c 

NASA Technical Memorandum 10 15413 

DEVELOPMENT OF I: IIGI IER-ORDER MODAL MET74ODS FOR 

TRANSIENT THERMAL AND STRUCTUIML ANALYSIS 

UIIARLES J. CAMARDA ANI) I~AI~IIAl~I~ ‘1’. I IAFI’KA 

(NASA-TB-101548) DEV ELOPHEIJT OF 

HIGHER-ORDER llODAL HETBODS FOE TRBNSIBNT 

N a 9- 2 46 77 

THERNAL AID STRUCTURAL h%hLYSIS [NASA. 

Langley 3esearch Center) 39 p CSCL 20K Unclas 

G3/39 0 217222 

nrnsn 

National Aeronautics and 

Space Administration 

bngley Research Cenler 

Hampton, Virginia 23665-5225 
ABSTRACT 

The paper presents and evaluates a force-derivative method which 

produces higher-order modal solutions to transient problems. These 

higher-order solutions converge to an accurate response using fewer 

°rees-of-freedom (eigenmodes) than lower-order methods such as the 

mode-displacemen t or mode-acceleration methods. Results are presented 

for non-proportionally damped structural problems as well as thermal 

problems modeled by finite elements. 
SYMBOLS 

C damping matrix 

E modulus of elasticity 

e 

K stiffness matrix 

spatial error norm (eq. (34)) 

E generalized stiffness matrix, conductance matrix 

M mass matrix, moment 

m subset of total number of degrees-of-freedom 

generalized mass matrix, capacitance matrix 

n total number of degrees-of-freedom 

Q force vector 

a generalized force or thermal load vector 

q modal coordinates of the second-order system 

t time 

U displacement 

X coordinate direction 

Y generalized displacement or temperature vector 

Z modal coordinates of the first-order system 

2 
ith damped eigenvalue (eqs. (3) and (4)) 

matrix of damped eigenvalues 

ith damped eigenvector 

matrix of damped eigenvectors 

ith normal eigenvector (eq. (21)) 

time-integrated error norm defined in eq. (33) 

matrix of frequencies squared (eq. (23)) 

ith circular natural frequency 

ith circular frequency of the damped free vibration 

matrix of damping coefficients (eq. (23)) 

density 

dummy variable of integration, temporal 

(33)) 

ith modal viscous damping factor 

Subscripts 

0 initial 

limit 

3 
Superscripts 

a approxi mate 

(i) ith derivative with respect to time 

* matrix of reduced number of eigenvectors or eigenvalues 

l or reduced number of modal coordinates 

4 
INTRODUCTION 

Transient thermal and structural analyses of complicated engineering 

problems which are modeled as discrete multidegree-of-freedom systems 

often require the solution of very large systems of equations. Reducing the 

order of such systems is highly desirable from the standpoint of increased 

computational efficiency. Some of the many methods for reducing the 

order of discrete multidegree-of-freedom structural dynamic systems 

include mass condensation methods (e.g., refs. 1 and 2) and reduced basis 

methods (e.g., refs. 3-7). 

set of basis vectors (e.g. eigenmodes, Ritz vectors, or Lanczos vectors) or a 

combination of basis vectors (e.g., eigenmodes and Ritz vectors (ref. 8)). 

Reduced basis methods have also been applied in solving transient thermal 

problems (refs. 9-1 1). Some problems, such as the thermal/structural 

analysis of space transportation systems or large space structures, require 

a large number of basis vectors to accurately represent the transient 

response. In addition, when singularities occur in the loading, convergence 

of a solution is not guaranteed. 

takes a large number of eigenmodes to accurately model the sharp thermal 

gradients within the domain (ref. 10). 

The reduced basis methods use either a truncated 

For most transient thermal problems, it 

When a reduced basis method uses the eigenmodes of a system of 

equations the method is referred to as a modal method. Two of the most 

widely used modal methods for transient structural analysis are the modedisplacement method (MDM) and the mode-acceleration method (MAM). 

