8.2. Example 2: Two Cantilever Beams with Frictionless Gap Contact

This example shows the nonlinear harmonic analysis of two cantilever beams with a gap contact between them using the harmonic balance method (HBM). Specifically, it demonstrates the following key points:

The example problem is presented in the following sections:

8.2.1. Problem Description and Modeling

The example is modeled in two ways:

  • Full model.

  • Model with CMS reduction.

8.2.1.1. Full Model

Two facing solid cantilever beams of length 0.16 m are meshed with SOLID185 elements. They are separated by a gap of size 1e-5 m and frictionless contact occurs on the last quarter of their length. The gap contact is defined with node-to-node CONTA178 elements.

Figure 8.4: Full Model

Full Model

An equivalent surface pressure is applied on the fourth quarter of the bottom beam to induce bending such that contact occurs.

The first bending frequency of the beams alone is f1 = 110.67 Hz. The HBM solution is computed in frequency range [0.80f1, 1.1f1] with NH = 9 harmonics to capture the contact response accurately.

8.2.1.2. Model with CMS Reduction

The size of the model is reduced by generating one superelement for each beam. The master nodes are composed of the following:

  • boundary condition nodes

  • loading nodes

  • nodes used for nonlinear contact

  • one “observation” node on the top beam.

Figure 8.5: Model with CMS Reduction

Model with CMS Reduction

Like any CMS model, the number of modes used to generate the superelements must be sufficient to adequately represent the dynamics in the frequency range of interest. Here, 6 modes are used.

The number of nonlinear equations is the same for both the full and CMS models: (30 contact nodes)*(3 DOFs/node) = 90 transient equations.

However, CMS reduction of the linear part of the model further saves computation time by reducing the number of linear equations. The number of linear equations are as follows:

  • Full model: (192 nodes)*(3 DOFs/node)*(2*9+1 harmonics) = 10944 harmonic equations

  • CMS model: ((46 nodes)*(3 DOFs/node)+(6 modes/substructure)*2 substructures) *(2*9+1 harmonics) = 2850 harmonic equations

The number of equations information is printed in the output when the SOLVE command is issued.

8.2.1.3. Model with Prestressed Superelement

This model is identical to that in Model with CMS Reduction except that a prestress load is applied on the tip of both beams. Linear perturbation substructuring analysis procedure is used to generate the prestressed superelements which are used in HBM analysis.

8.2.2. Input for the Analysis

To download the .dat files used for this example problem, click the links below.

hbm_example2a.dat - Full model
hbm_example2b.dat - CMS reduced model
hbm_example2c.dat - prestressed CMS reduced model

The contents of these files are listed below. The command listing shows in detail how to specify and run the HBM analysis. You can use them as templates and modify them to create a custom HBM analysis.

Input for the full model

/BATCH,LIST
/TITLE, Two cantilever beams with frictionless gap contact
/com =============================================================
/com                    DESCRIPTION:
/com  Harmonic Balance analysis of two facing solid cantilever 
/com  beams separated by a gap and interacting with frictionless 
/com  contact.
/com  - Case a:            full model
/com =============================================================

! Model Parameters
PI 	= ACOS(-1)
E   = 1E10                            ! Young's modulus
RHO = 2700                            ! Density
L   = 0.01                            ! Unit length
OMG = 110.67*2*PI                     ! single beam 1st eigenfrequency

KN   = -1E5                           ! contact normal stiffness
GAP  =  1E-5                          ! gap size
FEXT = 	40                            ! external force

! HBM and Continuation Parameters
NH     = 9                            ! number of harmonics


DS    = 2.0                           ! initial arc length (step length)
DSMIN = DS/100                        ! minimum arc length (step length)
DSMAX = 2*DS                          ! maximum arc length (step length)

FMIN = 0.80*OMG/(2*PI)                ! Starting frequency 
FMAX = 1.10*OMG/(2*PI)                ! Ending   frequency 

/PREP7

! element type
ET,1,185

ET,2,178
KEYOPT,2,1 ,0                         ! unidirectional
KEYOPT,2,2 ,1                         ! penalty-based method
KEYOPT,2,4 ,0                         ! gap size based on real constant GAP + node location
KEYOPT,2,10,0                         ! standard contact
R,2,KN

