This paper considers the response of a nonlinear string to random excitation. It is shown that, owing to the additional stress induced by the stretching of the string, the mean squared deflection at every point is smaller than that for the equivalent linear string

Caughey, T. K.

English

Caltech Authors - Main

Paper No. 
59-APMW-4 ( Response of a Nonlinear String 
T. K. CAUGHEY 1 to Random Loading I Y Assistant Professor of Applied Mechanics, Division of Engineering, California Institute of Technology, T h i s  paper considers the response of a nonli?zear string to random excitation. I t  i s  Pasadena, Calif. shown that, owing to the addi t imal  stress induced by the stretching of the string, the mean squared deflection at every $ s i l t  i s  smaller than that for the equivalent linear string. 
TnE nEsPoNsE of a nonlinear string to deterministic 
loading has been analyzed by several a ~ t h o r s . l - ~  G. F. Carrier1,* 
has analyzed the free oscillations of a nonlinear string, while 
Lee3 and the author4 have analyzed the response of a nonlinear 
string to deterministic loading. This paper is devoted to the 
study of a nonlinear string subjected to a random excitation. It 
is shown that, as in the case of deterministic loading, the addi- 
tional stress induced by the stretching of the string reduces the 
mean squared deflection a t  every point compared with that for the 
equivalent linear string. 
Formulation 
Consider an elastic string, clamped a t  its ends subjected to a 
random loading. The nomenclature used is as fol1oa.s: 
u = transverse deflection of string 
To = initial tension in string 
p = linear density of string 
p/3 = viscous damping coefficient 
A = cross-sectional area of string 
E = elastic modulus of string 
L = length of string 
f (r,  t ) = loading on string 
randomly varying force with a clipped Gaussian white spectrum 
and has the following cross correlation: 
sin W,T 
f(x, t l ) f ( q ,  tz) = 4D@x - q )  7 
where 
7 = t, - t2 
Since the eigenmodes for equation ( 1 )  are sine waves, let 
i r x  
f(x, t )  = x ai(t)  sin - 
i = l  L 
N i r x  
u(z, t )  = ba(t)  sin - 
i = l  L 
where N is chosen such that w, < w, < CON+,. 
As a consequence of the assumed form of the cross correlation 
(3), the ai in (4) are uncorrelated and the power spectrum gi(w) of 
ai is given by 
If the deflections and slopes are considered to be moderately 
= 0 w > w, 
small (consistent with the string remaining elastic) the equation 
of motion of the string is Substituting (4) into ( 1 )  
+f(x, t )  ( 1 )  
subject to the boundarv conditions that 
In  the analysis that follows it  will be assumed that f(x, t )  is a 
-
1 G. F. Carrier, "On the Nonlinear Vibration Problem of the 
Elastic String," Quarterly of Applied Mathematics, vol. 3, 1945, pp. 
157-165. 
2 G. F. Carrier, "-4 Note on the Vibrating Strine," Quarterlg of Ap- 
plied Mathematics, vol. 7 ,  1949, pp. 97-101. 
3 E. W. Lee, "Nonlinear Forced Vibrations of a Stretched String," 
British Journol of Apwlied Physics, vol. 8,  1957, pp. 411-413. 
4 T. K. Caughey, unpublished CalTech Report No. 85, 1956. 
For presentation at the West Coast Conference of the Applied 
Mechanics Division, Stanford, Calif., September 9-11, 1959, of THE 
AMERICAN SOCIETY OF MECHANICAL ENGINEERS. 
Discussion of this paper should be addressed to the Secretary, 
ASME, 29 West 39th Street, New York, N. Y., and will be accepted 
until one month after final publication of the paper itself in the 
JOURNAL OF APPLIED MECHANICS. 
NOTE: Statements and opinions advanced in papers are to be 
understood as individual expressions of their authors and not those 
of the Society. Manuscript received by ASME Applied Mechanics 
Division, November 18, 1957. Paper No. 59-APMW-4. 
