1. ( ) 1.1 t + t [m]{ü(t + t)} + [c]{ u(t + t)} + [k]{u(t + t)} = {f(t + t)} (1) m ü f c u k u 1.2 Newmark β (1) (2) ( [m] + t ) 2 [c] + β( t)2

212 1 6 1. (212.8.14) 1 1.1............................................. 1 1.2 Newmark β....................... 1 1.3.................................... 2 1.4 (212.8.19).................................. 4 2. 5 2.1............................................. 5 2.2.......................................... 6 3. (212.8.2) 8 3.1...................... 8 3.2....................................... 8 4. (212.8.23-212.8.27) 11 4.1............................................. 11 4.2.............................................. 23 4.3.............................................. 26

1. (212.8.14) 1.1 t + t [m]{ü(t + t)} + [c]{ u(t + t)} + [k]{u(t + t)} = {f(t + t)} (1) m ü f c u k u 1.2 Newmark β (1) (2) ( [m] + t ) 2 [c] + β( t)2 [k] {ü(t + t)} = {f(t + t)} [c]{v a (t)} [k]{v b (t)} (2) {v a (t)} = { u(t)} + t {ü(t)} (3) 2 ( ) 1 {v b (t)} = {u(t)} + t{ u(t)} + 2 β ( t) 2 {ü(t)} (4) { u(t + t)} ={ u(t)} + t t {ü(t)} + {ü(t + t)} (5) 2 ( 2 ) 1 {u(t + t)} ={u(t)} + t{ u(t)} + 2 β ( t) 2 {ü(t)} + β( t) 2 {ü(t + t)} (6) ( 1 β( t) 2 [m] + 1 ) [c] + [k] {u(t + t)} = {f(t + t)} + [m]{w a (t)} + [c]{w b (t)} (7) 2β t ) {w a (t)} = 1 β( t) 2 {u(t)} + 1 {w b (t)} = 1 2β t {u(t)} + ( 1 β t { u(t)} + ) ( 1 2β 1 { u(t + t)} = 1 ( 1 1 {u(t + t)} 2β t 2β t {u(t)} 2β 1 {ü(t + t)} = 1 β( t) 2 {u(t + t)} 1 2β( t) 2 {u(t)} 1 β t { u(t)} 2β 1 {ü(t)} (8) ( ) 1 { u(t)} + 4β 1 t{ü(t)} (9) ) ( ) 1 { u(t)} 4β 1 t{ü(t)} (1) ( ) 1 2β 1 {ü(t)} (11) β = 1 4 β = 1 6 1 t + t 1 m a = f k u = f k u = f 1

1.3 y N i, u i S i, v i M i, θ i l S j, v j M j, θ j N j, u j x 1 y Y V x v φ U u X 2 (X,Y ) (x,y) {f} [k] [T ] {u} [m] [k] T T E γ A g I φ l {f} = { N i S i M i N j S j M j } T (12) {u} = { u i v i θ i u j v j θ j } T (13) EA/l EA/l 12EI/l 3 6EI/l 2 12EI/l 3 6EI/l 2 [k] = 6EI/l 2 4EI/l 6EI/l 2 2EI/l EA/l EA/l 12EI/l 3 6EI/l 2 12EI/l 3 6EI/l 2 6EI/l 2 2EI/l 6EI/l 2 4EI/l [m] = γ A l g 1/3 1/6 13/35 11l/21 9/7 13l/42 11l/21 l 2 /15 13l/42 l 2 /14 1/6 1/3 9/7 13l/42 13/35 11l/21 13l/42 l 2 /14 11l/21 l 2 /15 (14) (15) 2

cos φ sin φ sin φ cos φ [T ] = 1 cos φ sin φ sin φ cos φ 1 (16) [T ] [K] [k] [K] = [T ] T [k][t ] (17) [M] [m] [M] = [T ] T [m][t ] (18) Rayleigh [C] = ζ m [M] + ζ k [K] (19) [C] [M] [K] ζ m ζ k 3

