Example 4: Dynamic Vibration Absorber (DVA)
A 4.1
Example 4: Dynamic Vibration Absorber (DVA)
Starting point for this exercise is the test of an oscillation damper with an eigenfrequency of 35 Hz.
Therefore a design should be found which has one or two natural modes in this region, so that the
relationship of the system with and without a dynamic vibration damper (also with the effects of a
second eigenmode) can be observed. In the present example the system has an eigenfrequency mode (A)
at 35Hz, where beam 2 acts as a rigid body in the y-direction on the beam 1 which is oscillating up and
down and bends in the shape of a half sine wave . By the next eigenfrequency mode (B), beam 2
executes a bending oscillation and twists beam 1 about its centre.
Also found on the vertical beam is a freely moveable mass m4, which can shift the frequency of (B) in
the region of 20 to 45 Hz. In order to make this into an FEM model, the vertical beam is made up of 50
very fine discrete elements. The mass element can be freely positioned in small steps, over the variable
nodes.
The eigenmodes for the system without a DVA can be calculated, after which the transient build up of
vibrations with and without the DVA should be considered. In each case the magnification functions are
to be determined, where the DVA is intentionally arranged with the shown starting parameters and
position, which are not optimal. In the course of the practical sessions Den Hartog’s theory should be
reviewed [2].
At the optimal arrangement of the DVA, the available ANSYS option to generate a unity modal
T
transformed mass matrix (see chapter 6),  M    I  should be used.
The reduced mass mnjred on connection node j of the DVA used to calm the nth natural frequency mode
can be calculated from mnjred   nj2 . For the case where the eigenfrequencies lie far enough apart, a
optimisation of the DVA can be performed according to Den Hartog’s theory.
1
Beam 2
/
ANSYS 5.0 A
FEB 9 1996
13:37:50
PLOT NO.
1
DISPLACEMENT
STEP= 1
SUB = 1
FREQ= 35.244
RSYS= 0
DMX = 0.268805
*DSCA= 0 99
ZV = 1
DIST= 0.65428
XF = 0.5948
YF = 0.183988
CENT OID HIDDEN
* DSCA= 0.99
ZV = 1
DIST= 0.783696
XF = 0.476482
YF
= 0.42721
CENTROID HIDDEN
Eigenmodes
Beam 1
\
Y
Z
ANSYS 5.0 A
FEB
9 1996
13:38:22
PLOT NO.
2
DISPLACEMENT
STEP= 1
SUB = 2
FREQ= 45.643
RSYS= 0
DMX = 0.83916
1
X
 A and B 
Y
Z
HARMONIC RESPONSE
HARMONIC RESPONSE
Fig. 4.1: Eigenmodes A and B
X
Example 4: Dynamic Vibration Absorber (DVA)
A 4.2
4.1 Profile Data:
Horizontal Beam 1 (20 elements)
A1
[10 -6 m2]
Iz1
[10 -12 m 4]
Iy2
[10 -12 m 4]
Hz
Hy
[m]
[m]
[m]
E1
[10 11 N/m 2]
1
[kg/m3]
[%]
1.1896
664
115.624
1 409.845
0.14
0.03
2.06
7 850
1
L1

Vertical Beam 2 (50 elements) with mass m2
A2
[10 -6 m2]
Iz1
[10 -12 m 4]
Iy2
[10 -12 m 4]
Hz
Hy
[m]
[m]
[m]
E2
[10 11 N/m 2]
2
[kg/m3]
[%]
0.9
704
217.717
1 600.298
0.14
0.04
2.06
7 850
1
L2
Hz, Hy : distance from neutral axis to outer edge of beam for stress calculations
Dynamic vibration absorber and mobile mass
k
c
m3
m4
m2 + m4
[N/m]
[kg/s]
[kg]
[kg]
[kg]
calculate
calculate
calculate
calculate
10.821
Fig. 4.2: Geometric layout of structure

Example 4: Dynamic Vibration Absorber (DVA)
A 4.3
4.2 Theoretical Calculation of the Fundamental Oscillation
The eigenfrequency is calculated from:
e 
k red
.
mred
With the reduced spring constant:
k red 
48  E  I Z 1
L1 ³
And the reduced mass:
mred  M red1  ( M 2  M 4 )  12  A1  L1    10.821kg
Resulting in:

e 
48  E  I Z 1
 220.87 1s
L1  6.2 kg  10.821 kg 
3

f e  35.15Hz .