It was shown in reference 12, that the MAM can be considered a higherorder modal method than the MDM and that the MAM converges to an 

accurate solution using fewer modes than the MDM (refs. 6 and 12). A 

method for generating improved or higher-order modal methods was 

developed in reference 7. This method was generalized to proportionallydamped structures (the damping matrix is a linear function of the stiffness 

and mass matrices) and evaluated for a variety of loads and damping 

ratios (ref. 12). This new method successively integrates-by-parts the 

convolution integral form of the solution and is called the force-derivative 

method (FDM) because it produces terms which are related to the forcing 

function and its time derivatives. The FDM was found to be more accurate 

than either the MDM or MAM. In particular, for problems in which there 

are a large number of closely-spaced frequencies (e.g., large truss-type 

space structures or multispan beams) the FDM is very effective in 

representing the important, but otherwise neglected, higher modes. 

work indicates that the FDM, which was based on reference 7, has several 

5 

Recent 
variants (refs. 13 and 14) for solving transient structural problems. One 

such variant is called the dynamic-correction method (DCM) (ref.13) which 

is useful when a particular solution exists for a given forcing function. 

The purpose of the present paper is to extend the work of reference 12 to 

include non-proportionally damped structural systems and to evaluate the 

usefulness of the FDM in solving non-proportionally damped struc turd , 

, problems as well as thermal problems. 

the MDM, MAM, FDM, and the DCM. 

Modal methods evaluated include I 

I 
THEORY 

First-Order or Damped-Mode Formulation 

. The equations of motion, in matrix form, of an n degrees of freedom 

system, together with the initial conditions are given by 

Mu+Cu + K u = Q(t) 

.. u(0) = UO, U(0) = uo 

where M, C, and K are the mass, damping, and stiffness matrices of the 

system; u and Q are the displacement and load vectors, respectively, and a 

dot denotes differentiation with respect to time. 

Transforming eq. (1) to first-order form results in the following system of 

equations: 

li$Y+Is.Y=Q Y(0) = Yo (2) 

where 

and 

7 
Assuming a solution to the homogenuous 

results in an eigenvalue problem 

form of eq. (2) as: 

For distinct eigenvalues (Ctr), the eigenvectors (Or) are normalized such 

that 

and then 

Equation (5) can also be written in matrix form as 

[@]'a [0] = [I] and [O]'K [Q] = [a] 

where [@I is a 2n by 2n modal matrix with its rrll coldmn equal to @, and 

[a] is a diagonal matrix consisting of the ar's. 

8 
A solution to eq. (2) is assumed in the form of the following modal 

summation 

r= 1 

T . Substituting eq. (6) into eq. (2) with premultiplication by @r results in the 

following, uncoupled, system of equations: 

The solution to eq. (7) is 

t 

Hence, the solution of eq. (2) becomes 

2n 

r= 1 

where 

9 
If the forcing function has continuous derivatives, the convolution integral 

of eq. (9) can be integrated by parts to produce higher-order modal 

methods (ref. 12). For example, if it is integrated by parts once, the 

following expression results 

2n 

r= 1 

t 

If all the modes are used in the second-to-last term in eq. (lo), this term 

can be written as 

2n 

r= 1 

I if equation (1 1) is substituted into equation (lo), the resulting expression 

is analogous to the MAM (ref. 6) and can be written as 

10 

I. 
2n 

r= 1 

-1 

+K Q(t) 

If the forcing function has continuous derivatives up to order N (CN), the 

convolution integral of eq. (9) can be integrated-by-parts N-times, 

resulting in the following expression 

N 

where the superscript (i-1) denotes tht: (i-l)th derivative. 

