This example demonstrates the use of stage clones, each of which is associated with a particular harmonic index, to perform a multiharmonic modal analysis of a simplified hollow cylinder modeled using two stages. Specifically, it demonstrates the following key points:
Creating a stage clone
Axial multistage modeling
Matched interstage boundaries
Applying interstage constraints to fully connect stages
Connecting all primary stages and stage clones together
The example problem is presented in the following sections:
The multistage system consists of two axially aligned cyclic stages of a hollow cylinder merged at a matched interstage boundary (Figure 7.21: Multistage Model of a Hollow Cylinder). Each stage is a sector of a hollow cylinder with different sector sizes. Stage 1 has 6 sectors while stage 2 has 12 sectors. While it is not possible to see, the model for stage 2 has two identical meshes, one for the primary stage and one for the stage clone. The interstage boundary conditions are applied in two separate steps. The first step applies interstage constraints for the primary stages (HI = 0 in this case), and the second step applies interstage constraints for the primary stage for stage 1 (HI = 0) and the stage clone for stage 2 (HI = 6). The modal analysis is run, and then a full 360° case is run to compare the modal frequencies.
Note: Multistage expansion is not currently supported for multiharmonic analyses.
The following table describes the key steps and related commands used for the example analysis. See Input for the Analysis for the detailed command listing.
| Step | Description | Mechanical APDL Commands |
| 1. | Create meshes |
CYLIND,... VMESH,... |
| 2. | Write .cbd files for each stage. | CDWRITE,... |
| 3. | Read in stage clones and primary stages. | CDREAD,... |
| 4. | Create stage components. | CM,... |
| 5. | Create stage and stage clones. | MSOPT,NEW,... |
| 6. | Apply boundary conditions. | D,... |
| 7. | Select interface nodes and elements for primary stages. | CMSEL,... |
| 8. | Apply interstage constraints for primary stages. | CEIMS,... |
| 9. | Select interface nodes and elements for stage clone for secondary harmonic and primary stage for primary harmonic. | CMSEL,... |
| 10. | Apply interstage constraints for stage clone for secondary harmonic and primary stage for primary harmonic. | CEIMS,... |
| 11. | Enter solution processor | /SOLU |
| 12. | Specify modal options | MODOPT,... |
| 13. | Solve the analysis | SOLVE |
| 14. | Enter/POST1 processor | /POST1 |
| 15. | List modal frequencies | SET,LIST |
To download the input files used for this example problem, click the link below.
/com =============================================================== /com Multistage multiharmonic modal analysis - hollow cylinder /com =============================================================== !create sector mesh for stage1 !------------------------------- /prep7 nu = 0.3 rho = 7800 e = 2e+11 ET,1,SOLID186 MP,EX,1,E MP,DENS,1,RHO MP,NUXY,1,NU TYPE,1 MAT,1 ro = 0.2 ri = 0.18 ltot = 1 elsz = 0.01 mratio = 3 !ratio between element size along thickness / along length theta1 = 60 nsect1 = 360/theta1 CYLIND, RI,RO, 0,ltot/2, 0,theta1 ! mesh !--------- NUMMRG,KP ESIZE,ELSZ LSEL,,LOC,Z,ltot/4 LESIZ,ALL,elsz*mratio LSEL,,LOC,Z,0 LSEL,A,LOC,Z,ltot/2 CSYS,1 LSEL,R,LOC,Y,theta1/2 LESIZ,ALL,,,20 ALLSEL CSYS,0 VMESH,ALL ! write stage1 sector to file !------------------------------------ cdwrite,db,sect1,cdb parsav,,sect,parm finish /clear, nostart ! create sector mesh for stage2 !----------------------------------------- /prep7 parres,,sect,parm ET,1,SOLID186 MP,EX,1,E MP,DENS,1,RHO MP,NUXY,1,NU TYPE,1 MAT,1 theta2 = 30 nsect2 = 360/theta2 CYLIND, RI,RO, ltot/2,ltot, 0,theta2 ! mesh !