4.3 Results with ANSYS
On the following page you will find a comparison of the natural frequency modes with frequency
specification for the undamped system, with mass m4 set high and low on beam 2 (see fig. 4.3).
Next presented are the magnification curves (see fig. 4.4, 4.5) in the x-, y- and z- directions for node 71,
(top of beam 2) for the system without and with DVA. The structural damping of the beam structure is
always 1%.
Lastly follows two diagrams showing the transient build up effects in the case of resonance (fe = 35 Hz).
Shown are the different movements of the top end of beam 2 over time without and with DVA. In both
cases the structural damping is 1%.
Example 4: Dynamic Vibration Absorber (DVA)
A 4.4
4.4 Modal analysis script: DVA with finite beam, mass and combin elements
Intro
finish
finish actual processing mode
/clear,nostart
clear database
Pre-Processing
/filname,p4modal,0
open file ‘p4modal.db’
/prep7
enter pre-processor
/title,dynamic vibration absorber (modal analysis)
define title
L1=1.1896
A1=664e-6
Iz1=115624e-12
Iy1=1409845e-12
Hz1=0.14
Hy1=0.03
EMOD1=2.06e11
DICHTE1=7850
L2=0.9
A2=704e-6
Iz2=217717e-12
Iy2=1600298e-12
Hz2=0.14
Hy2=0.04
EMOD2= 2.06e11
DICHTE2=7850
m4=10.821-A2*DICHTE2*L2
et,1,beam4
r,1,A1,Iz1,Iy1,Hz1,Hy1
mp,ex,1,EMOD1
mp,dens,1,DICHTE1
et,2,beam4
r,2,A2,Iz2,Iy2,Hz2,Hy2
mp,ex,2,EMOD2
mp,dens,2,DICHTE2
n,1
n,21,L1
fill
n,22,(L1/2),(L2/45)
n,71,(L1/2),L2
fill,22,71
type,1
real,1
mat,1
e,1,2
egen,20,1,1
type,2
real,2
mat,2
e,11,22
e,22,23
egen,49,1,22
KNOTEN=0
flag=1
*dowhile,flag
*ask,ou,'node for single mass on beam (22-71)?',1
*if,ou,ge,22,then
*if,ou,le,71,then
flag=0
*endif
*endif
*enddo
KNOTEN=ou
initialize all necessary input data for the first,
horizontal beam
initialize all necessary input data for the first, vertical
beam
calculate mass m4 (movable mass)
define all finite element data for horizontal beam
define all finite element data for vertical beam
create all nodes first
activate first finite element data set and generate all
horizontal beam elements
activate second finite element data set and generate
all vertical beam elements
input position of m4 due to an appropriate node
number between 22 and 71
Example 4: Dynamic Vibration Absorber (DVA)
et,4,mass21,,,2
r,4,m4,m4,m4
type,4
real,4
e,KNOTEN
K3=0
D3=0
m3=0
*ask,yn,'with damper? [y/n]','n'
A 4.5
create finite mass element m4 at selected position
initialize damper data with ‘0’
user input: model with/without damper
*if,yn,eq,'y',then
*ask,K3,'damper stiffness [N/m]?',56257.96
*ask,D3,'damping coefficient [Ns/m]?',103.01
*ask,m3,'absorber mass [kg]?',1.386
*ask,dir,'direction of discrete damper [x,y,z]','x'
*if,dir,eq,'x',then
dir=1
*elseif,dir,eq,'y',then
dir=2
*else
dir=3
*endif
et,3,combin40,,,dir,,,2
r,3,K3,D3,m3,,,
TYPE,3
REAL,3
KNOTEN=0
flag=1
*dowhile,flag
*ask,ou,'node for damper on beam (1-71)?',0
