Period/frequency estimation of a nonlinear oscillator

Introduction

Recently, García ¹ and Suárez ² gave some simple formulae for estimation of the period of a nonlinear oscillators. Considering a nonlinear oscillator in the form

u ″ ( t ) + f ( u ) = 0 , u ′ ( 0 ) = 0 , u ( 0 ) = A

(1)

T G a r c i a = 2 π A N + 2 ( N + 2 ) ∫ 0 A u N f ( u ) d u

(2)

T A n t o l a = 2 π ∫ 0 A W ( u ) u 2 d u ∫ 0 A W ( u ) f ( u ) u d u

(3)

Both formulae given in equations (2) and (3) are simple and useful for practical applications. This mini review article summarizes some even simpler period/frequency formulae mainly based on previous work on this special direction.

He’s frequency formulation

For the nonlinear oscillator given in equation (1), we can choose two arbitrary trial frequencies ω 1 and ω 2 for two trial solutions

u 1 ( t ) = A cos ⁡ ω 1 t

(4)

u 2 ( t ) = A cos ⁡ ω 2 t

(5)

R 1 ( t ) = − A ω 1 2 cos ⁡ ω 1 t + f ( A cos ⁡ ω 1 t )

(6)

R 2 ( t ) = − A ω 2 2 cos ⁡ ω 2 t + f ( A cos ⁡ ω 2 t )

(7)

He’s frequency formula reads ³

ω H e 2 = ω 1 2 R 2 ( 0 ) − ω 2 2 R 1 ( 0 ) R 2 ( 0 ) − R 1 ( 0 ) = f ( A ) A

(8)

For Duffing equation

u ″ + u + ε u 3 = 0 , u ′ ( 0 ) = 0 , u ( 0 ) = A

(9)

Choosing ω 1 = 1 and ω 2 = 2 , we have

ω 2 = 1 + ε A 2

(10)

Note that ω 1 and ω 2 can be arbitrarily chosen, for example ω 1 = 1 and ω 2 = 200 , and all cases end in the same result. When ε = 0 , equation (10) is the exact frequency; while when ε → ∞ , the exact frequency is ω e x = 0.9318 ε A 2 , the accuracy of equation (10) reaches 6.82%.

Geng and Cai ⁴ suggested a modification for equation (8)

ω G e n g − C a i 2 = ω 1 2 R 2 ( t 2 ) − ω 2 2 R 1 ( t 1 ) R 2 ( t 2 ) − R 1 ( t 1 ) = f ( A cos ⁡ π 6 ) A cos ⁡ π 6

(11)

The frequency formula was furthered improved in He^5,6

ω H e 2 = ω 1 2 R ˜ 2 − ω 2 2 R ˜ 1 R ˜ 2 − R ˜ 1 = 4 π A ∫ 0 π 2 f ( A cos ⁡ u ) ⋅ cos ⁡ u d u

(12)

R ˜ 1 = 4 T 1 ∫ 0 T 1 / 4 R 1 ( t ) cos ⁡ ( 2 π T 1 t ) d t

(13)

R ˜ 2 = 4 T 2 ∫ 0 T 2 / 4 R 2 ( t ) cos ⁡ ( 2 π T 2 t ) d t

(14)

For the Duffing equation

ω 2 = 1 + 3 4 ε A 2

(15)

The accuracy of the frequency given in equation (15) is 7% when ε → ∞ . Applications of the frequency are available in literature.^7–20

Max-min approach to nonlinear oscillators

We can re-write equation (1) in the form

u ″ + F ( u ) u = 0

(16)

If we search for an approximate solution in the form

u = A cos ⁡ ω t

(17)

Comparing equation (16) with a linear oscillator u ″ ( t ) + ω 2 u = 0 , we can obtain the following inequality for the square of the frequency

F min ⁡ < ω 2 < F max ⁡

(18)

We write Duffing equation in the form

u ″ + ( 1 + ε u 2 ) u = 0

(19)

Accordingly we have

1 1 < ω 2 < 1 + ε A 2 1

(20)

