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On the Lorentz linearization of a quadratically damped forced oscillator
Volume 89A, number 3
PHYSICS LETTERS
3 May 1982
ON THE LORENTZ LINEARIZATION OF A QUADRATICALLY DAMPED FORCED OSCILLATOR J.T.F. ZIMMERMAN Netherlands Institute for Sea Research, Texel, The Netherlands and Institute o f Meteorology and Oceanography, State University of Utrecht, The Netherlands Received 27 January 1982
The energy criterium used in the Lorentz linearization of a quadratically damped oscillator can be derived from the simplest nontrivial truncation of a renormalized perturbation expansion which gives the linearized friction coefficient as a function of forcing amplitude and frequency, together with the frequency-response functions of the fundamental harmonic and its odd multiples.
In order to overcome analytical difficulties in calculating the propagation of tidal waves, where frictional damping is parametrized by a nonlinear friction law, Lorentz [1 ] used an energy argument to replace the nonlinear friction term in the equation of motion by a linearized one. The method was also illustrated [2] by the more simple analogon o f a frictionally damped free oscillating point mass, the equation of motion of which reads, including a forcing term, + ~'l~l ~ + w 2 x = / ( t ) .
(1)
If eq. (1) is thought to describe the turbulent motion of an object immersed in a fluid, then 7 is a friction coefficient [ L - l ] inversely proportional to a relevant length, the size of the object say, and 60 is a frequency inversely proportional to the square root o f its apparent mass ("apparent" because part of the force excerted by the fluid is effectively a reduction of the object's inertia) [3]. Another interest in eq. (1) may come from certain processes in nuclear physics, provided quantization can be performed [4]. If the external force,f(t), is assumed to be sinusoidal with frequency o, the Lorentz linearization of eq. (1) consists in replacing 7 I~1 ~ by c~, such that over a complete cycle the work done by both expressions of the frictional force equals the work done by the external force:
7(I-~ ]3) = c(~2) = (.~f),
(2)
where brackets stand for averaging over the forcing period. Obviously c depends on the amplitude of the oscillation which is not known a priori. In Lorentz's view I1 ] c should be estimated iteratively starting with a guessed value. Here we show that one can quite sire. ply arrive at the Lorentz linearization by using a renormalized perturbation procedure which, without any guesswork, also gives an expression for the renormalized linear friction coefficient c in terms o f the amplitude and frequency of the forcing and for the corresponding frequency-response functions for the fundamental frequency and its higher harmonics. From the outset we suppose that a priori no "small parameter" can be associated with the quadratic friction term in eq. (1) - for, if so, then one can arrive at a solution by a simple perturbation expansion, using the method of averaging [5]. For strong nonlinear friction recourse can be had to a renormalized expansion. To this end we introduce a formal expansion parameter, X, and recast eq. (1): + XTI~I ~ + c~ + o~2x = f ( t ) ,
(3)
where now c = c o + Xc 1 + X2c2 + ... = 0 .
(4)
Looking for a solution of the form 0 031-9163/82/0000-0000/$02.75 © 1982 North-Holland
123
Volume 89A, number 3
PHYSICS LETTERS
x = x 0 + Xx 1 + ~2x2+ ....
(5)
3 May 1982
Thus, the frequency response function for X0 reads 7)?0 = f { ( 1 - ~2)2 + ~2~2(f, ~)} I/2
we have to zeroth order
(16)
which for f = f exp(iut) has the well-known solution
Substitution of this expression in the rhs of eq. (1 O) gives the solution for the frequency-response functions for the higher harmonics in the oscillation:
x0 = x0 exp [i(at + 00)] ,
TX]2n-1) = ( _ 1)n+l (8/'tr) f-'~a2
.20 + c02 0 + co2x0 = f ( t ) ,
(6)
(7)
where, with 6 = a/w and c0 = co~ w, ~0co2//= [(1 - a2) 2 + c'2o2]-1/2
(8)
00 = arctg ~0~/(~ 2 - 1).
(9)
To first order we have JCl + ¢0-'~1 + 602x1 = -')'l-~0[-~0 - Cl-~0 •
(10)
If the series for x is truncated at this lowest nontrivial order, eq. (5) gives, resetting ~ = 1, Cl.
(11)
By writing Ix01 x0 = (8/~) ~02 X~ ( - 1 ) n e x p i [ ( 2 n - 1 ) o t - 3 1~ + 0 o 1 n=l (2n-1)[(Zn-1) 2-4]
'
(12)
where x 0 denotes the modulus of 20, we see that an obvious requirement for renormalization is the vanishing of forcing at the fundamental frequency in the rhs o f e q . (10). This condition, with eq. (11) and the n = l term of (12) gives
c0 = (8-r/37r) ~ o .
7(Ig013) = c0<~o2).
