The Practical Step-by-Step Procedure of High-order Dynamic Response

Analysis for General Damped System

Ruifang Yu1,a, Xiyuan Zhou2,b and Meiqiao Yuan3,c 1Institute of Geophysics, China Earthquake Administration, Beijing, P.R. China

2 Beijing University of Technology, Beijing, P.R. China

3Institute of Earthquake Engineering, Chongqing, P.R. China

a yrfang126@126.com,

bzhouxy@bjut.edu.cn,

cyuanmeiqiao@yeah.net

Keywords: Damped system, Multiple eigenvalues, High-order response, Numerical integration; Step-by-Step procedure.

Abstract. For general damped linear systems with multiple eigenvalues, this study provides a

practical step-by-step procedure of the high-order dynamic response analysis. The method derived in

this study has clear physical concepts and is easily to be understood and mastered by engineering

designers. Moreover, a numerical example is used to analyze the correctness and the effectiveness of

the new method by comparing the calculation results with the theory solution.

Introduction

The modal superposition method is widely used in the dynamic response analysis of linear systems.

This method can simplify the dynamic response analysis through decoupling the vibration equation

based on the orthogonality of eigenvectors. The resulting complex multi-degree-of-freedom (MDOF)

system can be turned into the linear superposition of the independent dynamic responses of a series of

single degree of freedom (SDOF) systems subjected to identical ground motion. In the current seismic

design code for structures, it is usual to have design structures without equal or very close natural

frequencies. However, these issues change gradually not only with the growth in structural size and

configuration, but also the variety and complexity of the systems. Very close and multiple frequencies

are no longer unusual and sometimes even become inevitable. Moreover, it is worth considering

whether the earthquake responses corresponding to the two equal frequencies can be offset each other.

In fact, it is the basis on which the tuned mass damper (TMD) was developed[1,2]. In addition,

problems of multiple eigenvalues have become common issues of many electrical and control

systems. Inevitably, the methods of calculating the high-order dynamic response of system emerge. In

this paper, a practical step-by-step procedure of high-order dynamic response analysis for the general

damped system is obtained, which is suitable for engineering applications. The method derived in this

study has clear physical concepts and is easily to be understood and mastered by engineering

designers. Moreover, a numerical example is used to analyze the correctness and the effectiveness of

the new method by comparing the calculation results with the theory solution.

Dynamic Responses Analysis for a General Damped Linear System

It is well known that for a discrete system, with N degrees of freedom, the equations of motion in

terms of nodal displacements are expressed as

( )gy tMy + Cy + Ky = MI . (1)

Here M ,C and K are the NN mass, damping and stiffness matrices, which are symmetric matrices; y is a 1N nodal displacement vector which describes the dynamic response of the structure, and

N is an arbitrarily large integer; I is the displacement transformation vector that expresses the

Advanced Materials Research Vols. 378-379 (2012) pp 161-165Online available since 2011/Oct/27 at www.scientific.net (2012) Trans Tech Publications, Switzerlanddoi:10.4028/www.scientific.net/AMR.378-379.161

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displacement of each structures degree of freedom due to the static application of a unit support

displacement, and ( )ty g is the arbitrary time history of ground acceleration. Eq. (1) can be rewritten as a group of linear differential equations of first order, that is

( )gy t= +x Ax b . (2)

in which 1 1

=

M C M KA

I 0,

=

eb

0,

yx =

y. (3)

For a MDOF system with the multiple eigenvalues, it is assumed that there are z -pairs of

conjugate complex eigenvalues, ( ) ( ) ( )1 1 2 2, , , , , ,z z , whose multiplicity are 1 2, , , zk k k , respectively, in which

1 2 zk k k N+ + + = . Using the appropriate methods, such as the generalized

complex modal superposition method[3-5] and residue matrix decomposition method[6,7], the

displacement response ( )ty of system can be expressed as linear superposition of the z coupled systems based on the z multiple eigenvalues, i.e.

( ) ( ), , 1 , , , ,1 1

mkz

m i m i m i m i m i m i

m i

t q q q= =

= + +y E A B . (4)

in which: , , ,, ,

m i m i m iE A B are the generalized participation factors corresponding to the i -th order

response of m -th multiple eigenvalue m

, which can also be determined by consulting the relevant

references[4,6]. ( ),m iq t and ( ),m iq t are respectively the i -th order response of displacement and velocity corresponding to the m -th eigenvalues with multiplicity

mk , in which the order number i varies from 1

to m

k . Now we discuss the calculation methods of ( ),m iq t and ( ),m iq t . The First-order Dynamic Response Corresponding to Multiple Egenvalue. Suppose the order

number 1i = , then ( ),1mq t and ( ),1mq t can be calculated by solving the following equation

( ) ( ) ( )2,1 ,1 ,12 ( )m m m m m m gq t q t q t y t + + = . (5) in which

m and

m are the free vibration frequency and the corresponding critical damping ratio of

the m -th mode. And the displacement response ( ),1mq t can be expressed as Duhamel integration in terms of the sine, i.e.

