NASA/TP-2005-213943

The Effect of Basis Selection on Static and 

Random Acoustic Response Prediction Using a 

Nonlinear Modal Simulation 

Stephen A. Rizzi 

Langley Research Center, Hampton, Virginia 

Adam Przekop 

National Institute of Aerospace, Hampton, Virginia 

 

December 2005 
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NASA/TP-2005-213943

The Effect of Basis Selection on Static and 

Random Acoustic Response Prediction Using a 

Nonlinear Modal Simulation 

Stephen A. Rizzi 

Langley Research Center, Hampton, Virginia 

Adam Przekop 

National Institute of Aerospace, Hampton, Virginia 

National Aeronautics and 

Space Administration 

Langley Research Center 

Hampton, Virginia 23681-2199 

 

December 2005 
The use of trademarks or names of manufacturers in the report is for accurate reporting and does not constitute an

official endorsement, either expressed or implied, of such products or manufacturers by the National Aeronautics 

and Space Administration. 

Available from: 

NASA Center for AeroSpace Information (CASI) National Technical Information Service (NTIS) 

7121 Standard Drive 5285 Port Royal Road

Hanover, MD 21076-1320 Springfield, VA 22161-2171 

(301) 621-0390 (703) 605-6000
ABSTRACT 

An investigation of the effect of basis selection on geometric nonlinear response prediction using 

a reduced-order nonlinear modal simulation is presented. The accuracy is dictated by the 

selection of the basis used to determine the nonlinear modal stiffness. The scope of this 

investigation is limited to structures which do not exhibit linear bending-membrane coupling, but 

do exhibit nonlinear bending-membrane coupling. This study considers a suite of available bases 

including bending modes only, bending and membrane modes, coupled bending and companion 

modes, and uncoupled bending and companion modes. Companion modes represent an 

alternative to membrane modes and capture some of the membrane behavior resulting from 

bending-membrane coupling. The nonlinear modal simulation presented is broadly applicable 

and is demonstrated for nonlinear quasi-static and random acoustic response of flat beam and 

plate structures with isotropic material properties. Reduced-order analysis predictions are 

compared with those made using a numerical simulation in physical degrees-of-freedom to 

quantify the error associated with the selected modal bases. Bending and membrane responses

are separately presented to help differentiate the bases. 

1. INTRODUCTION 

The design of advanced aerospace vehicle components capable of withstanding high

vibroacoustic environments is hampered by a lack of accurate and computationally fast methods. 

Such methods are required in the design phase to quickly assess the impact of design changes on 

high-cycle fatigue life. Linear analysis methods are often inappropriate as structures may 

respond in a geometrically nonlinear fashion. Therefore, the use of a nonlinear analysis is 

required. Complicated structural geometries dictate the use of a finite element analysis (FEA). 

The traditional FEA employing numerical simulation in physical degrees-of-freedom (DoFs),

however, is computationally intensive and considered impractical in design environments where 

rapid prototyping is needed. For stochastic response, the computational burden is exacerbated by

the need to perform probabilistic analysis, such as Monte Carlo simulation [1, 2], which requires 

that multiple realizations of the response be computed to generate meaningful statistics.

Therefore, alternative methods are sought which retain the level of accuracy required, yet are 

computationally efficient. 

1 
Reduced-order nonlinear finite element analysis methods gain their computational advantage 

over direct numerical simulation by transforming the equations of motion in physical DoFs to 

modal coordinates. Consequently, system size is significantly reduced and can be solved in a 

time efficient manner by means of numerical simulation or by an equivalent linearization 

approach, as dictated by the desired fidelity of the analysis. In the recent past, equivalent 

linearization procedures [3, 4] have been shown to be applicable [5, 6] to this class of problems, 

albeit in an approximate sense. This study focuses on reduced-order numerical integration 

analysis. 

The problems of interest in this paper are those in which the structure responds to imposed loads 

in a geometrically nonlinear (large deflection) static and random fashion. This nonlinearity is

due to bending-membrane coupling and gives rise to membrane stretching when out-of-plane 

loading is applied. Consideration of structures which exhibit linear bending-membrane coupling 

is outside the scope of this work. Structures which exhibit linear bending-membrane coupling 

include curved structures and non-symmetrically laminated composites with a non-zero coupling 

stiffness matrix [B]. In the recent past, a significant amount of research has been performed in 

reduced-order methods development that is applicable to the class of problems of interest. These 

methods may be viewed as being in one of two categories; those in which the nonlinear modal

stiffness is directly evaluated from the nonlinear finite element stiffness matrix (so-called direct 

methods), and those in which the nonlinear modal stiffness is indirectly evaluated. Direct 

methods are typically implemented in special purpose finite element codes in which the 

nonlinear stiffness is known. The work by Mei et al [7] for flat configurations and the expansion 

of that approach for curved panels by Przekop et al [8, 9] are good examples of the direct 

method. In physical coordinates, the membrane displacements may be statically condensed into 

the bending displacements by neglecting in-plane inertia, and thus eliminate the need for

membrane modes in the modal basis. This approach is sometimes referred to as “direct physical 

condensation” [10]. Alternatively, modal in-plane inertia may be neglected once the system of 

equations is transformed to modal coordinates, in an approach subsequently referred to as “direct 

modal condensation.” In this variation, the membrane behavior must still be represented in the 

basis. 

2
For the analysis of complicated structures, commercial finite element codes are often required

due to their support for a large number of element types. Unfortunately, the nonlinear stiffness is 

not typically available, making implementation of the direct method not possible. The only 

known implementation of a direct method into a commercial code is due to Bathe and Gracewski 

[11], in the ADINA finite element program. Indirect stiffness evaluation methods arose from the 

desire to implement reduced-order nonlinear analyses within the context of any commercial 

finite element analysis. Examples of indirect stiffness evaluation approaches may be found in 

the work of McEwan et al [12, 13] and Muravyov and Rizzi [14]. The approach taken by 

McEwan involves solution of a series of nonlinear static problems, typically through application 

of static forces with distributions corresponding to bending modes only. Following a modal 

coordinate transformation, the nonlinear modal stiffness is determined by curve fitting the modal 

force – displacement relation. Because the nonlinear static problem uses prescribed forces, the 

displacement response is influenced by the effect of membrane stiffness resulting from nonlinear 

bending-membrane coupling. The effect of membrane modes is thus implicitly condensed into 

the bending modes. This is advantageous in that explicit inclusion of membrane modes in the

modal basis is eliminated. The implicit condensation of membrane modes however renders the 

approach unable to directly determine membrane displacements. Recently a post-processing

procedure involving a mapping technique, called an “estimated expansion basis” [15], was

developed to mitigate this problem. Nevertheless, the implicit condensation method is capable 

of producing only cubic nonlinear modal stiffness terms [15]. Its general applicability to nonplanar structures, in which the effects of quadratic stiffness and membrane inertia may be

significant, has yet to be explored. 

By contrast, the approach undertaken by the authors solves a series of simple algebraic equations

obtained from static nonlinear analyses using prescribed displacements obtained from a

combination of basis vectors. For the problems under consideration, the low-frequency bending 

modes obtained from the linear eigenvalue problem are uncoupled from the high-frequency 

membrane modes. In past works by the authors [5, 14], the basis vectors were formed from only 

low-frequency bending modes. However, to more accurately represent the effect of nonlinear 

bending-membrane coupling, the membrane response must be explicitly represented through 

inclusion of some form of membrane response in the basis. Inclusion of membrane modes may

be cumbersome because the task of identifying a particular high-frequency membrane mode 

3
amongst a great number of computed modes is labor intensive. In recognition of that, Hollkamp

et al [10, 15] and Mignolet et al [16] have developed “companion” or “dual” modes to help 

capture the effect of membrane response, without explicitly including membrane modes in the 

basis. 

For both direct and indirect stiffness evaluation approaches, the accuracy of the solution depends

on the selection of the modal basis, through which the nonlinear modal stiffness may be 

determined. If an insufficient basis is selected, then the predicted dynamic response of the 

reduced-order model may significantly differ from that of the full nonlinear model. Thus, there

is always a need to assess the appropriateness of the selected basis via comparison of the 

predicted reduced-order response with something other than a reduced-order method, e.g. 

experimental data or numerical simulation in physical DoFs. Recent work compared various 

reduced-order approaches with experimental data [10]. In this study, comparisons are made with 

numerical simulation in physical DoFs to permit identical specification of boundary conditions.

This paper assesses the effect of basis selection on the response obtained from a nonlinear modal

simulation, utilizing the authors’ indirect stiffness evaluation method. A suite of bases is 

considered including bending modes only, bending and membrane modes, coupled bending and 

companion modes, and uncoupled bending and companion modes. The effect of basis selection 

on the modal stiffness coefficients themselves is first investigated. Then, using these

coefficients, the nonlinear quasi-static and random response of simple planar aluminum beam

and plate structures under spatially uniform excitation is considered. These structures were 

selected to help keep the cost of the comparative physical DoFs simulation reasonable, yet retain 

the nonlinear bending-membrane coupling behavior of interest. The error associated with the

modal basis selection is quantified for both the displacement and stress response. Bending and 

membrane responses are separately presented to help differentiate the bases. 

