- Email: [email protected]
Calculation of single-mode approximation to the limit-cycle solution of a nonlinear wave equation
Mechanics Research Communications 37 (2010) 369–371
Contents lists available at ScienceDirect
Mechanics Research Communications journal homepage: www.elsevier.com/locate/mechrescom
Calculation of single-mode approximation to the limit-cycle solution of a nonlinear wave equation Ronald E. Mickens Department of Physics, Clark Atlanta University, Atlanta, GA 30314, USA
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 6 January 2010 Available online 10 May 2010
We construct an averaging procedure to calculate an approximation to the single mode limit-cycle solution to a nonlinear wave equation. The nonlinearity consists of two terms, one linear in the first-order time derivative, the second proportional to the third power of this derivative. An equation of this form appears in a model for growth of wind-induced vibrations in overhead power lines. © 2010 Elsevier Ltd. All rights reserved.
Keywords: Nonlinear wave equation Limit-cycle Approximation Averaging
1. Introduction The main goal of this paper is to propose a methodology for determining approximations to the limit-cycle solutions to the following nonlinear wave equation ∂2 y ∂2 y ∂y −ˇ = c2 2 + ˛ 2 ∂t dt ∂x
∂y ∂t
3 ,
(1.1)
where (˛, ˇ, c) are positive parameters. In particular, we will examine the “single-mode” solutions for which y = y(x, t) takes the form y(x, t) = A(t)B(x)
(1.2) 2. Methodology
with y(x, 0) = given ,
B(0) = 0,
B(L) = 0.
y(x, t) =
k=1
Ak (t) sin
kx L
The substitution of Eq. (1.2) into Eq. (1.1) gives
(1.3)
Eq. (1.1) was derived as a model for the growth of wind-induced oscillatory motions that may occur in overhead electrical transmission lines (Myerscough, 1973). The line or cable has supports at x = 0 and x = L, and under no force, except that of gravity, takes the shape of a shallow catenary type curve. The function y(x, t) is the displacement as a function of time at a point x. Previous work on this problem (Myerscough, 1973) assumed that the solution to Eq. (1.1) could be expressed as a Fourier series in x, i.e., ∞
and applied a Krylov–Bogoliubov averaging method (Andronov et al., 1966) to obtain an infinite set of coupled, second-order, ordinary differential equations for the Ak (t) functions. The one harmonic or single-mode approximation was taken to be the inclusion of only one term from the right-hand side of Eq. (1.4). Within this framework, the expression given above in Eq. (1.2) is a generalization of this one harmonic result. In this paper, we introduce an averaging methodology for calculating approximations to Eq. (1.1) using the assumed solution of Eqs. (1.2) and (1.3). Further, we show how to explicitly calculate this solution given the function B(x).
(1.4)
d2 B dA dA d2 A B = c2 A 2 + ˛ B−ˇ 2 dt dt dt dx
0093-6413/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechrescom.2010.04.010
B3 .
(2.1)
Let f (x) be a function of x, then its averaged value will be taken to be, over the interval L, 1 f¯ ≡ L
L
f (x)dx.
(2.2)
0
Therefore, averaging each term in Eq. (2.1) with respect to x gives
a1
dA d2 A dA = −c 2 |a2 |A + (˛a1 ) − (ˇa3 ) dt dt dt 2
3
,
(2.3)
where a1 ≡
E-mail address: [email protected].
3
1 L
0
L
B(x)dx > 0,
(2.4a)
370
a2 ≡
a3 ≡
R.E. Mickens / Mechanics Research Communications 37 (2010) 369–371
1 L
L
0
1 L
d2 B(x) dx < 0, dx2
(2.4b)
B3 (x)dx > 0,
(2.4c)
L
0
and the “sign” restrictions are necessary requirements on the function B(x) to yield a second-order differential equation such that A(t) has a limit-cycle solution (Andronov et al., 1966; Mickens, 1981, 1996). Dividing each term of Eq. (2.3) by a1 and defining ˝2 as c 2 |a2 | , ˝ ≡ a1 2
we obtain
(2.5)
d2 A + ˝2 A = ˛ − dt 2
ˇa3 a1
dA dt
2 dA dt
.