! material
MP,EX  ,1,E
MP,PRXY,1,0.3
MP,DENS,1,RHO
MP,BETD,1,0.00005

! model
BLOCK,0,L,0,16*L,0,2*L                ! bottom beam
BLOCK,L+gap,2*L+gap,12*L,28*L,0,2*L   ! top    beam

ESIZE,L
TYPE,1
REAL,1
MAT ,1
VMESH,ALL

! gap contact
VSEL,S,VOLU,,1,,,1
NSEL,R,LOC,X,L
NSEL,R,LOC,Y,12*L,16*L
CM,BEAM1_GAP,NODE
ALLSEL

VSEL,S,VOLU,,2,,,1
NSEL,R,LOC,X,L+GAP
NSEL,R,LOC,Y,12*L,16*L
CM,BEAM2_GAP,NODE
ALLSEL

CMSEL,S,BEAM1_GAP
CMSEL,A,BEAM2_GAP
TYPE,2
REAL,2
EINTF,1E-6,,,,GAP
ALLSEL

finish
  
/SOLU
ANTYPE,HARMIC

HROPT,HBM,NH                          ! HBM Solve and number of harmonics
HARFRQ,FMIN,FMAX

HBMOPT,NR,10
HBMOPT,CONTSET,,DS,DSMIN,DSMAX
HBMOPT,LIST

! boundary conditions
NSEL,S,LOC,Y,0
NSEL,A,LOC,Y,28*L
D,ALL,ALL
ALLSEL

! loading - surface pressure - harmonic 1 loading
NSEL,S,LOC,X,0
NSEL,R,LOC,Y,12*L,16*L
SF,ALL,PRES,FEXT
ALLSEL

ND_BOT  = NODE(0,16*L,L)
ND_TOP  = NODE(2*L+GAP,12*L,L)

KBC,1

SOLVE
FINISH

/com
/com _________ DIRECT POST-PROCESSING OF HARMONIC SOLUTION _________
/com

/POST1
*GET,JOBN,ACTIVE,,JOBNAM
FILE,%JOBN%_1hi0,rst
SET,LIST

SET,1,39
/VIEW,,0.1,0.25,1
/EDGE,,1
/DSCALE,,1E2
/TITLE, Two cantilever beams with frictionless gap contact
/SHOW,PNG,REV
	PLNSOL,U,SUM
/SHOW,CLOSE

FINISH

/com
/com _________ POST-PROCESSING USING MACRO 'HBM_EXPA' _________
/com

*GET,JOBN,ACTIVE,,JOBNAM
HBM_EXPA,JOBN,ND_BOT,'U','X',NH,'minmax'
*DIM ,_HBM_AMPL_BEAM     ,TABLE,_NSS,2
*VFUN,_HBM_AMPL_BEAM(1,1),COPY ,_HBM_AMPL(1)  ! save results for node %ND_BOT% (bottom beam)

HBM_EXPA,JOBN,ND_TOP,'U','X',NH,'minmax'
*VFUN,_HBM_AMPL_BEAM(1,2),COPY,_HBM_AMPL(1)   ! save results for node %ND_TOP% (top beam)

/SHOW,PNG,REV
	/AXLAB,X,FREQUENCY (HZ)
	/AXLAB,Y,AMPLITUDE
	/GCOLUMN,1,BOT_BEAM
	/GCOLUMN,2,TOP_BEAM
	*VPLOT,_HBM_FREQ(1),_HBM_AMPL_BEAM(1,1),2
/SHOW,CLOSE

/com
/com  Frequency (Hz)       Amplitude of the response
/com                     Bot. Beam     |   Top Beam
/com
*VWRITE,_HBM_FREQ(1),_HBM_AMPL_BEAM(1,1),_HBM_AMPL_BEAM(1,2)
(F16.8,2X,E16.8,2X,E16.8)

Input for the CMS model

/BATCH,LIST
/TITLE, Two cantilever beams with frictionless gap contact
/com =============================================================
/com                    DESCRIPTION:
/com  Harmonic Balance analysis of two facing solid cantilever 
/com  beams separated by a gap and interacting with frictionless 
/com  contact.
/com  - Case b:             CMS model
/com =============================================================

