Vibration Paper

1. The Elusive Free-Free Boundary Condition
by
Kenneth G. McConnell
Professor Emeritus, Vibration Engineering
Department of Aerospace Engineering
Iowa State University of Science and Technology
Ames, IA 50011
(mclken@msn.com)
ABSTRACT:
The application of substructuring concepts requires
the use of free-free FRF’s to describe the test item.
The free-free test environment is easily achieved in
theoretical calculations but is usually compromised
when simulated experimentally due to the
requirement to support the structure against gravity
forces. This paper looks at some interesting testing
issues through an analysis of a free-free beam
mounted on end springs as well as experimental
results from an actual beam. In other words, “Why
can’t I correct for these incorrect boundary conditions
except in a very special case that generally gives
unsatisfactory results?”
INTRODUCTION:
The importance of boundary conditions in
dealing with vibration problems was clearly brought
to my attention when dealing with an interesting
catapult excited vibration of the island structure of an
aircraft carrier as reported by Hagen, et el [1] and
McConnell [2]. In this case, the torsional stiffness of
the supporting sponson structure played an important
role in reducing the island’s fundamental natural
frequency from about 5 Hz for most modern aircraft
carriers to 3.2 Hz for the one in question. It turned
out that the island’s fundamental natural frequency
was highly dependent on the torsional stiffness of the
sponson structure.
Recently, vibration testing of space station
structures requires testing for many natural
frequencies and mode shapes that are below 1.0 Hz.
Foss [3] developed a method for achieving the free-
free boundary condition by suspending the structure
from a single point using a spring to support both the
exciter armature and the test structure and then
measuring the input force that consists of both the
spring support force and the excitation force. The
end result is the achievement of a real free-free
boundary condition.
However, this solution looked too good to
be true and it was for there can be some significant
problems. First, the single support point must be
above the mass center in order to cancel out the
gravity force, which limits the location of the
excitation force. For example, the support point may
not be the point where we need the driving point and
transfer FRF’s. Second, how do I test for the
potential 36 input-output driving point relationships
that may be needed to describe a structure for
connecting to another structure? Third, one may miss
important mode shapes and natural frequencies when
exciting at only one point in a single direction, etc.
THE ALMOST FREE-FREE BEAM
We shall try to explore some of the issues
that surround the so called free-free boundary
condition. A simple model is used as shown in Fig. 1

2. which is a cold rolled steel beam that has been tested
many times by my students over the years. The
characteristics of this beam are: length l = 92.0 inch
(233.7 cm); width b= 1.25 inch (3.18 cm); thickness
h = 1.00 inch (2.54 cm). It is supported by springs at
the left (k1) and right (kN) ends. The different
receptances are calculated using Mathcad software
and an influence coefficient beam formulation. This
calculation method is the same as a finite element
program in that the end springs influence both the
system modes shapes and natural frequencies.
This beam has two potential rigid body
modes of vibration dependent on the spring
constants. When the springs are very soft and equal
(like 0.140 lbs/in), these two rigid body modes are
straight up and down (y) for one mode and rotation
about the mass center g (θ) for the other mode.
Table 1 compares the calculated natural frequencies
with those measured when the highest rigid body
natural frequency is just over 1.0 Hz, giving a
frequency ratio greater than 20 to one. It is seen that
good agreement is achieved between the calculated
Table 1. Predicted vs Measured free-free Natural
Frequencies
Mode
Num
fp (Hz)-y
Theoretical
fp (Hz)-y
Measured
fp (Hz) - z
Theoretical
fp (Hz)-z
Measured
1 24.07 24.2 30.1
2 66.3 67.4 82.9 86.5
3 130.1 132.2 162.6 165.3
4 215.0 217.4 268.8 272.1
5 321.2 324.3 401.4 404.4
and measured natural frequencies. The first four
mode shapes are shown in Fig. 2 where the first mode
is Rigid-Body translation while the second mode is
Rigid-Body rotation about the mass center. The third
and fourth mode shapes match those of a free-free
beam. Hence, we obtained a reasonable simulation
of the free-free condition.
Fig. 1. Theoretical free-free beam on springs.
f
2
0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92
0.3
0.2
0.1
0
0.1
Mode 1 RB
Mode 2 RB
Mode 3 1st Free Free
Mode 4 2nd Free Free
First Four Mode Shapes
Position x
Amplitude
Fig. 2. First Four Mode Shapes for very soft springs of k1 = kN
= 0.140 lbs/in.
0 9.2 18.4 27.6 36.8 46 55.2 64.4 73.6 82.8 92
0.3
0.2
0.1
0
0.1
0.2
0.3
Mode 1 RB
Mode 2 RB
Mode 3 1st Free-Free
Mode 4 2nd Free-Free
First Four Mode Shapes
Position x
Amplitude
Fig. 3. First Four Mode Shapes for moderate springs
of k1 = kN = 140 lbs/in.
However, if we increase each spring rate by
1000 times to 140 lbs/in., we obtain the mode shapes
shown in Fig. 3. Here it is clear that the first and
second Rigid-Body mode shapes have changed and
are beginning to look like the first and second modes
b
h
/ / / / / /
θ
g
l
/ / / / / /
k1 kN
y