(2.6)
This equation is the Rayleigh equation and has a single and stable limit-cycle solution (Mickens, 1981; Nayfeh, 1981). Note that all of the parameters appearing in this Rayleigh differential equation are expressed in terms of the original parameters (c, ˛, ˇ) and quantities calculable from B(x). Since y(x, 0) is to be specified, then A(0) may be taken to be equal to one, i.e., y(x, 0) = A(0)B(x) = B(x).
(2.7)
Finally, we wish to point out that a similar, in format, but not applied to this specific problem, averaging technique has been used by Cveticanin (2008) to formulate averaged equations for nonlinear, one-dimensional, oscillators having fractional order forces.
In this section, we present the results of calculating approximations to the parameters of the limit-cycle for Eq. (2.6) using, respectively, the method of harmonic balance (Mickens, 1996) and first-order averaging (Andronov et al., 1966; Mickens, 1981). Since these two procedures are part of the standard methodology in the area of nonlinear oscillatory dynamics, no details of the calculations will be given. The harmonic balance method, in lowest order, assumes that the steady state limit-cycle, for Eq. (2.6), takes the form A(t) A∗ cos(ω, t),
(3.1)
where A∗
is the amplitude of the limit-cycle and ω is the angular frequency. If this procedure is applied to Eq. (2.6), then the following values are found for A∗ and ω:
4 ˛ 3
ω 2 = ˝2 =
a21
c 2 |a2 |a3
ˇ
,
(3.2)
c 2 |a2 | . a1
(3.3)
Therefore, the single mode, steady state approximation for the nonlinear PDE, given by Eq. (1.1) is y(x, t) A(t)B(x) = A∗ B(x) cos(˝t).
ˇ = ˇ1 ,
0 < 1,
(3.5)
where is a small dimensionless parameter, then Eq. (2.6) becomes
(3.7)
where the following restrictions must hold a(0, ) = 1,
(0, ) = 0.
(3.8)
Solving the averaged equations for a(t, ) and (t, ) gives (t, ) = 0, a(t, ) =
(3.9) A∗
(A∗ )2
exp(˛1 t/2)
+ [exp(˛1 t) − 1]
1/2 ,
(3.10)
where A∗ and ˝ are exactly the same values as determined by the harmonic balance calculation; see Eqs. (3.2) and (3.3). Therefore, this single mode, averaged approximation provides information on the transient behavior of the amplitude as it approaches the maximum limit-cycle value, A∗ . Thus, y(x, t) is y(x, t) A(t)B(x) = a(t, )B(x) cos(˝t).
(3.11)
Note that given the initial shape function B(x), the quantities (a1 , a2 , a3 ) must be calculated. For the lowest mode, B(x) should be zero at the ends, i.e., B(0) = 0 and B(L) = 0, but otherwise is non-zero in the interval 0 < x < L. We now calculate these three quantities for two typical functions satisfying these requirements. First, consider the function B(x) B(x) = B0 sin
x L
,
(3.12)
d2 A + ˝2 A = ˛1 − dt 2
ˇ1 a3 a1
dA dt
a1 =
a2 =
a3 =
1 L
L
0
1 L
B(x)dx =
0
1 L
L
L
d2 B =− dx2
2B 0
B(x)3 dx =
0
Next, take B(x) to be B(x) = B0
2B0 > 0,
x L
1−
L2 4B03 3
(3.13a)
< 0,
(3.13b)
> 0.
(3.13c)
x . L
(3.14)
A similar calculation produces the results a1 =
B0 > 0, 6
a2 = −
2B 0 L2
< 0,
a3 =
B03 140
> 0.
