Mathematics III TS 4353 Class B

Herlina SetiyaningsihCivil Engineering Department

Petra Christian University

PD Serentak (Simultan)f1(D)y + g1(D)z = h1(x)

f2(D)y + g2(D)z = h2(x)

PD: ∆y = ∆1 dan ∆z = ∆2

Jurusan Teknik Sipil Matematika III (TS 4353)

Fakultas Teknik Sipil dan Perencanaan Universitas Kristen Petra Bab 1

Example 1PD: Dy – z = ex

y + (D+2)z = 0

Δy = Δ1 (D2 + 2D +1) y= 3ex

yc = e-x(c1+c2x)



Example 1 (Lanjutan)

PUPD: y = yc + yp y = e-x (c1 + c2x) + ¾ ex

Δz = Δ2 (D2 + 2D +1)z = -ex

zc = e-x(c3+c4x)



PUPD: z = zc + zp z = e-x (c3 + c4x) -1/4ex

Example 1 (Lanjutan)y dan z masuk ke PD

D[e-x (c1 + c2x) + ¾ ex]- [e-x (c3 + c4x) -1/4ex]=ex

-e-x (c1 + c2x) + e-x c2 + ¾ ex -e-x c3 - e-x c4x + 1/4ex

= ex

e-x x (-c2 - c4) + e-x (-c1 + c2 - c3) = 0

-c2 - c4 = 0 c2 = - c4

-c1 + c2 - c3 = 0 c3 = -c1 + c2

PUPD: y = yc + yp y = e-x (c1 + c2x) + ¾ ex

PUPD: z = zc + zp z = e-x ((-c1 + c2 ) – c2x) -1/4ex



Example 2PD: (D+2)y = z

(D+1)z = 2y(D+2)y – z = 0-2y + (D+1)z = 0



= D2 + 3D

Example 2 (Lanjutan)Δy = Δ1 (D2 + 3D) y= 0

yc = c1+c2e-3x

Δz = Δ2 (D2 + 3D) z= 0

zc = c3+c4e-3x

y dan z masuk ke PD(D+2)y – z = 0(D+2) (c1+c2e-3x) – (c3+c4e-3x) = 0

D(c1+c2e-3x) + 2c1 + 2c2e-3x– c3- c4e-3x = 0

0 - 3c2e-3x + 2c1 + 2c2e-3x– c3- c4e-3x = 0



e-3x(-3c2+ 2c2 - c4) + (2c1 – c3) = 0

-c2 - c4= 0 c2 = -c4

2c1 – c3 = 0 2c1 = c3

PUPD: y = c1 + c2e-3x

PUPD: z = 2c1- c2e-3x



PD Euler(ax+b)2 y” + p(ax+b)y’ + qy = f(x); a, b, p, q

konstanSubstitusi: ax+b = et t = ln (ax+b)



Example 1(2x+1)2y” – 2(2x+1)y’ – 12y = 6xSubst: 2x+1 = et t=ln(2x+1)



Example 1 (Lanjutan)

(4D2 – 8D – 12)y = 3et - 3PR: (4D2 – 8D – 12)y = 0 PK : 4k2 – 8k – 12 = 0

k2 – 2k – 3 = 0 (k-3)(k+1) = 0k = 3 dan k = -1



yc = c1 e-t + c2 e3t

PUPD: y = yc + yp

= c1 e-t + c2 e3t -3/16et + ¼

= c1 (2x+1)-1 + c2 (2x+1)3 - 3/16(2x+1) + ¼



Aplikasi Bidang T. SipilPada struktur dinamik dikenal adanya getaran.

Persamaan umum getaran pada struktur dinamik :

- perpindahan (x)- kecepatan- percepatan

Sistem getaran dibagi menjadi 4:1. Getaran bebas2. Getaran bebas dan teredam3. Getaran terpaksa 4. Getaran teredam & terpaksa



Sistem getaran dibagi menjadi 4:1. Getaran bebas

2. Getaran bebas dan teredam

3. Getaran terpaksa

4. Getaran teredam & terpaksa



Getaran Terpaksamy” + ky = Fo cos ωtyc = e0t(c1 cos ω0t + c2 sin ω0t )

yp = B1 cos ωt + B2 sin ωt

y’p = -B1 ω sin ωt + B2 ω cos ωt

y”p = -B1 ω2 cos ωt - B2 ω2 sin ωt

m (-B1 ω2 cos ωt - B2 ω2 sin ωt) + k (B1 cos ωt + B2 sin ωt) = Fo cos ωt

B1 (k-m ω2) cos ωt + B2(k-m ω2) sin ωt = Fo cos ωtB1 (k-m ω2) = Fo B1 = Fo/ k-m ω2 B2(k-m ω2) = 0 Jurusan Teknik Sipil Matematika III (TS 4353)


PUPD:y = c1 cos ω0t + c2 sin ω0t + (Fo/ k-m ω2)cos ωt



Getaran Terpaksa & Teredammy” + cy’ + ky = Fo sin ωtyc = ??

yp = B1 cos ωt + B2 sin ωt

y’p = -B1 ω sin ωt + B2 ω cos ωt

y”p = -B1 ω2 cos ωt - B2 ω2 sin ωt

m(-B1 ω2 cos ωt - B2 ω2 sin ωt)+c(-B1 ω sin ωt + B2 ω cos ωt ) + k (B1 cos ωt + B2 sin ωt) = Fo sin ωt

B1 = [(k-m ω2 ) Fo] / [(k-m ω2)2 + (c ω) 2]B2 = [-c ω Fo] / [(k-m ω2)2 + (c ω) 2]Jurusan Teknik Sipil Matematika III (TS 4353)


Finish!!!!Bertemu di Integral

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Mathematics III TS 4353 Class B

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