2. NONLINEAR MODAL SIMULATION 

The nonlinear modal simulation analysis employed in this work consists of several parts. One or 

more methods, to be discussed, are first used to obtain a modal basis. Following a

transformation of the nonlinear system to modal coordinates, the modal stiffness coefficients are 

evaluated and the resulting coupled system of equations is numerically integrated to obtain the 

4
modal displacement time history. These are transformed back to physical coordinates for postprocessing, including stress recovery. 

2.1. MODAL COORDINATE TRANSFORMATION 

The equations of motion of the nonlinear system in physical DoFs may be written as 

( )t t + + ( ) ( (t)) = ( ) (1) MX CX FNL X F t

where M and C are the mass and proportional damping matrices, respectively, X is the

displacement response vector and F is the force excitation vector. As written, the nonlinear

restoring force vector, FNL , contains the linear force KX and nonlinear forces, where K is the

linear stiffness. 

A set of coupled modal equations with reduced DoFs is first obtained by applying the modal

coordinate transformation X = Φq to Equation (1), where q is the vector of modal coordinates. 

The modal basis matrix Φ is typically formed from the eigenvectors obtained from Equation (1) 

using only the linear stiffness. For flat isotropic structures, these may include any combination 

of bending and membrane modes. In lieu of membrane modes, the modal basis may include 

“companions” related to the membrane response, as discussed in the next section. Generally, a 

small set (L) of basis vectors are included giving 

 (2) 1 2 ( ) ( ) ( ( ), ( ), , ( )) ( ) L Mq   t t + + Cq  FNL q t q t ... q t = F t 

where, for mass-normalized eigenvectors, 

2

T

T

r r

T

NL NL

T

ζ ω

= = ⎡ ⎦

= = ⎡ ⎦

=

=

M M Ι

C C

F F

F F









Φ Φ

Φ Φ

Φ

Φ

(3)

and ω r are the undamped natural frequencies and r ζ are the viscous damping factors. 

5
2.2. MODAL BASIS SELECTION 

For the problems of interest in this paper, both bending and membrane behavior should be 

included in the basis selection since the large deflection nonlinearity couples their response. The 

basis vectors may be determined via several methods. Bases corresponding to the bending and 

membrane response may be determined through solution of the linear eigenvalue problem. Other 

basis vectors corresponding to the membrane response induced by bending-membrane coupling 

may be determined via alternative approaches, as discussed in Sec. 2.2.2. Lastly, while not the 

subject of this paper, basis vectors may also be determined via experiment or a hybrid scheme. 

2.2.1 Linear Eigenvectors 

Recall that for the problems of interest, the linear eigenvectors, obtained from Equation (1) using 

only the linear stiffness, are uncoupled and are either associated with low-frequency bending 

modes or high-frequency membrane modes. The selection of which bending modes to include 

depends on a number of factors including the excitation bandwidth, the spatial loading

distribution, and even geometric and material properties. The selection of which membrane

modes to include is less apparent than bending modes, as these high-frequency modes typically 

reside above the excitation bandwidth. Nevertheless, a reasonable starting point is to select the 

lowest membrane modes that are consistent with the spatial loading distribution and other 

physical properties. For example, for a uniform flat structure under uniform transverse loading,

the membrane displacement will be anti-symmetric, so anti-symmetric membrane modes would 

be selected. Inclusion of both bending and membrane eigenvectors in the modal basis, either 

independently or in pairs, is subsequently referred to as the bending and membrane mode (BM) 

basis. Inclusion of only the bending eigenvectors will be referred to as the bending mode only

(B) basis. In this study, the mass-normalized eigenvectors were obtained using 

MSC.NASTRAN normal modes analysis (solution 103). 

2.2.2 Companion Basis Vectors 

An alternative approach to using membrane modes is the use of so-called companion [10] or dual 

[16] modes. These modes are meant to represent the membrane behavior resulting from bending 

due to bending-membrane coupling. Previous authors utilized quasi-static approaches to 

determine the companion mode via nonlinear static analyses. For each companion mode, 

6
Hollkamp et al [10] prescribed a displacement field corresponding to the bending mode of 

interest. Mignolet et al [16] applied two loadings, each having the distribution of a particular 

bending mode shape but with different magnitudes, to obtain the companion for that bending 

mode. In either case, the resulting modes contained only the membrane related behavior, but not 

the bending. The companion modes obtained need to be mass-normalized prior to their use. 

A new method is now presented for computing the companion mode using a dynamic analysis. 

An initial stress-free, out-of-plane perturbation of the mesh is first introduced to couple the 

bending-membrane response. The magnitude of the perturbation is chosen to be very small such 

that a normal modes analysis yields virtually the same bending eigenvalues and eigenvectors as 

that of the flat structure. An engineering rationale for specifying the shape of the perturbation is 

not evident. It was found that the companion modes change with varying perturbation shapes.

In this investigation, the shape was chosen to be that of the first bending mode. The bending and 

high frequency membrane components remained identical to that of the flat structure, regardless 

of the shape of the perturbation. The MSC.NASTRAN normal modes analysis was again used to 

compute the mass-normalized eigenvectors, which now contain both the bending and membrane

behaviors, but at the natural frequencies of the original flat structure. Direct inclusion of these 

eigenvectors in the modal basis is subsequently referred to as the coupled bending and 

companion mode (CBC) basis. 

A more consistent usage of companion modes, with respect to references [10, 16], is to separate 

the DoFs associated with the bending and membrane behaviors. In practice, since the bending 

behavior is unchanged, the original low-frequency bending modes are retained. The bending

DoFs are set to zero in the newly obtained eigenvector to obtain the uncoupled dynamic

companion. Since the uncoupled dynamic companion is essentially obtained by partitioning the 

mass-normalized CBC mode, it is no longer itself mass-normalized. Therefore, an additional 

step of mass-normalization is necessary. Inclusion of both the original bending and uncoupled 

dynamic companion modes in the modal basis, independently or in pairs, is subsequently 

referred to as the uncoupled bending and companion mode (UBC) basis. 

7
2.2.3 Comparison of Membrane and Companion Modes 

Consider a clamped-clamped aluminum beam measuring 18-in. x 1-in. x 0.09-in (l x w x h) with 

the following material properties: 

6 6

2

10.6 10 4 , 4.0 10 , 2.588 10 f lb s

in E psi G psi ρ − − = × = × = × 4

The beam was modeled in MSC.NASTRAN using 144 CBEAM elements. At the clamped ends, 

all DoFs are constrained. For the uniformly distributed loadings to follow, the first six

symmetric bending modes (eigenvectors 1, 3, 7, 10, 14 and 19) at natural frequencies of 58, 312,

770, 1431, 2293 and 3354 Hz, respectively, were selected. The first six anti-symmetric 

membrane modes (eigenvectors 46, 81, 115, 153, 221 and 231) are at natural frequencies of 11.2, 

22.5, 33.7, 44.9, 56.1 and 67.3 kHz, respectively. The first four of these are plotted in Figure 1 – 

Figure 4. Also shown are the static companion modes obtained by Mignolet’s approach [16] and 

the dynamic companion modes obtained by the method outlined above. It is clear that both static 

and dynamic companion modes significantly differ in shape relative to the membrane mode. The 

effect of their inclusion in the modal basis on the stiffness coefficients is next considered. 

2.3. INDIRECT STIFFNESS EVALUATION METHOD 

The indirect stiffness evaluation method previously developed [14, 16] was used in this study. 

To summarize, the nonlinear force vector in Equation (2) may be written in the form

 (4) 1 2

1 1 1

( , , , ) 1,2, ,

L L L LLL r r r

NL L j j jk j k jkl j k l

j j k j j k j l k

F q  q ... ... q d q a q q b q q q r L = = = = = == + ∑ ∑∑ +∑∑∑ =

reducing the problem of determining the nonlinear stiffness from one in which a large set of 

simultaneous nonlinear equations must be solved to one involving simple algebraic relations. 

The linear, quadratic, and cubic nonlinear modal stiffness coefficients are written as , and

, respectively. 

j d a jk

jkl b

8
Dimensionless Beam Span (x/L)

N

ormalized

M

o

d

e

S

hap

e

0 0.25 0.5 0.75 1 -1.5

-1

-0.5

0

0.5

1

1.5 Membrane

Dynamic Companion

Static Companion

Figure 1: First anti-symmetric membrane and companion modes. 

Dimensionless Beam Span (x/L)

N

ormalized

M

o

d

e

S

hap

e

0 0.25 0.5 0.75 1 -1.5

-1

-0.5

0

0.5

1

1.5 Membrane

Dynamic Companion

Static Companion

Figure 2: Second anti-symmetric membrane and companion modes. 