Using the orthogonality of the circular functions, equation ( 6 )  be- 
comes 
where 
Equivalent Linear Form of Equation (7) 
The method of equivalent linearization has been applied suc- 
cessfully in the past to nonlinear differential equations with 
harmonic excitation. I t  will now be shown that, by a slightly 
different approach to the problem, this method may be applied 
to equation (7).  
Rewriting equation (7) in the form 
Journal of Applied Mechanics 
d2bn a ,  + a 2 + r;)' 5 K.), + ~ ( 4 ,  b2, . . . bN)  = - ( 9 )  The mean squared value of b, is readily computed from ( 1 7 ) .  P P Thus 
If in ( 9 ) ,  t(b,, bp, . . . b N )  is neglected, ( 9 )  is linear and can be solved 
u,2 = = 
readily. The smaller ~ ( b l ,  b2, . . . bN) is, the smaller the error lm Bn(w)dw 
in neglecting it. The logical choice of K,  is, therefore, that value 
of K ,  which minimizes ~ ( b l ,  bz, . . . 6,). = 22 D 
The choice of minimization procedure is somewhat arbitrary, - - {[(y)22 KVL - w 2 I 2  + [w,L?]2)-1 dw ( 1 8 )  PL n L P  
but by analogy with Galerkin's method, and for mathematical 
expediency, it  is desirable to use the minimization of the mean If /3 is small and 
squared error 
(7)' K n  < wo2 
{ [ i  + a 5 i2bi'] b, - Kd.)' dt ( 1 0 )  
i = l  
If the process is an ergodic one, then equation (10 )  may be re- The main contribution to the integral (18 )  coming from a small 
placed by the stochastic average. Further, since the ai are un- region close to 
correlated 
By the same token it  will be seen that if 
where p(b,) is the first probability density function of bi. the integral (18 )  will be very small, thus justifying the approxi- 
If in (9), e(b1, b2, . . . bN) is neglected, b, is Gaussian since a, is mate representation of u in (4). 
Gaussian. Substituting K ,  from (15 )  into (19 )  
where 
- 
- 
Minimizing e2 with respect to K,: Noting that 
N the mean squared deflection in the first mode of the linearized S" Sm [ l  + u x i2b$2] bn2 problem obtained by setting u = 0 in (7) .  Thus - m N fold - m i  = 1 
N u 2 n2 = { 1 +  a,.o2 [c i2 - + 2n2 2 II [exp ( -b,2/2u,2)dbil (141 u1,o2 i = I U I . O ~  I)-' ( 2 2 )  u1,02 
i = l  
Evaluating the integrals: Denoting 
If in equation ( 9 )  E( ) is neglected, then 
Equation ( 2 2 )  becomes 
The power spectrum of b, is therefore given by 
2zyn2 + [1 + 2S1yn - 1 = 0 ( 2 4 )  
Bn(m) = g,(@)p-' { [ ( ? f ) ' " K n - w 2 ] ' +  P L w / 3 1 2 ) - 1  ( 17 )  Solving: 
Y, = ( - ( I  + 2s) + [(I -I- zSI2 + 8 ~ ] ' / ~ }  [4z]-l ( 2 5 )  
where g,(w) is given by equation (5 ) .  5 See Appendix. 
2 Transactions of the ASME 
The positive root being taken in (25)  since y, by definition is Further, if in (28)  N i s  large, 
positive. Multiplying both sides of (25)  by 42, and summing on n 
zS 5 1/2[-1 + ( 1  + ~ N Z ) " ~ ]  ( 3 8 )  
42s = - N ( 1  + z S )  + N [ ( l  + zS)2 + 8z11/2 (26)  Using (38),  equation (37)  becomes 
- Rearranging terms and squaring U 2  
- 
--- A 211 + ( 1  + ~ N Z ) ~ / ~ ] - '  5 1 
UoZ 
( 3 9 )  
= ( 1 + ~ 5 ) ~ + 8 z  (27)  
Thus the mean squared deflection in the nonlinear case is always 
Solving for zS: less than the mean squared deflection in the equivalent linear 
string. 
