DKiPI_2_2019_s izradom -- Sage

Prisilno titranje sustava s jednim stupnjem slobode

Na prikazani okvir djeluje dinamička sila p(t). Treba odrediti funkciju odziva sustava za različite vrijednosti funkcije sile p(t). Rješenja odrediti za homogene početne uvjete. Također odrediti i funkciju odziva za prigušenje 5% za funkcije pobude pod a) i d).

Zadano je:

$EI = 15\cdot 10^3 kNm^2$

$L = 6 m$

$h = 3 m$

$m = 3 t$

$P_0 = 20 kN$

$t_1 = 0,6 s$

$t_2 = 1 s$

Matematički model prislinog neprigušenog titranja:

$m \ddot{u}(t) + k u(t) = P(t)$

Matematički model prisilnog prigušenog titranja:

$m \ddot{u}(t) + 2 \zeta \; m \; \omega_n \dot{u} (t) + k u(t) = P(t)$

EI = 15000;
L = 6;
h = 3;
m = 3;

k = 96*EI/(7*h^3)
k.n(digits=6)

$\newcommand{\Bold}[1]{\mathbf{#1}}7619.05$

Dinamičke karakteristike sustava su:

prirodna kružna frekvencija sustava: $\omega_n = \sqrt{\frac{k}{m}}$

period slobodnog titranja: $T_n = \frac{2\pi}{\omega}$

omega_n = sqrt(k/m)
T_n = 2*pi/omega_n
omega_n, T_n

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\frac{400}{3} \, \sqrt{\frac{1}{7}}, \frac{21}{200} \, \sqrt{\frac{1}{7}} \pi\right)$

omega_n.n(digits = 3), T_n.n(digits = 3)

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(50.4, 0.125\right)$

a) Konstantna sila $P(t) = P_0$

var('t')
u = function('u')(t)

P0 = 20

de1 = m*diff(u, t, 2) + k*u == P0
de1

$\newcommand{\Bold}[1]{\mathbf{#1}}\frac{160000}{21} \, u\left(t\right) + 3 \, \frac{\partial^{2}}{(\partial t)^{2}}u\left(t\right) = 20$

sol_1 = desolve( de1, u, ics = [0, 0, 0] )
sol_1

$\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{21}{8000} \, \cos\left(\frac{400}{21} \, \sqrt{7} t\right) + \frac{21}{8000}$

Vrijednost statičkog pomaka $u_{st} = \frac{P_0}{k}$

ust = P0/k
ust.n(digits=3)

$\newcommand{\Bold}[1]{\mathbf{#1}}0.00262$

s1s = plot( ust, [0,0.6], color='red')
s1d =  plot( sol_1, [0,0.6], color='blue')
show(s1s + s1d)

U trenutku $t_1 = 0.6$ s sila $P_0$ prestaje djelovati, pa za $t>0.6$ s imamo slobodno titranje

$\ddot{u}(t) + \omega_n^2 u(t) = 0$

de2 = diff(u, t, 2) + omega_n^2*u == 0

Početne uvjete za slobodno titranje određujemo iz funkcije odziva na konstantnu silu:

$u_{t1} = u(t=0.6)$

$v_{t1} = u'(t=0.6)$

u_t1 = sol_1.subs(t = 0.6)
v_t1 = diff(sol_1, t).subs(t = 0.6)
u_t1.n(digits = 5), v_t1.n(digits = 5)

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(0.0016220, -0.12225\right)$

sol_2 = desolve( de2, u, ivar = t, ics = [0.6, u_t1, v_t1] )
sol_2

$\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{21 \, {\left(\cos\left(11.42857142857143 \, \sqrt{7}\right) \cos\left(11.428571428571427 \, \sqrt{7}\right) + \sin\left(11.42857142857143 \, \sqrt{7}\right) \sin\left(11.428571428571427 \, \sqrt{7}\right) - \cos\left(11.428571428571427 \, \sqrt{7}\right)\right)} \cos\left(\frac{400}{21} \, \sqrt{7} t\right)}{8000 \, {\left(\cos\left(11.428571428571427 \, \sqrt{7}\right)^{2} + \sin\left(11.428571428571427 \, \sqrt{7}\right)^{2}\right)}} + \frac{21 \, {\left(\cos\left(11.428571428571427 \, \sqrt{7}\right) \sin\left(11.42857142857143 \, \sqrt{7}\right) - \cos\left(11.42857142857143 \, \sqrt{7}\right) \sin\left(11.428571428571427 \, \sqrt{7}\right) + \sin\left(11.428571428571427 \, \sqrt{7}\right)\right)} \sin\left(\frac{400}{21} \, \sqrt{7} t\right)}{8000 \, {\left(\cos\left(11.428571428571427 \, \sqrt{7}\right)^{2} + \sin\left(11.428571428571427 \, \sqrt{7}\right)^{2}\right)}}$