--------- NUMMRG,KP ESIZE,elsz LSEL,,LOC,Z,3*ltot/4 LESIZ,ALL,elsz*mratio LSEL,,LOC,Z,ltot/2 LSEL,A,LOC,Z,ltot CSYS,1 LSEL,R,LOC,Y,theta2/2 LESIZ,ALL,,,10 ALLSEL CSYS,0 VMESH,ALL ! write stage 2 sector to file !------------------------------------- cdwrite,db,sect2,cdb parsav,,sect,parm finish /clear, nostart /com =============================================================== /com Multistage simulation - hollow cylinder /com =============================================================== /filname,ms parres,,sect,parm cdread,db,sect2,cdb ! stage2 HI6 (stage clone) cdread,db,sect2,cdb ! stage2 HI0 cdread,db,sect1,cdb ! stage1 HI0 HI1 = 0 HI2A = 0 HI2B = 6 /prep7 csys,1 !-------------------------------------------------- ! Prepare stage1 !-------------------------------------------------- ! create components for stage1, HI = 0 !-------------------------------------- esel,,type,,1 cm, _STAGE1_BASE_ELM, ELEM nsle cm, _STAGE1_BASE_NOD, NODE allsel ESEL,,TYPE,,1 NSLE NSEL,R,LOC,Y,0 CM,_STAGE1_CYCLOW_NOD,NODE ESEL,,TYPE,,1 NSLE NSEL,R,LOC,Y,theta1 CM,_STAGE1_CYCHIGH_NOD,NODE ALLSEL ! create stage1 and cyclic CEs !-------------------------------- msopt,new,stage1,NSECT1,HI1 CECYCMS !-------------------------------------------------- ! Prepare stage2A, HI = 0 !-------------------------------------------------- ! create components for stage2A, HI = 0 !-------------------------------------- esel,,type,,2 cm,_STAGE2A_BASE_ELM,ELEM nsle cm,_STAGE2A_BASE_NOD,NODE allsel ESEL,,TYPE,,2 NSLE NSEL,R,LOC,Y,0 CM,_STAGE2A_CYCLOW_NOD,NODE ESEL,,TYPE,,2 NSLE NSEL,R,LOC,Y,theta2 CM,_STAGE2A_CYCHIGH_NOD,NODE ALLSEL ! create stage2A and cyclic CEs, HI = 0 !-------------------------------------- msopt,new,stage2A,NSECT2,HI2A CECYCMS !-------------------------------------------------- ! Prepare stage2B, HI = 6 !-------------------------------------------------- ! create components for stage2B !------------------------------ esel,,type,,3 cm, _STAGE2B_BASE_ELM, ELEM nsle cm, _STAGE2B_BASE_NOD, NODE allsel ESEL,,TYPE,,3 NSLE NSEL,R,LOC,Y,0 CM,_STAGE2B_CYCLOW_NOD,NODE ESEL,,TYPE,,3 NSLE NSEL,R,LOC,Y,theta2 CM,_STAGE2B_CYCHIGH_NOD,NODE ALLSEL ! create stage2B and cyclic CEs, HI = 6 !-------------------------------------- msopt,new,stage2B,NSECT2,HI2B,,,,,stage2A CECYCMS ! Apply boundary conditions !-------------------------------------- nsel,,loc,Z,0 nsel,a,loc,Z,ltot d,all,all allsel ! Select interstage nodes and elements for stage1 and stage2A !------------------------------------------------------------ ESEL,,TYPE,,1 NSLE NSEL,R,LOC,Z,ltot/2 CM,CM_INTF1_NODE,NODE NSEL,,LOC,Z,ltot/2 ESLN ESEL,R,TYPE,,2 CM,CM_INTF2_1_ELEM,ELEM ALLSEL CMSEL,,CM_INTF1_NODE CMSEL,,CM_INTF2_1_ELEM ! Apply interstage CEs - stage1, HI0 / stage2A, HI0 !---------------------------------------------------- CEIMS,,,stage1,stage2A msopt,list,all,1 ALLSEL ! Select interstage nodes and elements for stage1 and stage2B !------------------------------------------------------------ NSEL,,LOC,Z,ltot/2 ESLN ESEL,R,TYPE,,3 CM,CM_INTF2_2_ELEM,ELEM ALLSEL CMSEL,,CM_INTF1_NODE CMSEL,,CM_INTF2_2_ELEM ! Apply interstage CEs - stage1, HI0 / stage2B, HI6 !---------------------------------------------------- CEIMS,,,stage1,stage2B msopt,list,all,1 ALLSEL fini /solu antype,modal modopt,lanb,3,0,3150 solve finish /post1 /com ======================================= /com Multistage modal frequencies /com ======================================= set,list /com ======================================= finish /clear,nostart /com =============================================================== /com Full 360 degree reference simulation - hollow cylinder /com =============================================================== /filname,full parres,,sect,parm cdread,db,sect2,cdb ! stage 2 cdread,db,sect1,cdb ! stage 1 *get,nnode,node,,num,max /prep7 csys,1 ! form 360 stage 1 !