9
Dimensionless Beam Span (x/L)

N

ormalized

M

o

d

e

S

hap

e

0 0.25 0.5 0.75 1 -1.5

-1

-0.5

0

0.5

1

1.5 Membrane

Dynamic Companion

Static Companion

Figure 3: Third anti-symmetric membrane and companion modes. 

Dimensionless Beam Span (x/L)

N

ormalized

M

o

d

e

S

hap

e

0 0.25 0.5 0.75 1 -1.5

-1

-0.5

0

0.5

1

1.5 Membrane

Dynamic Companion

Static Companion

Figure 4: Fourth anti-symmetric membrane and companion modes. 

10
For the prescribed static displacement fields [16]

(5) 1 1 1 2 1 1 3 1 ˆ c c c X X = +φ φ q q = − X = + 1 φ q

q

q

]

]

the nonlinear forces evaluated using the MSC.NASTRAN nonlinear static solution (solution 106) 

are given as 

 (6) 

1

2

3

1 1 1 1 11 1 1 111 1 1 1

1 1 1 1 11 1 1 111 1 1 1

1 1 1 1 11 1 1 111 111

( )

( )

( ) ˆ ˆ ˆ ˆ ˆˆˆ

T r r r

NL NL

T r r r

NL NL

T r r r

NL NL

q d q a q q b q q

q d q a q q b q q

q d q a q q b qqq







= + = + ⎡ ⎤ + ⎡ ⎤ + ⎡ ⎤ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦

= − = − ⎡ ⎤ + ⎡ ⎤ − ⎡ ⎤ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦

= + = + ⎡ ⎤ + ⎡ ⎤ − ⎡ ⎤ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦

F F

F F

F F

Φ φ

Φ φ

Φ φ

The first two of Equation (6) are used to evaluate . The first and the third of Equation (6) 

are then used to determine and . It should be noted that the specified modal 

displacements in Equation (5) are single scalar quantities since the analysis is static. These 

modal displacements are specified such that the magnitude of the prescribed physical 

displacement field 

11 [ r a

1 [ ]

r d 111 [ r b

Xc is physically meaningful. It has been shown that the stiffness coefficients 

obtained via this approach are not sensitive to the particular value of modal displacement 

specified [14]. The last of Equation (5) are obtained by specifying a scaled modal displacement 

such that qˆ =αq . It was found in the course of this work that the stiffness coefficients obtained

were also insensitive to the particular value of α used. In this paper, values of 5 q 7 10− = × and 

α = 1.25 were used for all modes in the basis.

This procedure for evaluating the linear stiffness coefficients is advantageous over the

earlier implementation [14] using the MSC.NASTRAN linear static solution (solution 101), as 

the linear solution does not include in-plane displacements, and therefore is not capable of

handling thermal loads. The remaining nonlinear coefficients are evaluated through an additional 

series of nonlinear static solutions, as described in [14]. 

1 [ ]

r d

2.3.1 Comparison of Modal Stiffness Coefficients 

For the clamped-clamped beam structure previously considered, the effect of including a 

particular set of modes in the modal basis is shown in Table 1 – Table 5. Table 1 shows the 

11
diagonal terms of the linear stiffness coefficients for a set consisting of the first six symmetric 

bending modes ( - ) plus up to six membrane or companion modes ( - ). For each

basis, the first six linear stiffness coefficients are equal to the eigenvalues of the selected bending

modes. For the BM basis, the second six linear stiffness coefficients correspond to the 

eigenvalues of the selected membrane modes. The CBC basis lacks the second set of six

coefficients as only six modes are included in this basis, i.e. six coupled bending and companion 

modes. It is interesting to note that the stiffness coefficients for companion modes of both UBC 

bases differ from each other and from the membrane. Thus, there is no physical meaning to the 

linear stiffness coefficients - associated with the companion modes for the UBC bases.

Further, although comparable in magnitude to the membrane modes, the stiffness coefficients for 

companion modes of both UBC bases do not increase monotonically. For the CBC basis and 

both UBC bases, the off-diagonal terms are non-zero (not shown), indicating coupling of the

linear stiffness where none should exist. 

1

1 d 6

6 d 77 d 1212 d

7

7 d 1010 d

Table 1: Diagonal terms of beam linear stiffness obtained using five different modal bases. 

Shaded cells indicate bending eigenvalues. 

B BM CBC UBC UBC[16]

(static) 

1

1 d 1.318E+05 1.318E+05 1.318E+05 1.318E+05 1.318E+05 

2

2 d 3.845E+06 3.845E+06 3.845E+06 3.845E+06 3.845E+06 

3

3 d 2.341E+07 2.341E+07 2.341E+07 2.341E+07 2.341E+07 

4

4 d 8.080E+07 8.080E+07 8.080E+07 8.080E+07 8.080E+07 

5

5 d 2.075E+08 2.075E+08 2.075E+08 2.075E+08 2.075E+08 

6

6 d 4.442E+08 4.442E+08 4.442E+08 4.442E+08 4.442E+08 

7

7 d N/A 4.990E+09 N/A 7.090E+10 1.487E+10

8

8 d N/A 1.995E+10 N/A 7.621E+10 7.520E+10

9

9 d N/A 4.485E+10 N/A 2.587E+09 5.846E+10

10

10 d N/A 7.965E+10 N/A 2.057E+10 7.803E+10

11

11 d N/A 1.243E+11 N/A N/A N/A

12

12 d N/A 1.786E+11 N/A N/A N/A

Unlike direct physical and direct modal condensation, or the implicit condensation in McEwan’s 

indirect approach, the effect of quadratic stiffness in the present stiffness evaluation approach is 

12
not incorporated in the cubic stiffness, and must therefore be explicitly represented as indicated

in Equation (4). Diagonal terms for the quadratic stiffness coefficients are shown in Table 2. 

The diagonal terms are expected to be zero for this flat structure, and essentially are for the B

and BM bases. However, the CBC basis indicates significant coefficients for the first six modes

( - ). Stiffness coefficients for the companion modes of both UBC bases ( - ) are 

also significant, although their effect on the response is likely small for out-of-plane loadings. 

Interestingly, while the presence of anti-symmetric companion modes introduces significant 

diagonal terms for the CBC and UBC bases, the presence of anti-symmetric membrane modes 

does not affect the diagonal terms for the BM basis. The off-diagonal terms corresponding to the 

first (1

1

11 a

6

66 a

7

77 a 10

10 10 a

st symmetric bending) and seventh (1st membrane or companion) modal equations are 

shown in Table 3, and indicate a coupling between modes for both UBC bases and the BM basis. 

Table 2: Diagonal terms of beam quadratic stiffness obtained using five different modal bases. 

B BM CBC UBC UBC[16]

(static) 

1

11 1 a q -6.001E-11 -6.653E-11 6.937E+01 -3.673E-11 -1.737E-10

2

22 2 a q 4.044E-10 1.028E-10 -1.474E+03 1.874E-12 4.471E-10

3

33 3 a q -1.400E-09 1.061E-09 2.701E+03 -5.210E-10 7.226E-09

4

44 4 a q -2.658E-09 -1.834E-09 3.879E+03 2.047E-09 -1.192E-07

5

55 5 a q 7.535E-09 -6.914E-09 -5.006E+03 7.684E-10 5.356E-09

6

66 6 a q -3.101E-08 -2.980E-08 -6.295E+03 -4.212E-08 -2.676E-08

7

77 7 a q N/A 8.864E-06 N/A -7.245E-01 6.642E+00

8

88 8 a q N/A -4.808E-05 N/A 1.767E+00 1.570E+01

9

99 9 a q N/A -4.976E-05 N/A -5.094E-01 -7.283E+02

10

10 10 10 a q N/A -1.408E-04 N/A -5.227E-01 -4.782E+03

11

11 11 11 a q N/A -4.697E-05 N/A N/A N/A

12

12 12 12 a q N/A 5.307E-04 N/A N/A N/A

Relationships between quadratic modal stiffness coefficients were previously found in [14] and 

are given by 

1 7

17 11

1

77 17

2

2

a a

a a

=

7 = . (7)

13
Equation (7) should be satisfied for only the significant nonlinear coefficients. For the quadratic 

terms presented in Table 3, the BM basis satisfies the first of Equation (7). For both UBC bases, 

however, the significant terms do not satisfy the first of Equation (7). The second of Equation 

(7) is not satisfied for any basis because these terms are not significant. 

Table 3: Selected off-diagonal quadratic stiffness terms obtained using five modal bases. 