s2d =  plot( sol_2, [0.6,1], color='purple')
s2d

show( s1s + s1d + s2d )

reset('u_t1 v_t1')

a1) Konstantna sila $P(t) = P_0$ - prigušen sustav

Matematički model prislinog prigušenog titranja:

$\ddot{u}(t) + 2 \zeta \omega_n \dot{u}(t) + \omega_n^2 u(t) = \frac{P(t)}{m}$

gdje je $\zeta$ relativno prigušenje sustava (zadano 5%).

zeta = 0.05

Rješenje jednadžbe za $0 < t < t_1$

de3 = diff(u, t, 2) + 2*zeta*omega_n*diff(u, t) +  omega_n^2*u == P0/m

sol_3 = desolve( de3, u, ivar = t, ics = [0, 0, 0] )
sol_3

$\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{1}{152000} \, {\left(\sqrt{19} \sqrt{7} \sqrt{3} \sin\left(\frac{20}{3} \, \sqrt{19} \sqrt{3} t\right) + 399 \, \cos\left(\frac{20}{3} \, \sqrt{19} \sqrt{3} t\right)\right)} e^{\left(-\frac{20}{21} \, \sqrt{7} t\right)} + \frac{21}{8000}$

s3d =  plot( sol_3, [0,0.6], color='blue')
show(s1s + s3d)

numerical_approx(sol_3)

Traceback (click to the left of this block for traceback)
...
TypeError: cannot evaluate symbolic expression numerically

Traceback (most recent call last):
  File "<stdin>", line 1, in <module>
  File "_sage_input_38.py", line 10, in <module>
    exec compile(u'open("___code___.py","w").write("# -*- coding: utf-8 -*-\\n" + _support_.preparse_worksheet_cell(base64.b64decode("bnVtZXJpY2FsX2FwcHJveChzb2xfMyk="),globals())+"\\n"); execfile(os.path.abspath("___code___.py"))
  File "", line 1, in <module>
    
  File "/tmp/tmpiE9h6l/___code___.py", line 2, in <module>
    exec compile(u'numerical_approx(sol_3)
  File "", line 1, in <module>
    
  File "/opt/SageMath/local/lib/python2.7/site-packages/sage/misc/functional.py", line 1419, in numerical_approx
    return n(prec, algorithm=algorithm)
  File "sage/symbolic/expression.pyx", line 5981, in sage.symbolic.expression.Expression.numerical_approx (build/cythonized/sage/symbolic/expression.cpp:37400)
TypeError: cannot evaluate symbolic expression numerically

Rješenje jednadžbe za $t > t_1$

u_t1 = sol_3.subs(t = 0.6)
v_t1 = diff(sol_3, t).subs(t = 0.6)
u_t1.n(digits = 5), v_t1.n(digits = 5)

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(0.0024514, -0.027393\right)$