3. for a pinned end beam, which they will become if the
spring constants are increased by an additional factor
of 100.
The first five natural frequencies are shown
in Table 2 as a function of the frequency ratio β (=
f3/f2) where f3 is the beam’s first elastic or free-free
mode and f2 is the higher of the Rigid Body natural
frequencies. Here it is seen that the first elastic beam
mode f3 is quite sensitive to frequency ratio β. When
β decreases from 5.0 to 2.02, f3 increases from 24.75
to 30.57 Hz, while f3 varies from 24.06 (β = 152) up
to 24.75 Hz (β= 5). Hence, one concludes that β
should be on the order of ten or more in order for the
first free-free elastic mode to be reasonably correct.
We also note that the second and third elastic free-
free modes have little change in frequency and/or
mode shape as long as β is on the order of 10 or
more.
Table 2: Natural frequencies as a function of frequency
ratio β*
Freq
Ratio
f1 f2 f3 f4 f5
152 0.091 0.158 24.06 66.31 129.95
15.2 0.981 1.582 24.13 66.33 129.96
10.0 1.393 2.43 24.22 66.36 129.98
5.0 2.08 4.98 24.75 66.56 130.0
2.02 7.01 15.16 30.58 68.88 131.2
* Frequency ratio is first elastic mode natural
frequency divided by highest rigid body natural
frequency, i.e., β = f3/f2 in this table
WHERE DOES THIS CONTAMINATION OF
INFORMATION COME FROM?
Varoto [4] examined the dynamic
interaction between two structures, one called the test
item and the other called the vehicle. The resulting
equations show that the combined structural response
depends on the sum of the connector FRF’s as well as
the test item’s transfer FRF’s. The vehicle’s FRF’s
are as measured on the vehicle when the test item is
removed while all of the test item FRF’s must be
obtained under free-free boundary conditions.
There are potentially 36 required interface
FRF’s for each interface point; i.e.,
⎭
⎬
⎫
⎩
⎨
⎧
⎥
⎦
⎤
⎢
⎣
⎡
=
⎭
⎬
⎫
⎩
⎨
⎧
MDC
BA Fa
α
(1)
Note that a, α, F, and M can have up to three vector
components. Hence, each submatrix A, B, C, and D
has the potential to be 3x3 in size. Dong [5] has
developed an experimental technique to measure the
rotational as well as the linear FRF’s when using a T-
bar attachment.
Let us now look at the output motion at
location p on the beam in Fig. 1 due to an excitation
force of Fq applied at location q. The measured
motion yp (displacement, velocity, or acceleration) is
given by
NqNqqpqqpqp PVPVFVFMy ++== 11 (2)
where Mpq is the measured FRF between points p
and q,
Vpq is the actual free-free beam FRF between
points p and q,
Fq the excitation force at location q,
Vq1 and VqN are the actual free-free transfer
FRF’s between points q and 1 and
q and N,
while P1 = - k1H1q Fq is the spring force at location
1 and
PN = - kN HNqFq is the spring force at
location N due to the actual beam motion at
those points.
Substituting these last two spring forces intoEq. 2 and
canceling the common term Fq gives

4. NqNqNqqpqpq HkVHkVVM −−= 111 (3)
It is clear from Eq. (3) that the measured FRF Mpq is
contaminated by the two spring support terms
qq HkV 111− and NqNqN HkV− a
. It is seen that
this contamination involves both the free-free elastic
transfer FRF’s, Vq1 and VqN that we need to
determine as well as the measured transfer
receptances H1q and HNq that control the boundary
condition forces acting on the structure. Table 3 gives
the required units dependent on the type of FRF that
is measured; i.e., Mpq. In addition we see that all of
the V’s have the same units as Mpq while the H’s
always need to be receptances to go with the
corresponding spring so that the product of ksHsq is
dimensionless. We should suspect that each spring
support would contribute contamination terms like
those in Eq. 3. Hence, we can not always assume
that the contamination is negligible without checking
out the effect of different support conditions; i.e., use
different springs.
Table 3: Required Units that apply to Eq. 3
Mpq Vpq Vq1 VqN H1q HNq
Rec Rec Rec Rec Rec Rec
Mob Mob Mob Mob Rec Rec
Acc Acc Acc Acc Rec Rec
Rec = Receptance, Mob = Mobility,
Acc = Accelerance
EXAMPLE OF TWO DIFFERENT SPRINGS
Now let us consider an example where k1 =
140 lbs/in and kN = 0.140 lbs/in. The resulting mode
shapes are shown in Fig. 4 where it is seen that the
first mode appears like the left end is pinned to a
a
For “s” number of spring support points, these two
terms become ∑s
sqsqs HkV
rigid foundation while the second mode looks more
like a cantilever beam when the base moves up and
down. The third mode looks a lot like the first elastic
beam mode with the left end moving more than the
right end. The fourth mode looks nearly identical to
the second elastic beam mode.
Figure 5 shows the driving point receptance
for the left end at location 1. There is only one Rigid
Body mode showing in this plot since point 1 is
essentially a node point so that the rotational first
rigid body mode is not excited (see Fig. 4 for this
mode shape). When the Rigid Body effects are
subtracted from H11 it is seen that
0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92
0.3
0.2
0.1
0
0.1
0.2
0.3
Mode 1
Mode 2
Mode 3 1st Free-Free Mode
Mode 4 2nd Free-Free Mode
First Four Mode Shapes
Position x
Amplitude
Fig. 4. First four mode shapes, k1 = 140 and kN=
0.140 lbs/in
the first free-free natural frequency remains the same
around 30 Hz when the actual free-free natural
frequency is about 24 Hz.
If we apply Eq. (3) to this physical situation,
we obtain
11111111111 NNN HkVHkVVH −−= (4)
where H11 is the driving point receptance at the left
end of the beam. Since kN is very small compared to
k1 and V11 is the actual driving point receptance, We
use U11 in order to distinguish the predicted or
corrected receptance from the actual free-free driving
point receptance. Hence, we have that