1.4 (212.8.19) [c] 1 m ü + c u + k u = m φ ü + c m u + k m u = φ ü + 2 h ω u + ω 2 u = φ h = c/m 2 ω k ω = m Rayleigh c = ζ m m + ζ k k (2) h (ω f ) h = c/m 2 ω = ζ m m + ζ k k 2 ω m = ζ m 2 ω + ζ k ω = 1 2 4πf ζ m + πf ζ k i j h i h j f i f j ζ m ζ k h i = 1 ζ m + πf i ζ k 4πf i h j = 1 = ζ m + πf j ζ k 4πf j ζ m = 4π f i f j (f j h i f i h j ) f 2 j f 2 i ζ k = f j h j f i h i π(f 2 j f 2 i ) ζ m ζ k [c] = ζ m [m] + ζ k [k] 1 h 1 = h 2 = 5 1 2 f 1 f 2 2 f 1 f 2 2 h 1 = h 2 = 5( ) ζ m ζ k ( ) T (sec) f(hz) ω(rad/sec) T = 2π ω = 1 f f = ω 2π 4

2. 2.1 1 3 1 11 y 1 11 3 = 33 2 11 3 3 = 3 3 1 1 = 1 4 1 2 = 2 1 3 3 1 y 1 x ) x y 3 3 3 call getarg integer::negv character::fnamer*5,fnamew*5,fnamea*5!!! call getarg(1,dummy)! call getarg(2,fnamer)! call getarg(3,fnamew)! read(dummy,*) negv! dummy negv ( negv) del space Fortran http://www.nag-j.co.jp/fortran/index.html 5

! kk=mm if(negv<mm)kk=negv linebuf= mode. do i=1,kk write(dummy, (",",i) ) i linebuf=trim(linebuf)//dummy end do call del_spaces(linebuf) write (12, (a) ) trim(linebuf)! linebuf= eigen value do i=1,kk write(dummy, (",",e15.7) ) ev(i) linebuf=trim(linebuf)//dummy end do call del_spaces(linebuf) write (12, (a) ) trim(linebuf) Impulse dt,2.4e-3 ndata,1 1 1 1 1 1 1 2 3 4 3 4 2.2 (1) (2) 6

dt ndata 7

3. (212.8.2) 3.1 1 m ü + k u = u(t) = A sin(ω t) ü = ω 2 A sin(ω t) (k ω 2 m) u = u = A = k ω 2 m = k ω = m 1 ω ([k] ω 2 [m]) {u} = {} {u} ω 2 {u} ( 2 ω 2 ) [A] [B] λ ([A] λ [B]) {x} = {} 3.2 (1) 1m.5m.5m 1m.3m.3m E (t/m 2 ) 2,, A (m 2 ).25 I (m 4 ) 521 γ (tf/m 3 ) 2.35 l (m) 1 FEM 1 11 ( ) 8

E (t/m 2 ) 2,, A (m 2 ) 9 I (m 4 ) 675 γ (tf/m 3 ) 2.35 l (m) 1 (2) FEM 1 11 ( ) f n = 1 2π ( ) 2 nπ f n = 1 2π l EI ρa ( ) 2 α EI l ρa α cos(α) cosh(α) + 1 = - f n = 2n 1 E 4l ρ (3) 1 1 1 2 3 4 5 6 7 8 2.333 14.621 4.949 8.233 132.948 199.171 279.447 374.381 2.333 14.621 4.938 8.223 132.614 198.12 276.688 368.371 1. 1. 1. 1. 1.3 1.5 1.1 1.16 1 2 3 4 5 6 7 8 3.929 15.716 35.377 62.963 98.64 142.555 195.234 257.227 3.929 15.715 35.358 62.859 98.217 141.432 192.594 251.434 1. 1. 1.1 1.2 1.4 1.8 1.14 1.23 1 2 3 4 5 6 7 8 72.274 218.68 37.333 531.72 74.176 891.75 191.854 1293.716 72.199 216.598 36.997 55.396 649.795 794.194 938.593 182.992 1.1 1.9 1.26 1.51 1.84 1.123 1.163 1.195 9