de4 = diff(u, t, 2) + 2*zeta*omega_n*diff(u, t) +  omega_n^2*u == 0

sol_4 = desolve( de4, u, ivar = t, ics = [0.6, u_t1, v_t1] )
sol_4

$\newcommand{\Bold}[1]{\mathbf{#1}}\frac{7}{8000} \, {\left(\frac{{\left(3 \, \sqrt{19} \sqrt{7} \cos\left(4.0 \, \sqrt{19} \sqrt{3}\right) - {\left(3 \, \sqrt{19} \sqrt{7} \cos\left(4.0 \, \sqrt{19} \sqrt{3}\right) e^{\left(-0.5714285714285714 \, \sqrt{7}\right)} - \sqrt{3} e^{\left(-0.5714285714285714 \, \sqrt{7}\right)} \sin\left(4.0 \, \sqrt{19} \sqrt{3}\right)\right)} \cos\left(4.0 \, \sqrt{19} \sqrt{3}\right) - {\left(3 \, \sqrt{19} \sqrt{7} e^{\left(-0.5714285714285714 \, \sqrt{7}\right)} \sin\left(4.0 \, \sqrt{19} \sqrt{3}\right) + \sqrt{3} \cos\left(4.0 \, \sqrt{19} \sqrt{3}\right) e^{\left(-0.5714285714285714 \, \sqrt{7}\right)}\right)} \sin\left(4.0 \, \sqrt{19} \sqrt{3}\right) - \sqrt{3} \sin\left(4.0 \, \sqrt{19} \sqrt{3}\right)\right)} \cos\left(\frac{20}{3} \, \sqrt{19} \sqrt{3} t\right)}{\sqrt{19} \sqrt{7} \cos\left(4.0 \, \sqrt{19} \sqrt{3}\right)^{2} e^{\left(-0.5714285714285714 \, \sqrt{7}\right)} + \sqrt{19} \sqrt{7} e^{\left(-0.5714285714285714 \, \sqrt{7}\right)} \sin\left(4.0 \, \sqrt{19} \sqrt{3}\right)^{2}} - \frac{{\left({\left(3 \, \sqrt{19} \sqrt{7} e^{\left(-0.5714285714285714 \, \sqrt{7}\right)} \sin\left(4.0 \, \sqrt{19} \sqrt{3}\right) + \sqrt{3} \cos\left(4.0 \, \sqrt{19} \sqrt{3}\right) e^{\left(-0.5714285714285714 \, \sqrt{7}\right)}\right)} \cos\left(4.0 \, \sqrt{19} \sqrt{3}\right) - 3 \, \sqrt{19} \sqrt{7} \sin\left(4.0 \, \sqrt{19} \sqrt{3}\right) - {\left(3 \, \sqrt{19} \sqrt{7} \cos\left(4.0 \, \sqrt{19} \sqrt{3}\right) e^{\left(-0.5714285714285714 \, \sqrt{7}\right)} - \sqrt{3} e^{\left(-0.5714285714285714 \, \sqrt{7}\right)} \sin\left(4.0 \, \sqrt{19} \sqrt{3}\right)\right)} \sin\left(4.0 \, \sqrt{19} \sqrt{3}\right) - \sqrt{3} \cos\left(4.0 \, \sqrt{19} \sqrt{3}\right)\right)} \sin\left(\frac{20}{3} \, \sqrt{19} \sqrt{3} t\right)}{\sqrt{19} \sqrt{7} \cos\left(4.0 \, \sqrt{19} \sqrt{3}\right)^{2} e^{\left(-0.5714285714285714 \, \sqrt{7}\right)} + \sqrt{19} \sqrt{7} e^{\left(-0.5714285714285714 \, \sqrt{7}\right)} \sin\left(4.0 \, \sqrt{19} \sqrt{3}\right)^{2}}\right)} e^{\left(-\frac{20}{21} \, \sqrt{7} t\right)}$

s4d =  plot( sol_4, [0.6,1], color='blue')
show(s1s + s3d + s4d)

b) Linearna sila P(t)

Interval $0 < t < t_1$

t1=0.6

de5 = m*diff(u, t, 2) + k*u == P0/t1*t
de5

$\newcommand{\Bold}[1]{\mathbf{#1}}\frac{160000}{21} \, u\left(t\right) + 3 \, \frac{\partial^{2}}{(\partial t)^{2}}u\left(t\right) = 33.3333333333333 \, t$

sol_5 = desolve( de5, u, ics = [0, 0, 0] )
sol_5

$\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{21}{640000} \, \sqrt{7} \sin\left(\frac{400}{21} \, \sqrt{7} t\right) + \frac{7}{1600} \, t$

sol_5.simplify_full()

$\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{21}{640000} \, \sqrt{7} \sin\left(\frac{400}{21} \, \sqrt{7} t\right) + \frac{7}{1600} \, t$

Ako sila $P_0$ djeluje statički, pomak se također linearno povećava proporcionalno sili (nema titranja), $u_{st} = \frac{P_0}{k}\frac{t}{t_1}$

ust = P0*t/(t1*k)
ust

$\newcommand{\Bold}[1]{\mathbf{#1}}0.00437500000000000 \, t$

s5s = plot( ust, [0,t1], color='red')
s5d =  plot( sol_5, [0,t1], color='blue')
show(s5s + s5d)