! Model Parameters
PI 	= ACOS(-1)
E   = 1E10                          ! Young's modulus
RHO = 2700                          ! Density
L   = 0.01                          ! Unit length
OMG = 110.67*2*PI                   ! single beam 1st eigenfrequency

KN   = -1E5                         ! contact normal stiffness
GAP  =  1E-5                        ! gap size
FEXT =  40                          ! external force

! HBM and Continuation Parameters
NH     = 9                          ! number of harmonics
NMODE  = 6                          ! number of modes for CMS reduction

DS    = 2.0                         ! initial arc length (step length)
DSMIN = DS/100                      ! minimum arc length (step length)
DSMAX = 2*DS                        ! maximum arc length (step length)

FMIN = 0.80*OMG/(2*PI)              ! Starting frequency 
FMAX = 1.10*OMG/(2*PI)              ! Ending   frequency 

/PREP7

! element type
ET,1,185

! material
MP,EX  ,1,E
MP,PRXY,1,0.3
MP,DENS,1,RHO
MP,BETD,1,0.00005

! model
BLOCK,0,L,0,16*L,0,2*L              ! bottom beam
BLOCK,L+gap,2*L+gap,12*L,28*L,0,2*L ! top    beam

ESIZE,L
TYPE,1
REAL,1
MAT ,1
VMESH,ALL

FINISH

SAVE,ALL3D,DB
PARSAV,,ALL3D,PARM

/com
/com ______________ GENERATION PASS - BOTTOM BEAM _____________
/com
/FILNAME,BEAM1

/SOLU
ANTYPE,SUBSTR
SEOPT,BEAM1,3,,,NONE       ! generate stiffness, mass, and damping matrix 
                           !                 (NONE = no files kept for expansion)
CMSOPT,FIX,NMODE

! master dofs
VSEL,S,VOLU,,1,,,1         ! gap contact nodes - bottom beam
NSEL,R,LOC,X,L
NSEL,R,LOC,Y,12*L,16*L
CM,BEAM1_GAP,NODE
ALLSEL

NSEL,S,LOC,Y,0             ! boundary conditions - bottom beam
CM,BEAM1_BC,NODE
ALLSEL

NSEL,S,LOC,Y,12*L,16*L     ! loading - bottom beam
NSEL,R,LOC,X,0
CM,BEAM1_LOAD,NODE
ALLSEL

CMSEL,S,BEAM1_GAP
CMSEL,A,BEAM1_LOAD
CMSEL,A,BEAM1_BC
M,ALL,ALL
ALLSEL

VSEL,S,VOLU,,1,,,1         ! select only nodes and elements for bottom beam
CM,BEAM1_ELEM,ELEM

solve
finish

/com
/com ______________ GENERATION PASS - TOP BEAM _____________
/com
/FILNAME,BEAM2

/SOLU
ANTYPE,SUBSTR
SEOPT,BEAM2,3,,,NONE              ! generate stiffness, mass, and damping matrix 
                                  !                 (NONE = no files kept for expansion)
CMSOPT,FIX,NMODE

! master dofs
VSEL,S,VOLU,,2,,,1                ! gap contact nodes - top beam
NSEL,R,LOC,X,L+GAP
NSEL,R,LOC,Y,12*L,16*L
CM,BEAM2_GAP,NODE
ALLSEL

NSEL,S,LOC,Y,28*L                 ! boundary conditions - top beam
CM,BEAM2_BCLOAD,NODE
ALLSEL

TIP_NODE = NODE(2*L+GAP,12*L,L)   ! tip node

CMSEL,S,BEAM2_GAP
CMSEL,A,BEAM2_BCLOAD
NSEL,A,NODE,,TIP_NODE	
M,ALL,ALL
ALLSEL

VSEL,S,VOLU,,2,,,1                ! select only nodes and elements for top beam
CM,BEAM2_ELEM,ELEM
SOLVE
FINISH