(14)
Substitution of eqs. (7), (8) into eq. (13) finally gives the renormalized friction coefficient as a function of the dimensionless forcing amplitude f = 7f/co 2 and 6: c'8(~ o~) = [(8j~3rr) 2 + ¼(~-1 - ½ ( a -1 - b")2 .
×{(2n-1)[(2n
1)262c'2(f,o')) - l / 2 1
1)2__4]}
1,
n~>2.
(17)
Comparing these results with those for a genuine linearly damped oscillator, we observe that in the limit ~ 0 in both cases ~OW2ff -+ 1, in accordance with a balance between restoring force and driving force. For the same reason ~?t2n-l) (n > 2) ~ 0 for #-+ 0. In the limit ~ ~ oo a genuine linearly damped oscillator gives 2 _+fff-2 as does the linearized solution for x0- For the higher harmonics however we have in that limit x]2n-l)_+ f 2 ~ 4. Note that these limits for the higher harmonics imply a resonance peak for some value of 6" even in cases where the fundamental harmonic is overdamped and shows a response decreasing monotonuously for increasing 6. Finally, for 6 = 1, where the undamped oscillator has a resonance, the response of a genuine linearly damped oscillator is £ cof, whereas the linearized solution gives x0 cc f7/2, The response of the higher harmonics (or ~ = 1 gives (n ~> 2) .ft 2n 1) c~f f o r f ~ 1 and :f]2 n li c~),.q/2 for f > 1.
(13)
Evidently, Lorentz's energy criterium for linearization is now automatically satisfied, since
124
(2n--1)26212+(2n
× [(1 - a2) 2 + a 2 ~ ( L ~ ]
and
co =
× {[1
0-)4]1/2 (15)
Re¢erences l 11 H.A. Lorentz, Verslag Staatscommissie Zuiderzee 1918 1926 (Alg. Landsdrukkerij, Den Haag, 1926); not in Coll. Papers. [2] tt.A. Lorentz, De Ingenieur (1922) 695;also in: Coll. Papers, Vol. 4 (Nijhoff, Den Haag, 1937) p. 252. [3 ] G.H. Keulegan and LH. Carpenter, J. Res. Nat. Bureau Stand. 60 (1958) 423. [4] I:. Negro and A. Tartaglia, Phys. Lett. 77A (1980) l. [5] A.H. Nayfeh and D.T. Mook, Nonlinear oscillations (Wiley, New York, 1979).
PHYSICS LETTERS
3 May 1982
ON THE LORENTZ LINEARIZATION OF A QUADRATICALLY DAMPED FORCED OSCILLATOR J.T.F. ZIMMERMAN Netherlands Institute for Sea Research, Texel, The Netherlands and Institute o f Meteorology and Oceanography, State University of Utrecht, The Netherlands Received 27 January 1982
The energy criterium used in the Lorentz linearization of a quadratically damped oscillator can be derived from the simplest nontrivial truncation of a renormalized perturbation expansion which gives the linearized friction coefficient as a function of forcing amplitude and frequency, together with the frequency-response functions of the fundamental harmonic and its odd multiples.
In order to overcome analytical difficulties in calculating the propagation of tidal waves, where frictional damping is parametrized by a nonlinear friction law, Lorentz [1 ] used an energy argument to replace the nonlinear friction term in the equation of motion by a linearized one. The method was also illustrated [2] by the more simple analogon o f a frictionally damped free oscillating point mass, the equation of motion of which reads, including a forcing term, + ~'l~l ~ + w 2 x = / ( t ) .
(1)
If eq. (1) is thought to describe the turbulent motion of an object immersed in a fluid, then 7 is a friction coefficient [ L - l ] inversely proportional to a relevant length, the size of the object say, and 60 is a frequency inversely proportional to the square root o f its apparent mass ("apparent" because part of the force excerted by the fluid is effectively a reduction of the object's inertia) [3]. Another interest in eq. (1) may come from certain processes in nuclear physics, provided quantization can be performed [4]. If the external force,f(t), is assumed to be sinusoidal with frequency o, the Lorentz linearization of eq. (1) consists in replacing 7 I~1 ~ by c~, such that over a complete cycle the work done by both expressions of the frictional force equals the work done by the external force:
7(I-~ ]3) = c(~2) = (.~f),
(2)
where brackets stand for averaging over the forcing period. Obviously c depends on the amplitude of the oscillation which is not known a priori. In Lorentz's view I1 ] c should be estimated iteratively starting with a guessed value. Here we show that one can quite sire. ply arrive at the Lorentz linearization by using a renormalized perturbation procedure which, without any guesswork, also gives an expression for the renormalized linear friction coefficient c in terms o f the amplitude and frequency of the forcing and for the corresponding frequency-response functions for the fundamental frequency and its higher harmonics. From the outset we suppose that a priori no "small parameter" can be associated with the quadratic friction term in eq. (1) - for, if so, then one can arrive at a solution by a simple perturbation expansion, using the method of averaging [5]. For strong nonlinear friction recourse can be had to a renormalized expansion. To this end we introduce a formal expansion parameter, X, and recast eq. (1): + XTI~I ~ + c~ + o~2x = f ( t ) ,
(3)
where now c = c o + Xc 1 + X2c2 + ... = 0 .