*if,ou,ge,1,then
*if,ou,le,71,then
flag=0
*endif
*endif
*enddo
KNOTEN=ou
n,100,nx(KNOTEN),ny(KNOTEN)
e,KNOTEN,100
*endif
d,1,ux
d,1,uy
d,1,uz
d,1,rotx
d,1,roty
d,21,uy
d,21,uz
d,21,rotx
d,21,roty
if the model includes a damper the following steps are
done here:
input the stiffness, damping coefficient,
absorber mass and select the direction of action of the
damper
define finite element data set for damper
activate damper data set
select node of model the damper is acting
on
create finite damper element
define the boundary conditions for the model
/view,1,1,1,1
display is setting to isometric view
eplot
plot all finite elements
finish
leave pre-processor
save,p4modal,db,,all
save model data to ‘p4model.db’
Example 4: Dynamic Vibration Absorber (DVA)
A 4.6
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
Post-Processing
/post1
enter post1-processor
flag=1
set flag parameter to 1
*dowhile,flag
*ask,varMode,'show eigenform [0=exit]',1
*if,varMode,eq,0,then
flag=0
*elseif,varMode,le,numModes,then
set,,,,,,,varMode
pldisp,1
*endif
*enddo
do-loop showing the modes
(leave do-loop with ‘0’)
finish
leave post1-processor
Example 4: Dynamic Vibration Absorber (DVA)
4.5 Modal analysis results: vibrating beams with finite beam and mass elements
A 4.7
frequency is shifting to approx. 45Hz if m4 moves down
frequency is untouched by moving m4
frequency is shifting to approx. 75Hz when m4 moves down
frequency is untouched by moving m4
frequency is shifting to approx. 35Hz if m4 moves up
frequency is shifting to approx. 75 Hz when m4 moves up
Fig. 4.3 : By moving m4, you can shift the frequencies of the non-relevant modes and leave the frequency untouched near 35Hz
Example 4: Dynamic Vibration Absorber (DVA)
A 4.8
4.6 Harmonic analysis script: DVA with finite beam, mass and combin elements
Intro
see intro chap. 4.4
Pre-Processing
see pre-processing chap. 4.4
Solution-Processing
/solu
enter solution-processor
antype,harmic
define a harmonic simulation
hropt,full
define a calculation with full system matrices
user input:
start frequency. default=0
end frequency. default=100
number of substeps, default=100
structural damping coefficient,
default=0,01
*ask,f1,'frequency range: start [Hz]?',0
*ask,f2,'frequency range: end [Hz]?',100
*ask,parSteps,'substeps [-]?',100
*ask,dmprat,'structural damping [-]?',0.01
*ask,parKraft,'amplitude of force [N]?',1
kbc,1
KNOTEN=0
flag=1
*dowhile,flag
*ask,ou,'node on beam the force is acting on (1-71)?',16
*if,ou,ge,1,then
*if,ou,le,71,then
flag=0
*endif
*endif
*enddo
KNOTEN=ou
*ask,dir,'direction of force acting on beam [x,y,z]','x'
*if,dir,eq,'x',then
dir='fx'
*elseif,dir,eq,'y',then
dir='fy'
*else
dir='fz'
*endif
f,KNOTEN,dir,parKraft
user input:
select node number of structure on which the
force is acting. (default=16)
user input:
select the direction the force is acting in.