/com
/com ______________ USE PASS _____________
/com
/CLEAR,NOSTART
/FILNAME,USE

PARRES,,ALL3D,PARM

/PREP7

! load SE
ET,10,50
TYPE,10
SE,BEAM1
SE,BEAM2

! create NL elements
ET,2,178
KEYOPT,2,1 ,0                       ! unidirectional
KEYOPT,2,2 ,1                       ! penalty-based method
KEYOPT,2,4 ,0                       ! gap size based on real constant GAP + node location
KEYOPT,2,10,0                       ! standard contact
R,2,KN

! gap contact
SELTOL,1E-6
NSEL,S,LOC,X,L
NSEL,R,LOC,Y,12*L,16*L
CM,BEAM1_GAP,NODE
ALLSEL

NSEL,S,LOC,X,L+GAP
NSEL,R,LOC,Y,12*L,16*L
CM,BEAM2_GAP,NODE
ALLSEL

CMSEL,S,BEAM1_GAP
CMSEL,A,BEAM2_GAP
TYPE,2
REAL,2
EINTF,1E-6,,,,GAP
ALLSEL

FINISH
  
/SOLU
ANTYPE,HARMIC

HROPT,HBM,NH                        ! HBM Solve and number of harmonics
HARFRQ,FMIN,FMAX

HBMOPT,NR,10
HBMOPT,CONTSET,,DS,DSMIN,DSMAX
HBMOPT,LIST

! boundary conditions
NSEL,S,LOC,Y,0
NSEL,A,LOC,Y,28*L
D,ALL,ALL
ALLSEL

! loading - surface pressure mimicked with nodal load
!             FEXT is applied to nodes belonging to 1           loaded element
!           2*FEXT is applied to nodes belonging to 2 different loaded elements
!           4*FEXT is applied to nodes belonging to 4 different loaded elements

NSEL,S,LOC,X,0
NSEL,R,LOC,Y,12*L,16*L
NSEL,R,LOC,Z,L
NSEL,U,LOC,Y,12*L
NSEL,U,LOC,Y,16*L
CM,CENTER_NODE,NODE
F,ALL,FX,4*FEXT*(8*L**2)/32
ALLSEL

NSEL,S,LOC,Y,12*L
NSEL,A,LOC,Y,16*L
NSEL,R,LOC,X,0
NSEL,U,LOC,Z,L
CM,CORNER_NODE,NODE
F,ALL,FX,FEXT*(8*L**2)/32
ALLSEL

NSEL,S,LOC,X,0
NSEL,R,LOC,Y,12*L,16*L
CMSEL,U,CENTER_NODE
CMSEL,U,CORNER_NODE
CM,SIDE_NODE,NODE
F,ALL,FX,2*FEXT*(8*L**2)/32
ALLSEL

ND_BOT  = NODE(0,16*L,L)
ND_TOP  = NODE(2*L+GAP,12*L,L)

KBC,1

SOLVE
FINISH

/com
/com _________ POST-PROCESSING USING MACRO 'HBM_EXPA' _________
/com

/POST26
FILE,USE0,rst

JOBN='USE'
HBM_EXPA,JOBN,ND_BOT,'U','X',NH,'minmax'
*DIM ,_HBM_AMPL_BEAM     ,TABLE,_NSS,2
*VFUN,_HBM_AMPL_BEAM(1,1),COPY ,_HBM_AMPL(1) ! save results for node %ND_BOT% (bottom beam)

HBM_EXPA,JOBN,ND_TOP,'U','X',NH,'minmax'
*VFUN,_HBM_AMPL_BEAM(1,2),COPY,_HBM_AMPL(1)  ! save results for node %ND_TOP% (top beam)

/SHOW,PNG,REV
	/AXLAB,X,FREQUENCY (HZ)
	/AXLAB,Y,AMPLITUDE
	/GCOLUMN,1,BOT_BEAM
	/GCOLUMN,2,TOP_BEAM
	*VPLOT,_HBM_FREQ(1),_HBM_AMPL_BEAM(1,1),2
/SHOW,CLOSE

/com
/com  Frequency (Hz)       Amplitude of the response
/com                     Bot. Beam     |   Top Beam
/com
*VWRITE,_HBM_FREQ(1),_HBM_AMPL_BEAM(1,1),_HBM_AMPL_BEAM(1,2)
(F16.8,2X,E16.8,2X,E16.8)