(4)
Looking for a solution of the form 0 031-9163/82/0000-0000/$02.75 © 1982 North-Holland
123
Volume 89A, number 3
PHYSICS LETTERS
x = x 0 + Xx 1 + ~2x2+ ....
(5)
3 May 1982
Thus, the frequency response function for X0 reads 7)?0 = f { ( 1 - ~2)2 + ~2~2(f, ~)} I/2
we have to zeroth order
(16)
which for f = f exp(iut) has the well-known solution
Substitution of this expression in the rhs of eq. (1 O) gives the solution for the frequency-response functions for the higher harmonics in the oscillation:
x0 = x0 exp [i(at + 00)] ,
TX]2n-1) = ( _ 1)n+l (8/'tr) f-'~a2
.20 + c02 0 + co2x0 = f ( t ) ,
(6)
(7)
where, with 6 = a/w and c0 = co~ w, ~0co2//= [(1 - a2) 2 + c'2o2]-1/2
(8)
00 = arctg ~0~/(~ 2 - 1).
(9)
To first order we have JCl + ¢0-'~1 + 602x1 = -')'l-~0[-~0 - Cl-~0 •
(10)
If the series for x is truncated at this lowest nontrivial order, eq. (5) gives, resetting ~ = 1, Cl.
(11)
By writing Ix01 x0 = (8/~) ~02 X~ ( - 1 ) n e x p i [ ( 2 n - 1 ) o t - 3 1~ + 0 o 1 n=l (2n-1)[(Zn-1) 2-4]
'
(12)
where x 0 denotes the modulus of 20, we see that an obvious requirement for renormalization is the vanishing of forcing at the fundamental frequency in the rhs o f e q . (10). This condition, with eq. (11) and the n = l term of (12) gives
c0 = (8-r/37r) ~ o .
7(Ig013) = c0<~o2).
(14)
Substitution of eqs. (7), (8) into eq. (13) finally gives the renormalized friction coefficient as a function of the dimensionless forcing amplitude f = 7f/co 2 and 6: c'8(~ o~) = [(8j~3rr) 2 + ¼(~-1 - ½ ( a -1 - b")2 .
×{(2n-1)[(2n
1)262c'2(f,o')) - l / 2 1
1)2__4]}
1,
n~>2.
(17)
Comparing these results with those for a genuine linearly damped oscillator, we observe that in the limit ~ 0 in both cases ~OW2ff -+ 1, in accordance with a balance between restoring force and driving force. For the same reason ~?t2n-l) (n > 2) ~ 0 for #-+ 0. In the limit ~ ~ oo a genuine linearly damped oscillator gives 2 _+fff-2 as does the linearized solution for x0- For the higher harmonics however we have in that limit x]2n-l)_+ f 2 ~ 4. Note that these limits for the higher harmonics imply a resonance peak for some value of 6" even in cases where the fundamental harmonic is overdamped and shows a response decreasing monotonuously for increasing 6. Finally, for 6 = 1, where the undamped oscillator has a resonance, the response of a genuine linearly damped oscillator is £ cof, whereas the linearized solution gives x0 cc f7/2, The response of the higher harmonics (or ~ = 1 gives (n ~> 2) .ft 2n 1) c~f f o r f ~ 1 and :f]2 n li c~),.q/2 for f > 1.
(13)
Evidently, Lorentz's energy criterium for linearization is now automatically satisfied, since
124
(2n--1)26212+(2n
× [(1 - a2) 2 + a 2 ~ ( L ~ ]
and
co =
× {[1
0-)4]1/2 (15)
Re¢erences l 11 H.A. Lorentz, Verslag Staatscommissie Zuiderzee 1918 1926 (Alg. Landsdrukkerij, Den Haag, 1926); not in Coll. Papers. [2] tt.A. Lorentz, De Ingenieur (1922) 695;also in: Coll. Papers, Vol. 4 (Nijhoff, Den Haag, 1937) p. 252. [3 ] G.H. Keulegan and LH. Carpenter, J. Res. Nat. Bureau Stand. 60 (1958) 423. [4] I:. Negro and A. Tartaglia, Phys. Lett. 77A (1980) l. [5] A.H. Nayfeh and D.T. Mook, Nonlinear oscillations (Wiley, New York, 1979).