B BM CBC UBC UBC[16]

(static) 

1

11 1 a q -6.001E-11 -6.653E-11 6.937E+01 -3.673E-11 -1.737E-10

1

17 1 a q N/A -2.320E+05 N/A 4.625E+01 -2.398E+01

1

77 7 a q N/A -1.118E-06 N/A 3.312E-15 3.608E-15

7

11 1 a q N/A -1.160E+05 N/A 7.119E+10 -8.107E+10

7

17 1 a q N/A -3.946E-01 N/A 5.389E+01 3.757E-01

7

77 7 a q N/A 8.864E-06 N/A -7.245E-01 6.642E+00

Table 4 shows the diagonal terms of the cubic stiffness coefficients for the same modal bases as 

above. For each basis, all coefficients corresponding to the bending modes (1-6) are identical. 

Like the quadratic stiffness, coefficients corresponding to the companion modes of both UBC 

bases are large relative to those corresponding to the membrane modes of the BM basis. 

Selected off-diagonal stiffness coefficients are shown in Table 5. The most significant difference 

between the non-zero stiffness terms for the BM and UBC bases is that the and the 

terms are significant for the UBC bases and not for the BM basis. Conversely, the term is 

significant for the BM basis but not for either UBC basis. 

7

111 b 7777 b

1

177 b

Relationships between cubic modal stiffness coefficients were previously found in [14] and are 

given by 

. (8)

1 7

177 117

1 7

777 177

7 1

111 117

3

3

b b

b b

b b

=

=

=

Like the quadratic terms, Equation (8) should be preserved for only the significant nonlinear 

cubic coefficients. For the cubic terms presented in Table 5, the BM basis approximately

14
satisfies the first of Equation (8). For both UBC bases, however, neither the first nor the third of

Equation (8) are satisfied since terms on only one side of the equation are significant. 

Table 4: Diagonal terms of beam cubic stiffness obtained using five different modal bases. 

B BM CBC UBC UBC[16]

(static) 

1 2

111 1 b q 5.067E+02 5.067E+02 5.067E+02 5.067E+02 5.067E+02

2 2

222 2 b q 3.626E+04 3.626E+04 3.626E+04 3.626E+04 3.626E+04

3 2

333 3 b q 2.507E+05 2.507E+05 2.507E+05 2.507E+05 2.507E+05

4 2

444 4 b q 9.124E+05 9.124E+05 9.124E+05 9.124E+05 9.124E+05

5 2

555 5 b q 2.410E+06 2.410E+06 2.410E+06 2.410E+06 2.410E+06

6 2

666 6 b q 5.248E+06 5.248E+06 5.248E+06 5.248E+06 5.248E+06

7 2

777 7 b q N/A -4.897E-05 N/A 1.872E+02 -2.340E+02

8 2

888 8 b q N/A 3.534E-04 N/A -3.522E+00 -1.121E+03

9 2

999 9 b q N/A -3.199E-05 N/A 4.009E+01 1.121E+04

10 2

10 10 10 10 b q N/A 5.931E-07 N/A -1.499E+01 -9.658E+04

11 2

11 11 11 11 b q N/A 2.756E-03 N/A N/A N/A

12 2

12 12 12 12 b q N/A -7.361E-03 N/A N/A N/A

Table 5: Selected off-diagonal cubic stiffness terms obtained using five different modal bases. 

B BM CBC UBC UBC[16]

(static) 

1 2

111 1 b q 5.067E+02 5.067E+02 5.067E+02 5.067E+02 5.067E+02

1

117 1 7 b q q N/A -2.177E-06 N/A -3.244E-11 2.418E-09

1

177 1 7 b q q N/A -1.174E+03 N/A -2.829E-06 9.352E-07

1 2

777 7 b q N/A -6.761E-07 N/A 2.310E-14 9.921E-15

7 2

111 1 b q N/A 7.559E-02 N/A 5.522E+03 4.472E+04

7

117 1 7 b q q N/A -9.437E+02 N/A 8.816E+03 -3.782E+03

7

177 1 7 b q q N/A -3.496E-01 N/A 5.390E-01 1.671E+00

7 2

777 7 b q N/A -4.897E-05 N/A 1.872E+02 -2.340E+02

Further discussion regarding the choice of basis on the response prediction is reserved until 

Section 3. 

15
2.4. NUMERICAL INTEGRATION AND ELEMENT STRESS RECOVERY 

Having the nonlinear force vector in Equation (4) fully defined, the coupled modal nonlinear

equations of motion in Equation (2) may be solved. An equivalent linearization approach was 

presented in [5] to solve the system in an approximate manner. A more accurate approach

utilized in this paper numerically integrates Equation (2) using a fourth-order Runge-Kutta

method [17]. The resulting modal displacement time histories are transformed back to physical 

coordinates using the inverse modal transformation. 

Element stresses are recovered by post-processing the nodal physical DoFs directly within the 

finite element program. Since the stress post-processor of the finite element program is used, the 

stress calculation is identical to what would have been performed via a standard finite element

analysis in physical DoFs. For a particular output time step, the element physical DoFs, obtained

via the nonlinear modal simulation method, are applied to each element node as prescribed 

displacement fields in the MSC.NASTRAN nonlinear static solution and the element stresses are 

calculated. By repeating this operation for each output time step, the stress time history is 

determined. 

3. NONLINEAR STATIC AND RANDOM ACOUSTIC RESPONSE OF A BEAM 

The effect of modal basis selection on the nonlinear response is next considered for the same

beam structure previously examined. It should be noted that the nonlinear modal simulation 

approach considered herein is applicable to arbitrary spatial and temporal loading distributions. 

Since the problems of interest originate from uniform acoustic loadings, the cases selected are 

limited to structures exposed to spatially uniform loadings. Therefore, while the calculation of

nonlinear stiffness coefficients as described above is independent of the loading, conclusions 

drawn about the modal basis selection, specifically the choice of which modes to include for a 

particular basis type, are load-specific. The merits of one basis type (B, BM, CBC and UBC)

versus another, however, can be assessed independent of the loading distribution. For the 

uniform spatial distribution considered, two temporal variations are considered; a quasi-static 

loading and a band-limited random acoustic loading. Results from the nonlinear modal 

simulation are compared with those obtained via numerical simulation in physical coordinates. 

16
3.1. NUMERICAL SIMULATION IN PHYSICAL COORDINATES 

Numerical simulation analyses in physical coordinates serves as the basis for comparison of 

results from nonlinear modal simulation analyses. The finite element program ABAQUS 

(version 6.4) was used to generate nonlinear displacement and stress time histories. The double 

precision explicit integration scheme with an adaptive time integration step (referred to as

“element by element” in ABAQUS) was utilized for all analyses. The ABAQUS model 

consisted of 144 B21 elements. The choice of ABAQUS explicit analysis over alternative 

analysis methods, e.g. MSC.NASTRAN nonlinear transient solution (solution 129), was made

based on its superior ability to simulate very long response histories. This choice introduced an 

inconsistency of element formulations used between the nonlinear modal and physical

simulations. However, previous results (not shown) from a nonlinear static analysis in physical 

DoFs using MSC.NASTRAN (solution 106) and ABAQUS were nearly identical, indicating no 

significant difference between the different element formulations used. 

3.2. QUASI-STATIC RESPONSE 

In the limit of including all linear eigenvectors (the full set of BM modes) in the basis, results 

found using nonlinear modal simulation should be identical to those obtained via numerical 

simulation in physical coordinates because there is no modal truncation. The first purpose of the 

quasi-static response prediction is to investigate the effect of modal truncation. The second 

purpose is to investigate the spatial characteristics of results produced using each modal basis.

Because the quasi-static response is considered, dynamic effects due to the modal mass and 

damping are not significant. Therefore, only the modal stiffness coefficients dictate the accuracy 

of the solution for a particular basis. Further, as these coefficients are the same for the quasistatic and subsequent dynamic analyses, the quasi-static analysis provides a benchmark against

which the dynamic results may be judged. 

While it is possible to set the modal accelerations and velocities to zero in Equation (2) and solve 

for the modal displacements via Newton-Raphson (as per [14]), the more direct approach utilized 

here applies a quasi-static loading and numerically integrates to obtain the modal displacements. 

Once transformed, the physical displacements and stresses obtained from the reduced-order 

17
analysis are compared with those obtained from a nonlinear simulation in physical DoFs with the 

same quasi-static loading. 

3.2.1 Load Generation and Stress Post-Processing 

A uniformly distributed quasi-static pressure loading was applied to the beam to obtain the quasistatic response. The loading time history ramped from zero magnitude at time zero to its 

maximum level at 1.6384s, following which 0.5s of constant loading was applied. The response 

was obtained at 2.1384s, after the decay of any transient behavior. 

The bending and membrane stress components were separated by averaging the upper and lower 

surface stresses to obtain the membrane component, and then subtracting that from the surface 

stress to obtain the bending component. This separation allowed greater insight into the effect of 

the various modal bases on the stress response. 

3.2.2 Results 

The effect of modal truncation was first considered. Three BM modal bases were used; a 4-

mode basis consisting of the first two symmetric bending modes and first two anti-symmetric

membrane modes (2+2), an 8-mode basis consisting of the first four symmetric bending modes

and first four anti-symmetric membrane modes (4+4), and a 12-mode basis consisting of the first 

six symmetric bending modes and first six anti-symmetric membrane modes (6+6). 

