TIME-VARYING AND NON-LINEAR DYNAMICAL SYSTEM ...hitech.technion.ac.il/feldman/DETC2005-84644.pdf · Keywords: non-linear system, Hilbert transform, identification, envelope, instantaneous

1 Copyright © #### by ASME

Proceedings of ASME VIB 2005: 20th Biennial Conference on Mechanical Vibration and Noise

September 24-28, 2005 Long Beach, CA, USA

DETC2005-84644

TIME-VARYING AND NON-LINEAR DYNAMICAL SYSTEM IDENTIFICATION USING THE HILBERT TRANSFORM

Michael Feldman Faculty of Mechanical Engineering

Technion – Israel Institute of Technology Haifa, 32000, Israel

ABSTRACT

The objective of the paper is to explain a modern Hilbert transform method for analysis and identification of mechanical non-linear vibration structures in the case of quasiperiodic signals. This special kind of periodicity arises in experimental vibration signals. The method is based on the Hilbert transform of input and output signals in a time domain to extract the instantaneous dynamic structure characteristics. The paper focuses on the dynamic analysis and identification of three groups of dynamics systems: o Forced vibrations of linear and non-linear SDOF systems

excited with quasiperiodic force signal. o Combined forced vibrations of quasiperiodic time varying

linear and non-linear SDOF systems excited with harmonic signal.

o Combined self-excited and forced vibrations of non-linear SDOF systems excited with harmonic signal.

The study focuses on signal processing techniques for non-linear system investigation, which enable us to estimate instantaneous system dynamic parameters (natural frequencies, damping characteristics and their dependencies on a vibration amplitude and frequency) for different kinds of system excitation.

Keywords: non-linear system, Hilbert transform, identification, envelope, instantaneous frequency.

INTRODUCTION

Recent work in the area of time domain representations of vibration, such as the Hilbert Transform (HT) [ 1, 2], shows great promise for applications in dynamic system identification. The proposed methods, FREEVIB and FORCEVIB, for identifying instantaneous modal parameters (natural frequencies, damping characteristics and their dependencies on a vibration amplitude and frequency) prove to be very simple and effective. Of particular interest and importance is the use of the HT to interpret quasiperiodic non-linear systems motion. In the simplest case, a quasiperiodic signal is a signal that consists

of a sum of two sinusoidal signals with constant frequencies. The traditional Fourier transform enables estimating only two separate frequency points of the Frequency Response Function (FRF). The HT of the same signal contrastingly allows estimation of the FRF for a wide continuous frequency range.

Quasiperiodic motion The class of the so-called almost periodic motions is a particular sub-class of recurrent trajectories which is of interest in non-linear dynamics. The remarkable feature which reveals the origin of these trajectories is that each component of an almost periodic motion is an almost periodic function with well studied analytical properties. An almost periodic function is uniquely defined "in average'' by a trigonometric Fourier series

( ) ,ni tn

n

f t a e λ∞

=−∞

≈ ∑ where n

λ are real numbers. If all n

λ are

linear combinations (with integer coefficients) of a finite number of rationally independent elements from a basis of frequencies, then we have a particular case of almost periodic functions, namely quasiperiodic functions. A quasiperiodic signal, in this context, is a signal that consists of a sum of a given number of sinusoidal signals with known frequencies and unknown, time varying amplitudes and phase. This kind of quasi-periodicity arises in vibration signals. In many practical situations, it is desirable that parameters of a quasiperiodic signal be estimated in real time. A continuous estimation of these parameters can be used, for example, for measuring, monitoring, or diagnostics purposes.

Instantaneous characteristics of quasiperiodic vibration. Two-component signal representation Consider a vibration signal composed of two quasi-harmonics, each with a slow variable amplitude and frequency in the time domain. In this case, the signal can be modeled as a weighted sum of monocomponent signals, each with its own instantaneous frequency and amplitude function: that is,


1 20 0

1 2( )t t

i dt i dtF t a e a e

ω ω∫ ∫= + with 1 2 1 2, , and a a ω ω being unknown functions in the time domain. The envelope and the instantaneous frequency of the double-component vibration signal ( )tω are:

( )( )

122 2

1 2 1 2 2 1

22 1 2 1 2 2 1

1 2

( ) 2 cos ( )

( ) 2 cos ( )( )

