methods for simulation and hani characterization of nonlinear mechanical...

Blekinge Institute of TechnologyLicentiate Dissertation Series No. 2008:13

School of Engineering

Methods for siMulation and CharaCterization of nonlinear MeChaniCal struCtures

Martin Magnevall

Trial and error and the use of highly time-consu-ming methods are often necessary for modeling, simulating and characterizing nonlinear dynamical systems. However, for the rather common special case when a nonlinear system has linear relations between many of its degrees of freedom there are particularly interesting opportunities for more efficient approaches. The aim of this thesis is to develop and validate new efficient methods for the theoretical and experimental study of mecha-nical systems that include significant zero-memory or hysteretic nonlinearities related to only small parts of the whole system.

The basic idea is to take advantage of the fact that most of the system is linear and to use much of the linear theories behind forced response simula-tions. This is made possible by modeling the nonli-nearities as external forces acting on the underly-ing linear system. The result is very fast simulation routines where the model is based on the residues and poles of the underlying linear system. These residues and poles can be obtained analytically, from finite element models or from experimental

measurements, making these forced response rou-tines very versatile. Using this approach, a com-plete nonlinear model contains both linear and nonlinear parts. Thus, it is also important to have robust and accurate methods for estimating both the linear and nonlinear system parameters from experimental data.

The results of this work include robust and user-friendly routines based on sinusoidal and random noise excitation signals for characterization and description of nonlinearities from experimental measurements. These routines are used to create models of the studied systems. When combined with efficient simulation routines, complete tools are created which are both versatile and compu-tationally inexpensive.

The developed methods have been tested both by simulations and with experimental test rigs with promising results. This indicates that they are use-ful in practice and can provide a basis for future research and development of methods capable of handling more complex nonlinear systems.

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ISSN 1650-2140

ISBN 978-91-7295-156-32008:13

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artin Magnevall

2008:13

Methods for Simulation and Characterization

of Nonlinear Mechanical Structures

Martin Magnevall

Methods for Simulation and Characterization

of Nonlinear Mechanical Structures

Martin Magnevall

Blekinge Institute of Technology Licentiate Dissertation SeriesNo 2008:13

ISSN 1650-2140ISBN 978-91-7295-156-3

Department of Mechanical EngineeringSchool of Engineering

Blekinge Institute of TechnologySWEDEN

Acknowledgements

This work was carried out at the Department of Mechanical Engineering,School of Engineering, Blekinge Institute of Technology, Karlskrona, Swedenunder the supervision of Professor Kjell Ahlin and Professor Goran Broman.

I would like to extend a special thanks to my supervisors for their professionalsupport and guidance throughout the work.

To all my colleagues at the department; thank you for the pleasant and produc-tive working environment you help to create. A special thanks to my closestco-worker, Andreas Josefsson, for all fruitful discussions and great ideas. Iwould also like to extend a thanks to Tommy Gunnarsson at Sandvik Coro-mant for good co-operation and for making my stays in Sandviken very pleas-ant.

Financial support from the Swedish Knowledge Foundation, Sandvik Coro-mant and the Faculty Board of Blekinge Institute of Technology is gratefullyacknowledged.

Finally, I would like to extend my deepest gratitude to my family for all theirlove and support throughout the work.

Karlskrona, December 2008

Martin Magnevall

iii

Abstract

Trial and error and the use of highly time-consuming methods are often neces-sary for modeling, simulating and characterizing nonlinear dynamical systems.However, for the rather common special case when a nonlinear system has lin-ear relations between many of its degrees of freedom there are particularlyinteresting opportunities for more efficient approaches. The aim of this thesisis to develop and validate new efficient methods for the theoretical and exper-imental study of mechanical systems that include significant zero-memory orhysteretic nonlinearities related to only small parts of the whole system.

The basic idea is to take advantage of the fact that most of the system islinear and to use much of the linear theories behind forced response simula-tions. This is made possible by modeling the nonlinearities as external forcesacting on the underlying linear system. The result is very fast simulation rou-tines where the model is based on the residues and poles of the underlyinglinear system. These residues and poles can be obtained analytically, from fi-nite element models or from experimental measurements, making these forcedresponse routines very versatile. Using this approach, a complete nonlinearmodel contains both linear and nonlinear parts. Thus, it is also important tohave robust and accurate methods for estimating both the linear and nonlin-ear system parameters from experimental data.

The results of this work include robust and user-friendly routines based onsinusoidal and random noise excitation signals for characterization and de-scription of nonlinearities from experimental measurements. These routinesare used to create models of the studied systems. When combined with effi-cient simulation routines, complete tools are created which are both versatileand computationally inexpensive.

The developed methods have been tested both by simulations and with exper-imental test rigs with promising results. This indicates that they are usefulin practice and can provide a basis for future research and development ofmethods capable of handling more complex nonlinear systems.

v

Keywords: nonlinear dynamics, nonlinear parameter estimation, harmonicbalance, reverse-path, zero-memory nonlinear systems, hysteretic nonlineari-ties, dry-friction, geometric nonlinearities.

vi

Appended Papers

The appended papers have been reformatted from their original layouts to fitthe format of this thesis; the content is, however, unchanged.

