模态分析报告

模态分析实验报告

一实验原理模态分析方法是把复杂的实际结构简化成模态模型，来进行系统的参数识别（系统识别），从而大大地简化了系统地数学运算。通过实验测得实际响应来寻示相应的模型或调整预想的模型参数，使其成实际结构的最佳描述。工程实际中的振动系统都是连续弹性体，其质量与刚度具有分布的性质，只有掌握无限多个点在每瞬间时的运动情况，才能全面描述系统的振动。因此，理论上它们都属于无限多自由度的系统，需要用连续模型才能加以描述。但实际上不可能这样做，通常采用简化的方法，归结为有限个自由度的模型来进行分析，即将系统抽象为由一些集中质量块和弹性元件组中有n个集中质量，一般它便是一个n自由度的系统，需要n个独立坐标来描述它们的运动，系统的运动方程是n个二阶互相耦合（联立）的常微分方程。经化处理后，一个结构的动态特性可由N阶矩阵微分方程描述：成的模型。如果简化的系统模型离散MxCxKxft...(1)振向量；xx..分别为N维位移、速度和加速度响应.x，，式中ft为N维激向量；M、K、C分别为结构的质量、刚度和阻尼矩阵，通常为实对称N阶矩阵。设系统的初始状态为零，对方程式（1）两边进行傅里叶变换可得：KFMjCX2(2)式中的矩阵ZMjCK2(3)

反映了系统动态特性，称为系统动态矩阵或广义阻抗矩阵。其逆矩阵1K12MjC(4)HZ称为广义导纳矩阵，也就是传递函数矩阵。因此式(2)可以转化为X(5)FH矩阵中第i行第j列的元素为HHX(6)iFijj利用实际对称矩阵的加权正交性，有TKTMmrkr振型矩阵，其中矩阵称为n假设阻尼矩阵C也满足振型正12交性关系TCcr代入式(3)得ZTzr1(7)jcm式中zkr2rrr因此HZ1zrTmrirj22HN(8)j2ijr1rrrr上式中，kcmk分别为第r阶模态质量和模态刚度（又称r，r。,m2mrrrrrrr为广义质量和广义刚度）。,分别为第r阶模态频率、模态阻尼比和模态,rrr振型。不难发现，N自由度系统的频率响应，等于N个单自由度系统频率响应的线形叠加。为了确定全部模态参数,r,，实际上只需测量频率响应矩阵的,rr一列（对应一点激振，各点测量的H）或一行（对应依次各点激振，一点测H量的T）就够了。实验模态分析或模态参数识别的任务就是由一定频段内——模态频率，模态阻尼比的实测频率响应函数数据，确定系统的模态参数rr和振型。r为进行模态分析，首先要测得激振力及相应的响应信号，进行传递函数分析。i和j两点之间的传递函数表示在j点作用单在i点所引起的响应到i和j点之间的传递在j点加一个力信号激振，而在i点测量其引起的响应，就可得到计算传递函数曲线上的是连续变化的，分别测得其相应的响应，数曲线。然后建立结构模数和相应的模态振位力时，导纳，只要一个点。如果力信号就可以得到传递函采用适当的方法进行模态拟合，得到各阶模态参梁长（x向）0.68m，宽（y向）0.05m，高（z向）0.008m。分图1模态几何结构和节点分布图模态频率、阻尼和振型表1:模态频率和阻尼阶频率阻尼模态质量M模态刚度K模态阻尼C数(Hz)(%)12339.7982.2411.0000e+0006.2529e+0045.6036e+000163.0850.3941.0000e+0001.0500e+0064.0362e+000346.1530.3921.0000e+0004.7304e+0068.5347e+000表2:第一阶模态振型点号XYZ1234567890.0000e+0000.0000e+000-4.8670e-0030.0000e+0000.0000e+0002.0360e-0010.0000e+0000.0000e+0002.8036e-0010.0000e+0000.0000e+0005.8534e-0010.0000e+0000.0000e+0004.1341e-0010.0000e+0000.0000e+0004.3629e-0010.0000e+0000.0000e+0002.8959e-0010.0000e+0000.0000e+0002.7225e-0010.0000e+0000.0000e+0001.8071e-002100.0000e+0000.0000e+000-4.8670e-003110.0000e+0000.0000e+0002.0360e-001120.0000e+0000.0000e+0002.8036e-001130.0000e+0000.0000e+0005.8534e-001140.0000e+0000.0000e+0004.1341e-001150.0000e+0000.0000e+0004.3629e-001160.0000e+0000.0000e+0002.8959e-001170.0000e+0000.0000e+0002.7225e-001180.0000e+0000.0000e+0001.8071e-00219200.0000e+0000.0000e+000-4.8670e-0030.0000e+0000.0000e+0002.0360e-001210.0000e+0000.0000e+0002.8036e-001220.0000e+0000.0000e+0005.8534e-0012324250.0000e+0000.0000e+0004.1341e-0010.0000e+0000.0000e+0004.3629e-0010.0000e+0000.0000e+0002.8959e-001260.0000e+0000.0000e+0002.7225e-001270.0000e+0000.0000e+0001.8071e-0022829300.0000e+0000.0000e+000-4.8670e-0030.0000e+0000