( ) ( ) ( )( ) ( ) ( ) ( ),1 ,10 0

1sinm

t tt

m m g m g

m

q t e t y d h t y d

= = . (6)

here m m m

= and 21m m m

= are the damping coefficient and the damped frequency of the m -th

mode, and ( ),1mh t is the corresponding impulse response function, that is

( ) ( ) ( ),11

sinmt

m m

m

h t e t

= . (7)

and the velocity response ( ),1mq t can be obtained by taking the derivative of the preceding Eq. (7). As to the Duhamel integral expressed by formula (6), we can solve it through using simple

summation, trapezoidal rule and Simpsons rule[8]. What should be stated out is that the accuracy to

be expected from any of the above numerical procedures depends, of course, on the duration of time

interval t . In general, this duration must be selected short enough for both the load and the trigonometric functions used in the analysis to be well defined, and further, to provide the normal

engineering accuracy, it should also satisfy the condition 10t T = . Usually, the increased accuracy

obtained using Simpsons rule, rather than the simple summation or trapezoidal rule, justifies its use,

even though it is more complex. It is important to note that the response analysis procedures described

in above involve evaluation of many independent response contributions that are combined to obtain

the total response. Because superposition is applied to obtain the final result, neither of those methods

162 Applied Materials and Electronics Engineering

is suited for use in analysis of nonlinear response; therefore judgment must be used in applying them

in earthquake engineering where it is expected that a severe earthquake will induce inelastic

deformation in a code-designed structure.

The step-by-step procedure is a second general approach to dynamic response analysis of system

expressed by Eq. (5), and it is well suited to analysis of nonlinear response because it avoids any use

of superposition. These are many different step-by-step methods, but in all of them the loading and the

response history are divided into a sequence of the time intervals or step. The response during each

step then is calculated from the initial conditions (displacement and velocity) existing at the beginning

of the step and from the history of loading during the step. Thus the response for each step is an

independent analysis problem, and there is no need to combine response contributions within the step.

Step-by-step methods provide the only completely general approach to analysis of nonlinear response;

however, the methods are equally valuable in the analysis of linear response because the same

algorithms can be applied regardless of whether the structure is behaving linearly or not. The simplest

step-by-step method for analysis of SDOF systems is the so-called piecewise exact method[8],

which is based on the exact solution of the equation of motion for response of a linear structure to a

loading that varies linearly during a discrete time interval. The other step-by-step methods employ

numerical procedures to approximately satisfy the equations motion during each timeusing either

numerical differentiation or numerical integration. A vast body of literature has been written on these

subjects[8].

The High-order Dynamic Response Corresponding to Multiple Eigenvalue. For the i -th

response ( 1i > , and varies between 1 and m

k ) corresponding to the multiple eigenvalue m

, the

corresponding impulse response function can be written as

( ) ( )( ) ( ) ( )1,1

sinmi t

m i m m

m

h t t e t

= . (8)

If consider the input ( )gy of earthquake acceleration time history, the dynamic response can also be expressed in the general convolution integration form, i.e.

( ) ( ) ( ) ( )( ) ( )1,0

1sinm

ti t

m i m m g

m

q t t e t y d

= . (9)

The dynamic response expressed as Eq. (9) is named high-order dynamic response because Eq. (9)

has an additional dimensionless term ( )1i

m t

( 1i > ) compared to the case in the Eq. (6).

For the Duhamel integration showed in formula (9), we can solve the high-order dynamic response

through using the trapezoidal rule and Simpsons rule[8] mentioned afore. Correspondingly, the

velocity response of formula (9) can be expressed through Duhamel integration in terms of the sine

and cosine, i.e.

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )1, , 1 ,0

1 cosmt

i t

m i m m i m m i m m gq t i q t q t t e t y d

= . (10)

and can also be solved by using the trapezoidal rule and Simpsons rule. Then, can we solve the

high-order dynamic response through commonly used step-by-step method in the earthquake

engineering? The answer is Yes if the Eq. (9) can be transformed into the expression of a general

SDOF system. Now let us expand the term ( )1i

m t

of formula (9) into binomial expression, i.e.

( ) ( ) ( )( )

( )1

1 1 11 1

0

1 !

1 ! !

ii i i ji i j

m m m

j

it t t

i j j

=

= =

. (11)

After substituting this formula into formula (9), we obtain

( ) ( ) ( )( ) ( ),0

1sinm

tt i

m i m gG

m

q t e t y t d

= . (12)

Advanced Materials Research Vols. 378-379 163

in which ( ) ( ) ( ) ( )( )

( ) ( ) ( )1

1 11

0

1 !

