數(shù)學(xué)實(shí)驗(yàn)教程-綜合實(shí)驗(yàn)報(bào)告范例

單擺運(yùn)動(dòng)這是一個(gè)我們熟悉的物理模型，可看作工程技術(shù)中一些振動(dòng)問(wèn)題的簡(jiǎn)化。圖1中一根長(zhǎng)L的（無(wú)彈性的）細(xì)線，一端固定，另一端懸掛一質(zhì)量為m的小球，在重力作用下小球處于豎直的平衡位置。使小球偏離平衡位置一個(gè)初始角度，然后讓它自由落下，在不考慮空氣阻力的情況下，小球就會(huì)沿圓弧擺動(dòng)。問(wèn)題1建立該物理系統(tǒng)的數(shù)學(xué)模型；問(wèn)題2當(dāng)初始角度較小時(shí)，求解數(shù)學(xué)模型；問(wèn)題3當(dāng)初始角度較大時(shí)，還可用問(wèn)題2中的方法嗎？圖1中以θ=0位平衡位置，以右邊為正方向建立擺角θ的坐標(biāo)系。在小球擺動(dòng)過(guò)程中的任一位置θ，小球所受重力沿運(yùn)動(dòng)軌跡方向的分力為-mgsinθ（負(fù)號(hào)表示力的方向與θ的正方向相反），利用牛頓第二定律即得微分方程設(shè)小球初始偏離角度為θ0，且無(wú)初速，則方程的初始條件為求解(1)、(2)時(shí)，在θ0不大的條件下，可將方程（1）中的sinθ近似為θ，于是得到線性常系數(shù)微分方程容易算出方程（3）在初始條件（2）下的解為由解（4）顯然可知，簡(jiǎn)諧運(yùn)動(dòng)的周期為解：理論說(shuō)明部分描述單擺運(yùn)動(dòng)規(guī)律的微分方程（1）是2階微分方程，無(wú)解析解，但可用Matlab或其它軟件編程求其數(shù)值解，但都需要先將它化成方程組的形式。令則微分方程(1)化為初始條件轉(zhuǎn)化為在前面的兩式中，g=9.8，l=25，x10為10o=0.1745(弧度)及30o=0.5236(弧度)兩種情況.對(duì)于近似解(4)式，周期（2）方法描述令y=x1，z=x2，步長(zhǎng)h=0.1，則原方程組可改寫為①向前歐拉公式對(duì)n=0，1，2，…，向前歐拉計(jì)算公式如下：②改進(jìn)的歐拉公式對(duì)n=0，1，2，…，改進(jìn)的歐拉計(jì)算公式如下：③二階龍格—庫(kù)塔公式對(duì)n=0，1，2，…，二階龍格—庫(kù)塔公式計(jì)算公式如下：④四階龍格—庫(kù)塔公式對(duì)n=0，1，2，…，四階龍格—庫(kù)塔公式計(jì)算公式如下：（3）計(jì)算數(shù)據(jù)及數(shù)據(jù)圖形①M(fèi)atlab庫(kù)函數(shù)ode45(…)計(jì)算結(jié)果及圖形A.計(jì)算數(shù)據(jù)下面的數(shù)據(jù)表分別是對(duì)應(yīng)于兩種情況下的計(jì)算結(jié)果比較表。初始條件為θ=10o時(shí)的數(shù)值解與近似解計(jì)算表t數(shù)值解近似解t數(shù)值解近似解00.17450.17453.4606-0.0974-0.09790.00070.17450.17453.7106-0.1187-0.11930.00150.17450.17453.9606-0.1372-0.13770.00220.17450.17454.2106-0.1523-0.15270.00300.17450.17454.4606-0.1637-0.16400.00660.17450.17454.7106-0.1711-0.17130.01030.17450.17454.9606-0.1743-0.17440.01400.17450.17455.2106-0.1734-0.17320.01770.17450.17455.4606-0.1681-0.16780.03620.17450.17455.7106-0.1588-0.15830.05460.17440.17445.9606-0.1457-0.14500.07310.17430.17436.2106-0.1289-0.12810.09150.17420.17426.4606-0.1091-0.10800.18380.17340.17336.7106-0.0866-0.08530.27610.17190.17196.9606-0.0619-0.06050.36830.16990.16997.2106-0.0358-0.03430.46060.16730.16737.4606-0.0087-0.00720.71060.15760.15757.71060.01850.02010.96060.14400.14397.96060.04530.04681.21060.12690.12678.21060.07100.07251.46060.10680.10658.46060.09490.09631.71060.08400.08368.71060.11650.11781.96060.05910.05878