U trenutku $t_1 = 0.6$ s sila $P_0$ prestaje djelovati, pa za $t>0.6$ s imamo slobodno titranje

de2 #već ranije zadana jednadžba slobodnog titranja

$\newcommand{\Bold}[1]{\mathbf{#1}}\frac{160000}{63} \, u\left(t\right) + \frac{\partial^{2}}{(\partial t)^{2}}u\left(t\right) = 0$

Početne uvjete za slobodno titranje određujemo iz prethodno određene funkcije:

$u_{t1} = u(t=0.6)$

$v_{t1} = u'(t=0.6)$

u_t1 = sol_5.subs(t = 0.6)
v_t1 = diff(sol_5, t).subs(t = 0.6)
u_t1.n(digits = 5), v_t1.n(digits = 5)

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(0.0027052, 0.0027034\right)$

sol_6 = desolve( de2, u, ivar = t, ics = [0.6, u_t1, v_t1] )
sol_6

$\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{21 \, {\left(\sqrt{7} \cos\left(11.428571428571427 \, \sqrt{7}\right) \sin\left(11.42857142857143 \, \sqrt{7}\right) - \sqrt{7} {\left(\cos\left(11.42857142857143 \, \sqrt{7}\right) - 1\right)} \sin\left(11.428571428571427 \, \sqrt{7}\right) - 80 \, \cos\left(11.428571428571427 \, \sqrt{7}\right)\right)} \cos\left(\frac{400}{21} \, \sqrt{7} t\right)}{640000 \, {\left(\cos\left(11.428571428571427 \, \sqrt{7}\right)^{2} + \sin\left(11.428571428571427 \, \sqrt{7}\right)^{2}\right)}} - \frac{21 \, {\left(\sqrt{7} {\left(\cos\left(11.42857142857143 \, \sqrt{7}\right) - 1\right)} \cos\left(11.428571428571427 \, \sqrt{7}\right) + \sqrt{7} \sin\left(11.42857142857143 \, \sqrt{7}\right) \sin\left(11.428571428571427 \, \sqrt{7}\right) - 80 \, \sin\left(11.428571428571427 \, \sqrt{7}\right)\right)} \sin\left(\frac{400}{21} \, \sqrt{7} t\right)}{640000 \, {\left(\cos\left(11.428571428571427 \, \sqrt{7}\right)^{2} + \sin\left(11.428571428571427 \, \sqrt{7}\right)^{2}\right)}}$

s6d =  plot( sol_6, [0.6,1], color='blue')
s6d

show( s5s + s5d + s6d )

b) Linearna+konstantna sila P(t)

Interval $0 < t < t_1$ : rješenje je ranije određeno sol_5

Interval $t_1 < t < t_2$

u_t1 = sol_5.subs(t = 0.6)
v_t1 = diff(sol_5, t).subs(t = 0.6)
u_t1.n(digits = 3), v_t1.n(digits = 3)

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(0.00271, 0.00270\right)$

de7 = diff(u, t, 2) + omega_n^2*u == P0/m
de7

$\newcommand{\Bold}[1]{\mathbf{#1}}\frac{160000}{63} \, u\left(t\right) + \frac{\partial^{2}}{(\partial t)^{2}}u\left(t\right) = \left(\frac{20}{3}\right)$

sol_7 = desolve( de7, u, ics=[0.6, u_t1, v_t1] )
sol_7

$\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{21 \, {\left(\sqrt{7} \cos\left(11.428571428571427 \, \sqrt{7}\right) \sin\left(11.42857142857143 \, \sqrt{7}\right) - \sqrt{7} {\left(\cos\left(11.42857142857143 \, \sqrt{7}\right) - 1\right)} \sin\left(11.428571428571427 \, \sqrt{7}\right)\right)} \cos\left(\frac{400}{21} \, \sqrt{7} t\right)}{640000 \, {\left(\cos\left(11.428571428571427 \, \sqrt{7}\right)^{2} + \sin\left(11.428571428571427 \, \sqrt{7}\right)^{2}\right)}} - \frac{21 \, {\left(\sqrt{7} {\left(\cos\left(11.42857142857143 \, \sqrt{7}\right) - 1\right)} \cos\left(11.428571428571427 \, \sqrt{7}\right) + \sqrt{7} \sin\left(11.42857142857143 \, \sqrt{7}\right) \sin\left(11.428571428571427 \, \sqrt{7}\right)\right)} \sin\left(\frac{400}{21} \, \sqrt{7} t\right)}{640000 \, {\left(\cos\left(11.428571428571427 \, \sqrt{7}\right)^{2} + \sin\left(11.428571428571427 \, \sqrt{7}\right)^{2}\right)}} + \frac{21}{8000}$

s7d =  plot( sol_7, [0.6,1], color='blue')
s7s =  plot( P0/k, [0.6,1], color='red')

show(s5d + s5s + s7d + s7s)

c) harmonijska pobuda $P(t) = P_0 \sin(\omega t)$

Diferencijalna jednadžba:

