Example 1: Cantilever Beam
A1.1
Practical exercises with the FEM program ANSYS (to [35])
Example 1: Cantilever Beam
For a steel Cantilever Beam, built in at one end only, calculate the first three natural bending
frequencies f1 to f3, and determine the corresponding natural oscillation modes. For comparison,
calculate solutions for different numbers of elements and element types (beam, shell and solid
elements).
Fig. 1.1: Cantilever beam
l
b
h
E

600 mm
10 mm
20 mm
210 000 N/mm2
7850 kg/m3
1.1 Theoretical Calculation (see Lecture “Dynamics of Machinery” [2]):
The natural angular frequency is calculated from:
 K  AK 
E I
 L
4
 AK 
E I
( A   )  L4
with:
A1 = 3,52 ; A2 = 22,4 ; A3 = 61,7
Set as inputs, this results in:
 1  291,980 1 S
f1 = 46 ,470 Hz
 2  1858,06 1 S
f 2 = 295,40 Hz
 3  5117,96 1 S
f3 = 814,55 Hz
A1.2
Example 1: Cantilever Beam
1.2 Modal analysis script: cantilever beam with finite beam elements
Intro
finish
finish actual processing mode
/clear,nostart
clear the database
Pre-Processing
/filname,p1modalbeam,0
change the job name for the analysis
/prep7
enter pre-processing
/title,onesided clamped beam (BEAM)
title for actual problem
et,1,beam3
define finite element type 1
*ask,parHeight,'beam height [m] ?',0.02
user input: beam height
*ask,parWidth,'beam width [m] ?',0.01
user input: beam width
parArea = parHeight*parWidth
calculate cross section area
parADev =parWidth*parHeight*parHeight*parHeight/12
calculate moment of inertia
r,1,parArea,parADev,parHeight
define property set 1
*ask,parEModulus,'elastic modulus [N/m^2] ?',2.1e11
user input: elastic modulus
mp,ex,1,parEModulus
define elastic modulus for material set 1
*ask,parDensity,'density [kg/m^3] ?',7850
user input: density
mp,dens,1,parDensity
define density for material set 1
*ask,parLat,'lateral contraction [-] ?',0.3
user input: lateral contraction
mp,nuxy,1,parLat
define lateral contraction for material set 1
k,1,0,0,0,
define keypoint 1 with coordinates (0,0,0)
k,2,0.6,0,0,
define keypoint 2 with coordinates (0.6,0,0)
lstr,1,2
lmesh,1
define line 1 with keypoints 1 and 2
associate element attributes with the selected,
unmeshed lines
user input: number of finite elements
specify the divisions and spacing ratio on unmeshed
lines
generate nodes and line elements along lines
d,1,all,0
define DOF constraints at nodes
/view,1,1,1,1
define the viewing direction for the display
/pbc,all,,1
display option for all boundary symbols
eplot
produce an element display
finish
leave actual processor
save,p1modalbeam,db,,all
save database in p1modalbeam.db
latt,1,1,1,
*ask,numElements,'number of finite elements ?',1
lesize,1,,,numElements,,,,,1
Solution-Processing
/solu
enter the solution-processor
antype,modal
specify the analysis type and restart status
*ask,numModes,'number eigen -forms/-frequencies ?',3
modopt,subsp,numModes
user input: number of frequencies to calculate
specify the number of modes to expand/write for a
modal analysis
specify modal analysis options
solve
start a solution
finish
leave solution-processor
mxpand,anzModes
Example 1: Cantilever Beam
A1.3
Post-Processing
/post1
enter post1-processor
flag=1
set flag parameter to 1
*dowhile,flag
testing flag parameter: if 0=leave do-loop
*ask,varMode,'show eigenform [0=exit]',1
user input: eigenform to display
*if,varMode,eq,0,then
testing varMode parameter on equals 0
set,,,,,,,varMode
set flag parameter to 0
testing varMode parameter on lower equal
numModes
select calculated eigenform number to display
pldisp,1
display eigenform number
flag=0
*elseif,varMode,le,numModes,then
*endif
close * if query
*enddo
close *do-loop
finish
leave post1-processor
A1.4
Example 1: Cantilever Beam
1.3 Modal analysis Results: cantilever beam with finite beam elements
Fig. 1.2: 1st eigenmode 46.408 Hz
Fig. 1.3: 2nd eigenmode 290.468 Hz
Fig. 1.4: 3rd eigenmode 811.818 Hz
Example 1: Cantilever Beam
A1.5
1.4 Modal analysis script: cantilever beam with finite shell elements
Intro
See intro chap. 1.2
Pre-Processing
/filname,p1modalshell,0
change the jobname for the analysis
/prep7
enter pre-processing
/title,onesided clamped beam (SHELL)
title for actual problem
et,1,shell63
define finite element type 1