( )

a t a a a a dt

a a a dtt

a t

ω ω

ω ω ω ωω ω

⎡ ⎤= + + −⎣ ⎦⎡ ⎤− + −⎣ ⎦= +

∫

∫ (1)

Eq.(1) shows that the signal envelope ( )a t consists of two different parts, that is, a slow varying part including the sum of the component amplitudes squared 2 2

1 2a a+ and a rapidly varying (oscillating) part, the multiplication of these amplitudes with function cos of the relative phase angle between two components. The instantaneous frequency of the two constant frequency tones considered in Eq.(1) is generally time-varying and exhibits asymmetrical deviations about the frequency 1ω . For the two tones, not only are there time-varying deviations in the instantaneous frequency, but these deviations always force the instantaneous frequency beyond the frequency range of the signal components. It appears, then, that in general, the instantaneous frequency of a signal and the average frequency at each time of the signal are different quantities. The time varying difference between the two initial constant frequencies and the instantaneous frequency of the composed signal can be interpreted as frequency modulation within the signal. The developed HT technique could exploit this modulation property of the input excitation signal for modal analysis of dynamic systems. Non-linear viscous damping and frequency, or distributed linear frequencies and associated modal damping, are both made possible, in a computationally fast manner using HT identification. The Hilbert transfer identification The modern HT identification method, as a non-parametric method, is recommended for instantaneous modal parameters identification in a time domain, including determination of the system backbone, damping curves, and static force characteristics. The HT method is suggested for identification of SDOF linear and non-linear systems under free or forced vibration conditions. A second-order conservative system with a non-linear restoring force ( )k x and a solution ( ) cosx t A tω= can be represented in a general power series form

2 41 3 5

( ) 0

( ) ( ...)

x k x

k x x x xα α α

+ =

= + + + (2)

Applying the multiplication property of the HT for overlapping functions [ 1, 2] to Eq.(2), we obtain a new form of time varying equation of motion

20( ) ( ) 0x j t x t xδ ω+ + = (3)

where 0 ( )tω is the fast varying natural frequency function and

( )tδ is the fast varying fictitious damping function. If we consider only the mean value of the varying natural frequency function square, we get an important result

2 1 2 2 40 0 1 3 5

0

3 5( ) ...

4 8

T

T t dt A Aω ω α α α−= = + + +∫ (4)

which proves that the averaged natural frequency has, correct to the polynomial constant coefficients, the same expression as the initial non-linear restoring force ( )k x Eq.(2). This general result means that the estimated average natural frequency and

hence the system skeleton curve (backbone) ( )0A ω includes

the main information about the initial non-linear elastics characteristics and can be used for non-linear system identification. Clearly, after the averaging we get only the first term of the motion, so the HT identification method restores the initial non-linear forces approximately correct to the time varying first term of motion. In the first stage of the proposed identification technique, the envelope ( )A t and the instantaneous frequency ( )tω are extracted from the vibration and excitation signals on the base of the HT signal processing. In the next stage, the instantaneous undamped natural frequency and the instantaneous damping coefficient of the tested system are calculated according to formulas [ 1, 2]:

22 20 02

2( ) ; ( )

2

A A A At h t

A A A A

ω ωω ωω ω

= − + + = − − , where

( )A t and ( )tω are the envelope of the instantaneous frequency of the vibration. The mass parameter of each vibration model under identification is a priori taken as equal to 1. The obtained instantaneous functions are low-pass filtered to get the first term of vibration motion. In the final stage, the non-linear "frequency response function" [ 2] is constructed together with both the elastic and the damping static force characteristics, which are calculated according to the decomposition technique [ 2]:

20 0

200

( ) ( ), 0 ( ) ( ), 0( ) ; ( )

( ) ( ), 0( ) ( ), 0

x

x

t A t x h t a t xk x h x x

h t a t xt A t x

ω

ω

⎧ ⎧> >⎪ ⎪⎪ ⎪⎪ ⎪≈ ≈⎨ ⎨⎪ ⎪− <− <⎪ ⎪⎪⎩⎪⎩

It is convenient to represent the result of the HT identification in a standard form which includes the skeleton curve with the FRF and also the static force characteristics of the dynamics system.