-------------------- nsel,s,loc,z,0,ltot/2 esln,,1 egen,nsect1,nnode,all,,,,,,,,0,theta1,0 allsel esel,,type,,1 nsle nummrg,node esel,,type,,1 nsle nsel,r,loc,z,ltot/2 cm,stage1_node_intf,node allsel ! form 360 stage 2 !------------------- nsel,s,loc,z,ltot/2,ltot esln,,1 egen,nsect2,nnode,all,,,,,,,,0,theta2,0 allsel esel,,type,,2 nsle nummrg,node allsel nsel,,loc,z,ltot/2 esln esel,r,type,,2 cm,stage2_elem_intf,elem allsel ! tie stages cmsel,,stage1_node_intf cmsel,,stage2_elem_intf ceintf,,all allsel ! BC nsel,,loc,Z,0 nsel,a,loc,Z,ltot d,all,all allsel finish /solu antype,modal modopt,lanb,37,0,3150 solve finish /post1 /com ======================================= /com Full 360 reference modal frequencies /com ======================================= set,list /com ======================================= set,1,11 /show,png,rev /view,1,1,1,1 plnsol,u,sum set,1,26 plnsol,u,sum set,1,37 plnsol,u,sum /show,close finish
To list multistage cyclic symmetry settings and stage connections in the output, use the MSOPT,LIST command. The listing for this example is shown below.
MULTISTAGE OPTION: LIST STAGES DETAILED INFORMATION
TOTAL NUMBER OF STAGES = 3
COORDINATE SYSTEM = 1 (DEFAULT)
CURRENT STAGE FOR APPLYING CYCLIC CONSTRAINTS = STAGE2B
-------------------- STAGE = STAGE1 --------------------
NUMBER OF SECTORS = 6
HARMONIC INDEX = 0
NO DUPLICATE SECTOR
TOTAL NUMBER OF NODES = 4041
TOTAL NUMBER OF ELEMENTS = 680
BOUNDING BOX OF STAGE NODES
RADIAL DIRECTION MIN,MAX = 0.1800E+00 0.2000E+00
TANGENTIAL DIRECTION MIN,MAX = 0.00 60.00 (deg)
AXIAL DIRECTION MIN,MAX = 0.0000E+00 0.5000E+00
CYCLIC CONSTRAINT EQUATIONS
TOTAL NUMBER (ACTIVE) = 408
NUMBER OF FIRST EQUATION = 1
NUMBER OF LAST EQUATION = 423
STAGE IS CONNECTED WITH STAGE = STAGE2A
MULTISTAGE INTERFACE CONSTRAINT EQUATIONS
TOTAL NUMBER (ACTIVE) = 495
NUMBER OF FIRST EQUATION = 1270
NUMBER OF LAST EQUATION = 1764
STAGE IS CONNECTED WITH STAGE = STAGE2B
-------------------- STAGE = STAGE2A --------------------
NUMBER OF SECTORS = 12
HARMONIC INDEX = 0
NO DUPLICATE SECTOR
TOTAL NUMBER OF NODES = 2091
TOTAL NUMBER OF ELEMENTS = 340
BOUNDING BOX OF STAGE NODES
RADIAL DIRECTION MIN,MAX = 0.1800E+00 0.2000E+00
TANGENTIAL DIRECTION MIN,MAX = 0.00 30.00 (deg)
AXIAL DIRECTION MIN,MAX = 0.5000E+00 0.1000E+01
CYCLIC CONSTRAINT EQUATIONS
TOTAL NUMBER (ACTIVE) = 423
NUMBER OF FIRST EQUATION = 424
NUMBER OF LAST EQUATION = 846
STAGE IS CONNECTED WITH STAGE = STAGE1
STAGE IS THE FIRST HARMONIC OF STAGE = STAGE2B
-------------------- STAGE = STAGE2B --------------------
NUMBER OF SECTORS = 12
HARMONIC INDEX = 6
NO DUPLICATE SECTOR
TOTAL NUMBER OF NODES = 2091
TOTAL NUMBER OF ELEMENTS = 340
BOUNDING BOX OF STAGE NODES
RADIAL DIRECTION MIN,MAX = 0.1800E+00 0.2000E+00
TANGENTIAL DIRECTION MIN,MAX = 0.00 30.00 (deg)
AXIAL DIRECTION MIN,MAX = 0.5000E+00 0.1000E+01
CYCLIC CONSTRAINT EQUATIONS
TOTAL NUMBER (ACTIVE) = 423
NUMBER OF FIRST EQUATION = 847
NUMBER OF LAST EQUATION = 1269
STAGE IS CONNECTED WITH STAGE = STAGE1
STAGE IS A HARMONIC OF STAGE = STAGE2A
---------------------------------------------------------------------------
The input generates modal frequencies for the multistage analysis for the primary harmonic index 0 listed in Table 7.1: Multistage Modal Frequencies. These frequencies can be compared to the full 360° frequencies shown in Table 7.2: Full Model Modal Frequencies. Since the full 360° analysis generates results for the entire harmonic index spectrum, the values to compare are highlighted. The values from the multistage analysis and the full analysis match well.
