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Some comments on “Stability of forced periodic response in third order non-linear systems”
Journal
of‘Sound and Vibration (1979) 63(l), 145-147
LETTERS SOME
TO THE EDITOR
COMMENTS ON “STABILITY OF FORCED PERIODIC THIRD ORDER NON-LINEAR SYSTEMS”
RESPONSE
IN
Mittal in his paper [l] has dealt with the stability of the forced periodic response of third order non-linear systems governed by the equation of the type Yi’+ a? + b2 + cx + dx3 = Fcosot;(.)$d()/dt.
(1)
He has shown that the stability condition derived from the linearized version of equation (1) is the same as the result obtained by investigating the nature of the roots of the characteristic equation corresponding to the variational equation of (1) for small d. He concludes the paper with a note of suspicion regarding the applicability of the rule of vertical tangents to the frequency response curves for these systems. In a recent paper Christopher [2] has studied the stability of periodic solutions of a third order non-linear system arising in some servomechanisms. The equation is of the form Y + fijl + f,(x)a + ,fOx = F cos ot,
(2)
where fr(x) = m, + m2x2, and Christopher has discussed the stability of the forced response solution in terms of the asymptotic stability of the corresponding variational equation using a method due to Cesari. Taking the solution to have, in the first approximation, an amplitude A and frequency w he has shown that one of the asymptotic boundaries corresponds to the locus of vertical tangents of the frequency response curves in the A-<0 plane. Following the method described in reference [2] one can obtain similar results for equations of the type (1). The vector equivalent of equation (1) is f = J(t, x),
(3)
where ii = X2’
i3 = - (c + dx;)x,
ii-, = x3,
- bx, - ax3 + F cos or.
If equation (1) is assumed to have a solution x = p(t) = A cos(ot + f#l)
(4)
the variational equation of equation (3) with respect to equation (4) is
aJ(t,
4= i j=
1
(5)
pw)tj
axj
where J is the Jacobian matrix with respect to x. Using equations (3) and (5), one obtains
’ = [;
+ 3dx;)
-;
.-‘j-.
(6)
In scalar form equation (6) becomes ... r + at + b[ + (c + 3dx;)t = 0.
(7)
145 0022-460X/79/050145
+03 $02.00/O
@ 1979 Academic
Press Inc. (London)
Limited
146
LETTERSTOTHEEDITOR
Upon defining 2 = ot + 4, equation (7) reduces to 5”’ + h,5” + h,<’ + h,5: = -E(COS 2Z)5, where ( ‘)&d( )/dZ, h, = a/o, h, = b/w2, h, = c/o3 + E, E If
=
3/2(dA2/03).
now one looks upon E as a small parameter and lets h, = k,, h, = 1 + wI, h, = k, +
E02,
equation (8) takes the form t;“’+ kr<” + <’+ k,< = -&[Or5’ + (err + cos 22)5]. Following reference [2], one then finds that the asymptotic ponding to Case 1 and Case 2 of reference [2] reduce to (h, - V,)
(9)
stability conditions corres(10)
< 0
and (h, - h,)’ + (h, -
1)’ -
&*/4 > 0.
(j-1)
Upon substituting for ho, h,, h, and E these become c +sdA’
(12)
and (au.? -
c - 2da2)2 + 02(w2 2
-
b)2 - &d2A4
> 0.
(13)
Now
one can show that the boundary corresponding to expression (13) is the same as the locus of vertical tangents of the frequency response curves. The equation governing the frequency response curves is A’( c + zdA2
- aw2)’ + A202(02
- b)2 = F2.
(14)
The locus of vertical tangents is obtained from the condition do/dA = 0: i.e. W2(02 - b)2 + (ao2 -
c - $dA2)(aw2
-
c - $dA’)
= 0.
Rearranging, this expression yields 02(W2 - b)2 + (aw2 - c - %dA2)2 - &d2A4
= 0,
(15)
which is the same as the boundary corresponding to expression (13): i.e., the vanishing of the left-hand side of expression (13). Also one can show that the stability limit given by equation (9) of Tondl’s paper [3] is simply the locus of vertical tangents (15) with c = 1 and d = p. The locus of vertical tangents does not provide the complete picture of stability but it does represent one of the asymptotic stability boundaries for the class of systems considered herein. Department of Mechanical Engineering, University of Edinburgh, Edinburgh EH9 WL, U.K. (Received 5 August 1978)
M.