- 
Conclusions 
Since zS is positive, the positive radical is taken. From the foregoing analysis it  is clearly seen that the effect of 
substituting (28)  into (25) ,  y, may be calculated. will be the stress induced by the stretching of the string is to reduce the 
from (25)  that yn is independent of n, thus there is mean squared deflection a t  every point in the string compared 
equipartition of energy among the modes. with the corresponding mean squared deflection in the equivalent 
linear string. 
Calculation of Mean Squared Deflection i3 
Using equation ( 4 )  
N -  i r x  . j x r  
~2 = C bibi sin - sm - 
i, i L L 
Owing to the particular cross correlation function (3 ) ,  the bi are 
uncorrelated: thus 
b,bi = 0 i # j (31)  
Equation (30)  therefore becomes 
From equations (23)  and (25)  
Substituting (33)  into (32)  
- 
U 2  = a1,02( - ( I  + 2s) 
In  the linearized problem given by setting a = 0 in (7 ) ,  the 
mean squared deflection is given by 
N 1 i7Tx 
Uo2 = =".02 'C sin"' 
i = l  
Thus 
vz 
-= = { - (1  + z S )  + [(1  + Z S ) ~  + 8 ~ ] l / ~ ]  [4z]-l (36)  UoZ 
A P P E N D I X  
(a) Evaluaiion of integral (18). Let 
Thus 
un2 = 22D/pLr 1; { [Gn2 - w2I2 + [Pwl2] -ldw (40)  
Using integral 211.16~ of "Integraltaffel, Ester Teil, Unbestimmte 
Integral," by W. Grobner and N. Hofreiter (Springer, 1949), 
equation (40)  becomes 
asZ = 2D/pLrG,(sin 40 In [(Gn2 + wc2 + 2G,w, cos 4 0 ) /  
(GnZ + wez - 2ijnw, cos 40)l 
+ cos 40 tan+ [2Gnw, sin +o/(Gnz - W . Z ) ] ]  
{ P " ( s n 2  - Ps/4)] (41)  
where 
tan 2175~ = {P2(Gn2 - P2/4))L/2(ijn2 - P2/2} -l ( 4 2 )  
If p is small compared to G,, equation (41)  reduces to 
an2 11 2~/pL?rij,2P(tan-l[w,~/(G,2 - w 2 ) ]  
+ 0 [ p / a s n  In (26,/P) 11 ( 4 3 )  
For P/&, very small, the second term may be neglected entirely. 
Thus 
un2 A 2D/pLrG,2P(tan-1[w,P/(Gn" -3) ( 4 4 )  
If w, > i j ,  
u,2 A 2D/pLG,2P ( 4 5 )  
If Gn > w, 
u,2 2 ~ / p ~ r i j , ~ P { w , P / ( ~ , ~  - wc2)] 
= 2PDw,/pL?rPG,z(ijn2 - w2) 
2D/pL~fli j ,z  for P/G,< 1 ( 4 6 )  
This justifies the approximate representation of u in equation ( 4 )  
- 
where zS is given by equation (28). (b) Justification for neglecting term z 2 u n 2 / a l , 0 2 .  AS is 
If in (36)  z<< 1, (36)  reduces to  N +  1 
- shown in (46),  for 6 , 2  > wc2 U2 
-=: ' [l + 281-'5 1 (37)  
Uo2 gi2 2D0, LpLrij%Z(ij%z - w,z)] - 1  ( 4 7 )  
Journal of Applied Mechanics 3 
i.e., Comparing 
< @ ln [(N + 1 + W , / W ~ K N + ~ ' / ~ ) /  u.' < @e [In (1 i 2 / N ) ]  (51) 
- ~ . ~ ~ W O K N + I ' / ~  N + 1  2 i2z/$ I iz- "1.0~ - 00 
( N  + 1 - ~ , / ~ ~ o ~ n ~ + l ' " ) l  ( 5 1 )  
If @ is small and N is large, this ratio becomes very small and 
- ' ln [(fl -t 2 ) / N I  ( 5 2 )  may be neglected, thus justifying the treatment given in the 
~ T W ~ K N + ~ ' / ~  paper. 