I 

- If all the modes are used in the last N-terms of eq. (13), they can be 

represented as functions of &I and (ref. 12) as follows 
2n 

r= 1 

N 

-r N 1 

.L i= 1 J 

i= 1 

Alternate Damped-Mode Formulation 

I 

Equations 8 to 14 are analogous to those presented in reference 12, 

however, the present expressions solve a first-order system of equations, 

using the damped modes <Dr to decouple 

system (the damping matrix is not a linear function of the mass and 

stiffness matrices). 

doubled (2n). Equation 14 represents a means for developing higher-order 

modal methods than either the MDM or MAM, as demonstrated in 

reference 12, and is called the force-derivative method (FDM) because it 

produces terms which are related to the forcing function and its time 

derivatives. 

Equation 2 can also be considered to represent a heat conduction problem 

where, for that problem, fi would represent the capacitance matrix, 

a non-proportionally damped 

This means the order of the system of equations is 

would represent the conductance matrix, represents the thermal load 

vector, and Y is the vector of nodal temperatures. 

The MDM uses a subset, m (m e 2nj, of the eigenmodes to reduce the size of 

the problem and solves for Zr using eq. (8) and substitutes these values 

into a reduced modal summation in eq. (6) to approximate the response, 

Y(t). The MDM can be classified as a zeroth-order method because it is 

equivalent to using the FDM (eq. (14)) with N = 0. An analogous form of the 

MAM uses eq. (12), and a reduced modal summation to approximate Y(t) 

and can be classified as a first-order method (N = 1 in eq. (14)). The FDM 

12 
uses eq. (14) with N>1. 

with four integrations by parts offer improved approximations to the 

higher, neglected, modes for several structural problems. 

Reference 12 showed that the expressions obtained 

The FDM (eq. (14)) can be derived using an approach similar to that used 

in reference 13 which results in a form which is more suitable for inclusion 

into existing thermal and structural analysis codes. A numerical approach 

can be derived, similar to that presented in reference 13, which 

approximates the forcing function as a piecewise differentiable polynomial 

and which numerically integrates the reduced system of equations 

(eq. (8)). 

Equation 7 could be differentiated twice to produce the following equations 

For example, assume the forcing function is C2 differentiable. 

Re-arranging eq. (7) and substituting for 2, and 2, from eqs. (15a and 15b) 

results in 

Using eq. (6), the response can be written as 

13 
-1 -1 -1. -1 -1 -1 .. 

Y(t) =R Q(t) -it E Q(t) +it GlK MK Q(t) 

The last term can be evaluated using eq. (8) and Leibnitz's rule for 

differentiation of an integral to produce the following 

t 

Y(t) can be approximated using only a subset of the modes for 

eq. (17) and using eqs. (8) and (18), eq. (17) becomes 

last term in 

where the * denotes a reduced set of modes (m -e 2n) and Z can be 

calculated by numerically integrating eq. (7) 

This expression can be expanded, assuming a CN differentiable forcing 

function, to 

14 
N 

i=l 

Compared to eq. (14), the alternate formulation of the FDM (eq. (20)) does 

not require the solution of a convolution-type integral. In addition, the 

last term of eq. (20) is identical to a mode-displacement solution and as 

such the form of the FDM as given by eq. (20) is more suited for inclusion 

into existing computer codes. 

Second-Order or Natural-Mode Formulation 

Expressions which use the undamped natural modes can be developed in 

an analogous manner and result in an expression similar to eqs. (19) and 

(20). 

second-order system of equations (eq. (1)). The modes', Qr, are determined 

by solving the following eigenvalue problem 

This is accomplished by beginning with the undamped modes of the 

The modes are normalized as follows 

so that 

15 
Hence, the displacement response can be represented as 

Using eq. (22), and premultiplying eq. 

where 

and 

q +Aq+ 

T (1) by @r results in 

where [@I is the matrix of undamped eigenmodes, Q2 is a diagonal matrix 

whose diagonal terms can be represented as CI ii = o i and; for proportional 

damping, A 

represented as Aii = 26jWi. 

