IX International Conference on Recent Advances in Structural Dynamics
Southampton, UK, 17-19 July 2006
NONLINEAR ACOUSTIC RESPONSE OF AN AIRCRAFT FUSELAGE
SIDEWALL STRUCTURE BY A REDUCED-ORDER ANALYSIS
National Institute of Aerospace
Hampton, Virginia 23666, USA. Email: email@example.com
Stephen A. Rizzi
NASA Langley Research Center, Structural Acoustics Branch
Hampton, Virginia 23681, USA. Email: Stephen. a.rizzi(@nasa. gov
David S. Groen
The Boeing Company, Phantom Works Structures Technology
St. Louis, Missouri 63166, USA. Email: david.s.groen(@boeing.com
A reduced-order nonlinear analysis of a structurally complex aircraft fuselage sidewall panel
is undertaken to explore issues associated with application of such analyses to practical
structures. Of primary interest is the trade-off between computational efficiency and
accuracy. An approach to modal basis selection is offered based upon the modal participation
in the linear regime. The nonlinear static response to a uniform pressure loading and
nonlinear random response to a uniformly distributed acoustic loading are computed.
Comparisons of the static response with a nonlinear static solution in physical degrees-of-
freedom demonstrate the efficacy of the approach taken for modal basis selection. Changes in
the modal participation as a function of static and random loading levels suggest a means for
improvement in the basis selection.
Dynamic response analysis tools used by the aerospace industry rely heavily on a linear
approach. Commercial finite element codes make a linear modal frequency response of a
complicated structure straightforward, even for large degree-of-freedom (DoF) systems.
However, this approach is not suitable for predicting the response of aircraft structures when
they respond to high dynamic loading in a geometrically nonlinear manner. A nonlinear
dynamic analysis in physical DoFs is computationally prohibitive, especially for non-
deterministic problems involving long simulation times. It is common practice to reduce the
system size by modeling just a portion of the structure, e.g. a single bay in a multi-bay panel.
However, with such an approach, important global dynamics are lost because the boundary
conditions can not be accurately modeled in a nonlinear dynamic analysis.
In recognition of the above limitations, there has been significant effort in recent years in the
development of finite element based reduced-order nonlinear analysis methods [1-5]. These
methods have been numerically and experimentally validated for simple structures including
flat and curved beams and plates [6-8]. With the exception of a simple stiffened panel studied
Paper No. 135
in , the application of these reduced-order nonUnear analyses to a practical aerospace
structure has been almost non-existent. The purpose of this work is to explore the challenges
of applying such methods to typical aircraft structure, i.e. riveted skin-frame-longeron
metallic structures with system sizes ranging in the tens of thousands of DoFs. To take
advantage of the finite element libraries and pre- and post-processing capabilities of
commercial finite element codes, an indirect nonlinear stiffness evaluation procedure is used
. The procedure is implemented in the program RANSTEP, for "Reduced order Analysis
using a Nonlinear STiffness Evaluation Procedure," for use with MSC.NASTRAN and
ABAQUS finite element programs.
In the limit of including all linear eigenvectors in the modal basis, results found using the
reduced-order analysis should be identical to those obtained via an analysis in physical DoFs
because there is no modal truncation. However, the cost associated with this would likely
exceed the cost of the physical DoF analysis we wish to avoid. Since previous results for
simpler structures indicated that accurate reduced-order response predictions could be made
using an abbreviated modal basis, the focus of the present work is on finding a means of
methodically down-selecting the basis from the multitude of candidates. For this purpose, the
modal participation factor is considered. All modes need to initially be included in the
analysis to determine their modal participation, so that the most significant modes can be
retained in the subsequent analyses. In other words, we need to solve the problem the hard
way first to determine a more efficient solution for later use. Because of the high cost of
computing all nonlinear modal stiffness terms, the modal participation will be assessed using
a linear analysis for which the linear stiffness is readily available. Reduced-order nonlinear
static and random displacement responses are presented for an aluminum fuselage sidewall
panel. The static solution is compared with an analysis in physical DoFs to determine the
effect of modal truncation as the response increases from essentially linear to highly
nonlinear. The static comparison helps deduce the effect of modal truncation on the random
response where a physical DoF analysis is not available.