Printed in U. S. A 
Transactions of the ASME 


Response of a nonlinear string to random loading

Paper No. 
59-APMW-4 
( Response of a Nonlinear String 
T. K. CAUGHEY 1 to Random Loading 
I 
Y 
Assistant Professor of Applied Mechanics, 
Division of Engineering, 
California Institute of Technology, T h i s  paper considers the response of a nonli?zear string to random excitation. I t  i s  
Pasadena, Calif. shown that, owing to the addi t imal  stress induced by the stretching of the string, the 
mean squared deflection at every $ s i l t  i s  smaller than that for the equivalent linear 
string. 
TnE nEsPoNsE of a nonlinear string to deterministic 
loading has been analyzed by several a ~ t h o r s . l - ~  G. F. Carrier1,* 
has analyzed the free oscillations of a nonlinear string, while 
Lee3 and the author4 have analyzed the response of a nonlinear 
string to deterministic loading. This paper is devoted to the 
study of a nonlinear string subjected to a random excitation. It 
is shown that, as in the case of deterministic loading, the addi- 
tional stress induced by the stretching of the string reduces the 
mean squared deflection a t  every point compared with that for the 
equivalent linear string. 
Formulation 
Consider an elastic string, clamped a t  its ends subjected to a 
random loading. The nomenclature used is as fol1oa.s: 
u = transverse deflection of string 
To = initial tension in string 
p = linear density of string 
p/3 = viscous damping coefficient 
A = cross-sectional area of string 
E = elastic modulus of string 
L = length of string 
f (r,  t ) = loading on string 
randomly varying force with a clipped Gaussian white spectrum 
and has the following cross correlation: 
sin W,T 
f(x, t l ) f ( q ,  tz) = 4D@x - q )  7 
where 
7 = t, - t2 
Since the eigenmodes for equation ( 1 )  are sine waves, let 
i r x  
f(x, t )  = x ai(t)  sin - 
i = l  L 
N 
i r x  
u(z, t )  = ba(t)  sin - 
i = l  L 
where N is chosen such that w, < w, < CON+,. 
As a consequence of the assumed form of the cross correlation 
(3), the ai in (4) are uncorrelated and the power spectrum gi(w) of 
ai is given by 
If the deflections and slopes are considered to be moderately = 0 w > w, 
small (consistent with the string remaining elastic) the equation 
of motion of the string is Substituting (4) into ( 1 )  
+f(x, t )  ( 1 )  
subject to the boundarv conditions that 
In  the analysis that follows it  will be assumed that f(x, t )  is a 
-
1 G. F. Carrier, "On the Nonlinear Vibration Problem of the 
Elastic String," Quarterly of Applied Mathematics, vol. 3, 1945, pp. 
157-165. 
2 G. F. Carrier, "-4 Note on the Vibrating Strine," Quarterlg of Ap- 
plied Mathematics, vol. 7 ,  1949, pp. 97-101. 
3 E. W. Lee, "Nonlinear Forced Vibrations of a Stretched String," 
British Journol of Apwlied Physics, vol. 8,  1957, pp. 411-413. 
4 T. K. Caughey, unpublished CalTech Report No. 85, 1956. 
For presentation at the West Coast Conference of the Applied 
Mechanics Division, Stanford, Calif., September 9-11, 1959, of THE 
AMERICAN SOCIETY OF MECHANICAL ENGINEERS. 
Discussion of this paper should be addressed to the Secretary, 
ASME, 29 West 39th Street, New York, N. Y., and will be accepted 
until one month after final publication of the paper itself in the 
JOURNAL OF APPLIED MECHANICS. 
NOTE: Statements and opinions advanced in papers are to be 
understood as individual expressions of their authors and not those 
of the Society. Manuscript received by ASME Applied Mechanics 
Division, November 18, 1957. Paper No. 59-APMW-4. 
Using the orthogonality of the circular functions, equation ( 6 )  be- 
comes 
where 
Equivalent Linear Form of Equation (7) 
The method of equivalent linearization has been applied suc- 
cessfully in the past to nonlinear differential equations with 
harmonic excitation. I t  will now be shown that, by a slightly 
different approach to the problem, this method may be applied 
to equation (7).  