5. 111
11
11
1 Hk
H
U
−
≅ (5)
Now, we need to evaluate how much the rigid body
modes contaminate the prediction of Eq. 5 by
removing the Rigid Body modes from H11. Let B11
be H11 with the Rigid Body modes removed. Then,
Eq. 5 becomes
111
11
11
1 Bk
B
D
−
≅ (6)
The computed results using Eq. 5 and 6 are shown in
Fig. 6 where it is clearly seen that the first free-free
resonant frequency is essentially 24 Hz in both
corrections. However, it is seen that the correction
based on H11 and contains the Rigid Body modes and
gives strange results below the first elastic free-free
resonance. The correction based on B11, where the
rigid body modes were removed, gives a more
consistent result when compared to the actual
receptance at low frequencies. It is clearly seen in
both Figs. 5 & 6 that the notches are not completely
corrected. This error can have major significance
when the mating vehicle has low values as well in
this frequency range since it is the sum of the two
values that are involved in creating the combined
dynamic behavior.
It should be pointed out that using Eqs. 5
and 6 to correct the driving point receptance on the
right hand side does not cause a significant shift in
the first elastic mode frequency; i.e., f3. This is due
to kN being very soft compared to k1 which has a
major effect on the natural frequencies and mode
shapes. Hence, any simple correction scheme must
be applied with caution.
CONCLUSIONS
I hope that the reader sees that the free-free
boundary condition is difficult to obtain in practice
and we can be significantly fooled by the measured
results. We can achieve a reasonable simulation if
we can separate the first free-free resonance from the
suspension Rigid Body modes by a factor of nearly
10 to one; i.e., β = f3/f2 > 10. Then, removal of rigid
body modes from the measured data should give
reasonable results for driving point FRF’s where the
spring constant is largest. Other physical support
relationships need to be evaluated on an individual
basis. I hope that this little exercise will encourage
some bright person to think about all of these testing
issues, and they will come up with a really neat way
to correct the measured data, like measure all of the
support forces and then use multi-point input output
analysis.
REFERENCES:
1. Hagen, A and K. McConnell, “Catapult
Excited Mast, Island, and Hull Vibration
Studies on Three Aircraft Carriers: USS
America (CVA-66), USS Constellation
(CVA-64), and USSS John F. Kennedy
(CVA-67)”, Naval Ship Research and
Development Center Report 3181,
Washington, DC, 1969.
2. McConnell, K. G., “Tracking the Cause of
Large Shipboard Vibrations During
Catapult Launching,” Proc., SEM Fall
Conference, Savannah, GA, Nov. 1987.
3. Foss, Gary C., Free-Free Modal Testing
Without Suspension Modes, Proc. IMAC-
XIV, P. 437-441, Feb. 1996.
4. Varoto, Paulo S., The Rules for the
Exchange and Analysis of Dynamic
Information, Ph.D. Thesis, Iowa State
University, Ames, IA. 1996.

6. 5. Dong, Jianrong, Extraction Methods for
Multidirectional Driving Point Accelerance
and Transfer Point Accelerance Matrices,
Ph. D. Thesis, Iowa State University, Ames,
IA. 2000.
0 20 40 60 80 100 120 140 160 180 200
1 10
5
1 10
4
1 10
3
0.01
0.1
1
Measured DP Receptance H11
DP Receptance RB removed
Free-Free DP Receptance
Driving Point Receptance 11
Frequency (Hz)
Receptance(in/lb)
Fig. 5. Driving point receptance at point 1 as measured and corrected compared with free-free beam values, k1 = 140 and
kN = 0.140 lbs/in.
0 20 40 60 80 100 120 140 160 180 200
1 10
5
1 10
4
1 10
3
0.01
0.1
1
10
Correced LHS DP Receptance H11
LHS DP receptance B11
Actual Free-Free DP Receptance
Compare Measured and Corr DP Receptance
Frequency (Hz)
Receptance(in/lb)
Fig. 6. Comparison of corrected driving point receptance based on original and modified measured receptance using Eqs.
5 and 6.