*ask,parThick,'shell thickness [m] ?',0.01
user input: enter shell thickness
r,1,parThick
define property set 1
*ask,parEModulus,'modulus elasticity [N/m^2] ?',2.1e11
user input: enter modulus of elasticity
mp,ex,1,parEModulus
define material elasticity for set 1
*ask,parDensity,'density [kg/m^3] ?',7850.0
user input: enter material density
mp,dens,1,parDensity
define material density for set 1
*ask,parLat,'lateral contraction [-] ?',0.3
user input: enter lateral contraction
mp,nuxy,1,parLat
define material lateral contraction for set 1
*ask,parHeight,'beam height [m] ?',0.02
user input: beam height
*ask,parLength,'beam length [m] ?',0.6
aatt,1,1,1,0,
user input: beam length
defines rectangular area with parLenght and
parHeight
connect finite element attributes with area 1
*ask,parElemLength,'mean element length [m] ?',0.005
user input: mean element size
esize,parElemLength,0,
define previous entered mean element size
mshkey,1
switching to mapped meshing
amesh,1
create elements on area
mshkey,0
switching to free meshing
dl,4,,all,
define boundary conditions on line 4
da,1,uz,
define boundary conditions on area 1
da,1,rotx,
define boundary conditions on area 1
da,1,roty,
define boundary conditions on area 1
/view,1,1,1,1
define the viewing direction for the display
/pbc,all,,1
display option for all boundary symbols
eplot
produce an element display
finish
leave pre processor
save,p1modalshell,db,,all
save database in p1modalshell.db
blc4,0,0,parLength,parHeight
Solution-Processing
see Solution-Processing chap. 1.2
Post-Processing
see Post-Processing chap. 1.2
A1.6
Example 1: Cantilever Beam
1.5 Modal analysis results: cantilever beam with finite shell elements
Fig. 1.5: 1st eigenmode 46.405 Hz
Fig. 1.6: 2nd eigenmode 289.42 Hz
Fig. 1.7: 3rd eigenmode 804.238 Hz
Example 1: Cantilever Beam
A1.7
1.6 Modal analysis script: cantilever beam with finite solid elements
Intro
see intro chap. 1.2
Pre-Processing
/filname, p1modalsolid,0
change the jobname for the analysis
/prep7
enter preprocessing
/title,onesided clamped beam (SOLID)
title for actual problem
et,1,solid45
define finite element type 1
*ask,parEModulus,'modulus elasticity [N/m^2] ?',2.1e11
user input: enter modulus of elasticity
mp,ex,1,parEModulus
define material elasticity for set 1
*ask,parDensity,'density [kg/m^3] ?',7850.0
user input: enter material density
mp,dens,1,parDensity
define material density for set 1
*ask,parLat,'lateral contraction [-] ?',0.3
user input: enter lateral contraction
mp,nuxy,1,parLat
define material lateral contraction for set 1
*ask,parHeight,'beam height [m] ?',0.02
user input: beam height
*ask,parLength,'beam lenght [m] ?',0.6
user input: beam length
*ask,parWidth,'beam width [m] ?',0.01
user input: beam width
block,0,parLength,0,parHeight,0,parWidth,
create previously defined hexaedric volume
vatt,1,,1,0
connect finite element attributes with volume 1
*ask,parElemLength,'mean element length [m] ?',0.01
user input: mean element size
esize,parElemLength,0,
define previously entered mean element size
mshkey,1
switching to mapped meshing
vmesh,1
create elements on volume
mshkey,0
switching to free meshing
d,all,,,,,,uz,,,,,
boundary conditions for all nodes in z-direction
da,5,all,
boundary conditions on area 5
/view,1,1,1,1
define the viewing direction for the display
/pbc,all,,1
display option for all boundary symbols
eplot
produce an element display
finish
leave pre processor
save,p1modalsolid,db,,all
save database in p1modalsolid.db
Solution-Processing
see Solution-Processing chap. 1.2
Post-Processing
see Post-Processing chap. 1.2
A1.8
Example 1: Cantilever Beam
1.7 Modal analysis results: cantilever beam with finite solid elements
Fig. 1.8: 1st eigenmode 48.667 Hz
Fig. 1.9: 2nd eigenmode 303.424 Hz
Fig. 1.10: 3rd eigenmode 842.743 Hz
Example 2: Triangular Leaf Spring
A2.1
Example 2: Triangular Leaf Spring
In this example the first three natural oscillation modes of a Triangular Leaf Spring are to be investigated. The
natural frequency of the first eigenmode (natural mode) can be compared with the approximation calculation using
Rayleigh.
Fig. 2.1: Geometric layout of the triangular leaf spring
L
[mm]
B
[mm]
t
[mm]
E
[N/mm2]

