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Comments on “effect of creep deformations in mechanical systems under combined deterministic and random inputs”
Journal
of Sound and Vibration (1976) 44(2), 307-308
COMMENTS ON “EFFECT OF CREEP DEFORMATIONS IN MECHANICAL SYSTEMS UNDER COMBINED DETERMINISTIC AND RANDOM INPUTS”
The authors have attempted in their paper [l] to study the effect of creep in mechanical systems under random excitations. The title and the introduction are misleading since the excitation is strictly random. Since 0 is a random variable uniformly distributed in (0, 27r) t ,- 0) term, claimed to be deterministic by the authors, is in fact a random process. the COS(M’ If this point is overlooked the mean of the input and hence of the output cannot be zero. From the text it seems the aim of the paper is to study the nature of the moments of the response. According to equation (16a) of the authors the response is non-stationary. But this conclusion is based on an expression for Q which is valid only in the steady state. It would be more accurate to take Q = WXj h(7) Fi(t - 7) dr. 0
Also it may be mentioned here that the input auto-correlation R,,
(7) = +(F;
+
203 cos w, z
used by the authors is very approximate. Such an expression is true only for a single frequency input which one can obtain by making the bandwidth off(t) go to zero. Even granting that, for a qualitative study, the steady state result for Q with the above RFi (z) is sufficiently accurate, the stationarity of Q does not ensure that Q(t) and its integral, namely Qr(t), are uncorrelated. Unfortunately, the authors seem to have made this assumption in deriving equation (18) for R,(t, 7). Thus the consequent result on the mean square of x(t) is questionable. The lack of proper initial conditions on X makes the text very confusing. In particular, equation (19) conveys no meaning. In deriving an expression for (w,,(t)) equation (29) which states that (FiRj = (l/K,)(Fi
Q) + +(Fi
& = 0,
2
with Q = w;
j?z(r)F;(r - 7) dr, 0
has been used by the authors. This statement will be true only if the stationary process F,(t) is taken to be uncorrelated with itsjirst as well as second derivative processes at all times. This in general is not true for any stationary process. Consequently the expressions for (w,,(t)j and D(t) appear to be wrong. The correct expressions, with all the tedious algebra given by the authors avoided, can be obtained as follows. Since Fs = mg + Q,
(w,,(t)> = (l/K,)
i
0
(m2g2+ CQ'>)dT
0
w;(F; + 2a;)
308
LETTERSTOTHEEDITOR
The dissipated energy corresponding to this expression is D(t)
wi(F6 + 20;) =_I_ Kz [2[(wi - wf)’ + 4/F wf] I t. P.K. DASH
Department of Aeronautical Engineering, Indian Institute of Science, Bangalore-560012, India (Received 28 July 1975)
REFERENCE 1. H. R. SRIRANGARAJAN and
B. V. DASARATHY 1974 Journal of Sound and Vibration 3j, 453-458. Effect of creep deformations in mechanical systems under combined deterministic and random inputs.
AUTHORS’ REPLY The ensemble of the excitations, each member of which is the sum of sinusoidal and purely random functions of time, is indeed a random process as 8 is uniformly distributed between (0,2rc). However, each member of this ensemble, in a temporal sense, is a combination of a deterministic sinusoidal function of time, where 0 has some arbitrary fixed value in (0,27r), and a random function of time. This distinction between the ensemble and temporal variations has been lost sight of in the comments. The expression for R,, (2) is indeed correct as long asf(l) is a narrow band Gaussian process symmetric about w, and this is easily verified by using the analysis of Rice (Bell System Technical Journal 1944, Vol. 23) who was the first to study combined sinusoidal and narrow band random signals. While (Q) = 0, the mean of thesutput is not zero as can be seen in equation (16a). Equation (19) is an expression for the autocorrelation function of Qr(t) and is of course based on the assumption that Q(t) and Qr(t) are uncorrelated. The assumptions underlying determination of (w,,(t)) and (D(t)) are, as pointed out, questionable to some extent and the suggested alternative approach seems to be in order. Department of Mechanical Engineering, Indian Institute of Science, Bangalore-560012, India Computer Sciences Corporation, 8300 South Whitesburg Drive,
