随机响应分析
11.1 动力学环境分类
11.2 概述随机振动是统计意义下描述的振动，在任何瞬时大小未知，但其大小的概率超过一给定的值。
常见的例子如地震引起的地基运动、海洋波浪高度和频率、航天器和高耸建筑物受到的风压力、由于火箭与喷气发动机噪音引起的声波等。
MSC/NASTRAN 对随机响应分析是作为频率响应后处理进行的。输入包括频率响应的输出、用户给定的载荷条件（形式为自相关的谱密度）。输出为响应功率谱密度、自相关函数、响应的均方值。
MSC/NASTRAN 随机分析假设历经性随机过程随机动态环境例子
11.3 自相关与自谱自相关函数注：Rj(0)为均方值自谱函数
Fourier变换为
3）均方响应值
4）外观频率为N0
例子
11.4 各态历经性随机激励下线性系统响应计算
1) 线性系统单输入输出关系由频率响应分析得到其中，Hja(ω)为频率响应或输入到输出的传递函数对多输入单输出其矩阵形式为
输出自相关谱为其单个输入谱为
2）线性系统的多输入输出关系多输入输出谱关系其中，
输入互谱矩阵为其谱特性为
3）常用特殊情况单输入分析（完全相关输入）
不相关多输入
11.5 MSC/NASTRAN中随机分析的实现如果由频率响应计算结果为Hja(ω),但并不直接计算如需要Hja(ω),令 Fa(ω)=1
RANDPS卡片定义随机分析中使用的功率谱密度因子，频率相关形式为格式由情况控制卡RANDOM = SID选取自谱密度，J=K,X为大于0的整数，Y为0
TID=0,G(F)=0
11.5.2 TABRND1卡片用表格函数定义功率谱密度函数格式
3）
11.5.3 随机响应输入要求执行控制
2）情况控制
3） 模型数据
11.5.4 随机响应例子
1） 例1：单输入随机响应分析
（1）问题描述：
(a)矩形板如图；(b)基座运动（z方向）功率谱(PSD)表中给出；(c)整个频率范围的常临界阻尼比为0.03；(d)用log-log输入PSD;(e)使用模态求解法
(2) 使用具有大质量的模态法（在边界处用REB2单元）确定
9999点处的位移和加速度功率谱(PSD)
确定结点33和55的位移功率谱(PSD)
(3)输入文件
ID SEMINAR,PROB10
SOL 111
TIME 30
CEND
TITLE= RANDOM ANALYSIS - BASE EXCITATION
SUBTITLE= USING THE MODAL METHOD WITH LANCZOS
ECHO= UNSORTED
SPC= 101
SET 111= 33,55,9999
ACCELERATION(SORT2,PHASE)= 111
METHOD= 100
FREQUENCY= 100
SDAMPING= 100
RANDOM= 100
DLOAD= 100
$
OUTPUT(XYPLOT)
XTGRID= YES
YTGRID= YES
XBGRID= YES
YBGRID= YES
YTLOG= YES
XTITLE= FREQUENCY
YTTITLE= ACCEL RESPONSE BASE,MAGNITUDE
YBTITLE= ACCEL RESPONSE AT BASE,PHASE
XYPLOT ACCEL RESPONSE / 9999 (T3RM,T3IP)
YTTITLE= ACCEL RESPONSE AT TIP CENTER,MAGNITUDE
YBTITLE= ACCEL RESPONSE AT TIP CENTER,PHASE
XYPLOT ACCEL RESPONSE / 33 (T3RM,T3IP)
YTTITLE= ACCEL RESPONSE AT OPPOSITE CORNER,MAGNITUDE
YBTITLE= ACCEL RESPONSE AT OPPOSETE CORNER,PHASE
XYPLOT ACCEL RESPONSE / 55 (T3RM,T3IP)
$
$ PLOT OUTPUT IS ONLY MEANS OF VIEWING PSD DATA
$
XGRID= YES
YGRID= YES
XLOG= YES
YLOG= YES
YTITLE= ACCEL P S D AT LOADED CORNER
XYPLOT ACCEL PSDF / 9999(T3)
YTITLE= ACCEL P S D AT TIP CENTER
XYPLOT ACCEL PSDF / 33(T3)
YTITLE= ACCEL P S D AT OPPOSITE CORNER
XYPLOT ACCEL PSDF / 55(T3)
$
BEGIN BULK
PARAM,COUPMASS,1
PARAM,WTMASS,0.00259
$
INCLUDE ’plate.bdf’
$
GRID,9999,,0.,0.,0.