1 ! !

ii j i ji i

gG m g m m g

j

iy t t y t y

i j j

=

= =

. (13)

Then the form of Eq. (12) is identical to the Eq.(6) by changing the input of earthquake

acceleration. However, after comparing Eqs. (12) and (6), it can be seen that for the general SDOF

system subjected to ground motion, the earthquake input ( )gy is unrelated to varying time t , but the input ( )igGy t of formula (12) is related to time t . Supposed that input ( )igGy t is irrelevant to time t , that is, let ( ) ( )i igG gGy t y = , then the corresponding velocity response of formula (12) is

( ) ( ) ( ) ( )( ) ( )2, ,0

sin 1m mt

t i

m i m m m i m m gGq t q t e t y d =

. (14)

Substituting the Eq. (13) into Eq. (14), we get

( ) ( ) ( ) ( ) ( )( ) ( )11, ,0

sinmt

i ti

m i m m m i m m gq t q t t e t y d =

. (15)

Comparing Eq. (15) with Eq. (10), it can be seen that ( ) ( ), 11 m m ii q t is missing from Eq. (15). After analyzing the superposition integration formula for the high-order earthquake response, it can

be seen that if we adopt step-by-step procedure to calculate high-order earthquake response of system,

the calculation results of the displacement would not be influenced after changing the earthquake

input. However, the velocity response would be influenced because the earthquake input is function

relevant to time t , that is, we need to enclose term ( ) ( ), 11 m m ii q t into the velocity response after using step-by-step procedure, which means the real high-order velocity response ( ),m iq t is

( ) ( ) ( ) ( ), , , 11m i m i m m iq t q t i q t = +

. (16)

here ( ),m iq t is the velocity response obtained through step-by-step procedure after the earthquake input

is changed. The rest parameters are the same as afore.

Numerical Analysis and Examination

In order to verify the method mentioned above, we solve the high-order response of the damped

system through superposition numerical integration method and step-by-step procedure. And the

calculation results are compared with theory solution.

Suppose 6.2915m m

= =10.0735 , the input is ( ) sing my t t= (gal), and the time interval is 0.010t s = . Now let the order number 2i = , then the corresponding response is the second-order response. It is easy to get theory solution of displacement and velocity response described by Eq. (9) and Eq. (10)

under the Sine-Wave input. The Fig. 1 gives the theory solutions of displacement and velocity time

history. And the peak values of displacement and velocity responses are: 0.0136 cm and 0.1367cm/s,

respectively.

0 1 2 3 4 5 6 7 8-0.02

-0.01

0

0.01

0.02

Time (s)

Dis

p. (c

m)

0 1 2 3 4 5 6 7 8-0.2

-0.1

0

0.1

0.2

Time (s)

Vel. (

cm

/s)

Fig. 1 Theory solution of displacement and velocity response under the Sine-Wave input

We firstly solve the Duhamel integration of Eqs. (9) and (10) by using the trapezoidal rule. The

results show that the displacement response obtained from trapezoidal rule matches the theory data

well, but the peak value of velocity response is 0.1365 cm/s, as shown in Table 1, which is less than

theory solution. Besides, the step-by-step procedure the piecewise exact method[8] is selected to

164 Applied Materials and Electronics Engineering

calculate same example, the input is changed accordingly to the Eq.(13), and the velocity response is

calculated according to Eq. (16).The results show that piecewise exact method obtained identical

displacement response as that of theory, but the peak value of velocity response is 0.1366cm/s, as

shown in Table 1.

In order to find the difference of the second-order velocity responses between theory solutions and

numerical results, the time interval is changed to 0.005t s = . The results are shown in Table 1, it can be seen that the peak value of velocity response calculated by piecewise exact method perfectly matches

the theory solution. That is, the smaller the time interval t , the closer the results obtained from different methods. It can be seen from the above analysis that, in the engineering practice, the

high-order dynamic response of system can be calculated by using step-by-step method according to

the general SDOF system, in which the earthquake input is changed according to the Eq. (13) and the

velocity response is amended according to the Eq. (16). It should be pointed out that calculation of

velocity response using step-by-step method is sensitive to the time interval t .

Table 1 The peak value of velocity response

t [s] Trapezoidal rule [cm/s] Piecewise exact method [cm/s]

0.010 0.1365 0.1366

0.005 0.1366 0.1367

Conclusion

For the general damped linear system with multiple eigenvalues, a practical step-by-step procedure of

the high-order dynamic response analysis is obtained in this study, which is suitable for engineering

applications. The method derived in this study has clear physical concepts and is easily to be

understood and mastered by engineering designers. In addition, the correctness and the effectiveness

of the method derived in this study are verified by comparing the calculation results with the theory

solution. It should be pointed out that calculation of velocity response using step-by-step method is

sensitive to the time interval.

Acknowledgement

This research is funded by the project of DQJB11C22, IGPCEA.

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Advanced Materials Research Vols. 378-379 165

Applied Materials and Electronics Engineering 10.4028/www.scientific.net/AMR.378-379

The Practical Step-by-Step Procedure of High-Order Dynamic Response Analysis for General DampedSystem 10.4028/www.scientific.net/AMR.378-379.161 DOI References[1] Y. Fujino and M. Abe: Design formulas for tuned mass dampers based on a perturbation technique.Earthquake Engineering and Structural Dynamic Vol. 22 (1993), pp.833-854.http://dx.doi.org/10.1002/eqe.4290221002