(3.15)
Observe that the coefficients for both B(x) functions satisfy the sign conditions presented in Eq. (2.4). 4. Discussion and conclusions Based on the work of this paper, the following conclusions may be reached:
(3.4)
A somewhat more detailed result can be gotten by use of firstorder averaging. If the parameters ˛ and ˇ are rewritten as ˛ = ˛1 ,
A(t) = a(t, ) cos[˝t + (t, )]
where B0 is a constant. An easy and direct calculation gives
3. Applications
A∗ =
[This rewrite of ˛ and ˇ corresponds to the fact that typical values of these parameters, see Table 1 in Myerscough, 1973, correspond to = 0.01 with ˛1 = O(1) and ˇ1 = O(1).] The approximate solution to Eq. (3.6) is (Mickens, 1981)
2 dA dt
.
(3.6)
(i) Our generalized one mode approximation to the solution of Eq. (1.1) agrees in its major features with the results obtained by Myerscough (1973) who used a Fourier series representation for y(x, 0). (ii) The application of the methods of lowest order harmonic balance and first-order averaging produced exactly the same estimates for the amplitude and angular frequency of the expected limit-cycle solutions. However, the HB method only yielded these two quantities, while the averaging technique
R.E. Mickens / Mechanics Research Communications 37 (2010) 369–371
also allowed the transient (time-dependent) behavior of the approach to the limit-cycle solution to be calculated. (iii) Our results are based on an “averaging over space” procedure. This methodology clearly leads to a one mode solution to Eq. (1.1) that is in general agreement with the physical phenomena exhibited by vibrations in overhead lines (Myerscough, 1973). (iv) Calculations were done using two space functions B(x); see Eqs. (3.12) and (3.14). While both of these B(x) qualify as single mode functions, it should be noted that Eq. (3.12) represents a single Fourier harmonic, while Eq. (3.14) has contributions from all odd Fourier harmonics (Mickens, 1996). (v) Our major conclusion is that the space averaging procedure offers promise as a technique for calculating approximations to the solutions of nonlinear wave equations in one space dimensions (Debnath, 1997). The next step in this work is to consider a two mode representation for y(x, t), i.e., y(x, t) = A1 (t)B1 (x) + A2 (t)B2 (x)
(4.1)
where B1 (x) and B2 (x) satisfy the conditions B1 (0) = 0,
B1 (L) = 0,
B2 (0) = 0,
B2 (L) = 0,
(4.2)
with A1 (0) = 1,
A2 (0) = 1,
(4.3)
and, B1 (x) and B2 (x) have, respectively, no and one zero in the open interval 0 < x < L. We also assume that B1 (x) and B2 (x) satisfy the requirement
L
B1 (x)B2 (x)dx,
(4.4)
0
along with the condition Max|B2 (x)| Max|B1 (x)|,
0 < x < L.
(4.5)
371
The major task would be to see if some type of space averaging technique can be constructed such that given B1 (x) and B2 (x), appropriate ODE’s can be determined for A1 (t) and A2 (t), such that these equations, in turn, can either be exactly or approximately solved. Finally, there may be value in examining a generalization of Eq. (1.1). One such form is ∂2 y ∂2 y ∂y −ˇ = c 2 2 + f (y) + ˛ ∂t ∂t 2 ∂x
∂y ∂t
3
,
(4.6)
where f (y) is a polynomial function of y. Acknowledgements The author thanks Prof. Sandra Rucker (Clark Atlanta University, Department of Mathematical Sciences) for several productive discussions related to the reported results. This research was supported in part by funds from the School of Arts and Sciences Faculty Development Fund. This funding source had no direct involvement in carrying out this research. References Andronov, A.A., Vitt, A.A., Khaikin, S.E., 1966. Theory of Oscillators. Addison-Wesley, Reading, MA. Cveticanin, L., 2008. Oscillator with fraction order restoring force. Journal of Sound and Vibration 320, 1064–1082. Debnath, L., 1997. Nonlinear Partial Differential Equations for Scientists and Engineers. Birkhäuser, Boston. Mickens, R.E., 1981. An Introduction to Nonlinear Oscillations. Cambridge University Press, New York. Mickens, R.E., 1996. Oscillations in Planar Dynamic Systems. World Scientific, London. Myerscough, C.J., 1973. A simple model of the growth of wind-induced oscillations in overhead lines. Journal of Sound and Vibration 28, 699–713. Nayfeh, A.H., 1981. Introduction to Perturbation Techniques. Wiley-Interscience, New York.