Rewriting equation (7) in the form 
Journal of Applied Mechanics 
d2bn a ,  + a 2 + r;)' 5 K.), + ~ ( 4 ,  b2, . . . bN)  = - ( 9 )  The mean squared value of b, is readily computed from ( 1 7 ) .  
P P Thus 
If in ( 9 ) ,  t(b,, bp, . . . b N )  is neglected, ( 9 )  is linear and can be solved 
u,2 = = readily. The smaller ~ ( b l ,  b2, . . . bN) is, the smaller the error 
lm Bn(w)dw 
in neglecting it. The logical choice of K,  is, therefore, that value 
of K ,  which minimizes ~ ( b l ,  bz, . . . 6,). = 22 D 
The choice of minimization procedure is somewhat arbitrary, 
- - {[(y)22 KVL - w 2 I 2  + [w,L?]2)-1 dw ( 1 8 )  
PL n L P  
but by analogy with Galerkin's method, and for mathematical 
expediency, it  is desirable to use the minimization of the mean If /3 is small and 
squared error 
(7)' K n  < wo2 
{ [ i  + a 5 i2bi'] b, - Kd.)' dt ( 1 0 )  
i = l  
If the process is an ergodic one, then equation (10 )  may be re- The main contribution to the integral (18 )  coming from a small 
placed by the stochastic average. Further, since the ai are un- region close to 
correlated 
By the same token it  will be seen that if 
where p(b,) is the first probability density function of bi. 
the integral (18 )  will be very small, thus justifying the approxi- 
If in (9), e(b1, b2, . . . bN) is neglected, b, is Gaussian since a, is mate representation of u in (4). 
Gaussian. Substituting K ,  from (15 )  into (19 )  
where 
- 
- 
Minimizing e2 with respect to K,: Noting that 
N the mean squared deflection in the first mode of the linearized 
S" Sm [ l  + u x i2b$2] bn2 problem obtained by setting u = 0 in (7) .  Thus - m N fold - m i  = 1 
N u 2 n2 = { 1 +  a,.o2 [c i2 - + 2n2 2 II [exp ( -b,2/2u,2)dbil (141 u1,o2 i = I U I . O ~  I)-' ( 2 2 )  u1,02 
i = l  
Evaluating the integrals: Denoting 
If in equation ( 9 )  E( ) is neglected, then 
Equation ( 2 2 )  becomes 
The power spectrum of b, is therefore given by 
2zyn2 + [1 + 2S1yn - 1 = 0 ( 2 4 )  
Bn(m) = g,(@)p-' { [ ( ? f ) ' " K n - w 2 ] ' +  P L w / 3 1 2 ) - 1  ( 17 )  Solving: 
Y, = ( - ( I  + 2s) + [(I -I- zSI2 + 8 ~ ] ' / ~ }  [4z]-l ( 2 5 )  
where g,(w) is given by equation (5 ) .  5 See Appendix. 
2 Transactions of the ASME 
The positive root being taken in (25)  since y, by definition is Further, if in (28)  N i s  large, 
positive. Multiplying both sides of (25)  by 42, and summing on n 
zS 5 1/2[-1 + ( 1  + ~ N Z ) " ~ ]  ( 3 8 )  
42s = - N ( 1  + z S )  + N [ ( l  + zS)2 + 8z11/2 (26)  Using (38),  equation (37)  becomes 
- Rearranging terms and squaring U 2  
- --- A 211 + ( 1  + ~ N Z ) ~ / ~ ] - '  5 1 
UoZ 
( 3 9 )  
= ( 1 + ~ 5 ) ~ + 8 z  (27)  
Thus the mean squared deflection in the nonlinear case is always 
Solving for zS: less than the mean squared deflection in the equivalent linear 
string. - 
Conclusions 
Since zS is positive, the positive radical is taken. From the foregoing analysis it  is clearly seen that the effect of 
substituting (28)  into (25) ,  y, may be calculated. will be the stress induced by the stretching of the string is to reduce the 
from (25)  that yn is independent of n, thus there is mean squared deflection a t  every point in the string compared 
equipartition of energy among the modes. with the corresponding mean squared deflection in the equivalent 
linear string. 
