JOURNAL OFSOUND ANDVIBRATION

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Journal of Sound and Vibration 272 (2004) 1134–1136

Letter to the Editor

A superposition of exponential–trigonometric functions:generalized solution for several forced vibrations problems of a

one-degree-of-freedom system

P.A.A Lauraa,b,*, S.I. Simonettia,b, A. Juana,b

aDepartment of Engineering, Institute of Applied Mechanics, Universidad Nacional del Sur, Bah!ıa Blanca 8000, ArgentinabDepartment of Physics, Institute of Applied Mechanics, Universidad Nacional del Sur, Bah!ıa Blanca 8000, Argentina

Received 14 July 2003; accepted 23 July 2003

1. Introduction

Consider a one-degree-of-freedom vibrating system subjected to a forced excitation of the type

FðtÞ ¼XN

i¼1

Fieai t cosoit: ð1Þ

Accordingly the governing differential system is

M .x þ kx ¼XN

i¼1

Fieai tcosoit; xð0Þ ¼ xo; ’xð0Þ ¼ v0: ð2a–cÞ

Assuming now that ai ¼ oi ¼ 0 and taking i ¼ 1; Eq. (2a) reduces to

M .x þ kx ¼ F1; ð3Þ

which constitutes the situation where the system is subjected to a constant force of magnitude F1:On the other hand, if oi ¼ 0 one has the situation where the system is excited by exponentially

varying forces. Possibly the most important situation is the one where

a1 ¼ bio0 ð4Þ

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*Corresponding author. Department of Engineering, Institute of Applied Mechanics, Universidad Nacional del Sur,

Bah!ıa Blanca 8000, Argentina. Fax: +54-291-459-5157.

E-mail address: ima@criba.edu.ar (P.A.A. Laura).

0022-460X/$ - see front matter r 2003 Elsevier Ltd. All rights reserved.

doi:10.1016/j.jsv.2003.07.026

where the force values decrease exponentially with the time variable and constitutes an acceptableapproximation for blast loads [1]. The expression

F ðtÞ ¼XN

i¼1

Fiebi t ð5Þ

may be adjusted to accomodate to experimentally obtained force values during a blast situation.Finally, if ai ¼ 0; one has (adding the term F0=2)

F ðtÞ ¼ 12

F0 þXN

i¼1

Ficosoit; oi ¼ipT; ð6Þ

which constitutes the Fourier series for the case where the forcing excitation is a periodic force andthe Fi’s are determined using the well-known Fourier formula for the cosine series expansion,

Fi ¼2

T

Z T

0

F ðtÞcosipT

t dt: ð7Þ

Accordingly differential system (2) is a rather general one for three types of basic excitations.For the sake of simplicity it will be assumed that

xð0Þ ¼ ’xð0Þ ¼ 0: ð8Þ

On the other hand the consideration of viscous damping does not add any formal complicationsto the treatment presented herewith.No claim of originality is made by the authors but it is hoped that university students and

practitioners will find the results useful in their work. On the other hand, the present Noteconstitutes an extension of a recent educational article [2].

2. Mathematical development

Expressing Eq. (2a) in the form

.x þ o2nx ¼

XN

i¼1

F 0i e

ai t cosoit; ð9Þ

where o2n ¼ k=M and F 0

i ¼ Fi=M; assuming as particular solution of Eq. (9) the expression

xp ¼XN

i¼1

Aieai tcosoit þ Bie

ai tsinoit ð10Þ

and substituting in Eq. (9), one obtains equating coefficients of like terms,

Ai ¼Fi

M

a2i � o2i þ o2

n

ða2i � o2i þ o2

nÞ2 þ 4a2i o

2i

; ð11aÞ

Bi ¼Fi

M

2aioi

ða2i � o2i þ o2

nÞ2 þ 4a2i o

2i

: ð11bÞ

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P.A.A. Laura et al. / Journal of Sound and Vibration 272 (2004) 1134–1136 1135

Accordingly the general solution of Eq. (9) is

xðtÞ ¼C1cosont þ C2sinont þ1

M

XN

i¼1

Fi

ða2i � o2i þ o2

nÞ2 þ 4a2i o

2i

� ½ða2i � o2i þ o2

nÞeai tcosoit þ ð2aioiÞeai tsinoit: ð12Þ

Since

xð0Þ ¼ C1 þ1

M

XN

i¼1

Fiða2i � o2i þ o2

nÞ

ða2i � o2i þ o2

nÞ2 þ 4a2i o

2i

ð13Þ

one obtains

C1 ¼ �1

M

XN

i¼1

Fiða2i � o2i þ o2

nÞ

ða2i � o2i þ o2

nÞ2 þ 4a2i o

2i

:

Applying the second initial condition results in

C2 ¼ �1

onM

XN

i¼1

Fi

ða2i � o2i þ o2

nÞ2 þ 4a2i o

2i

½aiða2i � o2i þ o2

nÞ þ 2o2i ai: ð14Þ

Eqs. (12)–(14) describe the response of the system under the generalized excitation.For instance, if ai ¼ oi ¼ 0 and Fi ¼ 0 for i > 1 one obtains the well-known expression

xðtÞ ¼F1

kð1� cosontÞ: ð15Þ

On the other hand, if oi ¼ 0; the resulting expression yields

xðtÞ ¼ �1

Mcosont

XN

i¼1

Fiða2i þ o2nÞ

ða2i þ o2nÞ

�1

onMsinont

XN

i¼1

Fiaiða2i þ o2nÞ

a2i þ o2n

þ1

M

XN

i¼1

Fiða2i þ o2nÞ

ða2i þ o2nÞ

eai t: ð16Þ

Acknowledgements

The present study has been sponsored by CONICET and Secretar!ıa General de Ciencia yTecnolog!ıa of Universidad Nacional del Sur (Project Directors: Professor R.E. Rossi andC.A. Rossit). Miss S.I. Simonetti has been supported by a fellowship of Comisi !on deInvestigaciones Cient!ıficas (Buenos Aires Province).

References

[1] Ch.H. Norris, R.J. Hansen, M.J. Holley Jr., J.M. Biggs, S. Namyet, J.K. Minami, Structural Design for Dynamic

Loads, Mc Graw-Hill, New York, 1959.

[2] P.A.A. Laura, C.A. Rossit, D.V. Bambill, S.A. Vera, A note on the solution of very basic vibrations problems

obtained from a generalized situation, International Journal of Mechanical Engineering Education, in press.

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P.A.A. Laura et al. / Journal of Sound and Vibration 272 (2004) 1134–11361136