REDUCED-ORDER NONLINEAR FINITE ELEMENT ANALYSIS
The equations of motion of the nonlinear system in physical DoF can be written as
MX(t)^CX(t)^F^,(X(t)) = F(t) (1)
where M and C are the system mass and damping matrices, F^^^ is nonlinear restoring force,
X is the displacement response vector and F is the force excitation vector, respectively. A set
of coupled modal equations with reduced DoFs can be obtained by applying the modal
coordinate transformation X=Q>q to Equation (1), where q is the modal displacement
response vector. The modal basis matrix O is formed from the selected eigenvectors
obtained from a linear eigenanalysis. Generally, a small set of L basis functions are included
Mq(t) + Cq(t) + F^,(q,(t),q,(t),...,q,(t)) = F(t) (2)
where, for mass-normalized eigenvectors,
M=(D^M(D = riJ C^O^CO^[2C,coJ F^^^O^F^^ F=<D^F (3)
and 0)^ are the undamped natural frequencies and (^^ are the viscous damping factors. The
nonlinear modal restoring force F^^ can be expressed in terms of linear, d, quadratic
nonlinear, a, and cubic nonlinear, b, modal stiffness coefficients as 
j=l j=l k=j j=l k=j l=k
This form of F^^ reduces the problem of determining the hnear and nonUnear stiffness
coefficients to a series of nonUnear static problems with a series of different prescribed
displacement fields formed as a combination of eigenvectors. The number of required
displacement fields and nonlinear static solutions required to determine the nonlinear stiffness
coefficients is given by 
Number of Static Solutions = 3
Since the number of nonlinear static solutions increases rapidly as the number of basis
functions L increases, there exists a trade-off between the solution accuracy and
computational effort. These considerations will be discussed later in the paper.
Once the nonlinear stiffness terms are fully determined, Equation (2) is numerically integrated
using a 4^^ order Runge-Kutta scheme  to determine the modal displacement time history.
Physical displacements are reconstructed using the inverse modal coordinate transform.
FUSELAGE SIDEWALL PANEL
A section of a previously studied fuselage sidewall panel  served as the basis for the panel
considered in the present study. The MSC.NASTRAN FE model of the panel section is
presented in Figure 1. The skin panel measures 58.1 1 in. by 25.06 in. and is subdivided by a
riveted frame and longeron substructure into nine bays in 3-by-3 configuration. Each of the
nine bays measures 16.75 in. by 7.5 in. between rivet lines. The skin panel, frame and
longeron substructures are modeled using 14,832 CQUAD4 plate elements. The thickness of
the skin panel and frame L and Z-sections is 0.05 in., while the thickness of the longerons is
0.04 in. A total of 784 rivets attach the frame and longeron substructures to the skin panel,
attach the frame L and Z-sections to each other, and attach the longerons to the frames at the
intersection points, as indicated in Figure 1 . Each rivet is modeled using a single CBEAM
beam element. The edges of the skin panel are simply supported with three fixed
displacement DoFs and three free rotational DoFs. The entire model contains a total of
96,156 DoFs, 95,352 of which are unconstrained. The following material properties for
aluminum were used:
E = 10.5 X 10' psU ju = 0.33, p = 2.614 x 10"' ^^
Special modeling considerations. It is a common practice in FE model development to
model rivets as rigid kinematic couplings between the nodes being connected. The indirect
stiffness evaluation procedure as currently implemented in RANSTEP, however, precludes
such modeling, since it involves enforcing arbitrary displacement fields on the FE model .
Since it is not a valid operation to prescribe displacements at the both ends of a rigid element
, the rivets were modeled with beam elements as indicated above. A static analysis of
sidewall panels with rivets modeled as both kinematic couplings and CBEAM elements gave
essentially the same displacement response. Therefore, the use of CBEAM elements in lieu of
kinematic couplings for rivets did not significantly affect the results.
Modes classification. It is sometimes helpful to classify modes by their spatial characteristics
to facilitate the discussion. Several classifications of basis functions used in this reduced
order analysis approach have been previously offered for simple symmetric structures [7, 8].
For a complicated structure which does not preserve symmetry, these classifications are not
relevant. However, examination of the eigenvectors obtained for the fuselage sidewall panel
led to an observation that the modes can be classified into three other categories; global, local
and substructure modes. Global modes are generally low frequency modes whose shapes are
not significantly influenced by the substructure. Figure 2 shows the lowest global mode of the
panel. Local modes are ones in which the substructure is stiff relative to the individual bays.