Paper AJosefsson, A., Magnevall, M. and Ahlin,K. Control Algorithm for Sine Excita-tion on Nonlinear Systems, IMAC-XXIV, Conference & Exposition on Struc-tural Dynamics, St. Louis, Missouri, 2006

Paper BMagnevall, M., Josefsson, A. and Ahlin, K. On Nonlinear Parameter Estima-tion, ISMA 2006, Leuven, Belgium, 2006

Paper CMagnevall, M., Josefsson, A. and Ahlin, K. On Parameter Estimation andSimulation of Zero Memory Nonlinear Systems, IMAC-XXVI, Conference &Exposition on Structural Dynamics, Orlando, Florida, 2008

Paper DMagnevall, M., Josefsson, A., Ahlin, K. and Broman, G. Simulation and Char-acterization of a Nonlinear Hysteretic Damper, Submitted for publication,September 2008

vii

The Author’s Contribution to the Appended Papers

The appended papers were written together with co-authors. The present au-thor’s contributions to the individual papers are as follows:

Paper ATook part in planning and writing the paper.Responsible for the simulations and implementation.

Paper BResponsible for the planning and half of the writing.Responsible for the simulations concerning sinusoidal excitation.

Paper CResponsible for the planning and writing.Carried out all simulations and measurements.

Paper DResponsible for the planning and writing.Carried out all simulations and measurements.

viii

Other Publications

Magnevall, M., Josefsson, A. and Ahlin, K. Experimental Verification of aControl Algorithm for Nonlinear Systems, IMAC-XXIV, Conference & Expo-sition on Structural Dynamics, St. Louis, Missouri, 2006

Ahlin, K. Magnevall, M and Josefsson, A. Simulation of Forced Response inLinear and Nonlinear Mechanical Systems Using Digital Filters, ISMA 2006,Leuven, Belgium, 2006

Magnevall, M., Josefsson, A. and Ahlin, K. Parameter Estimation of Hystere-sis Elements Using Harmonic Input, IMAC-XXV, Conference & Expositionon Structural Dynamics, Orlando, Florida, 2007

Josefsson, A., Magnevall, M. and Ahlin, K. On Nonlinear Parameter Estima-tion with Random Noise Signals, IMAC-XXV, Conference & Exposition onStructural Dynamics, Orlando, Florida, 2007

Josefsson, A., Magnevall, M. and Ahlin, K. Estimating the Location of Struc-tural Nonlinearities from Random Data, IMAC-XXVI, Conference & Exposi-tion on Structural Dynamics, Orlando, Florida, 2008

ix

Contents

1 Introduction 31.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Aim and Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Nonlinear Dynamics 52.1 Linear and Nonlinear Systems . . . . . . . . . . . . . . . . . . . 5

2.1.1 Zero-Memory Nonlinearities . . . . . . . . . . . . . . . . 62.1.2 Hysteretic Nonlinearities . . . . . . . . . . . . . . . . . . 7

2.2 Detection of Nonlinear Systems . . . . . . . . . . . . . . . . . . 92.3 Measurements on Nonlinear Systems . . . . . . . . . . . . . . . 10

2.3.1 Broad-banded Excitation . . . . . . . . . . . . . . . . . 112.3.2 Harmonic Excitation . . . . . . . . . . . . . . . . . . . . 12

2.4 Nonlinear Parameter Estimation . . . . . . . . . . . . . . . . . 132.4.1 Using Random Noise . . . . . . . . . . . . . . . . . . . . 132.4.2 Using Harmonic Input . . . . . . . . . . . . . . . . . . . 16

3 Summary of Papers 193.1 Paper A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.2 Paper B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.3 Paper C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.4 Paper D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4 Conclusions and Future Research 21

References 23

Paper A 27

Paper B 41

Paper C 61

Paper D 79

1

Chapter 1

Introduction

The industrial demand for good simulation models is increasing. Since moststructures show some form of nonlinear behavior, linear models are not alwaysgood enough to describe the true dynamics. Therefore, nonlinear structuraldynamic analysis techniques are necessary. For example, a structure with abolted joint experiencing inter-facial movements between the surfaces of theconnected parts shows a hysteretic behavior due to friction [1]. A slenderbeam subjected to large amplitude vibrations also show clear nonlinear effects[2, 3, 4]. These are examples of relatively simple systems showing extensivenonlinear behavior. Another source of nonlinearity in mechanical structuresare material properties. For example, the use of rubber in vibration damp-ing is very common and makes the structure behave highly nonlinear, whichinvalidates any linear model trying to describe the dynamic behavior of thesystem [5].

1.1 Background

No general approach to detect and characterize the input-output relationshipof nonlinear systems exists. Instead, each individual case has to be treatedseparately. As an example, the most efficient way to treat a structure with ahysteretic nonlinearity might not be the same as the method used on a struc-ture with clearance. Numerous techniques for structural dynamic detectionand identification of nonlinear systems have been evolved in recent years [6, 7].What usually differentiate the methods is the expected type of nonlinearityand the excitation signal used.

In linear dynamics, a popular way to model a system is to use modal analysis.Modal analysis is used because it produces a small set of parameters describ-ing the behavior of the studied system for any input [8, 9, 10]. However, for

3

Chapter 1. Introduction

nonlinear systems, no general method for modeling and simulation exists. Inorder to successfully develop new methods for localization, characterizationand parameter estimation of nonlinear systems, fast and reliable simulationroutines are needed. In this thesis, a method where the nonlinearities aremodeled as external forces acting on an underlying linear system is used. Thismeans that much of the linear theory behind forced response simulation canbe used in the simulation of nonlinear systems. The result is new and fastsimulation routines where the model is based on the residues and poles of theunderlying linear system [11, 12]. Residues and poles can be obtained analyti-cally, from finite element models or from experimental measurements, makingthese forced response routines very versatile.

External factors which are hard to simulate and/or foresee can have a bigimpact on the results from experimental tests. Therefore, experimental testsare crucial to further validate the function and robustness of new methods.Experimental testing on nonlinear systems is challenging, therefore methodswhich help the test engineer to perform measurements on nonlinear systemsare important.