default=x
harfrq,f1,f2
set the frequency range for the calculation
nsubst,parSteps
set the sampling of the frequency range above
solve
start the calculation
finish
leave solution-processor
Post-Processing
/post26
enter time-history-post-processor
numvar,200,
plvar,2,3,4
define number of possible graphs
get data for x displacement for node 71 and map it on
curve with number 2
get data for x displacement for node 71 and map it on
curve with number 3
get data for x displacement for node 71 and map it on
curve with number 4
plot the curve numbers 2,3 and 4
finish
leave time-history-post-processor
nsol,2,71,u,x,displ_in_x_@71
nsol,3,71,u,y,displ_in_y_@71
nsol,4,71,u,z,displ_in_z_@71
Example 4: Dynamic Vibration Absorber (DVA)
4.7 Harmonic analysis results: DVA with finite beam, mass and combin elements
Fig. 4.4: Results of harmonic analysis without DVA
Fig. 4.5: Results of harmonic analysis with DVA
A 4.9
Example 4: Dynamic Vibration Absorber (DVA)
A 4.10
4.8 Transient analysis script: DVA with finite beam, mass and combin elements
Intro
see intro chap. 4.4
Pre-Processing
see pre-processing chap. 4.4
Solution-Processing
/solu
antype,trans
*ask,DAMPRAT,'structural damping [-]?',0.01
*ask,parFreq,'frequency of excitation [Hz]?',35
*ask,parAbtast,'sampling rate [-]?',8
*ask,parZeitEnde,'simulation end time [s]?',0.5
dmprat=DAMPRAT
fe=parFreq
te=1/fe
Set period duration
PI=2*asin(1)
define π
timint,off
set time integration effect off
time,1e-20
set small time step size for the first step
kbc,1
define stepping mode
lswrite,1
write the previous definition to load step file 1
timint,on
set time integration effect on
time,te
set simulation time to te
*ask,parKraft,'amplitude of force [N]?',1
user input: amplitude of force. default=’1’
KNOTEN=0
flag=1
*dowhile,flag
*ask,ou,'force at node (1-71)?',16
*if,ou,ge,1,then
*if,ou,le,71,then
flag=0
*endif
*endif
*enddo
KNOTEN=ou
*ask,dir,'direction of force [x,y,z]','x'
*if,dir,eq,'x',then
dir='fx'
*elseif,dir,eq,'y',then
dir='fy'
*else
dir='fz'
*endif
f,KNOTEN,dir,0
lswrite
ANZ_SUB=parAbtast
ANFANG=te+te/ANZ_SUB
ENDE=parZeitEnde
TIMESTEP=te/ANZ_SUB
*do,T,ANFANG,ENDE,TIMESTEP
Kraft=parKraft*sin(2*PI*fe*T)
f,KNOTEN,dir,Kraft
time,T
outres,all,1
lswrite
*enddo
lssolve,1,(2+(ENDE*fe-1)*ANZ_SUB),1
finish
user input: node on structure number the
exciting force is acting on. default=’16’
user input: select the direction the force is
acting in. default=’x’
write the first ‘real’ load step to the next load
step file
create the rest of the load step files
start the calculation for all the load steps
defined above
leave solution-processor
Example 4: Dynamic Vibration Absorber (DVA)
A 4.11
Post-Processing
/post26
enter time-history-post-processor
numvar,200,
plvar,2,3,4
define number of possible graphs
get data for x displacement for node 71 and map it on
curve with number 2
get data for x displacement for node 71 and map it on
curve with number 3
get data for x displacement for node 71 and map it on
curve with number 4
plot the curve numbers 2,3 and 4
finish
leave time-history-post-processor
nsol,2,71,u,x,displ_in_x_@71
nsol,3,71,u,y,displ_in_y_@71
nsol,4,71,u,z,displ_in_z_@71
Example 4: Dynamic Vibration Absorber (DVA)
4.9 Transient analysis results: DVA with finite beam, mass and combin elements
Fig. 4.6: Results for the transient simulation without DVA
Fig. 4.7: Results for the transient simulation with DVA
A 4.12