2 2 

is also a diagonal matrix whose diagonal terms can be 

16 
If we assume the forcing function is C2 differentiable, eq. (23) can be 

differentiated twice and back substituted into eq. (23) to produce the 

following expression, which is similar to eq. (16) 

Assuming proportional damping and zero initial conditions, the solution to 

eq. (23) can be written as 

where 

If eq. (25) is differentiated four times with respect to t and the expressions 

for q(3) and q(4) 

reduces to the following 

are substituted into eq. (24). the entire expression 

17 
Equation (26) agrees with resul 

in reference 13, 

systems. 

functions, the method above can produce successively higher-order modal 

methods and, as such, is just another formulation of the FDM. 

expression for an Nth-order modal method can be expressed as 

presented in reference 13 and, as shown 

is also valid for non-proportionally damped structural 

Assuming higher-order piecewise differentiable forcing 

The 

r= 1 

where 

and 

18 
The Dynamic-Correction Mmethod (DCM) 

The dynamic-correction method (DCM), reference 13, assumes a solution to 

eq. (2) in the form 

Y(t) = Yp(t) + Yc(t) 

where Yp(t) is a particular solution of eq. (2) and Yc(t) is the 

complimentary solution which represents the effects of initial conditions. 

In modal form, Yp(t) and Yc(t) can be represented as 

YP(t) = [@I ZP(t) 

Yc(0 = [a] Zc(t) 

and 

where Zp(t) and Zc(t) are the vectors of particular and complimentary 

solutions to the ’modal coordin’ate equations (eq. (7)). 

Using eqs. (6) and (28) 

The fundamental principle 

particular solution to eqs. (2) and (7), you can approximate the resp 

(eq. (30)) using a reduced set of modes as shown below 

of the DCM is that if you have an exact 

n 

19 
It can also be shown that in the limit as N goes to infinity, two terms in 

equation (20) can be written as 

and 

Hence, if an infinite number of integrations-by-parts are assumed in the 

FDM or if the convolution integral vanishes (e.g., for a polynomial forcing 

function of a lower order than the order of the FDM) the FDM would be 

equivalent to the DCM of reference 13. Also, if an exact solution to the 

convolution integrals of eqs. (8), (12), or (14) exists and is used; the 

response can be calculated without errors caused by the approximation of 

the forcing function. 

equations (32) will converge as shown above provided the frequency of 

the highest mode used in the approximation is larger than the forcing 

function frequency. 

For the case of a sinusoidal forcing function, 

20 
RESULTS AND DISCUSSION 

Structural Example: Two-Degree-Of-Freedom Problem 

A simple, two-degree-of-freedom spring-mass problem (fig. 1) with a 

sinusoidal forcing function was analyzed to compare the accuracy of the 

MDM, MAM, FDM, and DCM. This problem was also investigated in 

reference 13 and included in that reference are the particular solutions for 

polynomial as well as sinusoidal forcing functions. 

the sinusoidal forcing function, sin(mft), is applied to the second mass. The 

natural frequencies are 01= 19.54 rad/s and 02= 51.17 rad/s. The system 

is proportionally damped (A (eq. (23)) is diagonal) if a = K1/K2 (ref. 15). For 

the stiffnesses chosen, this corresponds to a value of a = 1.0. The accuracy 

of each method is assessed by a time-integrated error norm, which is 

defined as 

As shown in figure 1, 

a 

Results were 

where ui(t) is the calculated response using all the modes and ui(t) is the 

approximate response using a subset of the modes. 

calculated using both the real and damped modes (eqs. (20) and (27)) 

respectively. For this problem, the FDM used was of order four (N=4 in eqs. 

(20) and (27)). The time, T, selected for integrating the error was chosen to 

be Z= 16nlmf. 