$
RBE2,101,9999,12345,1,12,23,34,45
$
SPC1,101,12456,9999
$
CONM2,6000,9999,,1.0E8
$
$MAT1,1,.1,,.1,.286
$
$ EIGENVALUE EXTRACTION PARAMETERS
$
EIGRL,100,,2000.
$
$ SPECIFY MODAL DAMPING
$
TABDMP1,100,CRIT,
+,0.,.03,10.,.03,ENDT
$
$ POINT LOADING AT TIP CENTER
$
RLOAD2,100,600,,,310
$
TABLED1,310,
+,10.,1.,1000.,1.,ENDT
$
DAREA,600,9999,3,1.E8
$
$ SPECIFY FREQUENCY STEPS
$
FREQ,100,30.
FREQ1,100,20.,20.,50
FREQ4,100,20.,1000.,.03,5
$
$ SPECIFY SPECTRAL DENSITY
$
RANDPS,100,1,1,1.,0.,111
$
TABRND1,111,LOG,LOG
+,20.,0.1,30.,1.,100.,1.,500.,.1,
+,1000.,.1,ENDT
$
ENDDATA
（4）部分结果基座的加速度PSD(大小、相位)与频率关系结点55的加速度PSD(大小、相位)与频率关系
结点33的加速度PSD(大小、相位)与频率关系
例2：多输入随机响应分析问题：
输入文件
ID SEMINAR,PROB11
SOL 111
TIME 30
CEND
TITLE= FREQUENCY RESPONSE WITH PRESSURE AND POINT LOADS
SUBTITLE= USING THE MODAL METHOD WITH LANCZOS
ECHO= UNSORTED
SPC= 1
SET 111= 11,33,55
DISPLACEMENT(PLOT,PHASE)= 111
METHOD= 100
FREQUENCY= 100
SDAMPING= 100
RANDOM= 100
SUBCASE 1
LABEL= PRESSURE LOAD
DLOAD= 100
LOADSET= 100
SUBCASE 2
LABEL CORNER LOAD
DLOAD= 200
LOADSET= 100
$
OUTPUT (XYPLOT)
$
XTGRID= YES
YTGRID= YES
XBGRID= YES
YBGRID= YES
YTLOG= YES
YBLOG= NO
XTITLE= FREQUENCY (HZ)
YTTITLE= DISPLACEMENT RESPONSE AT LOADED CORNER,MAGNITUDE
YBTITLE= DISPLACEMENT RESPONSE AT LOADED CORNER,PHASE
XYPLOT DISP RESPONSE / 11 (T3RM,T3IP)
YTTITLE= DISPLACEMENT RESPONSE AT TIP CENTER,MAGNITUDE
YBTITLE= DISPLACEMENT RESPONSE AT TIP CENTER,PHASE
XYPLOT DISP RESPONSE / 33 (T3RM,T3IP)
YTTITLE= DISPLACEMENT RESPONSE AT OPPOSITE CORNER,MAGNITUDE
YBTITLE= DISPLACEMENT RESPONSE AT OPPOSITE CORNER,PHASE
XYPLOT DISP RESPONSE / 55 (T3RM,T3IP)
$
$ PLOT OUTPUT IS ONLY MEANS OF VIEWING PSD DATA
$
XGRID= YES
YGRID= YES
XLOG= YES
YLOG= YES
YTITLE= DISP P S D AT LOADED CORNER
XYPLOT DISP PSDF / 11(T3)
YTITLE= DISP P S D AT TIP CENTER
XYPLOT DISP PSDF / 33(T3)
YTITLE= DISP P S D AT OPPOSITE CORNER
XYPLOT DISP PSDF / 55(T3)
$
BEGIN BULK
PARAM,COUPMASS,1
PARAM,WTMASS,0.00259
$
$ MODEL DESCRIBED IN NORMAL MODES EXAMPLE
$
INCLUDE ’plate.bdf’
$
$ EIGENVALUE EXTRACTION PARAMETERS
$
EIGRL,100,10.,2000.