Hence, each bay responds independent of the global motion, as shown in Figure 3.
Substructure modes are those modes dominated by the motion of the substructure, as shown in
Figure 4. A great number of modes defy this simplistic classification, and may exhibit mixed
characteristics of two or more of the basic classification. For example, the mode shown in
Figure 5 exhibits characteristics of both a local and a substructure mode.
Figure 1: Finite element model of multi-bay fuselage sidewall panel. Rivet locations are
indicated with the symbol •.
As previously indicated, a nonlinear dynamic analysis of the panel in physical DoFs was
prohibitive because of the model size. Therefore, a nonlinear static analysis was first
performed so that results from the reduced order nonlinear analysis could be compared with
those obtained from a nonlinear static analysis in physical DoFs. The static response analysis
is useful because it helps to isolate differences between the physical and reduced-order
Figure 2: Global mode (Mode 1 - 68.13 Hz) Figure 3: Local mode (Mode 6 - 122.3 Hz)
Figure 4: Substructure mode
(Mode 7 -124.9 Hz)
Figure 5: Mixed local-substructure mode
(Mode 37 -271.4 Hz).
Quasi-static loading. A uniformly distributed pressure load was applied to the outside skin
surface. Five loading levels were considered, resulting in a nearly linear response at 0.01 psi
to highly nonlinear response at 1.00 psi. The load was applied in a quasi-static fashion by
ramping it up from zero magnitude at time zero to the specified maximum level at 1.6384s.
An additional 0.5s of constant loading was applied at the maximum level to allow any
transient behavior to decay, giving the quasi-static modal response at 2.1384s. This loading
scheme allowed the same solution procedure to be used for both static and random analyses.
Modal participation factor. For a simple structure, basis function selection is often quite
intuitive, given available closed-form solutions, symmetry of the structure and/or loading,
boundary conditions, low modal density, etc. For the non-symmetric multi-bay panel
considered, the choice is less obvious. Therefore, a more methodical approach using the
modal participation factor, v^, was utilized. The modal participation factor provides a
compact measure of the participation of the /^ mode in an expansion utilizing L modes and is
The cumulative modal participation is the sum of the largest S modal participation factors, i.e.,
By computing v^ for each of L modes, the S modes with the highest participation are easily
identified. Those at or above a particular level can be included in subsequent analyses. The
benefit of using this approach is that the modal basis selection is not affected by a physical
location or DoF of interest. It avoids the situation in which a poor choice of physical location
or DoF, e.g. on a nodal line, adversely influences the basis selection. On the other hand, since
the modal displacement response must be computed, v^ and Zs ^^^ load-specific, so a basis
selection made at one level and distribution of applied loading may not perform well at a
different level or distribution. As will be shown, such is the case when the basis selection is
made in the linear response regime and applied in the nonlinear regime.
Modal basis selection. The NASTRAN eigenanalysis (solution 103) indicated 89 modes in
the frequency range 0-500 Hz. This bandwidth was selected for consistency with the random
analysis that follows. A nonlinear modal reduction utilizing all 89 modes was not feasible,
since it would involve obtaining 125,759 nonlinear restoring force fields, as per Equation (5).
Therefore, a further modal reduction was sought. The following approach was undertaken. A
linear modal reduction was first performed utilizing all 89 modes. This operation is
computationally expedient as the linear stiffness is obtained directly from the linear
eigenanalysis. A uniform pressure of 0.01 psi was then applied in a quasi-static fashion and
modal participation factors v^^^ were computed for each mode. The modal participation factor
for the eight most significant modes was found to be roughly 1.0% or greater, giving a
cumulative participation of Zs^ =94.21%. The modal participation for the nineteen most
significant modes was found to be roughly 0.25%) or greater, giving a cumulative participation
of ^'f^ =98.35%. A plot of the cumulative participation zT for the highest participating
nineteen modes is shown in Figure 6. It is seen that there is a point beyond which the addition
of modes yields little added accuracy. At the same time, the number of nonlinear static
solutions grows rapidly with each additional mode. The difference between the nineteen and
eight-mode cumulative participation is a factor of 1.04, while the number of static nonlinear
problems required increases by a factor of 9.38 (from 164 to 1539). Lastly, note the
importance of selecting the highest contributing modes. For example, a selection of the
lowest nineteen modes versus the nineteen modes with the highest participation yields a
cumulative participation of 91.87%) compared to 98.35%).