Results of the error as a function of the forcing frequency for the 

undamped modal solution using one real mode for the proportionallydamped case (a =1.0) is shown in figure 2. In general, the forcing function 

21 
frequency must be lower than the highest natural frequency used in the 

approximate modal response for accurate results. As shown in figure 2, 

the accuracy increases as the order of the modal method increases. The 

results for the lower range of frequencies are shown more clearly in 

figure 3, which is an expanded error scale of figure 2, that the FDM (N=4) 

and DCM are similar and more accurate than the lower order methods such 

as the MDM (N=O) and the MAM (N=l) for of< 20 rad/s. For of> 20 rad/s, 

the DCM remains slightly more accurate than the FDM, however, as the 

forcing frequency approaches the second natural frequency (0 f= 5 1.17 

rad/s) all methods produce inaccurate results. It should be noted that 

results using one real mode or two damped modes are identical for the 

proportionally-damped case. 

~ 

Results for the non-proportionally-damped case (a = 20) using two 

damped modes, are shown in figure 4. 

proportionally-damped case with the exception that the DCM exhibits 

surprisingly good results at forcing function frequencies close to the second 

natural frequency. This result is unexplained at present and is believed to 

be fortuitous and, hence, it is recommended that all modal methods should 

include modes whose frequencies bound the frequency of the forcing 

function. 

modes (eq. (20)) and the undamped solution (using one real mode (eq. 

(27)) is shown in figure 5 for the non-proportionally-damped case (OC = 

20). 

modes (dashed lines) produces more accurate results than those using only 

one real mode (solid lines). 

DCM are nearly equivalent and result in the smallest error for frequencies 

as large as 30 rad/s. Hence, it may be beneficial, in some cases, to use the 

damped modes to obtain a more accurate solution. 

Results are similar to the 

A comparison of the damped-mode solution ( using two damped 

As shown in figure 5, the damped-mode solution using two damped 

The damped-mode solution for the FDM and 

The FDM produced similar results to the DCM for forcing frequencies below 

the first natural frequency. A comparison of the modal methods for a 

forcing frequency of= 10 rad/s is shown in figure 6. Once again, the 

higher-order modal methods result in more accurate solutions. 

relative errors near z = 0 are due to the zero initial conditions which cause 

the denominator of eq. (33) to approach zero at z = 0. 

is due to the addition of terms which are functions of the generalized 

stiffness and mass matrices and the force vector and its time derivatives. 

These additional terms approximate the effect of the higher modes which 

The large 

As explained in I reference 12, the increase in accuracy with the order of the modal method 

I 

I were neglected in the modal summation. 

22 
Thermal Example: Rod Heated At One End 

Results of references 10 and 11 indicate that Lanczos vectors can be 

effective reduced basis vectors for solving linear and non-linear thermal 

problems. Since the accuracy of the Lanczos vectors is comparable to that 

of the MAM for structural dynamic problems, it was expected that higherorder methods, such as the MAM, FDM and DCM, would be effective in 

solving complex thermal problems. 

similar to that presented in reference 10 with one exception; the present 

problem assumes the temperature at the right end of the rod is 

constrained to zero. 

value at time t = 10 sec and a ramp down to zero at time t = 20 sec as 

shown in figure 7. The error function used to evaluate convergence is a 

spatial error norm similar to that used in reference 12, namely 

' 

The problem selected to study is 

The forcing function is a ramp up from zero to a peak 

( T - Ta )'I' ( T-Ta) e= 

T~T (34) 

A total of twenty equally-spaced finite' elements were used to model the 

problem. 

number of modes for the MDM, MAM, and FDM are shown in figures 8a to 

8c for time t = 10 sec. 

(having an order of 2) is exactly equivalent to the DCM. The FDM 

converges with 5 modes to the exact solution as compared to 8 modes for 

the MAM and 18 modes for the MDM. A spatial error norm similar to that 

shown in eq. (34) is used to compare each method for time, t = 10 sec (fig. 

9). 

'the size and computational effort of thermal problems is illustrated in 

figure 9. The FDM or DCM require about 28-percent of the number of 

modes as compared to the MDM and about 63-percent of the number of 

modes as the MAM for an accurate thermal response. 

The temperature distributions in the rod as a function of the 

Because the forcing function is linear, the FDM 

The effectiveness of using higher-order modal methods for reducing . 