[kg/m3]
400
140 mm
4 mm
210 000 N/mm2
7850 kg/m3
2.1 Theoretical Calculation:
The angular natural frequency is calculated according to Rayleigh from:
L
R ² 
 E ( x)  I ( x)  y
0
  ²( x )  dx
0
L
  ( x)  A( x)  y
0
²( x )  dx
0
Where: E(x) = E = constant,
(x) =  = constant
x
x


A( x )  b  t   1    A0   1  


L
L
I ( x) 
b  t³ 
x
x

 1    I0  1  

12 
L
L
Approximate description of the 1st bending mode is:
²
 x  
 x  
y0 ( x)  1  cos
cos
  y0 ( x) 

4  L²
 L2
 L2
A2.2
Example 2: Triangular Leaf Spring
Fitting these into the above equation:
x   ²

 x  
0 E  I 0   1  L   4  L ² cos L  2    dx
 L
2
x 

 x  
0   A0   1  L  1  cos L  2    dx
2
L
 R2
Inserting the limits 0 and L in the above integrals:
 R2 
E  I0   4
E  I0   2

64  L3
16  L3
3    A0  L 7    A0  L

4
2
After simplification the equation has the form:
R
2
E  I 0   4  ( 2  4)

16    A0  L4  (3 2  28)
After inserting the values, the fundamental natural angular frequency, according to Rayleigh, results in:
 R = 270,42 1/s  f R1 = 43,04 Hz > f exact
Results from ANSYS: f1= 42,958 HZ
Example 2: Triangular Leaf Spring
A2.3
2.2 Modal analysis script: Triangular leaf spring with finite shell elements
Intro
see intro chap. 1.2
Pre-Processing
/filname,p2modalshell,0
change the jobname for the analysis
/prep7
enter preprocessing
/title,triangular leaf spring (SHELL)
title for actual problem
et,1,shell63
define finite element type 1
*ask,parLength,'triangular depth [m] ?',0.4
user input: height of triangle, see fig.2.1: L
L=parLength
define height of triangle
*ask,parWidth,'triangular base width [m] ?',0.14
user input: width of triangle, see fig.2.1: B
b=parWidth
define width of triangle
*ask,parEModulus,'modulus elasticity [N/m^2] ?',2.1e11
user input: enter modulus of elasticity
mp,ex,1,parEModulus
define material elasticity for set 1
*ask,parDensity,'density [kg/m^3] ?',7850.0
user input: enter material density
mp,dens,1,parDensity
define material density for set 1
*ask,parLat,'lateral contraction [-] ?',0.3
user input: enter lateral contraction
mp,nuxy,1,parLat
define material lateral contraction for set 1
*ask,parThick,'thickness of shell [m] ?',0.004
user input: thickness of triangular leaf
r,1,parThick
define thickness of triangular leaf
k,1,0,0,-b/2,
define keypoint 1 at specified coordinate
k,2,0,0,b/2,
define keypoint 2 at specified coordinate
k,3,l,0,0,
define keypoint 3 at specified coordinate
lstr,1,2
define line between keypoint 1 and 2
lstr,2,3
define line between keypoint 2 and 3
lstr,3,1
define line between keypoint 3 and 1
al,1,2,3
define area for triangular leaf spring
/view,1,1,1,1
define the viewing direction for the display
aatt,1,1,1,0,
connect finite element attributes with area 1
*ask,parElemLength,'mean element length [m] ?',0.01
user input: mean element size
esize,parElemLength,0,
define previous entered mean element size
mshkey,0
activate free meshing
amesh,1
mesh area 1
dl,1,,all
boundary condition for line 1
finish
leave pre processor
save,p2modalshell,db,,all
save database in p2modalshell.db
Solution-Processing
see Solution-Processing chap. 1.2
Post-Processing
see Post-Processing chap. 1.2
A2.4
Example 2: Triangular Leaf Spring
2.3 Modal analysis results: Triangular leaf spring with finite shell elements
Fig. 2.2: 1st eigenmode 42.958 Hz
Fig. 2.3: 2nd eigenmode 186.073 Hz
Fig. 2.4: 3rd eigenmode 414.073 Hz