Static displacement response. The static transverse response at the center of the center bay
is shown as a function of static pressure load in Figure 7. Three solutions are shown; a linear
modal solution including all 89 modes below 500 Hz, a reduced order nonlinear solution
(RANSTEP) utilizing nineteen modes, and a NASTRAN nonlinear static solution (solution
106) in physical DoFs. The linear modal solution is applicable at the lowest level of 0.01 psi.
As the loading increases, the linear solution becomes excessively conservative compared to
the nonlinear analysis in physical DoF. The results obtained via the RANSTEP analysis are
shown capable of capturing the nonlinear stiffening effect, although this effect is over-
pronounced and results in a non-conservative prediction relative to the physical DoF solution.
£ 20 -
No. Static Cases
- 1500 w
5 10 15
Number of Modes (S)
Figure 6: Cumulative participation jjf^ and
number of static cases as function of S modes.
^ Physical DoF
Static Pressure (psi)
Figure 7: Static transverse displacement
response at center of center bay.
To understand the cause of this difference, it is helpful to see how the modal participation
changes with increasing load. The nineteen selected mode numbers, frequencies and types
(G-global, L-local, S-substructure, LS-mixed local/substructure) are listed in Table 1. The
modal participation v^^ is used since computing v^^^ would have necessitated 125,759
nonlinear static solutions for the nonlinear regime. At the 1.0 psi loading, it is seen that the
participation of the first mode drops dramatically relative to the 0.01 psi loading. Further, the
contribution of the higher frequency modes generally increases with increasing load.
Therefore, it can be assumed that the modes not included in the original selection based on a
linear analysis also become increasingly important as the response becomes nonlinear. Since
the modal truncation error increases with increasing nonlinearity, it suggests that the basis
selection should be performed in the nonlinear regime for increased accuracy. Nevertheless,
the results shown demonstrate that the reduced order nonlinear modal simulation captures the
salient features of the nonlinear static response for a high-DoF stiffened structure.
Random acoustic loading. A uniformly distributed random pressure load was applied to the
outside skin surface. The loading spectrum is shown in Figure 8 and is based on
measurements made on a military aircraft at sea level. Loading time histories were generated
by summing sine waves with prescribed amplitudes and random phase, within the specified
bandwidth using a discrete inverse Fourier transform. A sharp roll-off of the input spectrum
at the cut-off frequency of fc = 500 Hz practically eliminates direct excitation of the structure
outside the frequency range of interest. Seven loading levels were considered to span the
response regime from essentially linear (106 dB OASPL or 5.567x10"^ psi RMS) to highly
nonlinear (160 dB OASPL or 0.29 psi RMS). At each level, ten separate loading time
histories, each with a 2.1384s duration, were generated for ensemble averaging.
Table 1: Modal participation factors v^^ obtained from the static analysis.
Pressure Loading (psi)
Modal participation and basis selection. A linear reduced order modal simulation was
performed at a very low excitation level of 106 dB OASPL to assess the modal participation.
The simulation utilized all 89 modes present in the bandwidth of interest. For the random
loading, the modal participation v^ may be written as
Like the static case, the same eight and nineteen modes were found to exceed 1.0% and 0.25%
of v^^^ , respectively, but with slightly different modal participation factors. The cumulative
modal contribution of the most significant eight modes was found to be zl^ = 92.78% , and
that of the most significant nineteen modes was found to be ^'f^ = 97.95% . In other words, at
the 106 dB excitation level, the truncated 81 modes for the eight-mode solution represented
7.22%) of cumulative contribution, while the 70 truncated for the nineteen-mode solution
represented 2.05% of cumulative contribution. The cumulative participation ^^^ for the most
significant nineteen modes is also shown in Figure 6 and resembles the behavior of the static
Random displacement response. For the random response analysis, a mass proportional
damping corresponding to 2.0% critical damping for the fundamental mode was specified.
The Runge-Kutta integration used a fixed integration time step between 50//S (for the lowest
excitation levels) and 5//s (for the highest excitation levels). The first 0.5s of each response
record was discarded to remove the initial transient response, resulting in response histories of
1.6384s in duration. For each simulation, the modal response was stored at every 50//S,
giving time records of 32,768 points.