Vibration modeling and numerical results This paper concentrates on dynamic analysis and identification of three groups of dynamics systems: (1) forced vibrations of linear and non-linear SDOF systems excited with quasiperiodic force signal; (2) combined forced vibrations of quasiperiodic time varying linear and non-linear SDOF systems excited with harmonic signal; and (3) combined self-excited and forced vibrations of non-linear SDOF systems excited with harmonic signal. For the given different equations of vibration motion, there is a combined analytical expression Eq.(5) for all the numerical for linear and non-linear modelling variants:

2 3

20

21 1 2 2 3 3

| | ( 1) ( )

(1 cos )

( ) cos2 cos2 cos2

x x x x x x kx x F t

k t

F t A f t A f t A f t

β

γ δ µ α

ω β ω

π π π

+ + + − + + =

= +

= + +

(5)


Consequently, we can construct a vibration motion by forming a combination of the following parameters: γ - the linear

viscous friction coefficient, δ - the non-linear "quadratic" friction coefficient, µ - the friction coefficient of the van-der-

Pol equation, k - the static elastic force coefficient, 20ω - the

linear undamped natural frequency square, α - the cubic coefficient of the Duffing equation, β - the amplitude

modulation coefficient of the elastic force coefficient, βω - the

frequency modulation coefficient of the elastic force coefficient, iA - the amplitude of the of excitation, if - the amplitude of the of excitation. Their corresponding numeric values are given in Table 1.

Model parameter Table 1

Model Parameters

1 2 3 4 5 6 7 8 γ 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02

δ 0 0.2 0 0.2 0 0 0 0

Dam

ping

µ 0 0 0 0 0 0.1 0.1 0.120ω 1 1 1 1 1 1 1 1 α 0 0.4 0 0 0 0.4 0 0

β 0 0 0.5 0.5 0.5 0 0 0

Syst

em

Ela

stic

ity

βω 0 0 0.03 0.03 1 0 0 0

1A 1 1 1 0 1 1e-3 1 0

1f 0.16 0.16 0.16 0 0.16 0.16 0.16 0

2A 1 1 0 0 0 0 1 0

Qua

sipe

riod

ic

2f 0.155 0.155 0 0 0 0 0.155 0

3A 0 0 0 1 0 0 0 0.1

Exc

itati

on

Swee

p

3f 0 0 0 2e-5 0 0 0 2e-5

The simulations of all differential equations of motion are performed with SIMULINK (MATLAB) with the permanent step value ODE4 (Runge-Kutta) solver.

Model 1 – Linear system

In the first test case (Table 1), we use two harmonics to generate a "beating" vibration regime. The frequency of the first harmonics 1 0.16f = Hz is close to the linear system

resonance frequency 0 / 2 0.159ω π ≈ Hz, and the frequency of

the second harmonics 2 0.155f = Hz is chosen to have three full period beatings during the total time of recorded vibration

1 2

1T

f f− (Figure 1). Naturally, the linear system does not

change the typical "two harmonics" wave and the spectrum form of the output (displacement) signal relative to the input (excitation) signal (Figure 1). Intermediate results of the HT identification, such as the time segment of the displacement with its envelope together with the

instantaneous frequencies, are given in Figure 2. The instantaneous frequency of the displacement (Figure 2, b, dashed line) varies in time around the constant value of the estimated instantaneous undamped natural frequency (bold line).

0 100 200 300 400 500−2

0

2

Exc

itatio

n

a

0 100 200 300 400 500

−50

0

50

Dis

plac

emen

t

Time, s

b

0 0.1 0.2 0.30

0.1

0.2

0.3

0.4

Frequency, HzP

ower

Spe

ctru

m M

agni

tude

Excitationc

0 0.1 0.2 0.30

100

200

300

400

Frequency, Hz

Pow

er S

pect

rum

Mag

nitu

de

Solution (Displacement)d

Figure 1. Model 1 - "Linear system": a) excitation, b) displacement, c) excitation spectrum, d) displacement spectrum.