Contents lists available at ScienceDirect
Mechanics Research Communications journal homepage: www.elsevier.com/locate/mechrescom
Calculation of single-mode approximation to the limit-cycle solution of a nonlinear wave equation Ronald E. Mickens Department of Physics, Clark Atlanta University, Atlanta, GA 30314, USA
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 6 January 2010 Available online 10 May 2010
We construct an averaging procedure to calculate an approximation to the single mode limit-cycle solution to a nonlinear wave equation. The nonlinearity consists of two terms, one linear in the first-order time derivative, the second proportional to the third power of this derivative. An equation of this form appears in a model for growth of wind-induced vibrations in overhead power lines. © 2010 Elsevier Ltd. All rights reserved.
Keywords: Nonlinear wave equation Limit-cycle Approximation Averaging
1. Introduction The main goal of this paper is to propose a methodology for determining approximations to the limit-cycle solutions to the following nonlinear wave equation ∂2 y ∂2 y ∂y −ˇ = c2 2 + ˛ 2 ∂t dt ∂x
∂y ∂t
3 ,
(1.1)
where (˛, ˇ, c) are positive parameters. In particular, we will examine the “single-mode” solutions for which y = y(x, t) takes the form y(x, t) = A(t)B(x)
(1.2) 2. Methodology
with y(x, 0) = given ,
B(0) = 0,
B(L) = 0.
y(x, t) =
k=1
Ak (t) sin
kx L
The substitution of Eq. (1.2) into Eq. (1.1) gives
(1.3)
Eq. (1.1) was derived as a model for the growth of wind-induced oscillatory motions that may occur in overhead electrical transmission lines (Myerscough, 1973). The line or cable has supports at x = 0 and x = L, and under no force, except that of gravity, takes the shape of a shallow catenary type curve. The function y(x, t) is the displacement as a function of time at a point x. Previous work on this problem (Myerscough, 1973) assumed that the solution to Eq. (1.1) could be expressed as a Fourier series in x, i.e., ∞
and applied a Krylov–Bogoliubov averaging method (Andronov et al., 1966) to obtain an infinite set of coupled, second-order, ordinary differential equations for the Ak (t) functions. The one harmonic or single-mode approximation was taken to be the inclusion of only one term from the right-hand side of Eq. (1.4). Within this framework, the expression given above in Eq. (1.2) is a generalization of this one harmonic result. In this paper, we introduce an averaging methodology for calculating approximations to Eq. (1.1) using the assumed solution of Eqs. (1.2) and (1.3). Further, we show how to explicitly calculate this solution given the function B(x).
(1.4)
d2 B dA dA d2 A B = c2 A 2 + ˛ B−ˇ 2 dt dt dt dx
0093-6413/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechrescom.2010.04.010
B3 .
(2.1)
Let f (x) be a function of x, then its averaged value will be taken to be, over the interval L, 1 f¯ ≡ L
L
f (x)dx.
(2.2)
0
Therefore, averaging each term in Eq. (2.1) with respect to x gives
a1
dA d2 A dA = −c 2 |a2 |A + (˛a1 ) − (ˇa3 ) dt dt dt 2
3
,
(2.3)
where a1 ≡
E-mail address: [email protected].
3
1 L
0
L
B(x)dx > 0,
(2.4a)
370
a2 ≡
a3 ≡
R.E. Mickens / Mechanics Research Communications 37 (2010) 369–371
1 L
L
0
1 L
d2 B(x) dx < 0, dx2
(2.4b)
B3 (x)dx > 0,
(2.4c)
L
0
and the “sign” restrictions are necessary requirements on the function B(x) to yield a second-order differential equation such that A(t) has a limit-cycle solution (Andronov et al., 1966; Mickens, 1981, 1996). Dividing each term of Eq. (2.3) by a1 and defining ˝2 as c 2 |a2 | , ˝ ≡ a1 2
we obtain
(2.5)
d2 A + ˝2 A = ˛ − dt 2
ˇa3 a1
dA dt
2 dA dt
.