A.V. RANGACHARYULU
147
LETTERS TO THE EDITOR REFERENCES 1. A. K. MITTAL 1978 Journal
of Sound and V&ration 58, 579-585. Stability of forced periodic response in third order non-linear systems. 2. P. A. T. CHRISTOPHER 1977 International Journal of Control xxvi, 901-916. Stability of periodic solutions of a non-linear differential equation arising in a Servomechanism theory. 3. A. TONDL 1976 Journal of Sound and Vibration 47, 133-135. Additional note on a third-order system.
A NOTE TO THE STABILITY OF FORCED PERIODIC RESPONSE ORDER NON-LINEAR SYSTEMS
IN THIRD
Dr Rangacharyulu in his contribution [l] dealing with the stability investigation of the forced periodic response in third order non-linear systems has derived two conditions of stability-his expressions (12) and (13). He has succeeded in proving that the boundary of
condition (13) can be generalized as the rule of vertical tangents. He has pointed out that this boundary fully agrees with that given by equation (9) in my note [2]. Before discussing condition (12) I would like to use this opportunity to correct mistakes in reference [2] (which appeared in conditions (4) and (5) of the final version due to some rearrangements). The correct version is as follows: 1 + $L42 > 0. ab - 1 - $P
(4)
> 0.
(5)
From these conditions the limit amplitude A can be determined: A = [2(ab - 1)/3~]“~
for
p > 0.
(6)
A = (-2/3,~)“~
for
11< 0.
(7)
The calculated limits in examples presented in Figures 1 and 2 in reference [2] are correct because the correct equations were used. The above mentioned condition (12) in reference [l] is fully equivalent with the condition (5) above (inasmuch as c = 1 in reference [2]). This condition is of importance only in cases where p > 0 (or d > 0 in reference [l]) because for p < 0, providing that a6 - 1 > 0, condition (5) is always fulfilled and then condition (4) must be taken into account. Therefore, the stability conditions in reference [l] are limited to the case c > 0. A.
National Research Institute for Machine Design, 250 97 Praha 9-Bzchovice, Czechoslovakia (Received
TONDL
10 October 1978) REFERENCES
1. M. A. V. RANGACHARWLU 1979 Journal of Sound and Vibration 63, 145-147.
‘Stability of forced periodic response in third order non-linear systems.” 2. A. TONDL 1976 .JournaI of Sound and Vibration 47, 133-135. Additional system.
Some comments on
note on a third order
of‘Sound and Vibration (1979) 63(l), 145-147
LETTERS SOME
TO THE EDITOR
COMMENTS ON “STABILITY OF FORCED PERIODIC THIRD ORDER NON-LINEAR SYSTEMS”
RESPONSE
IN
Mittal in his paper [l] has dealt with the stability of the forced periodic response of third order non-linear systems governed by the equation of the type Yi’+ a? + b2 + cx + dx3 = Fcosot;(.)$d()/dt.
(1)
He has shown that the stability condition derived from the linearized version of equation (1) is the same as the result obtained by investigating the nature of the roots of the characteristic equation corresponding to the variational equation of (1) for small d. He concludes the paper with a note of suspicion regarding the applicability of the rule of vertical tangents to the frequency response curves for these systems. In a recent paper Christopher [2] has studied the stability of periodic solutions of a third order non-linear system arising in some servomechanisms. The equation is of the form Y + fijl + f,(x)a + ,fOx = F cos ot,
(2)
where fr(x) = m, + m2x2, and Christopher has discussed the stability of the forced response solution in terms of the asymptotic stability of the corresponding variational equation using a method due to Cesari. Taking the solution to have, in the first approximation, an amplitude A and frequency w he has shown that one of the asymptotic boundaries corresponds to the locus of vertical tangents of the frequency response curves in the A-<0 plane. Following the method described in reference [2] one can obtain similar results for equations of the type (1). The vector equivalent of equation (1) is f = J(t, x),
(3)
where ii = X2’
i3 = - (c + dx;)x,
ii-, = x3,
- bx, - ax3 + F cos or.
If equation (1) is assumed to have a solution x = p(t) = A cos(ot + f#l)
(4)
the variational equation of equation (3) with respect to equation (4) is
aJ(t,
4= i j=
1
(5)
pw)tj
axj
where J is the Jacobian matrix with respect to x. Using equations (3) and (5), one obtains
’ = [;
+ 3dx;)
-;
.-‘j-.
(6)
In scalar form equation (6) becomes ... r + at + b[ + (c + 3dx;)t = 0.