1.2 Aim and Scope

The work presented in this thesis is part of a research project addressing”modeling and simulation of nonlinear mechanical systems, with emphasis ondamping”. The common approach used is to model and simulate the dynamicbehavior of nonlinear mechanical systems in the time domain. The simulationmethod used treats the nonlinearities as external forces acting on an under-lying linear system, taking advantage of the fact that the major parts of thestudied systems are linear with local nonlinear parts and/or boundary condi-tions. An important part of the project is to develop robust and user-friendlymethods to characterize and describe nonlinearities from experimental mea-surements. These methods are used to create nonlinear models of the studiedsystems. When combined with efficient simulation routines complete toolsfor treating nonlinear dynamical systems are created, which can assist in theproduct development process.

The specific aim of this thesis is to develop and validate new efficient methodsfor the theoretical and experimental study of mechanical systems that includesignificant zero-memory or hysteretic nonlinearities related to only small partsof the whole system.

4

Chapter 2

Nonlinear Dynamics

2.1 Linear and Nonlinear Systems

Systems used in engineering can generally be separated into two distinctgroups, linear and nonlinear. A system H is said to be linear if it fulfillsthe principle of superposition:

H{cx

}= cH

{x}

(2.1a)

H{x1 + x2

}= H

{x1

}+ H

{x2

}(2.1b)

A system which does not satisfy the properties of eq. (2.1) is defined as nonlin-ear. An illustrative example from [13] of a system which violates the principleof superposition is shown in figure 2.1. A static application of a beam which isrigidly clamped in both ends and subjected to a force in the center will shownonlinear behavior at large deflections due to elongation. At small deflectionsthe principle of superposition holds and the system behaves linearly.

An important property of a linear system is that when excited by a sinusoidalforce, the steady-state response will be sinusoidal with the same frequencyas the excitation force. Also, the amplitude and phase of the response arefunctions of the excitation frequency [8]. When a nonlinear system is excitedby a sinusoidal force the steady-state response generally contains higher har-monics, multiples of the excitation frequency. In some cases, the steady-stateresponse also contains subharmonic frequencies. In extreme cases a nonlinearsystem can show chaotic behavior, which is not a harmonic response but aresponse with a broad-banded frequency content [14]. Two different types ofnonlinearities are treated in this thesis, zero-memory and hysteretic.

5

Chapter 2. Nonlinear Dynamics

1F

2F

3F213 FFF +=

1x 2x 3x

213 xxx +≠

Figure 2.1: Example of a system violating the rule of superposition.

2.1.1 Zero-Memory Nonlinearities

A zero-memory nonlinear system is a system that is only dependent on thepresent input in some nonlinear fashion. Examples of systems with thesetype of nonlinearities are: bilinear systems, systems with clearance, clippedsystems, hardening spring and softening spring systems [6, 13, 15]. Examplesof nonlinear systems with hardening and softening nonlinearities are shown infigure 2.2. These systems are used in a simulation with sinusoidal excitationat different force levels. The effect these nonlinearities have on the frequencyresponse functions (FRFs) are displayed in figure 2.3, the FRFs are based onthe response of the fundamental harmonic. The nonlinear functions used inthe simulations are nlf(x) = 1.7e15x3 for the hardening spring and nlf(x) =5 arctan(4e5x) for the softening spring.

Figure 2.2: Examples of two nonlinear systems, hardening spring in the leftfigure and softening spring in the right.

6

2.1 Linear and Nonlinear Systems

27.5 28 28.5 29 29.5 30 30.510

−7

10−6

10−5

10−4

Frequency [Hz]

Rec

epta

nce

[m/N

]

Hardening Spring

F = 0.01NF = 0.2NF = 0.7N

28.5 29 29.5 30 30.5 31 31.5 32 32.510

−7

10−6

10−5

10−4

Softening Spring

Frequency [Hz]

Rec

epta

nce

[m/N

]

F = 0.05NF = 0.2NF = 1.0N

Figure 2.3: FRFs obtained with different excitation amplitudes.

It is evident that the nonlinearities have a significant effect on the FRFs.In the hardening spring case the resonance frequency increases as the forceincreases because the total stiffness of the system increases as a function ofdisplacement. The resonance frequency for the softening spring decreases asthe excitation force increases due to a decrease in the total stiffness as afunction of displacement. If the systems were linear, the FRFs would beindependent of the force level.

2.1.2 Hysteretic Nonlinearities

Hysteretic nonlinearities are encountered in, for example, different types of vi-bration isolators [16, 17], in sandwich composite structures [18] and in mechan-ical joints [1]. What differs the hysteretic nonlinearity from a zero-memorynonlinearity is its dependence on past values. The current nonlinear restor-ing force from a hysteretic nonlinearity is not only dependent on the presentdisplacement but also on past history displacements. This makes hystereticnonlinearities more difficult to handle compared with zero-memory nonlinear-ities. An example of a hysteretic nonlinearity is the stick-slip or Coulombmacro-slip model [19]. A single degree of freedom system with a stick-slipnonlinearity is shown in figure 2.4a.