FINISH

Input for the Prestress CMS Model

/BATCH,LIST
/TITLE, Two cantilever beams with frictionless gap contact
/com =============================================================
/com                    DESCRIPTION:
/com  Harmonic Balance analysis of two facing solid cantilever 
/com  beams separated by a gap and interacting with frictionless 
/com  contact.
/com  - Case c: prestressed CMS model
/com =============================================================

! Model Parameters
PI 	= ACOS(-1)
E   = 1E10                            ! Young's modulus
RHO = 2700                            ! Density
L   = 0.01                            ! Unit length
OMG = 110.67*2*PI                     ! single beam 1st eigenfrequency

KN   = -1E5                           ! contact normal stiffness
GAP  =  1E-5                          ! gap size
FEXT =  40                            ! external force

! HBM and Continuation Parameters
NH     = 9                            ! number of harmonics
NMODE  = 6                            ! number of modes for CMS reduction

DS    = 2.0                           ! initial arc length
DSMIN = DS/100                        ! minimum arc length
DSMAX = 2*DS                          ! maximum arc length

FMIN = 0.80*OMG/(2*PI)                ! Starting frequency 
FMAX = 1.10*OMG/(2*PI)                ! Ending   frequency 

/PREP7

! element type
ET,1,185

! material
MP,EX  ,1,E
MP,PRXY,1,0.3
MP,DENS,1,RHO
MP,BETD,1,0.00005

! model
BLOCK,0,L,0,16*L,0,2*L                ! bottom beam
BLOCK,L+gap,2*L+gap,12*L,28*L,0,2*L   ! top    beam

ESIZE,L
TYPE,1
REAL,1
MAT ,1
VMESH,ALL

FINISH

/com
/com ______________ PRESTRESS PASS _____________
/com

/SOLU
ANTYPE,STATIC

RESCONTROL,LINEAR,ALL,1

! boundary conditions
NSEL,S,LOC,Y,0
NSEL,A,LOC,Y,28*L
D,ALL,ALL
ALLSEL

! prestress load
NSEL,S,LOC,Y,16*L
NSEL,R,LOC,X,0,L
SF,ALL,PRES,2E6
ALLSEL

NSEL,S,LOC,Y,12*L
NSEL,R,LOC,X,L+GAP,2*L+GAP
SF,ALL,PRES,2E6
ALLSEL

SOLVE
FINISH

SAVE,ALL3D,DB
PARSAV,,ALL3D,PARM

/com
/com ______________ GENERATION PASS - BOTTOM BEAM _____________
/com
! /FILNAME,BEAM1

/SOLU
ANTYPE,STATIC,RESTART,,,PERTURB
PERTURB,SUBSTR,,,DZEROKEEP
SOLVE,ELFORM

SEOPT,BEAM1,3,,,NONE        ! generate stiffness, mass and damping matrix 
                            !                 (NONE = no files kept for expansion)
CMSOPT,FIX,NMODE

! master dofs
VSEL,S,VOLU,,1,,,1          ! gap contact nodes - bottom beam
NSEL,R,LOC,X,L
NSEL,R,LOC,Y,12*L,16*L
CM,BEAM1_GAP,NODE
ALLSEL

NSEL,S,LOC,Y,0              ! boundary conditions - bottom beam
CM,BEAM1_BC,NODE
ALLSEL

NSEL,S,LOC,Y,12*L,16*L      ! loading - bottom beam
NSEL,R,LOC,X,0
CM,BEAM1_LOAD,NODE
ALLSEL

CMSEL,S,BEAM1_GAP
CMSEL,A,BEAM1_LOAD
CMSEL,A,BEAM1_BC
M,ALL,ALL
ALLSEL

VSEL,S,VOLU,,1,,,1           ! select only nodes and elements for bottom beam
CM,BEAM1_ELEM,ELEM

SOLVE
FINISH

/com
/com ______________ GENERATION PASS - TOP BEAM _____________
/com
/CLEAR,NOSTART
! /FILNAME,BEAM2
RESUME,ALL3D,DB

/SOLU
ANTYPE,STATIC,RESTART,,,PERTURB
PERTURB,SUBSTR,,,DZEROKEEP
SOLVE,ELFORM

SEOPT,BEAM2,3,,,NONE              ! generate stiffness, mass and damping matrix 
                                  !                 (NONE = no files kept for expansion)
CMSOPT,FIX,NMODE

