5.781032 5.642646 5.515974

0.68 6.989599 6.836957 6.615328 6.375266 6.119502 5.986641 5.864644

0.72 7.253109 7.112603 6.90627 6.680371 6.437528 6.310636 6.193722

0.76 7.492388 7.364488 7.17414 6.963157 6.734047 6.613547 6.502107

0.8 7.706637 7.591769 7.418041 7.222681 7.008069 6.894364 6.78877

t (detik)

Respon Perpindahan, X(t)

0.84 7.895138 7.793687 7.637161 7.458075 7.258678 7.152149 7.052753

0.88 8.057264 7.969567 7.830765 7.668553 7.485037 7.386039 7.293173

0.92 8.192471 8.118823 7.998209 7.853411 7.686389 7.595253 7.509229

0.96 8.300309 8.240955 8.138932 8.012033 7.862063 7.779093 7.700197

1 8.380417 8.335556 8.252464 8.143887 8.011471 7.936945 7.86544

1.04 8.432527 8.402309 8.338426 8.248535 8.134114 8.06828 8.004407

1.08 8.456466 8.440991 8.396531 8.325625 8.229582 8.172661 8.116631

1.12 8.452153 8.451473 8.426584 8.374901 8.297557 8.249738 8.20174

1.16 8.419603 8.43372 8.428486 8.396198 8.337811 8.299255 8.259448

1.2 8.358925 8.387792 8.402229 8.389444 8.35021 8.321044 8.289562

1.24 8.27032 8.313841 8.347902 8.354663 8.334712 8.315035 8.291983

1.28 8.154086 8.212116 8.265686 8.29197 8.291369 8.281247 8.266701

1.32 8.010611 8.082955 8.155856 8.201576 8.220326 8.219792 8.213802

1.36 7.840373 7.92679 8.018778 8.083781 8.12182 8.130876 8.133462

1.4 7.643942 7.744143 7.854911 7.938979 7.996181 8.014797 8.025949

1.44 7.421974 7.535625 7.664803 7.767655 7.843828 7.871941 7.891623

1.48 7.17521 7.301931 7.449087 7.57038 7.66527 7.702787 7.730932

1.52 6.904475 7.043842 7.208485 7.347814 7.461104 7.507899 7.544414

1.56 6.610674 6.762222 6.943801 7.100701 7.232011 7.287928 7.332691

1.6 6.294787 6.458009 6.655919 6.829865 6.978758 7.04361 7.096471

1.64 5.957871 6.132222 6.3458 6.536211 6.702189 6.77576 6.836542

1.68 5.60105 5.785948 6.014482 6.220721 6.40323 6.485273 6.553774

1.72 5.225517 5.420344 5.66307 5.884449 6.082879 6.17312 6.249111

1.76 4.832527 5.036632 5.292738 5.528518 5.742205 5.840344 5.923571

-10

-8

-6

-4

-2

0

2

4

6

8

10

0 1 2 3 4 5 6 7

Res

po

n P

erp

ind

ah

an

(m

m)

waktu (detik)

BEBAN HARMONIS C0; =0.1 (Pengaruh Redaman terhadap Respon Struktur)

damping= 5%

damping= 20%

damping = 40%

damping = 60%

damping = 80%

damping = 90%

damping = 99%

=0.1 artinya frekuensi getaran

beban sangat kecil

dibandingkan dengan frekuensi

getaran struktur. Pada kasus ini

amplitudo getaran struktur lebih

dipengaruhi oleh nilai Po/K.

sedangkan pangaruh dan juga

sangat kecil

-15

-10

-5

0

5

10

15

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Pe

rpin

da

ha

n (

mm

)

Waktu Detik)


damping= 5%

dampng= 20%

damping= 40%

damping= 60%

damping= 80%

damping= 90%

damping= 99%

Pada kasus ini pengaruh

redaman sudah muai

nampak, dimana Amplitudi

selain dipengaruhi oleh

naiai Po/K, juga oleh nilai

dan

0.05 0.2 0.4 0.6 0.8 0.9 0.99

0.067 0.261 0.490 0.675 0.818 0.876 0.922

11.141 10.788 9.852 8.719 7.636 7.148 6.742

t (detik)

Respon Perpindahan, X(t); =0.5

-50

-40

-30

-20

-10

0

10

20

30

40

50

0 0.2 0.4 0.6 0.8 1 1.2

Pe

rpin

da

ha

n (

mm

)

Waktu (detik)


damping= 5%

damping= 20%

damping= 40%

damping= 60%

damping= 80%

damping= 90%

damping= 99%

Pada 1, atau

frekuensi getaran

struktur mendekati

frekuensi getaran

beban, pengaruh

sangat berarti

-15

-10

-5

0

5

10

15

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Pe

rpin

da

ha

n (

mm

)

waktu (detik)


damping= 5%

damping= 20%

damping= 40%

damping= 60%

damping= 80%

damping= 90%

damping= 99%

0.2 0.4 0.6 0.8 0.9 0.99

' 2.890398 5.780797 8.671195 11.56159 13.00679 14.30747

0.02083 0.047583 0.093477 0.218669 0.442374 1.37243

8.721042 9.957784 13.02728 22.70724 39.83114 82.92728

0 -0.18165 -0.47364 -1.21598 -4.92589 -17.0512 -81.3011

0.04 0.825392 1.818473 3.265532 5.481232 3.099609 -59.4959

0.08 1.821412 4.013791 7.358107 14.73681 22.43013 -18.7305

0.12 2.793113 5.995453 10.57432 20.89637 35.82492 28.00413

0.16 3.727519 7.657976 12.5311 22.66585 39.73931 65.81425

0.2 4.612155 8.912861 12.99541 19.67351 33.13741 82.65045

0.24 5.435209 9.693312 11.91193 12.54799 17.7663 73.14731

0.28 6.185691 9.957784 9.409716 2.7863 -2.30633 40.33332

0.32 6.853581 9.6922 5.786785 -7.56076 -21.7686 -5.33424

0.36 7.42996 8.910696 1.474636 -16.3194 -35.4703 -49.3019

0.4 7.907134 7.654873 -3.01315 -21.6495 -39.7853 -77.5578

0.44 8.27873 5.991578 -7.14205 -22.4313 -33.572 -81.0973

0.48 8.539787 4.009349 -10.4203 -18.5006 -18.4744 -58.7925

0.52 8.68682 1.813701 -12.4575 -10.6831 1.512134 -17.7514

0.56 8.717864 -0.47849 -13.011 -0.62123 21.09847 28.94672

0.6 8.632507 -2.74521 -12.0148 9.571167 35.10149 66.42001

0.64 8.431887 -4.86581 -9.58768 17.75278 39.81553 82.72634

0.68 8.118682 -6.72739 -6.01861 22.20475 33.99313 72.66914

0.72 7.697076 -8.23087 -1.73271 21.99176 19.17506 39.45347

0.76 7.172698 -9.29622 2.759558 17.15858 -0.71733 -6.33536

0.8 6.552548 -9.86673 6.923156 8.720576 -20.4199 -50.1052

Respon Perpindahan, X(t)

t (detik)

-100

-80

-60

-40

-20

0

20

40

60

80

100

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

pe

rpin

da

ha

n (

mm

)

Waktu (detik)

Respon Beban Harmonis ( dan : tetap, : bervariasi)

beta = 20%

beta = 40%

beta = 60%

beta = 80%

beta = 90%

beta = 99%

kuliah-3 beban harmonis

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