Calculation of Mean Squared Deflection i3 
Using equation ( 4 )  
N -  i r x  . j x r  
~2 = C bibi sin - sm - 
i, i L L 
Owing to the particular cross correlation function (3 ) ,  the bi are 
uncorrelated: thus 
b,bi = 0 i # j (31)  
Equation (30)  therefore becomes 
From equations (23)  and (25)  
Substituting (33)  into (32)  
- 
U 2  = a1,02( - ( I  + 2s) 
In  the linearized problem given by setting a = 0 in (7 ) ,  the 
mean squared deflection is given by 
N 
1 i7Tx 
Uo2 = =".02 'C sin"' 
i = l  
Thus 
vz 
-= = { - (1  + z S )  + [(1  + Z S ) ~  + 8 ~ ] l / ~ ]  [4z]-l (36)  UoZ 
A P P E N D I X  
(a) Evaluaiion of integral (18). Let 
Thus 
un2 = 22D/pLr 1; { [Gn2 - w2I2 + [Pwl2] -ldw (40)  
Using integral 211.16~ of "Integraltaffel, Ester Teil, Unbestimmte 
Integral," by W. Grobner and N. Hofreiter (Springer, 1949), 
equation (40)  becomes 
asZ = 2D/pLrG,(sin 40 In [(Gn2 + wc2 + 2G,w, cos 4 0 ) /  
(GnZ + wez - 2ijnw, cos 40)l 
+ cos 40 tan+ [2Gnw, sin +o/(Gnz - W . Z ) ] ]  
{ P " ( s n 2  - Ps/4)] (41)  
where 
tan 2175~ = {P2(Gn2 - P2/4))L/2(ijn2 - P2/2} -l ( 4 2 )  
If p is small compared to G,, equation (41)  reduces to 
an2 11 2~/pL?rij,2P(tan-l[w,~/(G,2 - w 2 ) ]  
+ 0 [ p / a s n  In (26,/P) 11 ( 4 3 )  
For P/&, very small, the second term may be neglected entirely. 
Thus 
un2 A 2D/pLrG,2P(tan-1[w,P/(Gn" -3) ( 4 4 )  
If w, > i j ,  
u,2 A 2D/pLG,2P ( 4 5 )  
If Gn > w, 
u,2 2 ~ / p ~ r i j , ~ P { w , P / ( ~ , ~  - wc2)] 
= 2PDw,/pL?rPG,z(ijn2 - w2) 
2D/pL~fli j ,z  for P/G,< 1 ( 4 6 )  
This justifies the approximate representation of u in equation ( 4 )  
- 
where zS is given by equation (28). (b) Justification for neglecting term z 2 u n 2 / a l , 0 2 .  AS is 
If in (36)  z<< 1, (36)  reduces to  N +  1 
- shown in (46),  for 6 , 2  > wc2 
U2 
-=: ' [l + 281-'5 1 (37)  
Uo2 gi2 2D0, LpLrij%Z(ij%z - w,z)] - 1  ( 4 7 )  
Journal of Applied Mechanics 3 
i.e., Comparing 
< @ ln [(N + 1 + W , / W ~ K N + ~ ' / ~ ) /  u.' < @e [In (1 i 2 / N ) ]  (51) 
- ~ . ~ ~ W O K N + I ' / ~  N + 1  2 i2z/$ I iz- "1.0~ - 00 
( N  + 1 - ~ , / ~ ~ o ~ n ~ + l ' " ) l  ( 5 1 )  
If @ is small and N is large, this ratio becomes very small and 
- ' ln [(fl -t 2 ) / N I  ( 5 2 )  may be neglected, thus justifying the treatment given in the 
~ T W ~ K N + ~ ' / ~  paper. 
Printed in U. S. A 
Transactions of the ASME 


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Response of a nonlinear string to random loading

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