The relationship between the system response and feedback force from thestick-slip nonlinearity follows the curve shown in figure 2.4b. Fnl is the feed-back force from the nonlinearity and Fd the force level at which the systemstarts to slip. As long as the feedback force is smaller than Fd, the stick-slip model acts like a linear spring with stiffness kd. When the displacementreaches the level where, Fnl = Fd, the feedback force from the stick-slip func-tion will remain at Fd until the motion changes direction. The damper willthen act as a linear spring again until the condition Fnl = −Fd is fulfilled. Thismeans that during stick-slip conditions, the feedback force from the damper

7


Figure 2.4: A SDOF system with a stick-slip nonlinearity.

will have two possible solutions for every displacement, except for the pointswhere the motion changes direction. Which of these two solutions it is de-pends on the history displacement of the damper. To further illustrate theeffect this nonlinearity has on the dynamics of the system in figure 2.4, resultsfrom simulations using sinusoidal excitation at different force levels are shownin figure 2.5. The coefficients used to define the stick-slip nonlinearity arekd = 1e7 N/m and Fd = 130 N .

20 25 30 35 40 4510

−8

10−7

10−6

10−5

10−4

Frequency [Hz]

Rec

epta

nce

[m/N

]

F=0.001*FdF=FdF=10*Fd

Figure 2.5: Effect of a stick-slip nonlinearity at different excitation levels.

As seen in figure 2.5 the stick-slip system shows a more complicated behaviorcompared with zero-memory nonlinearities. This system exhibits two linearregions. At low force levels, when the system never reaches the slip-level, thetotal stiffness is K + kd. At high force levels, large displacements, the systemis in constant slip meaning that there is no effect from the stick-slip function,thus the stiffness in the system is constituted by K alone. In between these two

8

2.2 Detection of Nonlinear Systems

regions there is a stick-slip region where the energy loss due to the hystereticeffect is very high. The stick-slip system is a simple example of a hystereticnonlinearity; more versatile and complex hysteresis models are described in[2, 20, 21].

2.2 Detection of Nonlinear Systems

It is essential to carry out a linearity check at the beginning of any dynamictest as the majority of the existent analysis methods are based on the assump-tion that the system is linear. There are several different ways to test if asystem is linear or not [8, 9, 13]. The techniques described in this section arethe ones used in this thesis. They have proven robust and usable in measure-ments on several test rigs and nonlinearities.

Harmonic distortion: This method is best used with sinusoidal input. Asmentioned previously, if a nonlinear system is excited by a sinusoidal signalthe response will generally contain additional frequency components. Theseare a clear indication that the studied system behaves nonlinearly.

The ordinary coherence function: This approach is mainly used withrandom or transient excitation. It is a quick and commonly used method toidentify nonlinear behavior. The ordinary coherence is a spectrum defined asthe linear relationship between any two signals [22]. The ordinary coherencefunction for a single input single output system is calculated by:

γ2 =| GXF |2

GFF GXX(2.2)

where GXF is the cross spectral density between the output X and input Fand GFF and GXX are the auto spectral densities of the input and outputsignals respectively. If a perfect linear relationship exists between two spectra,the ordinary coherence function will be equal to unity. To further illustratethis, results from a simulation using random noise excitation on the systemwith hardening nonlinearity from section 2.1.1 is shown in figure 2.6. Thesimulation is carried out at both low and high force levels. The FRFs andordinary coherence functions are calculated for both responses.

9


20 25 30 35 40 45 50

10−5

Frequency [Hz]

Rec

epta

nce

[m/N

]

Low ForceHigh Force

20 25 30 35 40 45 500

0.5

1

Frequency [Hz]

Coh

eren

ce

Low ForceHigh Force

Figure 2.6: FRFs and ordinary coherence functions of a nonlinear systemexcited at low and high force levels.

It is clear that the ordinary coherence function deviates from unity at higherforce levels, this is especially obvious around the resonance. This is becausethe nonlinearity is displacement-dependent and the displacement is large whenthe system resonates. Also seen is that at low force levels the nonlinearity isnot excited and the system behaves in a linear fashion.

Changes in the FRF with respect to the input force: The ratio betweenthe input force and the response of a linear system is constant, according to thelaw of superposition. For nonlinear systems superposition is not applicable,therefore an indication of nonlinear behavior can be found by comparing theFRFs obtained by excitation at different force levels. If the FRF changes withrespect to the input force, figures 2.3, 2.5 and 2.6, that is a clear indication ofnonlinearity.

As seen in the examples, a structural nonlinearity might be overlooked if theinput force is chosen in a way so that the nonlinearity is not excited. Therefore,it is important to make measurements both at different force levels and atdifferent locations before a system can be classified as linear or nonlinear.

2.3 Measurements on Nonlinear Systems

It is not uncommon for nonlinearities to be introduced in the measurementchain unintentionally due to insufficient checks on the test set-up. Somesources of nonlinear behavior due to a bad test set-up are: misalignments,looseness, pre-loads, cable rattle, poor transducer mounting and exciter prob-

10

2.3 Measurements on Nonlinear Systems

lems [13]. These nonlinear effects are not connected to the structure under testand should therefore be avoided, to be as sure as possible that the measure-ment reflects the intended behavior. Minimizing these effects are especiallyimportant if the measurement is to be carried out on an already nonlinearstructure and the recorded data used for nonlinear parameter estimation.

There are several different ways to excite a structure; the most commonly usedmethods in conventional modal analysis are: transient, random and harmonicexcitation.

2.3.1 Broad-banded Excitation

Transient and random excitation are broad-banded, meaning that the energyassociated with every single frequency is small, which can make it difficult toget enough energy into a system to excite structural nonlinearity. However,broad-banded excitation is still used in this thesis due to the fact that someof the most commonly used identification methods for nonlinear systems arebased on broad-banded excitation [15, 23].

Transient excitation is most commonly done by manually exciting the struc-ture with an impulse hammer. The frequency range can be determined tosome extent by the hardness of the tip used. The force level and excitationpoint is harder to control since it is difficult to hit the structure with exactlythe same force and in exactly the same point repeatedly.