(7)
145 0022-460X/79/050145
+03 $02.00/O
@ 1979 Academic
Press Inc. (London)
Limited
146
LETTERSTOTHEEDITOR
Upon defining 2 = ot + 4, equation (7) reduces to 5”’ + h,5” + h,<’ + h,5: = -E(COS 2Z)5, where ( ‘)&d( )/dZ, h, = a/o, h, = b/w2, h, = c/o3 + E, E If
=
3/2(dA2/03).
now one looks upon E as a small parameter and lets h, = k,, h, = 1 + wI, h, = k, +
E02,
equation (8) takes the form t;“’+ kr<” + <’+ k,< = -&[Or5’ + (err + cos 22)5]. Following reference [2], one then finds that the asymptotic ponding to Case 1 and Case 2 of reference [2] reduce to (h, - V,)
(9)
stability conditions corres(10)
< 0
and (h, - h,)’ + (h, -
1)’ -
&*/4 > 0.
(j-1)
Upon substituting for ho, h,, h, and E these become c +sdA’
(12)
and (au.? -
c - 2da2)2 + 02(w2 2
-
b)2 - &d2A4
> 0.
(13)
Now
one can show that the boundary corresponding to expression (13) is the same as the locus of vertical tangents of the frequency response curves. The equation governing the frequency response curves is A’( c + zdA2
- aw2)’ + A202(02
- b)2 = F2.
(14)
The locus of vertical tangents is obtained from the condition do/dA = 0: i.e. W2(02 - b)2 + (ao2 -
c - $dA2)(aw2
-
c - $dA’)
= 0.
Rearranging, this expression yields 02(W2 - b)2 + (aw2 - c - %dA2)2 - &d2A4
= 0,
(15)
which is the same as the boundary corresponding to expression (13): i.e., the vanishing of the left-hand side of expression (13). Also one can show that the stability limit given by equation (9) of Tondl’s paper [3] is simply the locus of vertical tangents (15) with c = 1 and d = p. The locus of vertical tangents does not provide the complete picture of stability but it does represent one of the asymptotic stability boundaries for the class of systems considered herein. Department of Mechanical Engineering, University of Edinburgh, Edinburgh EH9 WL, U.K. (Received 5 August 1978)
M.
A.V. RANGACHARYULU
147
LETTERS TO THE EDITOR REFERENCES 1. A. K. MITTAL 1978 Journal
of Sound and V&ration 58, 579-585. Stability of forced periodic response in third order non-linear systems. 2. P. A. T. CHRISTOPHER 1977 International Journal of Control xxvi, 901-916. Stability of periodic solutions of a non-linear differential equation arising in a Servomechanism theory. 3. A. TONDL 1976 Journal of Sound and Vibration 47, 133-135. Additional note on a third-order system.
A NOTE TO THE STABILITY OF FORCED PERIODIC RESPONSE ORDER NON-LINEAR SYSTEMS
IN THIRD
Dr Rangacharyulu in his contribution [l] dealing with the stability investigation of the forced periodic response in third order non-linear systems has derived two conditions of stability-his expressions (12) and (13). He has succeeded in proving that the boundary of
condition (13) can be generalized as the rule of vertical tangents. He has pointed out that this boundary fully agrees with that given by equation (9) in my note [2]. Before discussing condition (12) I would like to use this opportunity to correct mistakes in reference [2] (which appeared in conditions (4) and (5) of the final version due to some rearrangements). The correct version is as follows: 1 + $L42 > 0. ab - 1 - $P
(4)
> 0.
(5)
From these conditions the limit amplitude A can be determined: A = [2(ab - 1)/3~]“~
for
p > 0.
(6)
A = (-2/3,~)“~
for
11< 0.
(7)
The calculated limits in examples presented in Figures 1 and 2 in reference [2] are correct because the correct equations were used. The above mentioned condition (12) in reference [l] is fully equivalent with the condition (5) above (inasmuch as c = 1 in reference [2]). This condition is of importance only in cases where p > 0 (or d > 0 in reference [l]) because for p < 0, providing that a6 - 1 > 0, condition (5) is always fulfilled and then condition (4) must be taken into account. Therefore, the stability conditions in reference [l] are limited to the case c > 0. A.
National Research Institute for Machine Design, 250 97 Praha 9-Bzchovice, Czechoslovakia (Received
TONDL
10 October 1978) REFERENCES
1. M. A. V. RANGACHARWLU 1979 Journal of Sound and Vibration 63, 145-147.
‘Stability of forced periodic response in third order non-linear systems.” 2. A. TONDL 1976 .JournaI of Sound and Vibration 47, 133-135. Additional system.
Some comments on
note on a third order