.96060.13530.13652.21060.03290.03249.21060.15080.15172.46060.00580.00539.46060.16260.16332.7106-0.0214-0.02209.59550.16740.16792.9606-0.0482-0.04879.73030.17090.17133.2106-0.0737-0.07429.86520.17330.173510.00000.17440.1745初始條件為θ=30o時(shí)的數(shù)值解與近似解計(jì)算表t數(shù)值解近似解t數(shù)值解近似解00.52360.52363.5507-0.3030-0.31780.00030.52360.52363.8007-0.3649-0.37880.00050.52360.52364.0507-0.4181-0.43050.00080.52360.52364.3007-0.4614-0.47170.00100.52360.52364.5507-0.4937-0.50140.00230.52360.52364.8007-0.5145-0.51880.00360.52360.52365.0507-0.5233-0.52350.00490.52360.52365.3007-0.5198-0.51540.00620.52360.52365.5507-0.5042-0.49470.01260.52360.52365.8007-0.4768-0.46190.01900.52360.52366.0507-0.4381-0.41790.02540.52350.52356.3007-0.3891-0.36360.03180.52350.52356.5507-0.3308-0.30040.06380.52320.52326.8007-0.2646-0.22980.09590.52270.52277.0507-0.1919-0.15370.12790.52200.52197.3007-0.1146-0.07380.15990.52110.52107.5507-0.03450.00790.32010.51360.51317.80070.04640.08940.48030.50110.50018.05070.12620.16880.64050.48390.48218.30070.20290.24400.80070.46190.45928.55070.27470.31321.05070.41880.41438.80070.33990.37481.30070.36570.35949.05070.39690.42721.55070.30390.29569.30070.44450.46921.80070.23480.22469.55070.48150.49972.05070.16000.14819.80070.50720.51802.30070.08130.06809.85050.51090.52012.55070.0006-0.01379.90040.51420.52172.8007-0.0801-0.09519.95020.51690.52293.0507-0.1589-0.174210.00000.51920.52353.3007-0.2338-0.2491B．?dāng)?shù)值解圖形0012345678910-0.2-0.15-0.1-0.0500.050.10.150.2tthetaTheta=10o的數(shù)值解0012345678910-0.8-0.6-0.4-0.200.20.40.6tthetatheta=30的數(shù)值解C．近似解圖形0012345678910-0.2-0.15-0.1-0.0500.050.10.150.2tthetaTheta=10o的近似解0012345678910-0.8-0.6-0.4-0.200.20.40.6tthetatheta=30o的近似解（3）簡(jiǎn)要結(jié)論從上面的數(shù)據(jù)可以看出，處事角度為10o時(shí)精確（數(shù)值）解與近似解相差不大，而初始角度為30o時(shí)，隨著時(shí)間的增加差別就很大了。（4）計(jì)算程序①M(fèi)atlab5.0程序如下:%定義微分方程組functionxdot=danbai(t,x)g=9.8;l=0.25;xdot=zeros(2,1);xdot(1)=x(2);xdot(2)=-g/l*sin(x(1));%調(diào)用ode45(…)解此微分方程組%周期近似為10s,因此，時(shí)間t的取值范圍確定在[010]%微分方程組的初始解為：x0=[0.1745,0](即10o)，x0=[0.5236,0](即30o)，程序中只有第一種初始解；要得到第二種初始解，只需將a=0.1745改成0.5236重新運(yùn)行即可。t0=0;tf=1.0;a=0.1745;x0=[a,0];[t,x]=ode45('danbai',[t0tf],x0);g=9.8;l=0.25;w=sqrt(g/l)