Random excitation is usually done with a shaker, electrodynamic or hydraulic.Due to the randomness of the amplitude and phase of the excitation signal,random excitation creates a ”linearized” FRF; this linearization can make itdifficult to detect if a system behaves nonlinearly [13]. On the other hand,since the excitation signal is created in a computer, random excitation can becontrolled to a much higher extent than transient. For example, more energycan be added into the excitation signal around the resonance frequencies of thestudied systems and thereby facilitating excitation of a structural nonlinearity.

For the reasons stated above, random signals are the type of broad-bandedexcitation used in this thesis.

11


2.3.2 Harmonic Excitation

When using harmonic excitation only one frequency is excited at a time,meaning that all the energy in the signal is associated with one single fre-quency. This makes harmonic input suitable for exciting structural nonlinear-ities. Other benefits with sinusoidal excitation is that the FRFs obtained showthe true behavior of the system and are not linearized as with random signals.Also, it is easier to examine details in the studied systems response, such assteady-state hysteresis loops. These can be used to determine if a system hasa nonlinear hysteretic behavior, and in turn be used for parameter estimation.This is an important property of harmonic excitation which makes it favorableto use with systems showing hysteretic behavior.

There are two different ways to perform harmonic excitation, stepped-sine andsine-sweep. Using stepped-sine the system is excited with one frequency at atime and the measurement is taken when the system has reached a steady-state response. Sine-sweep is done by slowly and continuously varying thefrequency of the excitation signal over a specified interval. The benefit ofthe sine-sweep signal is its speed compared to stepped-sine excitation. Withstepped-sine it is easier to control the input signal, since measurements areperformed in steady-state allowing off-line adjustment of the input force. Thisis an important property when measurements are done on nonlinear systems,as further explained below.

Most theories developed around harmonic excitation of nonlinear systems as-sume that the system is excited with a pure sinusoidal signal [2, 19, 24]. Insimulations a pure sinusoidal excitation is easily accomplished. But in realapplications, the actual force applied to the structure is the reaction forcebetween the shaker and the structure [2, 25, 26], figure 2.7.

Figure 2.7: Sketch over reaction forces in the measurement chain.

As mentioned earlier the response from nonlinear systems generally containsadditional frequency components. These are transfered to the excitation signaland result in a distorted input force. This effect is usually most evident around

12

2.4 Nonlinear Parameter Estimation

the resonances when the system exhibits large deflections. The problem isfurther enhanced due to the fact that most shakers show nonlinear behaviorat large displacements [13]. Some form of control system is needed in orderto remove distortions from the force signal and maintain a constant excitationlevel. A control algorithm for pure sinusoidal excitation is developed in paperA. The algorithm uses an off-line nonlinear iteration approach to find a suitablevoltage signal as input to the shaker in order to obtain a pure sinusoidal forcesignal with a desired amplitude. Results from experimental tests [27] usingthis control algorithm are shown in figure 2.8.

4.005 4.01 4.015 4.02 4.025 4.03 4.035 4.04 4.045

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

Time [s]

For

ce [N

]

Reference sine0 iterationsAfter 5 iterations

4 4.01 4.02 4.03 4.04 4.05−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

Time [s]

For

ce [N

]

Reference sine0 iterationsAfter 4 iterations

Figure 2.8: Results from experimental tests of the control algorithm.

It is clear that the excitation force contains higher harmonics affecting thequality of the signal when only a sinusoidal signal is sent to the shaker (0iterations). After a few iterations a multi-harmonic input signal is obtainedwhich noticeably reduces the distortion in the force.


The overarching goal of parameter estimation is to find a mathematical modelthat describes the behavior of the observed system. The most popular methodused in linear theory is modal analysis. However, the methods developed forlinear systems break down if applied to nonlinear systems. Therefore, othermethods have to be used if the studied system is nonlinear. For a summaryof the most common nonlinear parameter estimation methods available todaythe reader is referred to the following survey articles [6, 7]. The differentapproaches used in this thesis can be divided into two groups dependent onthe input signal.

2.4.1 Using Random Noise

An important method for nonlinear parameter estimation based on randomnoise was initially developed by Julius S. Bendat [15]. This method is known as

13


Reverse-Path (RP) and treats the nonlinearities as extra feedback forces act-ing on an underlying linear system. The parameter estimation is performed inthe frequency domain, using least-square estimates, by conventional Multiple-Input-Single-Output (MISO) or Multiple-Input-Multiple-Output (MIMO) tech-niques known from linear analysis. A second method treated in this thesisbased on the same principles as RP is Nonlinear Identification through Feed-back of Outputs (NIFO) [23]. The basic principles of these methods are ex-plained in the following example. Consider Duffing’s equation:

mx + cx + kx + px3 = f (2.3)

Eq (2.3) can be expressed in the frequency domain as:

BX + pF (x3) = F (2.4)

where B is the impedance of the underlying linear system and F (.) is theFourier transform. If the nonlinearity is treated as an extra force acting onthe underlying linear system, this problem can be solved according to eitherRP or NIFO by conventional MISO techniques.

( )3x ( )3x

B

p

1−B

1−⋅Bp

X F XF

Figure 2.9: Solving Duffing’s equation using MISO techniques according toRP (left) and NIFO (right).

As seen in figure 2.9, using RP, eq 2.4 is formulated into a MISO system withdisplacement and nonlinear feedback as inputs and force as output. UsingNIFO, the inputs to the MISO model are force and nonlinear feedback, theoutput is displacement. An advantage with RP and NIFO is that both an esti-mate of the underlying linear system and the nonlinear coefficient are obtainedin one single analysis. This makes these methods very fast and attractive touse. However, as far as the author of this thesis knows, these methods are notyet applicable to systems with hysteresis.

