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

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1 ( ) Newmark β ( ) ( ) ( )

2 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 [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 = t + t 1 m a = f k u = f k u = f 1

3 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

4 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

5 1.4 ( ) [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 = 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

6 y = = = = y 1 x ) x y 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 5

7 ! 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) (2) 6

8 dt ndata 7

9 3. ( ) 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

10 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)

11 ( ) awk 2 cos BEGIN{ pi= 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

12 4. ( ) 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 t 13 t t = 1sec t

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

14 15 Acceleration Hz 13.7Hz 28.96Hz 37.99Hz 42.53Hz 44.95Hz 46.39Hz.5 Velocity Hz 13.7Hz 28.96Hz 37.99Hz 42.53Hz 15 Displacement Hz 13.7Hz 28.96Hz 5 Moment Hz 13.7Hz 28.96Hz 37.99Hz 42.53Hz 6 1gal dt=1sec 13

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

16 5 Moment dt=1 sec Hz 13.7Hz 28.96Hz 37.99Hz 42.53Hz 5 Moment dt=5 sec Hz 14.38Hz 36.38Hz dt=2 sec Moment 2.33Hz 14.59Hz 48Hz dt=1 sec Moment 2.33Hz 14.62Hz 4.73Hz 8 1 gal 15

17 dt=1 sec dt=5 sec dt=2 sec dt=1 sec gal 16

18 15 Acceleration Hz 26.68Hz 46Hz 44.85Hz 48.5Hz 49.7Hz.1 Velocity Hz 26.68Hz 46Hz 44.85Hz 48.5Hz 5 Displacement Hz 26.68Hz.5 Shear force Hz 26.68Hz 46Hz 44.85Hz 1 1gal dt=1sec 17

19 15 Acceleration dt=1 sec Hz 26.68Hz 46Hz 44.85Hz 48.5Hz 49.7Hz 15 Acceleration dt=5 sec Hz 32.3Hz 15 Acceleration dt=2 sec Hz 34.81Hz 15 Acceleration dt=1 sec Hz 35.23Hz 11 1 gal 18

20 .5 Shear force dt=1 sec Hz 26.68Hz 46Hz 44.85Hz.5 Shear force dt=5 sec Hz 32.3Hz.5 Shear force dt=2 sec Hz 34.81Hz.5 Shear force dt=1 sec Hz 35.23Hz 12 1 gal 19

21 dt=1 sec dt=5 sec dt=2 sec dt=1 sec Shear force (t) Shear force (t) Shear force (t) Shear force (t) gal 2

22 (4) f n( t=1s) f n ( t=5s) (Hz) f n ( t=2s) f n ( t=1s) t=1s t=5s f n t = t/t n t=2s t=1s t=1s t=5s ( 2 / 1 ) t=2s t=1s f n ( t=1s) f n ( t=5s) (Hz) f n( t=2s) f n( t=1s) t=1s t=5s f n t = t/t n t=2s t=1s t=1s t=5s ( 2 / 1 ) t=2s t=1s T n t 5% 1/1 14 gnuplot 21

23 Ratio of EV by THA to EV by EVA Cantilever Simple beam Ratio of time step to natural period t/t n 14 ( / ) ( / ) (EV EVA THA) 22

24 Hz 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) sec sin Case-1 dt=1 sec Acceleration (m/s) sec sin Case-2 dt=1 sec Case-2 step Case-3 dt=1 sec Case-2 Acceleration (m/s) Acceleration (m/s) Original (dt=1s) Case 1 (dt=1s) Case 2 (dt=1s) Case 3 (dt=1s) sec 4.73Hz 1gal 23

25 16 Case-1 Case-2 Case-3 Case-2 Case-2 Case-3 Case-2 Original 1/1 step 24

26 Original dt=1 sec Acceleration (m/s) Original (dt=1s) Original (dt=1s) Case-1 dt=1 sec Acceleration (m/s) Case 1 (dt=1s) Case 1 (dt=1s) Case-2 dt=1 sec Acceleration (m/s) Case 2 (dt=1s) Case 2 (dt=1s) Case-3 dt=1 sec Acceleration (m/s) Case 3 (dt=1s) Case 3 (dt=1s) Hz 1gal 25

27 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 ) ( ) 14.62Hz(2 ) Case-1 Case-2 Case-1 Case No damping Case-1 ζ m = ζ k = Zm=1.464 (f=2.33hz, h=5).8 Case-2 ζ m = ζ k = 4354 (f=14.62hz, h=2) Zk= sec 1gal dt=1sec 26

28 (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 Case C h f δ p[] p[1] p[2] p[3] Case E E E E+ Case E E E E+ 2 1 Case C h f δ p[] p[1] p[2] p[3] Case E E E E+1 Case E E E E Case C h f δ p[] p[1] p[2] p[3] Case E E E E+ Case E E E E+ 27

29 Case-1 ζ m = ζ k =.8 Zm= Case-1 ζ m = ζ k =.8 Zm= Case-1 ζ m = ζ k =.8 Zm= Case-2 ζ m = ζ k = Zk= Case-2 ζ m = ζ k = Zk= Case-2 ζ m = ζ k = Zk= sec 1gal dt=1sec 28

30 (4) h ζ m ζ k f h = 1 4πf ζ m h = πf ζ k for Case-1 for Case-2 19 Case Case Case-1 Case-2 ζ ζ m =1.464 ζ k = (Hz) Case-1 Theory Case-1 Analysis Case-2 Theory Case-2 Analysis Damping factor h Frequency f (Hz) 19 ( gnuplot ) 29

. (.8.). t + t m ü(t + t) + c u(t + t) + k u(t + t) = f(t + t) () m ü f. () c u k u t + t u Taylor t 3 u(t + t) = u(t) + t! u(t) + ( t)! = u(t) + t u(

. (.8.). t + t m ü(t + t) + c u(t + t) + k u(t + t) = f(t + t) () m ü f. () c u k u t + t u Taylor t 3 u(t + t) = u(t) + t! u(t) + ( t)! = u(t) + t u( 3 8. (.8.)............................................................................................3.............................................4 Nermark β..........................................