- 

23 

I 
CONCLUDING REMARKS 

of the mode-displacement, mode-acceleration, force-derivative, and the 

dynamic-correction methods (MDM, MAM, FDM, and DCM respectively) in 

solving proportionally and non-proportionally damped structural dynamic 

problems as well as transient thermal problems. The higher-order modal 

methods, such as the FDM or DCM, are very effective in solving structural 

Results of a two-degree-of-freedom spring-mass-damper system indicate 

that, for the proportionally damped problem, a solution using two damped 

modes produces identical results as one using only one natural mode. 

Hence, there is no advantage in using a damped modal solution to solve a 

proportionally damped problem. However, for the non-proportionally 

damped problem, the use of two damped modes produces more accurate 

results than the single natural .mode case. The DCM has the lowest 

percentage error of all the mode-superposition methods over the 

frequency range of 2 to 50 rad/s. The FDM, having an order of four (four 

integrations-by-parts (N=4)), produced similar results to the DCM up to a 

forcing frequency of about 35-40 rad/s. For the proportionally damped 

problem, all the methods were inaccurate near a frequency of 50 rad/s 

(close to the second natural frequency of the system). 

A one-dimensional heat conduction problem (rod heated at one end) was 

also investigated to evaluate the usefulness of using the FDM to solve 

thermal problems. The higher-order modal methods such as the FDM were 

very effectiye in solving the thermal problem. 

were inefficient in solving thermal problems because the nature of the 

problem required the inclusion of the higher modes for an accurate 

solution. 

but neglected, modes resulted in an accurate solution using only five 

modes out of a total of twenty as compared to the MDM which required 18 

modes for an accurate solution. 

Until now, modal methods 

The ability of the FDM to approximate the effects of the higher, 

24 
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26 
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Report Documentation Page 

. Repon No. 2. Government Accession No. 

NASA TM- 10 1548 

1. Title and Subtitle 

Development of Higher-Order Modal Methods forTransient 

Thermal and Structural Analysis 

7. A~thorkl 

Charles J. Camarda and Raphael 1. Haftka 

3. Recipient's Catalog No. 

5. Report Date 

February 1989 - 6. Performing Organization Code 

8. Performing Organization Report No. 

____ __ - - . - - -. . . . - .. . - - - -- 10. Work Unit No. I ~ 

9. Performing Organizatton Name and Address 

NASA. Langley Research Center 

Hampton, Virginia 23665-5225 . 

2. Sponsoring Agency Name and Address 

National Aeronautics and Space Administration 

Washington, DC 20546-0001 

505-63-31-03 

11. Contract or Grant No. 

13. Type of Report and Period Covered 

Technical Memorandum 

14 Sponsoring Bgency Code 

I 

5. Supplementary Notes 

17. Key Words lsuggcsted by Authorls)l 

Reduced Basis Methods , Transient Thermal Analysis, Transient 

Structural Analysis , Modal Methods , Mode-SuDerDosi tion Methods. Force-derivative 

Char1 es J . Camarda , NASA Langl ey Research Center , Hampton , VA. 

Raphael T. Haftka, Professor of Aerospace and Ocean Engineering, 

Virginia Polytechnic and State University, Blacksburg, VA. 

18. Distribution Statement 

Unclassified - Unlimited 

Subject Category 39 

6. Abstract 

The paper presents and evaluates a force-derivative method which 

produces higher-order modal solutions to transient problems. These 

higher-order solutions converge to an accurate response using fewer 

degrees-of-freedom (eigenmodes) than 1 ower-order methods such as 

the mode-displ acement or mode-acceleration methods. Results are 

presented for non-proportionally damped structural problems as 

well as thermal problems modeled by finite elements. 

. 

I- 1-- 

Method, higher-order Modal -Methods 

19. Security Classif. (of this report) 1 20. Securlty Classif. (of thin page) 121. No. of pages 122. Price 1 A03 I 38 

Uncl ass i f i ed I Unclassified 

NASA FORM 167S OCT 86