A very useful tool available using MISO and MIMO analysis is the multiplecoherence function. This function indicates how much of an output that canbe explained by all the inputs, in a linear fashion. Thus the multiple co-herence function is an effective measurement of how well a nonlinear modelcorresponds to a measurement. An indication of the goodness-of-fit is neededin real applications. For example, the nonlinear functional form of the testedstructure or the exact location of the nonlinearity might not be fully known

14


beforehand. In that case an iterative approach is required to obtain the bestpossible fit to the available data [3, 4, 28]. The multiple coherence can becalculated as:

γ2m =

{GXF }[GFF ]−1{GXF}H

GXX(2.5)

{GXF } is the cross spectral density between the output and input vector and[GFF ] is the auto spectral density matrix of the input vector described in figure2.9. GXX is the auto spectral density of the output and a scalar since onlyone output is selected at a time. Results from a nonlinear analysis using RP,from Paper C, are shown in figures 2.10 and 2.11. The nonlinear response dataused in the analysis comes from a simulation of a cantilever beam with twononlinear elements. Raw-FRF and Raw-Coherence are the estimates obtainedby treating the system as linear.

10 20 30 40 50 60 70 80

10−5

10−4

10−3

10−2

Frequency [Hz]

Rec

epta

nce

[m/N

]

Raw−FRFEstimated Linear FRFTrue Linear FRF

100 200 300 400 500 600 700 800

107

108

109

Frequency [Hz]

Rea

l Par

t of N

onlin

ear

Coe

ffici

ents

P1

P2

Figure 2.10: FRFs and nonlinear coefficients estimated from nonlinear analysisusing RP.

100 200 300 400 500 600 7000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency [Hz]

Coh

eren

ce

Raw−CoherenceMultiple Coherence

Figure 2.11: Coherence from linear and nonlinear analysis using RP.

15


If the system is treated as linear, both the estimation of the FRF and thecoherence function between the input and output will be very bad. After thenonlinear analysis, a good estimate of the underlying linear system as wellas the nonlinear coefficients are obtained. The multiple coherence is close tounity for all frequencies indicating a very good fit of the nonlinear model.

2.4.2 Using Harmonic Input

An equivalent method to RP and NIFO does not exist for harmonic excita-tion, that is a method which estimates both the nonlinear coefficients andthe underlying linear system in one iteration from a single measurement. Asexplained previously, harmonic excitation is suitable when dealing with hys-teretic systems. In this thesis two parameter estimation methods based onharmonic input are suggested. Both these methods are applied to hystereticnonlinearities.

The first method is performed in the frequency domain utilizing the fact that itis possible to obtain well defined FRFs showing the true behavior of the studiedsystem when using harmonic input. It is possible to calculate an analyticalnonlinear FRF by the methods of harmonic-balance (HB) or multi-harmonic-balance (MHB) [2, 19, 21, 24]. HB and MHB are the analytical analogue ofstepped-sine testing [13]. HB uses the fact that a nonlinear system excitedby a sinusoidal signal will respond with the excitation frequency as well ashigher harmonics multiples of the excitation frequency. This means that boththe response of the system as well as the nonlinear force can be describedby Fourier series. As an example, Duffing’s oscillator from eq. (2.3) can, ifexcited by a sinusoidal signal, be expressed by a one term Fourier expansion:

BX1 − F0 − Fpx3 = 0 (2.6)

where B is the impedance of the underlying linear system, X1 refers to theFourier coefficient of the displacement at the excitation frequency, F0 is theFourier coefficient of the excitation force and Fpx3 is the Fourier coefficient ofthe nonlinear feedback force (dependent on the total displacement x). Thisnonlinear equation needs to be solved at every frequency step and the solutionis the frequency response of the system. To extend HB to MHB, additional har-monics need to be considered [2, 13, 24] which means that additional equationsare added and instead a nonlinear system of equations need to be solved atevery frequency step. These equations are usually solved by a frequency/timedomain alteration method. In this thesis the Newton-Raphson or Broyden’smethods are used to solve the equations. Examples of methods used byother authors are: the incremental harmonic balance method (IHB), [29], the

16


Galerkin/Newton-Raphson (GNR) method [30] and the Galerkin/Levenberg-Marquardt (GLM) method [31]. The computational cost of MHB is generallylarger than HB, but provides a more accurate estimate of the frequency re-sponse. Once the analytical nonlinear FRF is calculated it can be used inconjunction with a measured nonlinear FRF to estimate the nonlinear coeffi-cients [20, 21].

In the second method, proposed in paper D, the parameter estimation is per-formed in the time domain. Since the signal obtained from harmonic excitationtests contains a small number of frequency components, analytical signals ofboth the excitation force and response can be estimated. These analytical ex-pressions provide a straightforward and accurate integration into both velocityand displacement. In the proposed method these analytical signals are usedto estimate the nonlinear feedback force. This force is then used as referenceand the nonlinear feedback from the simulation is fitted by a genetic algorithm.

Both methods used in this thesis are similar in the sense that the output gen-erated by the nonlinear model is compared to a reference measurement. Thenonlinear parameters are then changed and a new model output is calculated,the optimal set of parameters is obtained by nonlinear optimization.

17

Chapter 3

Summary of Papers

3.1 Paper A

This paper presents an algorithm for reducing harmonic distortion in the forcesignal during experimental tests on nonlinear systems with sinusoidal excita-tion. The distortions are attenuated by sending a multi-harmonic voltagesignal with the correct amplitudes and phases to the shaker. Since the rela-tion between the input voltage signal and the measured force is nonlinear, aniterative approach is required to find the correct set of harmonic componentsof the input voltage signal. The control algorithm uses the Newton-Raphsonand Broyden method as nonlinear solvers. Simulations show that the con-trol algorithm is capable of obtaining a non-distorted force signal even in thepresence of a significant nonlinearity.