(2.6)
This equation is the Rayleigh equation and has a single and stable limit-cycle solution (Mickens, 1981; Nayfeh, 1981). Note that all of the parameters appearing in this Rayleigh differential equation are expressed in terms of the original parameters (c, ˛, ˇ) and quantities calculable from B(x). Since y(x, 0) is to be specified, then A(0) may be taken to be equal to one, i.e., y(x, 0) = A(0)B(x) = B(x).
(2.7)
Finally, we wish to point out that a similar, in format, but not applied to this specific problem, averaging technique has been used by Cveticanin (2008) to formulate averaged equations for nonlinear, one-dimensional, oscillators having fractional order forces.
In this section, we present the results of calculating approximations to the parameters of the limit-cycle for Eq. (2.6) using, respectively, the method of harmonic balance (Mickens, 1996) and first-order averaging (Andronov et al., 1966; Mickens, 1981). Since these two procedures are part of the standard methodology in the area of nonlinear oscillatory dynamics, no details of the calculations will be given. The harmonic balance method, in lowest order, assumes that the steady state limit-cycle, for Eq. (2.6), takes the form A(t) A∗ cos(ω, t),
(3.1)
where A∗
is the amplitude of the limit-cycle and ω is the angular frequency. If this procedure is applied to Eq. (2.6), then the following values are found for A∗ and ω:
4 ˛ 3
ω 2 = ˝2 =
a21
c 2 |a2 |a3
ˇ
,
(3.2)
c 2 |a2 | . a1
(3.3)
Therefore, the single mode, steady state approximation for the nonlinear PDE, given by Eq. (1.1) is y(x, t) A(t)B(x) = A∗ B(x) cos(˝t).
ˇ = ˇ1 ,
0 < 1,
(3.5)
where is a small dimensionless parameter, then Eq. (2.6) becomes
(3.7)
where the following restrictions must hold a(0, ) = 1,
(0, ) = 0.
(3.8)
Solving the averaged equations for a(t, ) and (t, ) gives (t, ) = 0, a(t, ) =
(3.9) A∗
(A∗ )2
exp(˛1 t/2)
+ [exp(˛1 t) − 1]
1/2 ,
(3.10)
where A∗ and ˝ are exactly the same values as determined by the harmonic balance calculation; see Eqs. (3.2) and (3.3). Therefore, this single mode, averaged approximation provides information on the transient behavior of the amplitude as it approaches the maximum limit-cycle value, A∗ . Thus, y(x, t) is y(x, t) A(t)B(x) = a(t, )B(x) cos(˝t).
(3.11)
Note that given the initial shape function B(x), the quantities (a1 , a2 , a3 ) must be calculated. For the lowest mode, B(x) should be zero at the ends, i.e., B(0) = 0 and B(L) = 0, but otherwise is non-zero in the interval 0 < x < L. We now calculate these three quantities for two typical functions satisfying these requirements. First, consider the function B(x) B(x) = B0 sin
x L
,
(3.12)
d2 A + ˝2 A = ˛1 − dt 2
ˇ1 a3 a1
dA dt
a1 =
a2 =
a3 =
1 L
L
0
1 L
B(x)dx =
0
1 L
L
L
d2 B =− dx2
2B 0
B(x)3 dx =
0
Next, take B(x) to be B(x) = B0
2B0 > 0,
x L
1−
L2 4B03 3
(3.13a)
< 0,
(3.13b)
> 0.
(3.13c)
x . L
(3.14)
A similar calculation produces the results a1 =
B0 > 0, 6
a2 = −
2B 0 L2
< 0,
a3 =
B03 140
> 0.
(3.15)
Observe that the coefficients for both B(x) functions satisfy the sign conditions presented in Eq. (2.4). 4. Discussion and conclusions Based on the work of this paper, the following conclusions may be reached:
(3.4)
A somewhat more detailed result can be gotten by use of firstorder averaging. If the parameters ˛ and ˇ are rewritten as ˛ = ˛1 ,
A(t) = a(t, ) cos[˝t + (t, )]
where B0 is a constant. An easy and direct calculation gives
3. Applications
A∗ =
[This rewrite of ˛ and ˇ corresponds to the fact that typical values of these parameters, see Table 1 in Myerscough, 1973, correspond to = 0.01 with ˛1 = O(1) and ˇ1 = O(1).] The approximate solution to Eq. (3.6) is (Mickens, 1981)
2 dA dt
.