$m \ddot{u}(t) + k u(t) = P_0 \sin (\omega t)$

c1) Reznancija $\omega = \omega_n$

omega = 1.*omega_n
t_p=4*T_n

de8 = m*diff(u, t, 2) + k*u == P0*sin(omega*t)
de8

$\newcommand{\Bold}[1]{\mathbf{#1}}\frac{160000}{21} \, u\left(t\right) + 3 \, \frac{\partial^{2}}{(\partial t)^{2}}u\left(t\right) = 20 \, \sin\left(133.333333333333 \, \sqrt{\frac{1}{7}} t\right)$

sol_8 = desolve( de8, u, ics = [0, 0, 0] )
sol_8

$\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{1}{40} \, \sqrt{7} t \cos\left(\frac{400}{21} \, \sqrt{7} t\right) + \frac{21}{16000} \, \sin\left(\frac{400}{21} \, \sqrt{7} t\right)$

s8d =  plot( sol_8, [0,t_p], color='blue')
show(s8d)

c2) Pulsiranje (eng. bating) $\omega \approx \omega_n$

Zadajemo npr. $\omega = 0.95\omega_n$

omega = 0.95*omega_n
t_p = 40*T_n

de9 = m*diff(u, t, 2) + k*u == P0*sin(omega*t)
de9

$\newcommand{\Bold}[1]{\mathbf{#1}}\frac{160000}{21} \, u\left(t\right) + 3 \, \frac{\partial^{2}}{(\partial t)^{2}}u\left(t\right) = 20 \, \sin\left(126.666666666667 \, \sqrt{\frac{1}{7}} t\right)$

sol_9 = desolve( de9, u, ics = [0, 0, 0] )
sol_9

$\newcommand{\Bold}[1]{\mathbf{#1}}-\frac{133}{5200} \, \sin\left(\frac{400}{21} \, \sqrt{7} t\right) + \frac{7}{260} \, \sin\left(\frac{380}{21} \, \sqrt{7} t\right)$

s9d =  plot( sol_9, [0,t_p], color='blue')
show(s9d)

c3) Prigušeno titranje

Diferencijalna jednadžba je:

$m \ddot{u}(t) + c \dot{u}(t) + k u(t) = P_0 \sin (\omega t)$

$\ddot{u}(t) + 2 \zeta \omega_n \dot{u}(t) + \omega_n^2 u(t) = \frac{P_0}{m} \sin (\omega t)$

Rezonancija $\omega = \omega_n$

omega = 1.*omega_n
t_p=20*T_n

de10 = diff(u, t, 2) + 2*omega_n*zeta*diff(u, t) + omega_n^2*u == P0/m*sin(omega*t)
de10

$\newcommand{\Bold}[1]{\mathbf{#1}}13.3333333333333 \, \sqrt{\frac{1}{7}} \frac{\partial}{\partial t}u\left(t\right) + \frac{160000}{63} \, u\left(t\right) + \frac{\partial^{2}}{(\partial t)^{2}}u\left(t\right) = \frac{20}{3} \, \sin\left(133.333333333333 \, \sqrt{\frac{1}{7}} t\right)$

sol_10 = desolve( de10, u, ics = [0, 0, 0] )
sol_10

$\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{15200} \, {\left(\sqrt{19} \sqrt{7} \sqrt{3} \sin\left(\frac{20}{3} \, \sqrt{19} \sqrt{3} t\right) + 399 \, \cos\left(\frac{20}{3} \, \sqrt{19} \sqrt{3} t\right)\right)} e^{\left(-\frac{20}{21} \, \sqrt{7} t\right)} - \frac{21}{800} \, \cos\left(\frac{400}{21} \, \sqrt{7} t\right)$

s10d =  plot( sol_10, [0,t_p], color='blue',plot_points=500)
show(s10d)

DKiPI_2_2019_s izradom

2263 days ago by mdemsic@grad.hr