( ) awk 2 cos BEGIN{ pi=3.14159265358979323846 E=1 I=1 m=1 al=1 } for(iii=1;iii<=1;iii++){ x1=(iii-1)*pi x2=x1+pi do{ xi=.5*(x1+x2) f1=cos(x1)*.5*(exp(x1)+exp(-x1))+1 f2=cos(x2)*.5*(exp(x2)+exp(-x2))+1 fi=cos(xi)*.5*(exp(xi)+exp(-xi))+1 if(f1*fi<)x2=xi if(f2*fi<)x1=xi if(fi==)exit if(f1*fi<)eps=xi-x1 if(f2*fi<)eps=x2-xi }while(1e-6<eps) f1=(iii*pi/al)^2*sqrt(e*i/m)/2/pi f2=(xi/al)^2*sqrt(e*i/m)/2/pi printf "%.3f,%.3f\n",f1,f2 } 1

4. (212.8.23-212.8.27) 4.1 a. b. c. (1) 1sec 1sec 1gal 6 t = 1sec 7 8 t 9 t t = 1sec 1 t = 1 (2) 1sec 1sec 1gal 1 t = 1sec 11 12 t 13 t t = 1sec t 1 7 11

(3) 9 13 t gnuplot Acc. (gal) Acc. (gal) Acc. (gal) Acc. (gal) 12 1 t=1s 8 6 4 2 1 2 3 4 5 Time (sec) 12 1 t=5s 8 6 4 2 1 2 3 4 5 Time (sec) 12 1 t=2s 8 6 4 2 1 2 3 4 5 Time (sec) 12 1 t=1s 8 6 4 2 1 2 3 4 5 Time (sec) 5 12

15 Acceleration 1 5 2.32Hz 13.7Hz 28.96Hz 37.99Hz 42.53Hz 44.95Hz 46.39Hz.5 Velocity.4.3.2.1 2.32Hz 13.7Hz 28.96Hz 37.99Hz 42.53Hz 15 Displacement 1 5 2.32Hz 13.7Hz 28.96Hz 5 Moment 4 3 2 1 2.32Hz 13.7Hz 28.96Hz 37.99Hz 42.53Hz 6 1gal dt=1sec 13

15 Acceleration dt=1 sec 1 5 2.32Hz 13.7Hz 28.96Hz 37.99Hz 42.53Hz 44.95Hz 46.39Hz 15 Acceleration dt=5 sec 1 5 2.34Hz 14.38Hz 36.38Hz 15 Acceleration dt=2 sec 1 5 2.33Hz 14.59Hz 48Hz 15 Acceleration dt=1 sec 1 5 2.33Hz 14.62Hz 4.73Hz 7 1 gal 14

5 Moment dt=1 sec 4 3 2 1 2.32Hz 13.7Hz 28.96Hz 37.99Hz 42.53Hz 5 Moment dt=5 sec 4 3 2 1 2.34Hz 14.38Hz 36.38Hz dt=2 sec 5 4 3 2 1 Moment 2.33Hz 14.59Hz 48Hz dt=1 sec 5 4 3 2 1 Moment 2.33Hz 14.62Hz 4.73Hz 8 1 gal 15

dt=1 sec dt=5 sec dt=2 sec dt=1 sec.8.8.5 1. 1.5 2..8.8.5 1. 1.5 2..8.8.5 1. 1.5 2..8.8.5 1. 1.5 2. 9 1 gal 16

15 Acceleration 1 5 3.91Hz 26.68Hz 46Hz 44.85Hz 48.5Hz 49.7Hz.1 Velocity 8 6 4 2 3.91Hz 26.68Hz 46Hz 44.85Hz 48.5Hz 5 Displacement 4 3 2 1 3.91Hz 26.68Hz.5 Shear force.4.3.2.1 3.91Hz 26.68Hz 46Hz 44.85Hz 1 1gal dt=1sec 17