Table 7.1: Multistage Modal Frequencies
| SET | TIME/FREQ | LOAD STEP | SUBSTEP | CUMULATIVE |
|---|---|---|---|---|
| 1 | 1570.2 | 1 | 1 | 1 |
| 2 | 2508.9 | 1 | 2 | 2 |
| 3 | 3140.4 | 1 | 3 | 3 |
Table 7.2: Full Model Modal Frequencies
| SET | TIME/FREQ | LOAD STEP | SUBSTEP | CUMULATIVE |
|---|---|---|---|---|
| 1 | 672.35 | 1 | 1 | 1 |
| 2 | 672.35 | 1 | 2 | 2 |
| 3 | 1020.2 | 1 | 3 | 3 |
| 4 | 1020.2 | 1 | 4 | 4 |
| 5 | 1100.2 | 1 | 5 | 5 |
| 6 | 1100.2 | 1 | 6 | 6 |
| 7 | 1275.5 | 1 | 7 | 7 |
| 8 | 1275.5 | 1 | 8 | 8 |
| 9 | 1394.1 | 1 | 9 | 9 |
| 10 | 1394.1 | 1 | 10 | 10 |
| 11 |
1570.2 | 1 | 11 | 11 |
| 12 | 1868.5 | 1 | 12 | 12 |
| 13 | 1868.5 | 1 | 13 | 13 |
| 14 | 1969.0 | 1 | 14 | 14 |
| 15 | 1969.0 | 1 | 15 | 15 |
| 16 | 1980.5 | 1 | 16 | 16 |
| 17 | 1980.5 | 1 | 17 | 17 |
| 18 | 1984.3 | 1 | 18 | 18 |
| 19 | 1984.3 | 1 | 19 | 19 |
| 20 | 2150.5 | 1 | 20 | 20 |
| 21 | 2150.5 | 1 | 21 | 21 |
| 22 | 2445.8 | 1 | 22 | 22 |
| 23 | 2445.8 | 1 | 23 | 23 |
| 24 | 2471.2 | 1 | 24 | 24 |
| 25 | 2471.2 | 1 | 25 | 25 |
| 26 |
2508.9 | 1 | 26 | 26 |
| 27 | 2662.6 | 1 | 27 | 27 |
| 28 | 2662.6 | 1 | 28 | 28 |
| 29 | 2905.1 | 1 | 29 | 29 |
| 30 | 2905.1 | 1 | 30 | 30 |
| 31 | 2918.5 | 1 | 31 | 31 |
| 32 | 2918.5 | 1 | 32 | 32 |
| 33 | 3071.7 | 1 | 33 | 33 |
| 34 | 3071.7 | 1 | 34 | 34 |
| 35 | 3093.2 | 1 | 35 | 35 |
| 36 | 3093.2 | 1 | 36 | 36 |
| 37 |
3140.4 | 1 | 37 | 37 |
The contour plots for the full 360° results are presented in Figure 7.22: Fundamental Harmonic Index 0 Response at 1570.2 Hz, Figure 7.23: Fundamental Harmonic Index 0 Response at 2508.9 Hz, and Figure 7.24: Fundamental Harmonic Index 0 Response at 3140.4 Hz. The plots show that the response is truly a harmonic index 0 response, that is, all sectors have the same response. It is noted that in the multistage response, the inclusion of harmonic index 6 for stage 2 is necessary to get the correct response of the system.