Using He Chengtian’s average, ²¹ we have

ω 2 = m + n ( 1 + ε A 2 ) m + n = 1 + k ε A 2

(21)

Taylor series ³⁴

To describe its trajectory of a nonlinear oscillator is extremely difficult for the whole solution domain, but it is relatively easy to give an accurate prediction of its trajectory at its initial stage [0, T/4], where T is its period. According to equation (1), we have

u ′ ( 0 ) = 0 , u ( 0 ) = A and u ″ ( 0 ) = − f ( u ( 0 ) ) = − f ( A )

(22)

Differentiating equation (1) with respect to time, we can easily obtain the value for u ( n ) ( 0 ) , so we can obtain its initial trajectory by Taylor series

u ( t ) = u ( 0 ) + u ′ ( 0 ) t + 1 2 ! u ″ ( 0 ) t 2 + 1 3 ! u ‴ ( 0 ) t 3 + 1 4 ! u ( 4 ) ( 0 ) t 4 + ⋅ ⋅ ⋅ , t ∈ [ 0 , T / 4 ]

(23)

A higher order Taylor series predicts a higher accuracy of its initial trajectory. Considering its periodic property, we have

u ( T / 4 ) = 0

(24)

where T is the period of equation (1).

To elucidate the effectiveness of this simple approach, we consider a nonlinear oscillator in the form ³⁴

u ″ + s i g n ( u ) = 0 , u ( 0 ) = A , u ′ ( 0 ) = 0

(25)

When 0 < t < T 4 , the solution is monotonely decreasing, and equation (25) becomes

u ( 0 ) = A , u ′ ( 0 ) = 0 , u ″ ( 0 ) = − 1

(26)

Solving equation (26) results in

u ( t ) = A − 1 2 t 2 , 0 < t < T 4

(27)

Matching u ( T / 4 ) = 0 , we have

A − 1 2 ( T 4 ) 2 = 0

(28)

Its period can be easily solved from equation (28), which reads

T = 4 2 A = 5.656854 A

(29)

which happens to be the exact period.

The simpler, the better

In He, ³⁵ a fast estimation of frequency was given, which is

ω 2 = d f ( u ( t ) ) d u | u = u i

(30)

or

ω 2 = d f ( u ( t ) ) d u | t = T / N

(31)

The usual location point is chosen as u i = A / 2 or t = T / 12 . This is the simplest formula with relatively high accuracy, and can be used as a fast look into the periodic property of a nonlinear oscillator. For the Duffing equation, when we choose u i = A / 2 , we have

ω 2 = 1 + 3 ε u 2 | u = A / 2 = 1 + 3 4 ε A 2

(32)

Another simple estimation is

ω 2 = F ( u ( 0 ) ) = f ( u ( 0 ) ) u ( 0 ) = f ( A ) A

(33)

Using a weighting function, equation (33) can be modified as ²

ω 2 = ∫ 0 A W ( u ) f ( u ) d u ∫ 0 A W ( u ) u d u

(34)

If we choose W ( u ) = u n , we obtain the following result

ω 2 = ∫ 0 A u n ( u + ε u 3 ) d u ∫ 0 A u n ⋅ u d u = 1 n + 2 A n + 2 + 1 n + 4 ε A n + 4 1 n + 2 A n + 2 = 1 + n + 2 n + 4 ε A 2

(35)

The optimal value for n for Duffing equation is 25 < n < 26 .

When n = 25 , ω 2 = 1 + 27 29 ε A 2 with relative error of 3.55%,

When n = 26 , ω 2 = 1 + 14 15 ε A 2 with relative error of 3.68%.

The García number is defined as a natural number. However, the optimal value of n in equation (35) is n = 25.3255.

Conclusions

This paper gives some simple formulae for fast prediction of frequency of a nonlinear oscillator, and all obtained results given in this paper are valid for the whole solution domain 0 < ε < ∞ with a relatively high accuracy of less than 7%, while the perturbation method depends upon a small parameter, its solution process is more complex than all formulae given in this paper, and further its results are only valid for the case when ε ≪ 1 .