15 Acceleration dt=1 sec 1 5 3.91Hz 26.68Hz 46Hz 44.85Hz 48.5Hz 49.7Hz 15 Acceleration dt=5 sec 1 5 3.93Hz 32.3Hz 15 Acceleration dt=2 sec 1 5 3.92Hz 34.81Hz 15 Acceleration dt=1 sec 1 5 3.92Hz 35.23Hz 11 1 gal 18

.5 Shear force dt=1 sec.4.3.2.1 3.91Hz 26.68Hz 46Hz 44.85Hz.5 Shear force dt=5 sec.4.3.2.1 3.93Hz 32.3Hz.5 Shear force dt=2 sec.4.3.2.1 3.92Hz 34.81Hz.5 Shear force dt=1 sec.4.3.2.1 3.92Hz 35.23Hz 12 1 gal 19

dt=1 sec dt=5 sec dt=2 sec dt=1 sec Shear force (t) Shear force (t) Shear force (t) Shear force (t) 5 5.5 1. 1.5 2. 5 5.5 1. 1.5 2. 5 5.5 1. 1.5 2. 5 5.5 1. 1.5 2. 13 1 gal 2

(4) 2 1 2.33 14.62 4.95 8.23 132.95 199.17 279.45 2 f n( t=1s) 2.33 13.7 28.96 37.99 42.53 44.95 46.39 f n ( t=5s) 2.34 14.38 36.38 57.32 71.56 8.3 85.74 (Hz) f n ( t=2s) 2.33 14.59 48 74.37 11.76 142.7 167.6 f n ( t=1s) 2.33 14.62 4.73 76.66 125.93 177.98 229.34 t=1s 23.137.29.38.425.45.464 t=5s 12 72.182.287.358.42.429 f n t = t/t n t=2s 5 29 8.149.222.285.335 t=1s 2 15 41 77.126.178.229 t=1s 1..937.77.474.32.226.166 t=5s 1.4.984.888.714.538.43.37 ( 2 / 1 ) t=2s 1..998.979.927.833.716.6 t=1s 1. 1..995.956.947.894.821 1 3.93 35.38 98.6 195.23 328.99 516.37 2 f n ( t=1s) 3.91 26.68 46 44.85 f n ( t=5s) 3.93 32.3 63.5 79.93 (Hz) f n( t=2s) 3.92 34.81 88.27 141.14 f n( t=1s) 3.92 35.23 95.63 175.13 255.25 324.16 t=1s 39.267.41.449 t=5s 2.162.318.4 f n t = t/t n t=2s 8 7.177.282 t=1s 4 35 96.175.255.324 t=1s.995.754.46.23 t=5s 1..913.644.49 ( 2 / 1 ) t=2s.997.984.895.723 t=1s.997.996.97.897.776.628 14 T n t 5% 1/1 14 gnuplot 21

Ratio of EV by THA to EV by EVA 1.2 1..8.6.4.2 Cantilever Simple beam.1.2.3.4.5 Ratio of time step to natural period t/t n 14 ( / ) ( / ) (EV EVA THA) 22

4.2 3 4.73Hz dt=1sec 4.73Hz 1gal 2 15 dt=1sec Original dt=1sec 1sec Case-1 1sec 4hz Case-2 dt=1sec dt=1sec step Case-3 dt=1sec dt=1sec Original dt=1 sec Acceleration (m/s).15.1 5 5.1.15 1sec sin Case-1 dt=1 sec Acceleration (m/s).15.1 5 5.1.15 1sec sin Case-2 dt=1 sec Case-2 step Case-3 dt=1 sec Case-2 Acceleration (m/s) Acceleration (m/s).15.1 5 5.1 Original (dt=1s) 1 2 3 4 5 6 7 8 9.1 Case 1 (dt=1s) 1 2 3 4 5 6 7 8 9.1 Case 2 (dt=1s).15 1 2 3 4 5 6 7 8 9.1.15.1 5 5.1 Case 3 (dt=1s).15 1 2 3 4 5 6 7 8 9.1 15.1sec 4.73Hz 1gal 23