.0000e+0002.0360e-0010.0000e+0000.0000e+0002.8036e-001310.0000e+0000.0000e+0005.8534e-0010.0000e+0000.0000e+0004.1341e-001330.0000e+0000.0000e+0004.3629e-001323435360.0000e+0000.0000e+0002.8959e-0010.0000e+0000.0000e+0002.7225e-0010.0000e+0000.0000e+0001.8071e-002表3:第二阶模态振型点号1XYZ0.0000e+0000.0000e+000-1.3104e-0010.0000e+0000.0000e+0003.2022e-0010.0000e+0000.0000e+000-2.0210e-0020.0000e+0000.0000e+0003.0863e-0020.0000e+0000.0000e+0003.1234e-0020.0000e+0000.0000e+000-3.2342e-0020.0000e+0000.0000e+0001.2481e-0010.0000e+0000.0000e+000-1.9983e-0010.0000e+0000.0000e+0001.5804e-0010.0000e+0000.0000e+000-1.3104e-0010.0000e+0000.0000e+0003.2022e-0010.0000e+0000.0000e+000-2.0210e-0020.0000e+0000.0000e+0003.0863e-0020.0000e+0000.0000e+0003.1234e-0020.0000e+0000.0000e+000-3.2342e-0020.0000e+0000.0000e+0001.2481e-0010.0000e+0000.0000e+000-1.9983e-0010.0000e+0000.0000e+0001.5804e-0010.0000e+0000.0000e+000-1.3104e-0010.0000e+0000.0000e+0003.2022e-0010.0000e+0000.0000e+000-2.0210e-0020.0000e+0000.0000e+0003.0863e-0020.0000e+0000.0000e+0003.1234e-0020.0000e+0000.0000e+000-3.2342e-0020.0000e+0000.0000e+0001.2481e-0010.0000e+0000.0000e+000-1.9983e-0010.0000e+0000.0000e+0001.5804e-0010.0000e+0000.0000e+000-1.3104e-0010.0000e+0000.0000e+0003.2022e-001234567891011121314151617181920212223242526272829303132333435360.0000e+0000.0000e+000-2.0210e-0020.0000e+0000.0000e+0003.0863e-0020.0000e+0000.0000e+0003.1234e-0020.0000e+0000.0000e+000-3.2342e-0020.0000e+0000.0000e+0001.2481e-0010.0000e+0000.0000e+000-1.9983e-0010.0000e+0000.0000e+0001.5804e-001表4:第三阶模态振型点号XYZ10.0000e+0000.0000e+000-1.0935e-00220.0000e+0000.0000e+000-1.0899e-00130.0000e+0000.0000e+000-6.0115e-00240.0000e+0000.0000e+000-7.4510e-00350.0000e+0000.0000e+0001.0507e-00160.0000e+0000.0000e+0002.5104e-00270.0000e+0000.0000e+000-2.7690e-00280.0000e+0000.0000e+000-9.4133e-00290.0000e+0000.0000e+000-3.0500e-003100.0000e+0000.0000e+000-1.0935e-002110.0000e+0000.0000e+000-1.0899e-001120.0000e+0000.0000e+000-6.0115e-002130.0000e+0000.0000e+000-7.4510e-003140.0000e+0000.0000e+0001.0507e-001150.0000e+0000.0000e+0002.5104e-002160.0000e+0000.0000e+000-2.7690e-002170.0000e+0000.0000e+000-9.4133e-002180.0000e+0000.0000e+000-3.0500e-003190.0000e+0000.0000e+000-1.0935e-002200.0000e+0000.0000e+000-1.0899e-001210.0000e+0000.0000e+000-6.0115e-002220.0000e+0000.0000e+000-7.4510e-003230.0000e+0000.0000e+0001.0507e-001240.0000e+0000.0000e+0002.5104e-002250.0000e+0000.0000e+000-2.7690e