100 150 200 250 300 350 400 450 500

−60

−40

−20

0

20

40

60

Displacement and Envelope

Am

plitu

de

a

100 150 200 250 300 350 400 450 5000.15

0.16

0.17

0.18

0.19

0.2

Instantaneous Frequencies

Undamped natural frequency

Time, s

Fre

quen

cy,

Hz

b

Figure 2. Model 1 – Linear system: a) displacement and envelope; b) instantaneous frequencies. This frequency modulation performance allows reconstruction of the tested dynamic structure in a wide frequency range. The obtained final results of the HT identification, including the skeleton curve, the FRF, the elastic static force characteristics, the damping curve, and the friction force, are shown in Figure 3 with the bold line. In the same figure, the initial characteristics from Table 1, namely the linear skeleton line

0 10.159

2 2 2

k ωπ π π= = ≈ Hz, the linear elastic static force


kx x= , the linear damping curve 0.012

γ= , and the linear

friction force 0.02x xγ = , are shown with a dashed line. However, the difference between the initial linear and the estimated characteristics is less than 0.1%, so it can not be distinguished in Figure 3.

0.14 0.15 0.16 0.17 0.18

20

30

40

50

60

Frequecy, Hz

Dis

plac

emen

t Am

plitu

de

Skeleton curve and FRFa

0 0.005 0.01

20

30

40

50

60

Damping coefficient

Vel

ocity

Am

plitu

de

Damping curvec

−50 0 50−60

−40

−20

0

20

40

60Elastic Static Force

For

ce

Displacement

b

−50 0 50

−1

−0.5

0

0.5

1

Friction Force

Velocity

For

ce

d

Figure 3. Model 1 – Linear system. Identification results: a) skeleton curve and FRF; b) elastic static force; c) damping curve; d – friction force characteristics.

Model 2 – "Non-Linear" system

Let us now examine the identification results for the non-linear model 2, which contains two different types of non-linearity: non-linear "quadratic" damping and also a non-linear cubic elastic component inherent to the Duffing equation (Table 1). Again, a forced vibration regime is produced by the same quasiperiodic force input signal. The existence of non-linearity

0 100 200 300 400 500−2

0

2

Exc

itatio

n

a

0 100 200 300 400 500−2

0

2

Dis

plac

emen

t

Time, s

b

0 0.2 0.4 0.6 0.8 1

10−6

10−4

10−2

Frequency, Hz

Pow

er S

pect

rum

Mag

nitu

de

Excitationc

0 0.2 0.4 0.6 0.8 1

10−6

10−4

10−2

Frequency, Hz

Pow

er S

pect

rum

Mag

nitu

de


Figure 4. Model 2 - "Non-linear" system: a) excitation, b) displacement, c) excitation spectrum, d) displacement spectrum.

can be noticed immediately from the time data (Figure 4, a, b), whose input and output wave shapes differ, and also from the output spectrum (Figure 4, d), which has high frequency multiple harmonics. The HT identification results are given in Figures 5 and 6. The comparison between the estimated (bold line) and the initial (dashed line) skeleton curve, taken as the first term of Eq.(4),

22 21 30

0

3( ) 1 34( )

2 2 2

AA Af A

α αωπ π π

+ += = = , shows

210 220 230 240 250 260 270 280 290 300

−1.5

−1

−0.5

0

0.5

1

1.5


Am

plitu

de

a

210 220 230 240 250 260 270 280 290 3000.15

0.16

0.17

0.18

0.19

0.2

0.21

0.22


Natural undamped frequency

Time, s

Fre

quen

cy,

Hz

b

Figure 5. Model 2 - "Non-linear" system: a) displacement and envelope; b) instantaneous frequencies.

0.15 0.2 0.250.6

0.8

1

1.2

1.4

1.6

1.8

Frequecy, Hz

Dis

plac

emen

t Am

plitu

de


0 0.05 0.1 0.15 0.2

0.8

1

1.2

1.4

1.6

1.8

Damping coefficient

Vel

ocity

Am

plitu

de

Damping curvec

−1 0 1

−3

−2

−1

0

1

2

3

Elastic Static Force

For

ce

Displacement

b

−1 0 1

−0.5

0

0.5

Friction Force

Velocity

For

ce

d

Figure 6. Model 2 - "Non-linear" system. Identification results: a) skeleton curve and FRF; b) elastic static force; c) damping curve; d – friction force characteristics. that these skeleton curves are in good close agreement. The identified (bold line) and the initial (dashed line) static force characteristics 3 31 0.4kx x xα+ = + (Figure 6, b) are very close together. The identified (bold line) and the initial (dashed line)


friction force characteristics | | 0.02 0.2 | |x x x x x xγ δ+ = + are also close to each other (Figure 6, d). This model illustrates that HT identification makes it possible to restore non-linear characteristics even in the case of combined non-linearity in both the elastics and the friction parts of the equation of motion. Note that the identified static force characteristics have a small deviation from the initial characteristics to the "linear" direction. In other words, they are slightly less non-linear than the initial force characteristics. This means that the proposed HT identification restores only the main first term of the motion.