(3.6)
(i) Our generalized one mode approximation to the solution of Eq. (1.1) agrees in its major features with the results obtained by Myerscough (1973) who used a Fourier series representation for y(x, 0). (ii) The application of the methods of lowest order harmonic balance and first-order averaging produced exactly the same estimates for the amplitude and angular frequency of the expected limit-cycle solutions. However, the HB method only yielded these two quantities, while the averaging technique
R.E. Mickens / Mechanics Research Communications 37 (2010) 369–371
also allowed the transient (time-dependent) behavior of the approach to the limit-cycle solution to be calculated. (iii) Our results are based on an “averaging over space” procedure. This methodology clearly leads to a one mode solution to Eq. (1.1) that is in general agreement with the physical phenomena exhibited by vibrations in overhead lines (Myerscough, 1973). (iv) Calculations were done using two space functions B(x); see Eqs. (3.12) and (3.14). While both of these B(x) qualify as single mode functions, it should be noted that Eq. (3.12) represents a single Fourier harmonic, while Eq. (3.14) has contributions from all odd Fourier harmonics (Mickens, 1996). (v) Our major conclusion is that the space averaging procedure offers promise as a technique for calculating approximations to the solutions of nonlinear wave equations in one space dimensions (Debnath, 1997). The next step in this work is to consider a two mode representation for y(x, t), i.e., y(x, t) = A1 (t)B1 (x) + A2 (t)B2 (x)
(4.1)
where B1 (x) and B2 (x) satisfy the conditions B1 (0) = 0,
B1 (L) = 0,
B2 (0) = 0,
B2 (L) = 0,
(4.2)
with A1 (0) = 1,
A2 (0) = 1,
(4.3)
and, B1 (x) and B2 (x) have, respectively, no and one zero in the open interval 0 < x < L. We also assume that B1 (x) and B2 (x) satisfy the requirement
L
B1 (x)B2 (x)dx,
(4.4)
0
along with the condition Max|B2 (x)| Max|B1 (x)|,
0 < x < L.
(4.5)
371
The major task would be to see if some type of space averaging technique can be constructed such that given B1 (x) and B2 (x), appropriate ODE’s can be determined for A1 (t) and A2 (t), such that these equations, in turn, can either be exactly or approximately solved. Finally, there may be value in examining a generalization of Eq. (1.1). One such form is ∂2 y ∂2 y ∂y −ˇ = c 2 2 + f (y) + ˛ ∂t ∂t 2 ∂x
∂y ∂t
3
,
(4.6)
where f (y) is a polynomial function of y. Acknowledgements The author thanks Prof. Sandra Rucker (Clark Atlanta University, Department of Mathematical Sciences) for several productive discussions related to the reported results. This research was supported in part by funds from the School of Arts and Sciences Faculty Development Fund. This funding source had no direct involvement in carrying out this research. References Andronov, A.A., Vitt, A.A., Khaikin, S.E., 1966. Theory of Oscillators. Addison-Wesley, Reading, MA. Cveticanin, L., 2008. Oscillator with fraction order restoring force. Journal of Sound and Vibration 320, 1064–1082. Debnath, L., 1997. Nonlinear Partial Differential Equations for Scientists and Engineers. Birkhäuser, Boston. Mickens, R.E., 1981. An Introduction to Nonlinear Oscillations. Cambridge University Press, New York. Mickens, R.E., 1996. Oscillations in Planar Dynamic Systems. World Scientific, London. Myerscough, C.J., 1973. A simple model of the growth of wind-induced oscillations in overhead lines. Journal of Sound and Vibration 28, 699–713. Nayfeh, A.H., 1981. Introduction to Perturbation Techniques. Wiley-Interscience, New York.