Huntsville, Alabama 35802, U.S.A. (Received 24 September 1975)
H.R. SRIRANGARAJAN
B.V. DASARATHY
of Sound and Vibration (1976) 44(2), 307-308
COMMENTS ON “EFFECT OF CREEP DEFORMATIONS IN MECHANICAL SYSTEMS UNDER COMBINED DETERMINISTIC AND RANDOM INPUTS”
The authors have attempted in their paper [l] to study the effect of creep in mechanical systems under random excitations. The title and the introduction are misleading since the excitation is strictly random. Since 0 is a random variable uniformly distributed in (0, 27r) t ,- 0) term, claimed to be deterministic by the authors, is in fact a random process. the COS(M’ If this point is overlooked the mean of the input and hence of the output cannot be zero. From the text it seems the aim of the paper is to study the nature of the moments of the response. According to equation (16a) of the authors the response is non-stationary. But this conclusion is based on an expression for Q which is valid only in the steady state. It would be more accurate to take Q = WXj h(7) Fi(t - 7) dr. 0
Also it may be mentioned here that the input auto-correlation R,,
(7) = +(F;
+
203 cos w, z
used by the authors is very approximate. Such an expression is true only for a single frequency input which one can obtain by making the bandwidth off(t) go to zero. Even granting that, for a qualitative study, the steady state result for Q with the above RFi (z) is sufficiently accurate, the stationarity of Q does not ensure that Q(t) and its integral, namely Qr(t), are uncorrelated. Unfortunately, the authors seem to have made this assumption in deriving equation (18) for R,(t, 7). Thus the consequent result on the mean square of x(t) is questionable. The lack of proper initial conditions on X makes the text very confusing. In particular, equation (19) conveys no meaning. In deriving an expression for (w,,(t)) equation (29) which states that (FiRj = (l/K,)(Fi
Q) + +(Fi
& = 0,
2
with Q = w;
j?z(r)F;(r - 7) dr, 0
has been used by the authors. This statement will be true only if the stationary process F,(t) is taken to be uncorrelated with itsjirst as well as second derivative processes at all times. This in general is not true for any stationary process. Consequently the expressions for (w,,(t)j and D(t) appear to be wrong. The correct expressions, with all the tedious algebra given by the authors avoided, can be obtained as follows. Since Fs = mg + Q,
(w,,(t)> = (l/K,)
i
0
(m2g2+ CQ'>)dT
0
w;(F; + 2a;)
308
LETTERSTOTHEEDITOR
The dissipated energy corresponding to this expression is D(t)
wi(F6 + 20;) =_I_ Kz [2[(wi - wf)’ + 4/F wf] I t. P.K. DASH
Department of Aeronautical Engineering, Indian Institute of Science, Bangalore-560012, India (Received 28 July 1975)
REFERENCE 1. H. R. SRIRANGARAJAN and
B. V. DASARATHY 1974 Journal of Sound and Vibration 3j, 453-458. Effect of creep deformations in mechanical systems under combined deterministic and random inputs.
AUTHORS’ REPLY The ensemble of the excitations, each member of which is the sum of sinusoidal and purely random functions of time, is indeed a random process as 8 is uniformly distributed between (0,2rc). However, each member of this ensemble, in a temporal sense, is a combination of a deterministic sinusoidal function of time, where 0 has some arbitrary fixed value in (0,27r), and a random function of time. This distinction between the ensemble and temporal variations has been lost sight of in the comments. The expression for R,, (2) is indeed correct as long asf(l) is a narrow band Gaussian process symmetric about w, and this is easily verified by using the analysis of Rice (Bell System Technical Journal 1944, Vol. 23) who was the first to study combined sinusoidal and narrow band random signals. While (Q) = 0, the mean of thesutput is not zero as can be seen in equation (16a). Equation (19) is an expression for the autocorrelation function of Qr(t) and is of course based on the assumption that Q(t) and Qr(t) are uncorrelated. The assumptions underlying determination of (w,,(t)) and (D(t)) are, as pointed out, questionable to some extent and the suggested alternative approach seems to be in order. Department of Mechanical Engineering, Indian Institute of Science, Bangalore-560012, India Computer Sciences Corporation, 8300 South Whitesburg Drive,
Huntsville, Alabama 35802, U.S.A. (Received 24 September 1975)
H.R. SRIRANGARAJAN
B.V. DASARATHY