16 Case-1 Case-2 Case-3 Case-2 Case-2 Case-3 Case-2 Original 1/1 step 24

Original dt=1 sec Acceleration (m/s).15.1 5 5.1 Original (dt=1s).15.1.2.3.4.5.6.7.8.9 1..4 Original (dt=1s).2.2.4.1.2.3.4.5.6.7.8.9 1. Case-1 dt=1 sec Acceleration (m/s).15.1 5 5.1 Case 1 (dt=1s).15.1.2.3.4.5.6.7.8.9 1..4 Case 1 (dt=1s).2.2.4.1.2.3.4.5.6.7.8.9 1. Case-2 dt=1 sec Acceleration (m/s).15.1 5 5.1 Case 2 (dt=1s).15.1.2.3.4.5.6.7.8.9 1..4 Case 2 (dt=1s).2.2.4.1.2.3.4.5.6.7.8.9 1. Case-3 dt=1 sec Acceleration (m/s).15.1 5 5.1 Case 3 (dt=1s).15.1.2.3.4.5.6.7.8.9 1..4 Case 3 (dt=1s).2.2.4.1.2.3.4.5.6.7.8.9 1. 16 4.73Hz 1gal 25

4.3 (1) 1gal 1 dt=1sec 2 [c] ζ m ζ k [c] = ζ m [m] + ζ k [k] ζ m ζ k h f h = 1 4πf ζ m + πf ζ k (2) Case f h ζ m ζ m 1 ( ) 2.33Hz(1 ) 5 1.464 2 ( ) 14.62Hz(2 ) 2 4354 17 Case-1 Case-2 Case-1 Case-1 2.8 No damping Case-1 ζ m = 1.4649 ζ k =.8 1 2 3 4.8 Zm=1.464 (f=2.33hz, h=5).8 Case-2 ζ m = ζ k = 4354 (f=14.62hz, h=2).8 1 2 3 4.8 Zk=4354 1 2 3 4 17 1sec 1gal dt=1sec 26

(3) Simplex u(t) = C exp( h 2πf t) sin(2πf t + δ) Simplex 4 p[i] u(t) = p[] exp( p[1] 2π p[2] t) sin(2π p[2] t + p[3]) Simplex 1 2 18 1 Case C h f δ p[] p[1] p[2] p[3] Case-1 2 -.3942434E+.4998969E-1.2323818E+1 -.1581718E+ Case-2 2 -.3912423E+.319515E-2.2332857E+1 -.26563E+ 2 1 Case C h f δ p[] p[1] p[2] p[3] Case-1 2.238415E+.7693699E-2.14649E+2.1865452E+1 Case-2 2.217669E+.189449E-1.1456833E+2.1951257E+1 3 2 Case C h f δ p[] p[1] p[2] p[3] Case-1 2.9948E-1.2681717E-2.47246E+2 -.531464E+ Case-2 2.646299E-1.3773392E-1.3938786E+2.1865192E+ 27

Case-1 ζ m = 1.4649 ζ k =.8 Zm=1.464.8 1 2 3 4 Case-1 ζ m = 1.4649 ζ k =.8 Zm=1.464 1.8 1 2 3 4 Case-1 ζ m = 1.4649 ζ k =.8 Zm=1.464 1 2.8 1 2 3 4 Case-2 ζ m = ζ k = 4354.8 Zk=4354.8 1 2 3 4 Case-2 ζ m = ζ k = 4354.8 Zk=4354 1.8 1 2 3 4 Case-2 ζ m = ζ k = 4354.8 Zk=4354 1 2.8 1 2 3 4 18 1sec 1gal dt=1sec 28

(4) h ζ m ζ k f h = 1 4πf ζ m h = πf ζ k for Case-1 for Case-2 19 Case-2 3 18 1 2 Case Case-1 Case-2 ζ ζ m =1.464 ζ k =4354 1 2 3 1 2 3 (Hz) 2.324 14.64 4.72 2.333 14.568 39.388 5 77 27 32 189 377 51 8 29 32 199 539.998.963.931 1..95.699 1 Case-1 Theory Case-1 Analysis Case-2 Theory Case-2 Analysis Damping factor h.1 1 1 Frequency f (Hz) 19 ( gnuplot ) 29