$
$ SPECIFY MODAL DAMPING
$
TABDMP1,100,CRIT,
+,0.,.03,10.,.03,ENDT
$
$ FIRST LOADING
$
RLOAD2,100,300,,,310
$
TABLED1,310,
+,10.,1.,1000.,1.,ENDT
$
$ UNIT PRESSURE LOAD TO PLATE
$
LSEQ,100,300,400
$
PLOAD2,400,1.,1,THRU,40
$
$ SECOND LOADING
$
RLOAD2,200,600,,,310
$
$ POINT LOAD AT TIP CENTER
$
DAREA,600,11,3,1.
$
$ SPECIFY FREQUENCY STEPS
$
FREQ1,100,20.,20.,49
$
$ SPECIFY SPECTRAL DENSITY
$
RANDPS,100,1,1,1.,0.,100
RANDPS,100,2,2,1.,0.,200
RANDPS,100,1,2,1.,0.,300
RANDPS,100,1,2,0.,1.,400
$
TABRND1,100,
+,20.,0.1,30.,1.,100.,1.,500.,.1,
+,1000.,.1,ENDT
$
TABRND1,200,
+,20.,0.5,30.,2.5,500.,2.5,1000.,0.,
+,ENDT
$
TABRND1,300,
+,20.,-.099619,100.,-.498097,500.,.070711,1000.,0.,
+,ENDT
$
TABRND1,400,
+,20.,.0078158,100.,.0435791,500.,-.70711,1000.,0.,
+,ENDT
$
ENDDATA
11.1 动力学环境分类
11.2 概述随机振动是统计意义下描述的振动，在任何瞬时大小未知，但其大小的概率超过一给定的值。
常见的例子如地震引起的地基运动、海洋波浪高度和频率、航天器和高耸建筑物受到的风压力、由于火箭与喷气发动机噪音引起的声波等。
MSC/NASTRAN 对随机响应分析是作为频率响应后处理进行的。输入包括频率响应的输出、用户给定的载荷条件（形式为自相关的谱密度）。输出为响应功率谱密度、自相关函数、响应的均方值。
MSC/NASTRAN 随机分析假设历经性随机过程随机动态环境例子
11.3 自相关与自谱自相关函数注：Rj(0)为均方值自谱函数
Fourier变换为
3）均方响应值
4）外观频率为N0
例子
11.4 各态历经性随机激励下线性系统响应计算
1) 线性系统单输入输出关系由频率响应分析得到其中，Hja(ω)为频率响应或输入到输出的传递函数对多输入单输出其矩阵形式为
输出自相关谱为其单个输入谱为
2）线性系统的多输入输出关系多输入输出谱关系其中，
输入互谱矩阵为其谱特性为
3）常用特殊情况单输入分析（完全相关输入）
不相关多输入
11.5 MSC/NASTRAN中随机分析的实现如果由频率响应计算结果为Hja(ω),但并不直接计算如需要Hja(ω),令 Fa(ω)=1
RANDPS卡片定义随机分析中使用的功率谱密度因子，频率相关形式为格式由情况控制卡RANDOM = SID选取自谱密度，J=K,X为大于0的整数，Y为0
TID=0,G(F)=0
11.5.2 TABRND1卡片用表格函数定义功率谱密度函数格式
3）
11.5.3 随机响应输入要求执行控制
2）情况控制
3） 模型数据
11.5.4 随机响应例子
1） 例1：单输入随机响应分析
（1）问题描述：
(a)矩形板如图；(b)基座运动（z方向）功率谱(PSD)表中给出；(c)整个频率范围的常临界阻尼比为0.03；(d)用log-log输入PSD;(e)使用模态求解法