Model 3 – "Parametric" system

As a next test, let us consider a structure that describes a slow modulated elastic force 2

0 (1 cos )k tβω β ω= + =

1 0.5cos 0.03t+ of the trivial dynamics system under external

0 100 200 300 400 500−1

0

1

Exc

itatio

n

a

0 100 200 300 400 500−20

0

20

Dis

plac

emen

t

Time, s

b

0 0.1 0.2 0.30

0.05

0.1

0.15

0.2

Frequency, Hz

Pow

er S

pect

rum

Mag

nitu

de

Excitationc

0 0.1 0.2 0.30

1

2

3

4

5

6

Frequency, Hz

Pow

er S

pect

rum

Mag

nitu

de


Figure 7. Model 3 - "Parametric" system: a) excitation, b) displacement, c) excitation spectrum, d) displacement spectrum.

50 100 150 200 250 300 350 400 450 500 550−20

−10

0

10

20Displacement and Envelope

Am

plitu

de

a

50 100 150 200 250 300 350 400 450 500 550

0.12

0.14

0.16

0.18

0.2


Time, s

Fre

quen

cy,

Hz

b

Figure 8. Model 3 - "Parametric" system: a) displacement and envelope; b) instantaneous frequencies.

harmonics excitation (Table 1). The generated vibration has a rather complicated form over time and also in the frequency domain (Figure 7). But HT identification restores this modulation in close detail. Thus, Figure 8, b shows that the identified instantaneous undamped natural frequency of the vibration (bold line) completely coincides with the varying

initial elastic force modulation function 1 0.5cos 0.03

2

tf

π+

=

−0.01 0 0.01 0.02 0.03

5

10

15

20

Damping coefficient

Vel

ocity

Am

plitu

de

Damping curvea

−20 −10 0 10 20

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

Friction Force

Velocity

For

ce

b

Figure 9. Model 3 - "Parametric" system: Identification results: a) damping curve; b) friction force characteristics. (dashed line).The obtained damping characteristics (Figure 9, a, b, bold line) demonstrate the linear type of friction force, which is a good match for the initial linear type of the "parametric"

model 3 (dashed line) with 0.012

γ= and 0.02x xγ = . This

example demonstrates that the HT method enables identification of a slow modulation of the system parameters, even in the case of the simplest monoharmonic excitation.

0 100 200 300 400 500−1

0

1

Exc

itatio

n

a

0 100 200 300 400 500

−2

0

2

Dis

plac

emen

t

Time, s

b

0 0.1 0.2 0.30

0.002

0.004

0.006

0.008

0.01

0.012

Frequency, Hz

Pow

er S

pect

rum

Mag

nitu

de

Excitationc

0 0.1 0.2 0.30

0.02

0.04

0.06

0.08

0.1

Frequency, Hz

Pow

er S

pect

rum

Mag

nitu

de


Figure 10. Model 4 - "Parametric + non-linear + sweep" system: a) excitation, b) displacement, c) excitation spectrum, d) displacement spectrum.


Model 4 – "Parametric + non-linear + sweep" system

The next system as well as the previous one has a slow modulated elastic force, but in addition to the linear viscous, it also has a non-linear “quadratic” friction member. The tested model has a different external force excitation with sweeping increasing frequency instead of single harmonics (Table 1). The generated vibration in time and also in the frequency domain (Figure 10) takes a rather complicated form. The results of HT identification are shown in Figures 11 and 12.

50 100 150 200 250 300 350 400 450 500 550

−3

−2

−1

0

1

2

3


Am

plitu

de

a

50 100 150 200 250 300 350 400 450 500 5500.1

0.12

0.14

0.16

0.18

0.2


Time, s

Fre

quen

cy,

Hz

b

Figure 11. Model 4 - "Parametric + non-linear + sweep" system: a) displacement and envelope; b) instantaneous frequencies. The identified instantaneous undamped natural frequency (bold line) completely coincides with the initial elastic force

modulation 1 0.5cos0.03

2

tf

π+

= (Figure 11, b, dashed line).