3.2 Paper B

Paper B presents two parameter estimation methods; one based on randomnoise input and another based on sinusoidal excitation. The method basedon random excitation treats the nonlinearities as force feedback terms actingon an underlying linear system. The parameter estimation is performed inthe frequency domain by using conventional MIMO/MISO techniques knownfrom linear theory. The paper also shows that the result from parameterestimation using partially correlated inputs gives the same result as whenusing uncorrelated inputs. The benefit of using partially correlated inputsis that the problem formulation becomes much simpler. The method basedon sinusoidal excitation uses a combination of multi-harmonic-balance (MHB)and stepped-sine excitation. The parameter estimation is performed in thefrequency domain by matching the measured and simulated frequency responsefunctions with each other. The paper shows that parameter estimation using

19

Chapter 3. Summary of Papers

random noise signals is efficient and can easily be extended to multi-degree-of-freedom systems with several nonlinearities. The benefit of the MHB/stepped-sine method is its versatility. It can handle both zero-memory nonlinearitiesand nonlinearities with memory, such as systems with hysteresis.

3.3 Paper C

This paper compares two methods for parameter estimation using randomnoise excitation, Nonlinear Identification through Feedback of Outputs (NIFO)and Reverse-Path (RP). Also, the RP method is extended to general nonlin-ear systems with an arbitrary number of nonlinearities. Comparisons of theperformance between these two methods is made both in simulations and onan experimental test rig. The conclusion drawn is that both methods performwell in simulations, but in the experimental case RP performs much betterthan NIFO. The underlying linear system and the nonlinear coefficient ob-tained from the analysis by RP was used to create a model of the test rig.Simulations showed that the model was able to predict the response of thetest rig within a small error margin.

3.4 Paper D

Paper D presents a simulation and parameter estimation routine for a hys-teretic nonlinear passive damper. A new forced response routine for an arbi-trary multi-degree-of-freedom system with one Bouc-Wen nonlinearity is de-veloped. This routine is then used as basis in a two-stage parameter estimationprocess performed in time domain. The new method proved very successfulon simulated data, even when contaminated by noise. It also worked wellwith experimental data, although the Bouc-Wen model could not completelydescribe the observed hysteresis behavior seen in the test rig.

20

Chapter 4

Conclusions and FutureResearch

The present thesis focuses on development of methods for simulation and char-acterization of nonlinear mechanical structures.

Two methods for nonlinear parameter estimation using random excitation,Reverse-Path (RP) and Nonlinear Identification through Feedback of Outputs(NIFO) are applied to structures with zero-memory nonlinearities with goodresults. Both methods perform equally well in simulations but RP is preferredin experimental tests due to its ability to get good estimates of the underlyinglinear systems. The results indicate that the routines developed for nonlin-ear identification using random excitation are efficient and have the potentialto be applied on complex nonlinear structures without further modifications.Another benefit with random excitation is that both the nonlinear coefficientand underlying linear system can be obtained by a single analysis. Therefore,this type of nonlinear analysis of measurement data can be useful even if thepurpose is not to build a nonlinear model but, for example, to obtain betterestimates of the linear frequency response functions for correlation with finiteelement models. However, as far as the author of this thesis knows, there isstill no way to apply these methods to systems with hysteresis.

An important part of this thesis is parameter estimation and simulation of sys-tems with hysteretic behavior. The methods proposed are based on harmonicexcitation. A routine for pure sinusoidal excitation is developed. This routinehas proven successful in both simulations and experimental tests. Also, twodifferent parameter estimation procedures based in harmonic input are sug-gested, both of them have been applied to systems with hysteresis with promis-ing results. This indicates that the developed methods concerning harmonic

21

Chapter 4. Conclusions and Future Research

excitation are useful in practice and have potential to be further developedand extended to deal with more complex nonlinear structures than treated inthis thesis.

Future research includes application of the methods developed in this thesison machine tools and tools for metal cutting. Interesting topics within thisfield are simulation and characterization of different types of nonlinear pas-sive dampers for vibration reduction during cutting. Simulation of nonlinearphenomenon during turning and milling are also possible application areas forthe methods used and developed in this thesis.

Another important question that needs attention is to try to find a way ofapplying the nonlinear parameter estimation methods based on random ex-citation, used in this thesis, on systems with hysteresis. In most cases thehysteretic nonlinearities affect the system response, even when excited at verylow force levels. This means that it is difficult to obtain good estimates of theunderlying linear systems. Therefore, the methods based on random excita-tion are not only beneficial due to their speed but also due to their ability toobtain estimates of the underlying linear systems.

22

References

[1] D Smallwood, D Gregory, and R Coleman. Damping investigations of asimplified frictional shear joint. In Proceedings of Shock and VibrationSymposium, 2000.

[2] Janito V. Ferreira. Dynamic Response Analysis of structures with non-linear components. PhD thesis, Imperial College, London, 1998.

[3] A. Josefsson, M. Magnevall, and K. Ahlin. On nonlinear parameter esti-mation with random noise signals. In Proceedings of IMAC XXV, 2007.

[4] M. Magnevall, A. Josefsson, and K. Ahlin. On parameter estimation andsimulation of zero memory nonlinear systems. In Proceedings of IMACXXVI, 2008.

[5] C.M. Richards and R. Singh. Experimental characterization of nonlinearrubber mounts. Proceedings of the International Modal Analysis Confer-ence - IMAC, 2:1678 – 1684, 2000.