(2) 使用具有大质量的模态法（在边界处用REB2单元）确定
9999点处的位移和加速度功率谱(PSD)
确定结点33和55的位移功率谱(PSD)
(3)输入文件
ID SEMINAR,PROB10
SOL 111
TIME 30
CEND
TITLE= RANDOM ANALYSIS - BASE EXCITATION
SUBTITLE= USING THE MODAL METHOD WITH LANCZOS
ECHO= UNSORTED
SPC= 101
SET 111= 33,55,9999
ACCELERATION(SORT2,PHASE)= 111
METHOD= 100
FREQUENCY= 100
SDAMPING= 100
RANDOM= 100
DLOAD= 100
$
OUTPUT(XYPLOT)
XTGRID= YES
YTGRID= YES
XBGRID= YES
YBGRID= YES
YTLOG= YES
XTITLE= FREQUENCY
YTTITLE= ACCEL RESPONSE BASE,MAGNITUDE
YBTITLE= ACCEL RESPONSE AT BASE,PHASE
XYPLOT ACCEL RESPONSE / 9999 (T3RM,T3IP)
YTTITLE= ACCEL RESPONSE AT TIP CENTER,MAGNITUDE
YBTITLE= ACCEL RESPONSE AT TIP CENTER,PHASE
XYPLOT ACCEL RESPONSE / 33 (T3RM,T3IP)
YTTITLE= ACCEL RESPONSE AT OPPOSITE CORNER,MAGNITUDE
YBTITLE= ACCEL RESPONSE AT OPPOSETE CORNER,PHASE
XYPLOT ACCEL RESPONSE / 55 (T3RM,T3IP)
$
$ PLOT OUTPUT IS ONLY MEANS OF VIEWING PSD DATA
$
XGRID= YES
YGRID= YES
XLOG= YES
YLOG= YES
YTITLE= ACCEL P S D AT LOADED CORNER
XYPLOT ACCEL PSDF / 9999(T3)
YTITLE= ACCEL P S D AT TIP CENTER
XYPLOT ACCEL PSDF / 33(T3)
YTITLE= ACCEL P S D AT OPPOSITE CORNER
XYPLOT ACCEL PSDF / 55(T3)
$
BEGIN BULK
PARAM,COUPMASS,1
PARAM,WTMASS,0.00259
$
INCLUDE ’plate.bdf’
$
GRID,9999,,0.,0.,0.
$
RBE2,101,9999,12345,1,12,23,34,45
$
SPC1,101,12456,9999
$
CONM2,6000,9999,,1.0E8
$
$MAT1,1,.1,,.1,.286
$
$ EIGENVALUE EXTRACTION PARAMETERS
$
EIGRL,100,,2000.
$
$ SPECIFY MODAL DAMPING
$
TABDMP1,100,CRIT,
+,0.,.03,10.,.03,ENDT
$
$ POINT LOADING AT TIP CENTER
$
RLOAD2,100,600,,,310
$
TABLED1,310,
+,10.,1.,1000.,1.,ENDT
$
DAREA,600,9999,3,1.E8
$
$ SPECIFY FREQUENCY STEPS
$
FREQ,100,30.