! master dofs
VSEL,S,VOLU,,2,,,1                ! gap contact nodes - top beam
NSEL,R,LOC,X,L+GAP
NSEL,R,LOC,Y,12*L,16*L
CM,BEAM2_GAP,NODE
ALLSEL

NSEL,S,LOC,Y,28*L                 ! boundary conditions - top beam
CM,BEAM2_BCLOAD,NODE
ALLSEL

TIP_NODE = NODE(2*L+GAP,12*L,L)   ! tip node

CMSEL,S,BEAM2_GAP
CMSEL,A,BEAM2_BCLOAD
NSEL,A,NODE,,TIP_NODE	
M,ALL,ALL
ALLSEL

VSEL,S,VOLU,,2,,,1                ! select only nodes and elements for top beam
CM,BEAM2_ELEM,ELEM
SOLVE
FINISH

/com
/com ______________ USE PASS _____________
/com
/CLEAR,NOSTART
/FILNAME,USE

PARRES,,ALL3D,PARM

/PREP7

! load SE
ET,10,50
TYPE,10
SE,BEAM1
SE,BEAM2

! create NL elements
ET,2,178
KEYOPT,2,1 ,0                   ! unidirectional
KEYOPT,2,2 ,1                   ! penalty-based method
KEYOPT,2,4 ,0                   ! gap size based on real constant GAP + node location
KEYOPT,2,10,0                   ! standard contact
R,2,KN

! gap contact
SELTOL,1E-6
NSEL,S,LOC,X,L
NSEL,R,LOC,Y,12*L,16*L
CM,BEAM1_GAP,NODE
ALLSEL

NSEL,S,LOC,X,L+GAP
NSEL,R,LOC,Y,12*L,16*L
CM,BEAM2_GAP,NODE
ALLSEL

CMSEL,S,BEAM1_GAP
CMSEL,A,BEAM2_GAP
TYPE,2
REAL,2
EINTF,1E-6,,,,GAP
ALLSEL

FINISH
  
/SOLU
ANTYPE,HARMIC

HROPT,HBM,NH                     ! HBM Solve and number of harmonics
HARFRQ,FMIN,FMAX

HBMOPT,NR,10
HBMOPT,CONTSET,,DS,DSMIN,DSMAX
HBMOPT,LIST

! boundary conditions
NSEL,S,LOC,Y,0
NSEL,A,LOC,Y,28*L
D,ALL,ALL
ALLSEL

! loading - surface pressure mimicked with nodal load
!             FEXT is applied to nodes belonging to 1           loaded element
!           2*FEXT is applied to nodes belonging to 2 different loaded elements
!           4*FEXT is applied to nodes belonging to 4 different loaded elements

NSEL,S,LOC,X,0
NSEL,R,LOC,Y,12*L,16*L
NSEL,R,LOC,Z,L
NSEL,U,LOC,Y,12*L
NSEL,U,LOC,Y,16*L
CM,CENTER_NODE,NODE
F,ALL,FX,4*FEXT*(8*L**2)/32
ALLSEL

NSEL,S,LOC,Y,12*L
NSEL,A,LOC,Y,16*L
NSEL,R,LOC,X,0
NSEL,U,LOC,Z,L
CM,CORNER_NODE,NODE
F,ALL,FX,FEXT*(8*L**2)/32
ALLSEL

NSEL,S,LOC,X,0
NSEL,R,LOC,Y,12*L,16*L
CMSEL,U,CENTER_NODE
CMSEL,U,CORNER_NODE
CM,SIDE_NODE,NODE
F,ALL,FX,2*FEXT*(8*L**2)/32
ALLSEL

ND_BOT  = NODE(0,16*L,L)
ND_TOP  = NODE(2*L+GAP,12*L,L)

KBC,1

SOLVE
FINISH

/com
/com _________ DIRECT POST-PROCESSING _________
/com

/POST26
NUMVAR,200
FILE,USE_1hi0,rst                ! select result file containing harmonic 1 results

NSOL,102,ND_BOT,u,x              ! store and print frequency history results for harmonic 1
ABS ,102,102