Displacement results are presented at the 1⁄4 span (4.5-in. from the clamped end), and element 

stresses near the 1⁄4 span (4.4375-in. from the clamped end). These locations will help to

elucidate the benefits and liabilities of the various modal bases under severe conditions as the 

membrane stress component is more significant there relative to the bending stress component

than at other locations along the length, e.g., the clamped end or mid-span. 

The transverse displacement response error is shown in Figure 5 for a range of loadings. Results

shown in this and in the subsequent three figures are relative to the physical DoFs solution from 

the ABAQUS explicit analysis, i.e., 

% Physical Reduced Order Error

Physical

⎛ − =⎜

⎝ ⎠

100 ⎞

⎟× . (9)

18
Pressure (psi)

Tran

s

v

ers

e

Dis

plac

e

ment Error (%)

10-3 10-2 10-1 100 -1

0

1

2

3

4

2 Bending + 2 Membrane

4 Bending + 4 Membrane

6 Bending + 6 Membrane

Figure 5: Transverse displacement error at beam 1⁄4 span for three BM bases. 

It is seen that all bases except for 2+2 agree very well with the physical DoFs solution and

asymptote to a small error. Since the subsequent random analysis in physical DoFs could only 

be performed using the ABAQUS explicit solution, the same solution was used here for 

consistency. Note that if the MSC.NASTRAN nonlinear static solution was used instead of the 

ABAQUS explicit analysis, a different set of small errors would be obtained. For example, for 

the 6+6 basis, at the highest level of 0.4608 psi, the error would be -0.07% using 

MSC.NASTRAN instead of +0.3%. Since the error is very small, Figure 5 should not be 

misinterpreted to indicate that the 4+4 basis is superior to the 6+6 basis at the highest loading 

level. Rather, the error in the 4+4 and 6+6 reduced-order analyses falls into an error band that is

insignificant, regardless of the physical DoFs solution used for comparison. 

The membrane displacement response error is shown in Figure 6. Here the 2+2 basis

substantially under-predicts the response across the loading range, the 4+4 basis slightly overpredicts, and the 6+6 basis compares well across the range. The errors in bending and membrane 

stress components are shown in Figure 7 and Figure 8, respectively. In both cases, the 6+6 basis 

compares the most favorably across the loading range. The above results show that it is possible 

to achieve highly accurate displacement and stress response predictions using a truncated BM

basis. 

19
Pressure (psi)

M

e

m

bra

n

e

Dis

pla

c

e

m

e

nt Error (%)

10-3 10-2 10-1 100 -5

0

5

10

15

20

25

30

2 Bending + 2 Membrane

4 Bending + 4 Membrane

6 Bending + 6 Membrane

Figure 6: Membrane displacement error at beam 1⁄4 span for three BM bases. 

Pressure (psi)

Bending

Stre

s

s

Error (%)

10-3 10-2 10-1 100 -10

-5

0

5

2 Bending + 2 Membrane

4 Bending + 4 Membrane

6 Bending + 6 Membrane

Figure 7: Bending stress error at beam 1⁄4 span for three BM bases. 

20
Pressure (psi)

M

e

m

bra

n

e Stre

s

s Error (%)

10-3 10-2 10-1 100 -5

0

5

10

2 Bending + 2 Membrane

4 Bending + 4 Membrane

6 Bending + 6 Membrane

Figure 8: Membrane stress error at beam 1⁄4 span for three BM bases. 

Next considered are the effects of different modal basis variants having a comparable number of 

included modes. The following modal bases were considered; the first six symmetric bending 

and first six anti-symmetric membrane modes (BM 6+6), the first six symmetric bending modes 

(B 6), the first six coupled bending-companion modes (CBC 6), and the dynamic version of UBC 

having the first six symmetric bending modes and first four companion modes (6+4). A 

numerical instability for both UBC bases was encountered, limiting the number of companion 

modes for a particular excitation level. Radu et al [18] attributed this problem to the high 

frequency content of the companion modes. However, since the problem was not observed for 

the BM basis having comparable frequency content, a more likely reason has to do with the high 

degree of coupling between some bending and companion modes, as previously discussed. A 

static condensation in modal coordinates was proposed and demonstrated to mitigate this

behavior [18], although no attempt was made to implement this scheme in the present work. The 

instability was severe enough, however, that results for the UBC static basis could not be

obtained for the quasi-static loading case. For this reason, only the (6+4) UBC dynamic basis 

was considered. 

A single loading level (0.0576 psi) was applied and results are presented at locations across the

beam semi-span to show how accuracy varies with location. Transverse displacements for each

21
modal basis are shown in Figure 9. The BM and UBC bases compare very well with the 

physical DoFs solution, while the B and CBC somewhat under-predict the response. By 

comparison, only the BM basis compares well with the membrane displacements shown in

Figure 10. The bending-only basis B is incapable of predicting any membrane displacement, 

while the CBC basis correctly predicts the shape, but significantly under-predicts the magnitude. 

The UBC basis both over-predicts the magnitude and incorrectly predicts the shape. 

Beam Span (in)

Tran

s

v

ers

e

Dis

plac

e

ment (in)

012345 6789 0.000

0.005

0.010

0.015

0.020

0.025

Physical DoF

BM (6+6)

B (6)

CBC (6)

UBC (6+4)

Figure 9: Quasi-static transverse displacements along beam semi-span for 4 modal bases. 

The bending and membrane stress behaviors are shown in Figure 11 and Figure 12, respectively. 

Bending stress results from all bases compare well with the physical DoFs solution. The nearly 

constant membrane stress is accurately captured by only the BM basis. The other bases predict 

incorrect magnitudes and spatial distributions. With regard to the bending-only basis, the 

membrane stress arises from the bending-membrane coupling in the stress recovery calculation,

even though this basis predicts a zero membrane displacement. The fact that there are several 

points along the beam where membrane stress results from the B, CBC and UBC agree with the 

physical DoFs solution highlights the need to consider a number of spatial locations in assessing 

the accuracy of the modal basis. Note that only the BM basis accurately predicts bending and 

membrane displacement and stress response at all span-wise locations. 

22
Beam Span (in)

M

e

m

brane

Dis

pla

c

e

m

ent (in)

012345 6789 -1E-05

0E+00

1E-05

2E-05 Physical DoF

BM (6+6)

B (6)

CBC (6)

UBC (6+4)

Figure 10: Quasi-static membrane displacements along beam semi-span for 4 modal bases. 

Beam Span (in)

Bending

Stre

s

s (p

si)

012345 6789 -1200

-1000

-800

-600

-400

-200

0

200

400

600 Physical DoF

BM (6+6)

B (6)

CBC (6)

UBC (6+4)

Figure 11: Quasi-static bending stress along beam semi-span for 4 modal bases. 

23
Beam Span (in)

M

e

m

bra

n

e

Stress (psi)

012345 6789 -40

-20

0

20

40

60

80

100

120 Physical DoF

BM (6+6)

B (6)

CBC (6)

UBC (6+4)

Figure 12: Quasi-static membrane stress along beam semi-span for 4 modal bases. 

A compact measure of an error along the span of the beam may be found from

( )

2

2

1

1 % 100

N

i i

i i

Physical Reduced Order

Error

N Physical =

⎡ ⎤ − ⎢ = ⎢

⎣ ⎦

∑ ⎥ ×

⎥ (10) 

where N is a number of points along the beam span used for the estimation. Figure 13 and 

Figure 14 present the transverse and membrane displacement errors, respectively. It is seen that

BM basis provide the best reduced-order approximation through the entire range of excitation 

levels studied. From Figure 14 it can be also concluded that while the CBC basis outperforms

the UBC basis, both incur unacceptably large errors. 

Figure 15 and Figure 16 present the bending and membrane stress errors, respectively. For the 

bending stress, the superiority of the BM basis is most evident at the highest loading. For the 

membrane stress, only the BM basis achieves errors less than 5%, while the B, CBC and UBC 

bases incur unacceptably large errors. 

24
Pressure (psi)

Tran

s

v

ers

e

Dis

plac

e

ment Error (%)

10-3 10-2 10-1 100 0

2

4

6

8

10

BM

B

CBC

UBC

Figure 13: Quasi-static transverse displacement error integrated along the beam span. 

Pressure (psi)

M

e

m

bra

n

e

Dis

pla

c

e

m

e

nt Error (%)

10-3 10-2 10-1 100 0

100

200

300

400

500

BM

CBC

UBC

Figure 14: Quasi-static membrane displacement error integrated along the beam span. 

25
Pressure (psi)

Bending

Stre

s

s

Error (%)

10-3 10-2 10-1 100 0

20

40

60

80

100

BM

B

CBC

UBC

Figure 15: Quasi-static bending stress error integrated along the beam span. 