−0.1 0 0.1 0.2 0.3 0.4

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

Damping coefficient

Vel

ocity

Am

plitu

de

Damping curvea

−2 −1 0 1 2

−1

−0.5

0

0.5

1

Friction Force

Velocity

For

ce

b

Figure 12. Model 4 - "Parametric + non-linear + sweep" system: Identification results: a) damping curve; b) friction force characteristics.

The obtained non-linear friction force characteristics (bold line) are in close agreement with the initial non-linear type of friction force (dashed line) | | 0.02 0.2 | |x x x x x xγ δ+ = + (Figure 12, b).

Model 5 – "Parametric instability" system

The static elastic force for this case is now modulated with a high modulation frequency equal to the constant natural frequency of the system 0 1βω ω= = (Table 1). The amplitude

of the solution of the system under external harmonic excitation increases infinitely (Figure 13, b). This behavior illustrates the parametric instability of the tested system.

0 100 200 300 400 500−1

0

1

Exc

itatio

n

a

0 100 200 300 400 500−2

0

2x 10

4

Dis

plac

emen

t

Time, s

b

0 0.1 0.2 0.30

0.05

0.1

0.15

0.2

Frequency, Hz

Pow

er S

pect

rum

Mag

nitu

deExcitationc

0 0.1 0.2 0.30

1

2

3

4x 10

6

Frequency, Hz

Pow

er S

pect

rum

Mag

nitu

de


Figure 13. Model 5 - "Parametric instability" system: a) excitation, b) displacement, c) excitation spectrum, d) displacement spectrum.

0.05 0.1 0.15 0.2 0.25

2000

4000

6000

8000

10000

12000

14000

16000

Frequecy, Hz

Dis

plac

emen

t Am

plitu

de


−0.03 −0.02 −0.01 0 0.01

2000

4000

6000

8000

10000

12000

14000

16000

Damping coefficient

Vel

ocity

Am

plitu

de

Damping curvec

−2 −1 0 1 2

x 104

−2

−1

0

1

2x 10

4 Elastic Static Force

For

ce

Displacement

b

−2 −1 0 1 2

x 104

−500

0

500Friction Force

Velocity

For

ce

d

Figure 14. Model 5 - "Parametric instability" system: a) skeleton curve and FRF; b) elastic static force; c) damping curve; d – friction force characteristics. In this case, the input harmonics excitation has practically no influence on the observed unstable oscillation. Therefore, we


will use the HT identification FREVIB method that analyses only the vibration output of the structure. The HT identification method restores only the correct skeleton curve and the elastic force characteristics (Figure 14, a, b). It is clear that in this case, the HT instead of the initial linear damping restores a negative increment and the corresponding negative (in the opposite direction) friction characteristics (Figure 14, c, d). In some special cases, the obtained increment can be used for a quality analysis of the instability growth rate of such unstable vibration solutions.

Model 6 – "Self excited + non-linear" system

The next test combines the non-linear friction part common to the van-der-Pol oscillator and the non-linear cubic elastic force part typical to the Duffing equation. Model 6 being tested has a very low level harmonic input excitation (Table 1). As expected, the tested model 6 displays a known self-excited regime of non-linear vibration in time (Figure 15, b).

0 100 200 300 400 500−1

0

1x 10

−3

Exc

itatio

n

a

0 100 200 300 400 500−1

0

1

Dis

plac

emen

t

Time, s

b

0 0.2 0.4 0.6 0.8 1

10−12

10−10

10−8

Frequency, Hz

Pow

er S

pect

rum

Mag

nitu

de

Excitationc

0 0.2 0.4 0.6 0.8 1

10−6

10−4

10−2

Frequency, Hz

Pow

er S

pect

rum

Mag

nitu

de


Figure 15. Model 6 - "Self excited + non-linear" system: a) excitation, b) displacement, c) excitation spectrum, d) displacement spectrum. The corresponding spectrum of the self-excited vibration that shows high frequency multiple harmonics confirms the existence of the non-linear elastic part (Figure 15, d). Again, the observed self-excited oscillation regime practically does not depend on the input excitation signal. The HT identification method FREEVIB used here restores in full details both the non-linear friction and the non-linear elastics parts. Thus, the identified (bold line) and the initial (dashed line) skeleton curve, as the first term of Eq.(4), practically coincide (Figure 17, a). Similarly, the identified (bold line) and the initial (dashed line) static force characteristics 3 31 0.4kx x xα+ = + practically coincide as well (Figure 17, b). The identified non-linear friction force characteristics (bold line) are in close agreement with the initial non-linear type of the friction force (dashed line)

2 2( 1) 0.1 ( 1)x x x xµ − = − (Figure 17, d).