FREQ1,100,20.,20.,50
FREQ4,100,20.,1000.,.03,5
$
$ SPECIFY SPECTRAL DENSITY
$
RANDPS,100,1,1,1.,0.,111
$
TABRND1,111,LOG,LOG
+,20.,0.1,30.,1.,100.,1.,500.,.1,
+,1000.,.1,ENDT
$
ENDDATA
（4）部分结果基座的加速度PSD(大小、相位)与频率关系结点55的加速度PSD(大小、相位)与频率关系
结点33的加速度PSD(大小、相位)与频率关系
例2：多输入随机响应分析问题：
输入文件
ID SEMINAR,PROB11
SOL 111
TIME 30
CEND
TITLE= FREQUENCY RESPONSE WITH PRESSURE AND POINT LOADS
SUBTITLE= USING THE MODAL METHOD WITH LANCZOS
ECHO= UNSORTED
SPC= 1
SET 111= 11,33,55
DISPLACEMENT(PLOT,PHASE)= 111
METHOD= 100
FREQUENCY= 100
SDAMPING= 100
RANDOM= 100
SUBCASE 1
LABEL= PRESSURE LOAD
DLOAD= 100
LOADSET= 100
SUBCASE 2
LABEL CORNER LOAD
DLOAD= 200
LOADSET= 100
$
OUTPUT (XYPLOT)
$
XTGRID= YES
YTGRID= YES
XBGRID= YES
YBGRID= YES
YTLOG= YES
YBLOG= NO
XTITLE= FREQUENCY (HZ)
YTTITLE= DISPLACEMENT RESPONSE AT LOADED CORNER,MAGNITUDE
YBTITLE= DISPLACEMENT RESPONSE AT LOADED CORNER,PHASE
XYPLOT DISP RESPONSE / 11 (T3RM,T3IP)
YTTITLE= DISPLACEMENT RESPONSE AT TIP CENTER,MAGNITUDE
YBTITLE= DISPLACEMENT RESPONSE AT TIP CENTER,PHASE
XYPLOT DISP RESPONSE / 33 (T3RM,T3IP)
YTTITLE= DISPLACEMENT RESPONSE AT OPPOSITE CORNER,MAGNITUDE
YBTITLE= DISPLACEMENT RESPONSE AT OPPOSITE CORNER,PHASE
XYPLOT DISP RESPONSE / 55 (T3RM,T3IP)
$
$ PLOT OUTPUT IS ONLY MEANS OF VIEWING PSD DATA
$
XGRID= YES
YGRID= YES
XLOG= YES
YLOG= YES
YTITLE= DISP P S D AT LOADED CORNER
XYPLOT DISP PSDF / 11(T3)
YTITLE= DISP P S D AT TIP CENTER
XYPLOT DISP PSDF / 33(T3)
YTITLE= DISP P S D AT OPPOSITE CORNER
XYPLOT DISP PSDF / 55(T3)
$
BEGIN BULK
PARAM,COUPMASS,1
PARAM,WTMASS,0.00259
$
$ MODEL DESCRIBED IN NORMAL MODES EXAMPLE
$
INCLUDE ’plate.bdf’
$
$ EIGENVALUE EXTRACTION PARAMETERS
$
EIGRL,100,10.,2000.
$
$ SPECIFY MODAL DAMPING
$
TABDMP1,100,CRIT,
+,0.,.03,10.,.03,ENDT
$
$ FIRST LOADING
$
RLOAD2,100,300,,,310
$
TABLED1,310,
+,10.,1.,1000.,1.,ENDT
$
$ UNIT PRESSURE LOAD TO PLATE
$
LSEQ,100,300,400
$
PLOAD2,400,1.,1,THRU,40
$
$ SECOND LOADING
$
RLOAD2,200,600,,,310
$
$ POINT LOAD AT TIP CENTER
$
DAREA,600,11,3,1.
$
$ SPECIFY FREQUENCY STEPS
$
FREQ1,100,20.,20.,49
$
$ SPECIFY SPECTRAL DENSITY
$
RANDPS,100,1,1,1.,0.,100
RANDPS,100,2,2,1.,0.,200
RANDPS,100,1,2,1.,0.,300
RANDPS,100,1,2,0.,1.,400
$
TABRND1,100,
+,20.,0.1,30.,1.,100.,1.,500.,.1,
+,1000.,.1,ENDT
$
TABRND1,200,
+,20.,0.5,30.,2.5,500.,2.5,1000.,0.,
+,ENDT
$
TABRND1,300,
+,20.,-.099619,100.,-.498097,500.,.070711,1000.,0.,
+,ENDT
$
TABRND1,400,
+,20.,.0078158,100.,.0435791,500.,-.70711,1000.,0.,
+,ENDT
$
ENDDATA