Pressure (psi)

M

e

m

bra

n

e Stre

s

s Error (%)

10-3 10-2 10-1 100 0

20

40

60

80

100

BM

B

CBC

UBC

Figure 16: Quasi-static membrane stress error integrated along the beam span. 

To summarize the findings from the quasi-static analysis, only the BM basis was able to capture 

the transverse and membrane displacements, and the bending and membrane stresses with high 

accuracy. The errors associated with the B, CBC, and UBC bases are generally smaller for the 

26
transverse displacement and bending stress than they are for the membrane displacement and

membrane stress. The errors associated with the membrane displacement and stress are so gross

that the B, CBC and UBC bases are deemed unsuitable for their prediction for structures 

exhibiting nonlinear bending-membrane coupling. Consequently, if the membrane contribution 

is significant relative to that of bending, the B, CBC and UBC bases will also not be suitable. 

Therefore, some assessment of the membrane contribution must be made via an alternative 

analysis before blind application of the B, CBC or UBC bases. 

3.3. RANDOM ACOUSTIC RESPONSE 

For the nonlinear random response investigation, the beam was subjected to a uniformly 

distributed acoustic loading with a bandwidth of 1500 Hz. Three loading levels were considered 

to span the response regime from essentially linear to highly nonlinear; a low overall sound 

pressure level of 128 dB (0.0072 psi RMS), a medium level of 146 dB (0.0576 psi RMS), and a

high level of 164 dB (0.4608 psi RMS). Damping was chosen to be sufficiently high so that 

good comparisons could be made at the peaks of the PSD. A level of mass proportional damping 

was specified corresponding to 2.0% critical damping for the first symmetric bending mode. The 

damping specified was the same for both the reduced-order and physical DoFs simulation 

analyses. For the nonlinear modal simulation, a fixed time integration step of between 50 μs

(for the lowest excitation levels) and 2 μs (for the highest excitation levels) was used. 

3.3.1 Load Generation and Ensemble Averaging 

The same loading time history and ensemble averaging was used for the numerical simulation 

analysis in physical and modal coordinates. The loading time histories were generated by 

summing equal amplitude sine waves, each with random phase, within the specified bandwidth 

using a discrete inverse Fourier transform. This procedure was identical to that used in previous

work [5] by the authors, so further details are omitted for brevity. The loading produced by this 

method has a Gaussian distribution. A sharp roll-off of the input spectrum practically eliminates 

excitation of the structure outside the frequency range of interest. 

For each load level, ten ensembles of displacement and stress response were generated lasting

2.1384s each. The first 0.5s of each response record was discarded to remove the initial transient 

27
response [5], resulting in response histories of 1.6384s in duration. For each simulation, the 

displacement and stress response was stored at every 50 μs, giving time records of 32,768 

points. A 32,768-point FFT was subsequently used to compute the power spectral density (PSD) 

function. 

3.3.2 Results 

The results shown are for the same bases considered in the quasi-static response analysis. The 

spectral results have been visually smoothed to more clearly observe the behavior. Transverse 

displacement PSDs are shown in Figure 17 for the medium loading of 146 dB. For this level, all 

modal bases agree well with the numerical simulation in physical DoFs, with the most significant 

differences at the third and fourth mode for the B, CBC and UBC bases. At this level, the 

transverse displacements obtained by the B and CBC bases are essentially the same. 

Frequency (Hz)

Tran

s

v

ers

e

Dis

placement P

S

D (in2/Hz)

0 500 1000 1500 10-14

10-12

10-10

10-8

10-6

10-4

Physical DoF

B (6)

BM (6+6)

CBC (6)

UBC (6+4)

Figure 17: Quarter-span transverse displacement PSD of beam at 146 dB. 

The membrane displacement PSDs for this load level, shown in Figure 18, offer a different

perspective. The BM basis is the only one to compare well with the physical DoFs simulation 

results. The CBC basis is inaccurate on two accounts; the magnitude is clearly incorrect and the 

shape of the frequency response mimics that of the bending response in Figure 17. Like the 

quasi-static response, the UBC basis over-predicts the response amplitude. Though the loading 

28
spectrum falls off sharply at 1500 Hz, the membrane displacement response obtained using

physical DoFs simulation and nonlinear modal simulations with B, BM, and UBC bases, extend 

to twice that frequency because of period doubling (not shown). The membrane displacement 

response computed using the CBC basis does not exhibit period doubling. 

Frequency (Hz)

M

e

m

bra

n

e

Displace

m

e

nt P

S

D (in2/Hz)

0 500 1000 1500 10-22

10-20

10-18

10-16

10-14

10-12

10-10 Physical DoF

BM (6+6)

CBC (6)

UBC (6+4)

Figure 18: Quarter-span membrane displacement PSD of beam at 146 dB. 

The bending stress PSDs at 146 dB are shown in Figure 19. The BM basis compares very well 

across the frequency range, even at the anti-resonances. The B and CBC stress PSDs are

essentially the same, and under-predict the third and fourth resonances. The UBC basis is 

similar, but also under-predicts the fundamental. The membrane stress PSDs are shown in 

Figure 20. The BM basis compares remarkably well with the physical DoFs simulation. Again,

the B and CBC bases predict nearly the same response to each other. Below about 630 Hz, the 

B, CBC, and UBC results fall on either side of the physical DoFs simulation, while above that 

frequency they all over-predict the response. 

At the highest loading of 164 dB, the transverse displacement PSD exhibits the peak spreading 

and shifting characteristic of spring hardening nonlinearity, see Figure 21. For this case, the 

bending-only and CBC bases indicate a higher degree of nonlinearity than the physical DoFs

simulation results, as evidenced by the first and second peaks shifted to higher frequencies, 

greater broadening, and smaller magnitudes. Results from the UBC basis were not available due 

29
to the aforementioned numerical integration problems. The BM basis results compare well 

across the frequency range, with some loss of peak response in the third peak. 

Frequency (Hz)

B

ending

Stre

s

s

P

S

D (psi

2/Hz)

0 500 1000 1500 10-3

10-2

10-1

100

101

102

103

104

Physical DoF

B (6)

BM (6+6)

CBC (6)

UBC (6+4)

Figure 19: Quarter-span bending stress PSD of beam at 146 dB. 

Frequency (Hz)

M

e

m

bra

n

e

Stre

s

s

P

S

D (p

si

2/H

z)

0 500 1000 1500 10-5

10-4

10-3

10-2

10-1

100

101

102

103

Physical DoF

B (6)

BM (6+6)

CBC (6)

UBC (6+4)

Figure 20: Quarter-span membrane stress PSD of beam at 146 dB. 

30
The membrane behavior shown in Figure 22 follows the observations made at 146 dB. The

excellent agreement between BM basis and physical DoFs simulation results demonstrate the 

efficacy of the BM basis in the highly nonlinear dynamic response regime. 

Frequency (Hz)

Tran

s

v

ers

e

Dis

plac

e

m

e

nt P

S

D (i

n2/H

z)

0 500 1000 1500 10-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

Physical DoF

B (6)

BM (6+6)

CBC (6)

Figure 21: Quarter-span transverse displacement PSD of beam at 164 dB. 

Frequency (Hz)

M

e

m

bra

n

e

Displace

m

e

nt P

S

D (in2/H

z)

0 500 1000 1500 10-18

10-16

10-14

10-12

10-10

10-8

Physical DoF

BM (6+6)

CBC (6)

Figure 22: Quarter-span membrane displacement PSD of beam at 164 dB. 

Bending and membrane stress PSDs at 164 dB are shown in Figure 23 and Figure 24, 

respectively. The BM basis most closely compares with the physical DoFs simulation bending 

31
stress across the frequency range, with some loss in amplitude response in the peak near 850 Hz. 

The bending-only and CBC bases significantly under-predict the bending stress of the first and 

second modes and, above about 950 Hz, indicate different character than the BM basis and 

physical DoFs results. 

Frequency (Hz)

Bending

Stre

s

s

P

S

D (psi

2/H

z)

0 500 1000 1500 101

102

103

104

105 Physical DoF

B (6)

BM (6+6)

CBC (6)

Figure 23: Quarter-span bending stress PSD of beam at 164 dB. 

Frequency (Hz)

M

e

m

bra

n

e Stre

s

s PS

D (p

si

2/H

z)

0 500 1000 1500 10-1

100

101

102

103

104

105

106

Physical DoF

B (6)

BM (6+6)

CBC (6)

Figure 24: Quarter-span membrane stress PSD of beam at 164 dB. 

32
In Figure 24, the BM basis results compare favorably with the physical DoFs simulation results 

across the frequency range. The bending-only and CBC bases over-predict the membrane stress 

magnitude and width of the first peak. Above about 750 Hz, the character substantially differs 

from the physical DoFs results. 