100 120 140 160 180 200 220 240−1

−0.5

0

0.5

1Displacement and Envelope

Am

plitu

de

a

100 120 140 160 180 200 220 240

0.155

0.16

0.165

0.17

0.175

0.18

0.185


Time, s

Fre

quen

cy,

Hz

b

Figure 16. Model 6 - "Self excited + non-linear" system: a) displacement and envelope; b) instantaneous frequencies.

0.12 0.14 0.16 0.18 0.20.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Frequecy, Hz

Dis

plac

emen

t Am

plitu

deSkeleton curve and FRFa

−0.05 −0.04 −0.03 −0.02 −0.01 0 0.010.2

0.4

0.6

0.8

1

Damping coefficient

Vel

ocity

Am

plitu

de

Damping curvec

−1 −0.5 0 0.5 1

−1

−0.5

0

0.5

1


For

ce

Displacement

b

−2 −1 0 1 2−0.04

−0.02

0

0.02

0.04Friction Force

Velocity

For

ce

d

Figure 17. Model 6 - "Self excited + non-linear" system: a) skeleton curve; b) elastic static force; c) damping curve; d) friction force characteristics.

Model 7 – Self + forced excited system

Let us now examine the next system with the combined non-linear friction part common to the van-der-Pol oscillator and high level quasiperiodic excitation (Table 1). The chosen forced vibration regime becomes dominant, and the vibration shape shown in Figure 18, a, b is similar in appearance to the forced vibration of the non-linear system in Figure 4, a, b. The HT, however, detects the actual properties of the tested system. The identified skeleton curve and the elastic static force characteristics (bold line) take a trivial linear form corresponding to the initial linear elastics part (dashed line) of the tested system (Figure 20, a, b). The identified non-linear friction force characteristics (bold line) are in close agreement


with the initial non-linear type of the friction force (dashed line) 2 2( 1) 0.1 ( 1)x x x xµ − = − (Figure 20, d).

0 100 200 300 400 500−2

0

2

Exc

itatio

n

a

0 100 200 300 400 500

−2

0

2

Dis

plac

emen

t

Time, s

b

0 0.1 0.2 0.30

0.1

0.2

0.3

0.4

Frequency, Hz

Pow

er S

pect

rum

Mag

nitu

de

Excitationc

0 0.1 0.2 0.30

0.2

0.4

0.6

0.8

1

1.2

Frequency, Hz

Pow

er S

pect

rum

Mag

nitu

de


Figure 18. Model 7 - "Self + forced excited" system: a) excitation, b) displacement, c) excitation spectrum, d) displacement spectrum.

100 150 200 250 300 350 400 450 500

−3

−2

−1

0

1

2

3


Am

plitu

de

a

100 150 200 250 300 350 400 450 500

0.15

0.16

0.17

0.18

0.19

0.2



Time, s

Fre

quen

cy,

Hz

b

Figure 19. Model 7 - "Self + forced excited" system: a) displacement and envelope; b) instantaneous frequencies.

Model 8 – Self excited + sweep system

The system under consideration repeats the previous system with the non-linear friction part common to the van-der-Pol oscillator, but now it involves sweeping frequency forced excitation (Table 1). The obtained wave shape in time and also