-002260.0000e+0000.0000e+000-9.4133e-002270.0000e+0000.0000e+000-3.0500e-003280.0000e+0000.0000e+000-1.0935e-002290.0000e+0000.0000e+000-1.0899e-001300.0000e+0000.0000e+000-6.0115e-002310.0000e+0000.0000e+000-7.4510e-003320.0000e+0000.0000e+0001.0507e-0013334350.0000e+0000.0000e+0002.5104e-0020.0000e+0000.0000e+000-2.7690e-0020.0000e+0000.0000e+000-9.4133e-002360.0000e+0000.0000e+000-3.0500e-003四Matlab数值计算计算频响函数首先从实验中提取出时域激励信号（F-t）和响应信号（x-t），已知采样频率fs=1000Hz，采样量N=1024，采样时间间隔t=0.001s，则由采样分辨率公式ff/Ns（4-1）f计算得对时域信号进行快速傅里叶变换，由频响函数的定义式=0.977，H(f)X(f)/F(f)（4-2）或估计式H(f)G/GF(f)X(f)/F(f)F(f)（4-3）mm1fxffk1k1即可计算出频响函数。在第二点测量响应，通过下面程序得到各点敲击后的传递函数幅频曲线：clearclcn=1024fs=1020.24/n;tch1=load('n1.txt');tch2=load('n2.txt');fch1=fft(tch1,n);fch2=fft(tch2,n);图3.1传函H21的幅频曲线图3传函H22的幅频曲线传函幅频x104876543210)2ss/m(noitarelecca020040060080010001200frequency(HZ)图4传函H23的幅频曲线传函幅频x10432.52)2ss/m(no1.51itarelecca0.50020040060080010001200frequency(HZ)图5传函H24的幅频曲线传函幅频x10443.53)22.52ss/m(noitarele1.51cca0.50020040060080010001200frequency(HZ)图6传函H25的幅频曲线传函幅频x1043.532.52)2ss/m(noitar1.51elecca0.50020040060080010001200frequency(HZ)图7传函H26的幅频曲线传函幅频x1043.532.52)2ss/m(noitar1.51elecca0.50020040060080010001200frequency(HZ)图8传函H27的幅频曲线传函幅频500045004000350030002500200015001000500)2ss/m(noitarelecca0020040060080010001200frequency(HZ)图9传函H28的幅频曲线传函幅频x10443.53)22.52ss/m(noitarele1.51cca0.50020040060080010001200frequency(HZ)图10传函H29的幅频曲线取前三阶模态，将九个峰值对应的横坐标平均后得到各阶的振动频率f1=41.328Hzf2,=164.241Hz，f3=353.162Hz,取峰值得到振型，其中纵坐标的正负与对应的相位的正负一致。表5拟合的频率和振型阶数123频率41.328振型164.241353.1621234567890.2240831.4504222.3504223.4070484.9502663.4296822.4504221.0532590.3189152.59819119.30355429.72815912.229255-3.316171-19.972926-24.314696-14.768748-1.0152290.1605024.3756843.167132-1.097602-5.7807220.3051934.0443443.6723630.96981五理论值简支梁长（x向）0.68m，宽（y向）0.05m，高（z向）0.008m。欧拉梁（不考虑剪切）EIi22ml4EiE=2.06e+011Pa；k=5/6；G=0.79e+011Pa，单位其中，i指的是模态的阶数，长度质量m=3.12kg/m^3,截面惯性矩I=2.13e-009m^4,长l=0.68m,厚h=0.008m，计算得表6：模态频率模态阶数1阶2阶3阶欧拉梁频率（Hz）40.4914161.9673364.4257六有限元计算采用有限元分析软件计算简支梁的模态参数，用shell63单元进行模拟，将几何模型划分网格，得出简支梁模型如下图1ELEMENTSAUG31201416:53:36UYZX图11单元划分模型模态计算结果1NODALSOLUTIONAUG31201416:54:34STEP=1SUB=1FREQ=40.258USUM(AVG)RSYS=0DMX=.969082SMX=.969082UXMNYZMX0.215351.430703.646054.861406.107676.323027.538379.75373.969082图12一阶振型1NODALSOLUTIONAUG31201416:54:49STEP=1SUB=2FREQ=161.84USUM(AVG)RSYS=0DMX=.972594SMX=.972594UMXXMNYZ0.216132.432264.648396.864528.756462.972594.108066.324198.54033图13二阶振型1NO