To quantify the error in the root-mean-square of the response, the following measure was used: 

( ) ( ) % 100

( )

RMS Physical DoFs RMS Reduced Order Error RMS Physical DoFs

⎛ − = × ⎜

⎝ ⎠

⎞

⎟ . (11) 

The error in RMS response is an indicator of the error over the entire frequency range, and hence 

is not capable of representing the differences in the spectral character. Figure 25 and Figure 26 

present the error estimates for transverse and membrane displacements, respectively, and Figure 

27 and Figure 28 present the error estimates for bending and membrane stresses, respectively.

While the RMS transverse displacement is estimated with least accuracy by the BM basis, it 

should be noted that this is primarily attributable to differences in the fundamental mode, which 

dominates the response. The 7% error in the RMS displacement transverse displacement is 

considered acceptable for a random analysis, particularly at the high nonlinear levels where the 

width of the confidence intervals is large [5]. 

Pressure (psi)

Tran

s

v

ers

e

Dis

plac

e

ment Error (%)

10-3 10-2 10-1 100 -10

-5

0

5

BM

B

CBC

UBC

Figure 25: Error in random acoustic RMS transverse displacement response at beam 1⁄4 span. 

33
On the other hand, the RMS membrane displacement is captured most accurately by the BM 

basis. As shown in Figure 18 and Figure 22, the membrane displacement is dominated by the 

zero frequency component, which results from the fact that the membrane displacement 

oscillates between zero and some positive value. For the error in RMS membrane displacement, 

the CBC and UBC bases can be off by as much as one order of magnitude. 

Pressure (psi)

M

e

m

bra

n

e

Displace

m

e

nt Error (%)

10-3 10-2 10-1 100 -300

-200

-100

0

100

BM

CBC

UBC

Figure 26: Error in random acoustic RMS membrane displacement response at beam 1⁄4 span. 

Pressure (psi)

Bending

Stre

s

s

Error (%)

10-3 10-2 10-1 100

0

20

40

BM

B

CBC

UBC

Figure 27: Error in random acoustic RMS bending stress response at beam 1⁄4 span. 

34
With regard to the stress response in Figure 27 and Figure 28, the BM basis provides the most 

accurate stress response, with a somewhat higher error in the bending stress at the medium load 

level. Again, this is likely a reflection of differences in the peak values. Bases B and CBC 

provide the lowest quality of stress estimation. For the bending stress, the RMS value is affected 

by several peaks of comparable magnitude, while the RMS membrane stress is dominated by the 

zero frequency component (see Figure 20 and Figure 24). 

Pressure (psi)

M

e

m

bra

n

e Stre

s

s Error (%)

10-3 10-2 10-1 100 -100

-50

0

50

100

BM

B

CBC

UBC

Figure 28: Error in random acoustic RMS membrane stress response at beam 1⁄4 span. 

4. NONLINEAR STATIC AND RANDOM ACOUSTIC RESPONSE OF A PLATE 

A plate was next considered to help understand the complexities of modal basis selection for a 

two-dimensional structure. Since the beam results presented in Sec. 3 indicated that the BM

basis was superior over all the other basis variants considered, the nonlinear reduced-order 

results for the plate were computed using only the BM basis. 

The plate measured 14-in. x 10-in. x 0.04-in. (x x y x h) with the aluminum material properties 

introduced in Sec. 2.2.3. The boundary conditions were simply supported on all sides.

Specifically, along the 14-in. sides (in the x-direction), the transverse displacement DoF (w), the 

membrane displacement DoF in the y-direction (v), and the rotational DoF about the y-axis were 

35
constrained. Along the 10-in. sides (in the y-direction), the w DoF, the membrane displacement

DoF in the x-direction (u), and the rotational DoF about the x-axis were constrained. 

The ABAQUS model used for simulation in physical DoFs consisted of 8960 S4R elements 

measuring 0.125-in. x 0.125-in. The S4R element has one integration point, compared to four 

for the S4 element, and is required in the ABAQUS explicit analysis. The double precision 

explicit integration scheme with adaptive time step was used for both quasi-static and random 

acoustic response analyses. 

For the reduced-order analysis, the MSC.NASTRAN model used consisted of 8960 CQUAD4 

elements measuring 0.125-in. x 0.125-in. Up to the first twelve (eigenvectors 1, 4, 8, 11, 12, 19, 

22, 23, 28, 31, 35, and 40) doubly symmetric bending modes, i.e., symmetric in both x-y planform directions, were included in the basis. These occurred at natural frequencies 58, 216, 367, 

524, 531, 838, 985, 1003, 1140, 1308, 1452, and 1633 Hz, respectively. For the membrane 

modes, the selection was guided by the previous experience gained from the beam analysis, that 

is, membrane displacement components were either both anti-symmetric, or one was antisymmetric and the other was zero. Up to the first twelve membrane modes were included 

(eigenvectors 426, 438, 618, 638, 778, 798, 894, 917, 935, 1142, 1152, and 1184). These 

occurred at natural frequencies of 15.2, 15.4, 21.2, 21.9, 26.1, 26.6, 29.6, 30.3, 30.8, 36.7, 36.9, 

and 37.9 kHz, respectively. 

4.1. QUASI-STATIC RESPONSE 

The effect of modal truncation on the quasi-static displacement results was first considered. The 

applied loading had a quasi-steady characteristic, as described in Sec. 3.2.1, and was uniformly

distributed over the plate surface. Three BM modal bases were used; an 8-mode basis consisting

of the first four symmetric bending modes and first four anti-symmetric membrane modes (4+4),

a 16-mode basis consisting of the first eight symmetric bending modes and first eight antisymmetric membrane modes (8+8), and a 24-mode basis consisting of the first twelve symmetric 

bending modes and first twelve anti-symmetric membrane modes (12+12). By analogy to the 

beam results, plate displacement results are presented at the 1⁄4 - 1⁄4 span location (3.5-in. from the 

boundary in the x-direction, and 2.5-in. from the boundary in the y-direction). 

36
The transverse displacement (w) error and both membrane displacement (u in the x-direction, v

in the y-direction) errors were computed as per Equation (9), and are shown in Table 6 for a 

range of loadings. It is seen that for the lowest quasi-static loading of 5.801 x10-4 psi, the modal

truncation has little effect on the displacement accuracy, and results computed with eight,

sixteen, and twenty-four basis vectors are essentially the same. The center displacement at this 

level was = 0.016. As the quasi-static loading increases, the effect of nonlinearity 

increases, with a center displacement of = 0.408 for the 1.856 x10

max w / h

h

h

max w / -2 psi load, and

=1.545 for the 0.297 psi load. The number of basis vectors used plays an increasingly 

important role as the load increases. The effect is more pronounced in the membrane 

displacement component errors than it is in the transverse displacement error. 

max w /

Table 6: Plate quasi-static displacement errors at (1⁄4, 1⁄4) location, %. 

w u v

5.801 x10-4 psi 

BM (12+12) -2.16 -4.32 -4.79

BM (8+8) -2.17 -4.39 -0.18

BM (4+4) -2.55 -2.89 -1.31

1.856 x10-2 psi 

BM (12+12) -1.52 -3.52 -3.41

BM (8+8) -1.41 -2.55 2.54 

BM (4+4) 1.34 4.23 7.44

0.297 psi 

BM (12+12) -0.48 -3.89 0.56 

BM (8+8) 1.06 5.34 13.81 

BM (4+4) 4.28 17.25 16.79 

In the process of selecting membrane modes for the basis, the lowest anti-symmetric modes were 

included without regard for the number of periods in the x and y-directions. Since the ydimension is shorter, it is also stiffer. This means that membrane modes corresponding to this 

direction will generally have higher natural frequencies than the ones associated with membrane 

motion in the longer plate dimension. Of the first 12 membrane modes used, the maximum 

number of periods in the x-direction was four, while the maximum number in the y-direction was 

two. Consequently, it is expected that modal reduction utilizing 12 membrane modes for the 

plate is less accurate than the best modal reduction for the beam utilizing 6 membrane modes, as 

the latter had six periods along the beam’s span. The conclusion is that the reduced-order

37
analysis for a two-dimensional structure requires an expanded modal basis relative to a onedimensional structure of comparable characteristic length and excitation bandwidth. 

4.2. RANDOM ACOUSTIC RESPONSE 

For the random analysis, a uniformly distributed acoustic load with a bandwidth of 1024 Hz was

applied. Two loading levels were considered; a low overall sound pressure level of 106 dB 

(5.801 x10-4 psi RMS) and a high level of 164 dB (0.297 psi RMS). Similar to the beam studies, 

a mass proportional 2% critical damping corresponding to the fundamental mode was prescribed. 

The displacement error corresponding to the random plate response in an RMS sense, as per 

Equation (11), is presented in Table 7 for the two loadings levels. 