0.1 0.12 0.14 0.16 0.18 0.2

1

1.5

2

2.5

3

Frequecy, Hz

Dis

plac

emen

t Am

plitu

de


−0.1 0 0.1 0.2 0.3

1

1.5

2

2.5

3

Damping coefficient

Vel

ocity

Am

plitu

de

Damping curvec

−2 0 2−3

−2

−1

0

1

2

3Elastic Static Force

For

ce

Displacement

b

−2 0 2

−1.5

−1

−0.5

0

0.5

1

1.5

Friction Force

Velocity

For

ce

d

Figure 20. Model 7 - "Self + forced excited" system: a) skeleton curve; b) elastic static force; c) damping curve; d) friction force characteristics. the corresponding spectrum shape demonstrate the typical resonance performance of the structure (Figure 21). Again the characteristics identified on the base of the HT skeleton curve and the elastic static force characteristics (bold line) take a trivial linear form that corresponds to the initial linear elastics part (dashed line) of the tested system (Figure 23, a, b). The identified non-linear friction force characteristics (bold line) are in close agreement with the initial non-linear type of the friction force (dashed line) 2 2( 1) 0.1 ( 1)x x x xµ − = − (Figure 23, d).

0 100 200 300 400 500−1

0

1

Exc

itatio

n

a

0 100 200 300 400 500

−2

0

2

Dis

plac

emen

t

Time, s

b

0 0.1 0.2 0.3 0.4 0.50

1

2

3

4

5

6

x 10−3

Frequency, Hz

Pow

er S

pect

rum

Mag

nitu

de

Excitationc

0 0.1 0.2 0.3 0.4 0.50

0.01

0.02

0.03

0.04

Frequency, Hz

Pow

er S

pect

rum

Mag

nitu

de


Figure 21. Model 8 - "Self excited + sweep" system: a) excitation, b) displacement, c) excitation spectrum, d) displacement spectrum.


50 100 150 200 250 300 350 400 450 500 550

−2

−1

0

1

2


Am

plitu

de

a

50 100 150 200 250 300 350 400 450 500 550

0.15

0.2

0.25



Time, s

Fre

quen

cy,

Hz

b

Figure 22. Model 8 - "Self excited + sweep" system: a) displacement and envelope; b) instantaneous frequencies.

0.12 0.14 0.16 0.18 0.20.5

1

1.5

2

2.5

Frequecy, Hz

Dis

plac

emen

t Am

plitu

de


−0.1 0 0.1 0.2

1

1.5

2

2.5

Damping coefficient

Vel

ocity

Am

plitu

de

Damping curvec

−2 −1 0 1 2

−2

−1

0

1

2


For

ce

Displacement

b

−2 −1 0 1 2−1

−0.5

0

0.5

1Friction Force

Velocity

For

ce

d

Figure 23. Model 8 - "Self excited + sweep" system: a) skeleton curve and FRF; b) elastic static force; c) damping curve; d) friction force characteristics.

Conclusions This paper has considered the details and obtained results of estimating the parameters of quasiperiodic systems consisting of a sum of a given number of harmonics signals. The HT identification methods used, FREEVIB and FORCEVIB, enable detailed reconstruction and separation of the actual combined non-linear elastic and the friction force of the equation of motion of an SDOF system. The proposed HT technique allows identification of linear, non-linear, and modulated parameters under different kinds of excitation including the quasiperiodic input signal with only two harmonics components. HT identification allows reconstruction of the tested dynamic structure in a wide continuous frequency range around the resonance, whereas the traditional Fourier transform enables estimation only of the two corresponding discrete frequency points on the FRF. A further continuous real-time estimation of the initial parameters of non-linear dynamical systems can be used, for example, for measurement, monitoring and diagnostics purposes.

ACKNOWLEDGMENTS I would like to thank Dr. Oded Gottlieb for his help in

choosing the quasiperiodic vibration models and their parameters.

REFERENCES 1. M. Feldman, "Non-linear system vibration analysis using

Hilbert transform -- II. Forced vibration analysis method "FORCEVIB"". Mechanical Systems and Signal Processing, 1994, 8(3), pp. 309-318.

2. M. Feldman, "Non-Linear Free Vibration Identification via the Hilbert Transform," Journal of Sound and Vibration. v 208, n 3, December 4, 1997, pp. 475-489.

3. M. Feldman, "Hilbert Transforms", Encyclopedia of Vibration, Academic Press, pp. 642-648, 2001.

4. M. Feldman. "Matlab programs for the HT identification", October 2003, from Technion. Web site: http://hitech.technion.ac.il/feldman/HT_identification.zip

Documents

TIME-VARYING AND NON-LINEAR DYNAMICAL SYSTEM ...hitech.technion.ac.il/feldman/DETC2005-84644.pdf · Keywords: non-linear system, Hilbert transform, identification, envelope, instantaneous