The RMS transverse displacement response at the center of the center bay is shown in Figure
9. The linear modal results were obtained using all 89 modes, and the reduced order nonlinear
RANSTEP results were obtained using nineteen modes. The nonlinear behavior is as
expected, with the onset of geometrically nonlinear response emerging in the range between
136 dB OASPL (0.0178 psi) and 142 dB OASPL (0.0356 psi).
Figure 8: Random pressure loading spectrum.
1 innir MnHnI
1 1 1
0.05 0.1 0.15
RMS Pressure (psi)
Figure 9: RMS transverse displacement
response at center of center bay.
Transverse displacement PSDs at the center of the center bay were obtained from linear and
nonlinear modal simulations at low (106 dB), moderate (136 dB), and high (154 dB OASPL
or 0.145 psi) levels of acoustic excitations, see Figure 10 to Figure 13. Figure 10
demonstrates that the only difference in the linear response regime is attributable to modal
truncation above the highest selected mode at 304.5 Hz. Figure 1 1 illustrates why an analysis
of a single panel is problematic even in the linear response regime. The linear modal response
shown was obtained from the full multi-bay panel analysis and is identical to that presented in
Figure 10. Overlaid are responses obtained from physical DoF analyses of a single panel
having the same dimensions as the center bay of the full panel, but without substructure.
Either fully clamped or simply supported boundary conditions were specified at what would
have been the rivet lines. Clearly, neither of the simplified analyses is capable of predicting
the response across the full frequency range. While a frequency dependent boundary
condition could be developed for the linear regime, no such remedy exists for the nonlinear
Center SS Panel Only
Center CL Panel Only
Linear Modal (Multi-Bay)
J I L
Figure 10: Linear and nonlinear transverse Figure 1 1 : Comparison of multi-bay panel and
displacement PSD at 106 dB OASPL. center bay analyses at 106 dB OASPL.
Figure 12 shows that as the excitation level increases to 136 dB, the PSD peaks resulting from
the nonlinear analysis begin to shift toward higher frequencies and to broaden relative to the
linear analysis. Nonlinear versus linear PSD peaks cross-identification, however, remains
possible at this level. This level of nonlinearity is comparable to the highest level studied for
a much thicker panel in reference . In the highly nonlinear response regime. Figure 13,
peak cross-identification is no longer possible. The response is dominated by one broad
washed-out peak, which likely originates from several modes included in the analysis.
Figure 12: Linear and nonlinear transverse
displacement PSD at 136 dB OASPL.
Figure 13: Linear and nonlinear transverse
displacement PSD at 154 dB OASPL.
It is helpful to investigate how the modal participation changes as a function of response level.
Table 2 provides the modal participations v^^ and v^^ at three load levels. As the loading
increases, the participation of the fundamental mode decreases in favor of greater participation
of higher frequency modes. For example, v^ = 0.35% at 106 dB, but increases to 6.42% at
154 dB. Furthermore, in the linear response regime (106 dB), v^^ and v^^ are very similar for
the modes that are included in both analyses, indicating that modal convergence has been
achieved. However, as the loading increases, v^^ and v^^ begin to vastly differ due to modal
truncation. As in the static case, this suggests that the basis selection should be performed in
the nonlinear regime for increased accuracy. Unfortunately, this requires solving the problem
once the hard way, i.e., solving for all nonlinear modal stiffness terms and integrating the
larger coupled system to obtain the modal participation, so that subsequent analyses can be
performed more efficiently with a reduced basis.
Table 2: Modal participation factors v^^ and v^^ obtained at different loading levels.
A nonlinear reduced-order static and random response analysis of a typical aircraft fuselage
sidewall panel was presented. The reduced-order static analysis captured the essential
features of the nonlinear behavior, but over-predicted the effect of nonlinearity compared with
a solution in physical DoFs. The reduced-order random response analysis was shown to
predict the expected behavior, but quantitative comparisons with a physical DoF solution were
not available due to their prohibitive computational cost.
The modal participation was found to change as a function of the nonlinear response. The
participation of the lowest, most dominant mode decreased with increasing nonlinearity, while
that of the higher frequency modes increased. Therefore, a converged modal basis obtained
via a linear analysis produced truncation error in the nonlinear response regime.
Consequently, an alternative means of identifying the modal participation in the nonlinear
regime is needed without the expense of computing the full set of nonlinear modal stiffness
terms. This is an area requiring further study.
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