Table 7: Plate random displacement errors at (1⁄4, 1⁄4) location, %. 

w u v

106 dB 

BM (12+12) -0.32 -10.17 -2.55

BM (8+8) -0.32 -10.13 -2.57

BM (4+4) -0.40 -10.97 -2.56

160 dB 

BM (12+12) 4.00 -38.68 -26.43 

BM (8+8) 14.63 46.35 48.02

BM (4+4) 20.64 63.04 52.00

For the 106 dB level, which results in a linear response regime with = 0.024, the effect of 

modal truncation is negligible; 8, 16, and 24 basis vectors analyses yield essentially the same

RMS error. As the excitation level is increased to 160 dB, the response becomes highly 

nonlinear with = 2.373. At this level, modal truncation starts playing a very important 

role and its effect is significantly magnified when compared to a quasi-static case. The quasistatic analysis performed utilizing 8 basis vectors (4+4) for the 0.297 psi load yielded a 

transverse displacement error of about 4%, while the random response analysis for the 160 dB

load (0.297 psi RMS) required 24 basis vectors (12+12) to obtain comparable results. This trend 

was also observed in the results for the beam. Consistent with observations based on the quasistatic analysis, the error in the random transverse displacement is substantially less than the error 

in either component of the random membrane displacement. 

max w / h

w / h max

38
5. CONCLUSIONS 

Nonlinear modal simulation using an indirect nonlinear stiffness evaluation method has been

shown to provide accurate predictions of nonlinear quasi-static and random response, when an 

appropriate basis is selected. When used in conjunction with direct numerical simulation for

validation, the approach constitutes one of the few high-fidelity design options for nonlinear 

random vibration and high-cycle fatigue. 

The following conclusions are limited to the class of problems considered, i.e. those exhibiting 

nonlinear bending-membrane coupling. Of the four modal basis variants considered, the bending 

and membrane modal basis was found to be the only one to accurately predict transverse and

membrane displacement, and bending and membrane stress at any location on the structure. Its

main drawback is that identification and selection of membrane modes is labor intensive even for 

the simple beam and plate structures considered. 

The bending-only basis was the simplest amongst the variants in terms of the modal basis 

selection. However, the bending-only basis was incapable of computing membrane

displacement response, and the accuracy of the membrane stress prediction was highly 

dependent on location. Further, it over-predicted the effects of nonlinearity on the random 

response. 

For the combined bending-companion modal basis, there was no issue with regard to identifying

the companion modes, and their inclusion did not increase the system size. However, there was

no engineering rationale for selecting which initial imperfection shape to apply when computing

the dynamic companion. It was found that results obtained using this basis were comparable to 

those obtained using the bending-only basis, except for the membrane displacement, which the

latter was unable to predict. The random membrane displacement response did not exhibit 

period doubling, and was significantly reduced in amplitude relative to the physical DoFs

simulation. 

For the uncoupled bending-companion modal basis, there was also no engineering rationale for 

selecting which initial imperfection shape to apply when computing the dynamic companion. 

When computing the static companion, there was no engineering rationale for selecting which

displacement or loading to apply. Both static and dynamic UBC bases indicated different 

39
coupling between bending and companion modes than the coupling between bending and 

membrane modes for the BM basis. For both static and dynamic UBC bases, a numerical 

instability problem limited the number of companions that could be included in the basis,

without the use of static condensation in modal coordinates. Finally, for the UBC dynamic basis, 

the membrane displacement and stress response prediction was highly dependent upon location. 

Finally, with regard to two-dimensional structures, it was found that the number of basis vectors 

needed for an accurate response prediction in a highly nonlinear regime had to be considerably

expanded as compared to one-dimensional structures. 

REFERENCES 

[1] Vaicaitis, R., "Recent advances of time domain approach for nonlinear response and 

sonic fatigue," Structural Dynamics: Recent Advances, Proceedings of the 4th 

International Conference, pp. 84-103, Southampton, UK, 1991. 

[2] Green, P.D. and Killey, A., "Time domain dynamic finite element modelling in acoustic

fatigue design," Structural Dynamics: Recent Advances, Proceedings of the 6th

International Conference, Vol. 2, pp. 1007-1025, Southampton, UK, 1997. 

[3] Caughy, T.K., "Equivalent linearization techniques," Journal of the Acoustical Society of 

America, Vol. 35, pp. 1706-1711, 1963. 

[4] Roberts, J.B. and Spanos, P.D., Random vibration and statistical linearization. New 

York, NY, John Wiley & Sons, 1990. 

[5] Rizzi, S.A. and Muravyov, A.A., "Comparison of nonlinear random response using 

equivalent linearization and numerical simulation," Structural Dynamics: Recent

Advances, Proceedings of the 7th International Conference, Vol. 2, pp. 833-846,

Southampton, UK, 2000. 

[6] Rizzi, S.A., "On the use of equivalent linearization for high-cycle fatigue analysis of

geometrically nonlinear structures," Structural Dynamics: Recent Advances, Proceedings 

of the 8th International Conference, Southampton, UK, 2003. 

[7] Mei, C., Dhainaut, J.M., Duan, B., Spottswood, S.M., and Wolfe, H.F., "Nonlinear

random response of composite panels in an elevated thermal environment," Air Force

Research Laboratory, Wright-Patterson Air Force Base, OH, AFRL-VA-WP-TR-2000-

3049, October 2000. 

[8] Przekop, A., "Nonlinear response and fatigue estimation of aerospace curved surface 

panels to combined acoustic and thermal loads," Ph.D. Dissertation, Old Dominion 

University, 2003. 

[9] Przekop, A., Guo, X., Azzouz, M.S., and Mei, C., "Reinvestigation of nonlinear random 

response of shallow shells using finite element modal formulation," Proceedings of the 

45th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials 

Conference, AIAA-2004-1553, Palm Springs, CA, 2004. 

[10] Hollkamp, J.J., Gordon, R.W., and Spottswood, S.M., "Nonlinear sonic fatigue response 

prediction from finite element modal models: a comparison with experiments," 

40
Proceedings of the 44th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics,

and Materials Conference, AIAA-2003-1709, Norfolk, VA, 2003. 

[11] Bathe, K.J. and Gracewski, S., "On nonlinear dynamic analysis using substructuring and 

mode superposition," Computers and Structures, Vol. 13, pp. 699-707, 1981. 

[12] McEwan, M.I., Wright, J.R., Cooper, J.E., and Leung, Y.T., "A finite element/modal 

technique for nonlinear plate and stiffened panel response prediction," Proceedings of the 

42nd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials 

Conference, AIAA-2001-1595, Seattle, WA, 2001. 

[13] McEwan, M.I., Wright, J.R., Cooper, J.E., and Leung, Y.T., "A combined modal/finite 

element analysis technique for the dynamic response of a non-linear beam to harmonic

excitation," Journal of Sound and Vibration, Vol. 243, No. 4, pp. 601-624, 2001. 

[14] Muravyov, A.A. and Rizzi, S.A., "Determination of nonlinear stiffness with application 

to random vibration of geometrically nonlinear structures," Computers and Structures, 

Vol. 81, No. 15, pp. 1513-1523, 2003. 

[15] Hollkamp, J.J. and Gordon, R.W., "Modeling membrane displacements in the sonic 

fatigue response prediction problem," Proceedings of the 46th

AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, 

Austin, TX, 2005. 

[16] Mignolet, M.P., Radu, A.G., and Gao, X., "Validation of reduced order modeling for the 

prediction of the response and fatigue life of panels subjected to thermo-acoustic effects," 

Structural Dynamics: Recent Advances, Proceedings of the 8th International Conference, 

Southampton, UK, 2003. 

[17] Press, W.H., Teukolsky, S.A., Vetterling, W.T., and Flannery, B.P., "Numerical recipes, 

the art of scientific computing," CDROM v2.10, Cambridge University Press, 2002. 

[18] Radu, A.G., Yang, B., Kim, K., and Mignolet, M.P., "Prediction of the dynamic response 

and fatigue life of panels subjected to thermo-acoustic loading," Proceedings of the 45th 

AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, 

AIAA-2004-1557, Palm Springs, CA, 2004. 

41
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The Effect of Basis Selection on Static and Random Acoustic Response

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Stephen A. Rizzi (NASA)

Adam Przekop (NIA)

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14. ABSTRACT

An investigation of the effect of basis selection on geometric nonlinear response prediction using a reduced-order nonlinear

modal simulation is presented. The accuracy is dictated by the selection of the basis used to determine the nonlinear modal

stiffness. This study considers a suite of available bases including bending modes only, bending and membrane modes,

coupled bending and companion modes, and uncoupled bending and companion modes. The nonlinear modal simulation

presented is broadly applicable and is demonstrated for nonlinear quasi-static and random acoustic response of flat beam and

plate structures with isotropic material properties. Reduced-order analysis predictions are compared with those made using a

numerical simulation in physical degrees-of-freedom to quantify the error associated with the selected modal bases. Bending

and membrane responses are separately presented to help differentiate the bases.

15. SUBJECT TERMS

Reduced-order numerical simulation; Geometric nonlinear response; Nonlinear stiffness evaluation

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