NSOL,103,ND_TOP,u,x              ! store and print frequency history results for harmonic 1
ABS ,103,103

/SHOW,png,rev
	/AXLAB,X,Frequency (Hz)
	/AXLAB,Y,Harmonic 1 - amplitude
	PLVAR,102,103
/SHOW,close

PRVAR,102,103
FINISH

/com
/com _________ POST-PROCESSING USING MACRO 'HBM_EXPA' _________
/com

/POST26
FILE,USE0,rst

JOBN='USE'
HBM_EXPA,JOBN,ND_BOT,'U','X',NH,'minmax'
*DIM ,_HBM_AMPL_BEAM     ,TABLE,_NSS,2
*VFUN,_HBM_AMPL_BEAM(1,1),COPY ,_HBM_AMPL(1) ! save results for node %ND_BOT% (bottom beam)

HBM_EXPA,JOBN,ND_TOP,'U','X',NH,'minmax'
*VFUN,_HBM_AMPL_BEAM(1,2),COPY,_HBM_AMPL(1)  ! save results for node %ND_TOP% (top beam)

/SHOW,PNG,REV
	/AXLAB,X,FREQUENCY (HZ)
	/AXLAB,Y,AMPLITUDE
	/GCOLUMN,1,BOT_BEAM
	/GCOLUMN,2,TOP_BEAM
	*VPLOT,_HBM_FREQ(1),_HBM_AMPL_BEAM(1,1),2
/SHOW,CLOSE

/com
/com  Frequency (Hz)       Amplitude of the response
/com                     Bot. Beam     |   Top Beam
/com
*VWRITE,_HBM_FREQ(1),_HBM_AMPL_BEAM(1,1),_HBM_AMPL_BEAM(1,2)
(F16.8,2X,E16.8,2X,E16.8)
FINISH

8.2.3. Results

Output contains information on the convergence, the arc length, and the maximum amplitude of each harmonic at each substep. The example output message below shows that the solver needed two iterations to converge.

RESIDUAL INFINITE NORM VALUE =     0.003732 
     ITERATION 1 COMPLETED 
   RESIDUAL INFINITE NORM VALUE =    8.863e-08 
     ITERATION 2 COMPLETED 
    >>> SOLUTION CONVERGED
                        CURRENT STEP SIZE   =              10 

      ** MAXIMUM AMPLITUDE OF DISPLACEMENT SOLUTIONS **
            HARMONIC     REAL          IMAG
                0      0.7646E-06
                1      0.1620E-04    0.1836E-04
                2      0.4504E-06    0.9648E-07
                3      0.4688E-07    0.1278E-06
                4      0.6442E-07    0.3625E-07
                5      0.4176E-07    0.3909E-07
                6      0.5943E-07    0.1138E-06
                7      0.1696E-07    0.1060E-06
                8      0.4490E-07    0.3106E-07
                9      0.2315E-07    0.2209E-07
 *** LOAD STEP     1   SUBSTEP    37  COMPLETED.  FREQUENCY=   108.811  

You can create displacement plots for a given harmonic at a selected frequency using direct postprocessing in /POST1 as shown below.

Figure 8.6: Displacement

Displacement


You can calculate the response amplitude using the HBM_EXPA macro, which performs postprocessing in a similar manner to example 1.

The following figure shows the linear harmonic results for the bottom beam.

Figure 8.7: Bottom Beam - Linear Harmonic Results

Bottom Beam - Linear Harmonic Results


Effects of the contact on the bottom beam can be seen in the following figure which shows a reduction in amplitude as well as an asymmetry of the peak. Contact between the two beams occurs around 96 Hz when the top beam starts to respond to harmonic excitation.

Figure 8.8: Effects of Contact on the Bottom Beam - Full model (A), CMS model (B)

Effects of Contact on the Bottom Beam - Full model (A), CMS model (B)

Compared to the full model, the CMS model gives similar results, but it reduces the overall computation time by a factor of about 6.

The prestressed CMS model also gives quite similar results. The difference lies in the localization of the peak which is shifted toward the high frequencies and the amplitude of the peak which is smaller (stiffening behavior).

Figure 8.9: Effects of Prestress Load on Response Peak

Effects of Prestress Load on Response Peak