[6] Gaetan Kerschen, Keith Worden, Alexander F. Vakakis, and Jean-ClaudeGolinval. Past, present and future of nonlinear system identificationin structural dynamics. Mechanical Systems and Signal Processing,20(3):505 – 592, 2006.

[7] Douglas E. Adams and Randall J. Allemang. Survey of nonlinear detec-tion and identification techniques for experimental vibrations. In Pro-ceedings of the 23rd International Conference on Noise and VibrationEngineering, ISMA, pages 517 – 529, Leuven, Belgium, 1998.

[8] Nuno Manuel Mendes Maia and Julio Martins Montalvao e Silva. Theo-retical and experimental modal analysis. Research Studies Press, Taunton,1997.

[9] David John Ewins. Modal testing : theory, practice and application. Re-search Studies Press, Baldock, 2. ed. edition, 2000.

23

REFERENCES

[10] Randall J. Allemang. Analytical and experimental modal analysis. Struc-tural Dynamics Research Laboratory, University of Cincinnati, Cincin-nati, Ohio 45221-0072, 1994. UC-SDRL-CN-20-263-662.

[11] K. Ahlin, M. Magnevall, and A. Josefsson. Simulation of forced responsein linear and nonlinear mechanical systems using digital filters. In Pro-ceedings of ISMA 2006, 2006.

[12] Y. Challeecharan and K. Ahlin. Fast simulation of nonlinear mechanicalsystems. In Proceedings of IMAC XXII, 2004.

[13] K. Worden and G. R. Tomlinson. Nonlinearity in Structural Dynamics:Detection, Identification and Modelling. IOP Publishing Ltd, Bristol, UK,2001.

[14] Lawrence N. Virgin. Introduction to experimental nonlinear dynamics :a case study in mechanical vibration. Cambridge University Press, Cam-bridge, 2000.

[15] Julius S. Bendat. Nonlinear system analysis and identification from ran-dom data. Wiley, New York, 1990.

[16] A. Al Majid and R. Dufour. Harmonic response of a structure mountedon an isolator modelled with a hysteretic operator: Experiments andprediction. Journal of Sound and Vibration, 277(1-2):391 – 403, 2004.

[17] Y.Q. Ni, J.M. Ko, and C.W. Wong. Identification of non-linear hystereticisolators from periodic vibration tests. Journal of Sound and Vibration,217(4):737 – 756, 1998.

[18] K. H. Hornig. Parameters characterization of the bouc/wen mechanicalhysteresis model for sandwich composite materials by using real coded ge-netic algorithms. Technical report, Mechanical Engineering Department,Ross Hall, Auburn, Alabama, 2000.

[19] Giovanna Girini and Stefano Zucca. Multi-harmonic analysis of a sdoffriction damped system. In Proceedings of 3rd Youth Symposium on Ex-perimental Solid Mechanics, Poretta Terme, Italy, 2004.

[20] M. Magnevall, A. Josefsson, and K. Ahlin. Parameter estimation of hys-teresis elements using harmonic input. In Proceedings of IMAC XXV,2007.

[21] M. Magnevall, A. Josefsson, and K. Ahlin. On nonlinear parameter esti-mation. In Proceedings of ISMA 2006, 2006.

24

REFERENCES

[22] Julius S. Bendat and Allan G. Piersol. Random data : analysis andmeasurement procedures. Wiley, New York, 2., rev. and expanded ed.edition, 1986.

[23] D.E. Adams and R.J. Allemang. Frequency domain method for esti-mating the parameters of a non-linear structural dynamic model throughfeedback. Mechanical Systems & Signal Processing, 14(4):637 – 656, 2000.

[24] Hugo Ramon Elizalde Siller. Non-linear modal analysis methods for en-gineering structures. PhD thesis, Imperial College, London, 2004.

[25] S Rossmann. Development of force controlled modal testing on rotor sup-ported by magnetic bearings. Master’s thesis, Imperial College, London,1999.

[26] I. Bucher. Exact adjustment of dynamic forces in presence of non-linearfeedback and singularity - theory and algorithm. Journal of Sound andVibration, 218(1):1 – 27, 1998.

[27] M. Magnevall, A. Josefsson, and K. Ahlin. Experimental verification of acontrol algorithm for nonlinear systems. In Proceedings of IMAC XXIV,2006.

[28] A. Josefsson, M. Magnevall, and K. Ahlin. Estimating the location ofstructural nonlinearities from random data. In Proceedings of IMACXXVI, 2008.

[29] S. L. Lau, Y. K. Cheung, and S. Y. Wu. Incremental harmonic balancemethod with multiple time scales for aperiodic vibration of nonlinearsystems. Journal of Applied Mechanics, Transactions ASME, 50(4a):871– 876, 1983.

[30] Hideyuki Tamura, Yoshihiro Tsuda, and Atsuo Sueoka. Higher approx-imation of steady oscillations in nonlinear systems with single degreeof freedom - suggested multi-harmonic balance method. Bulletin of theJSME, 24(195):1616 – 1625, 1981.

[31] C.W. Wong, Y.Q. Ni, and J.M. Ko. Steady-state oscillation of hystereticdifferential model. ii: Performance analysis. Journal of Engineering Me-chanics, 120(11):2299 – 2324, 1994.

25

Paper A

Control Algorithm For SineExcitation On Nonlinear

Systems

27

Paper A is published as:

Josefsson, A., Magnevall, M. and Ahlin,K. Control Algorithm for SineExcitation on Nonlinear Systems, IMAC-XXIV Conference & Exposi-tion on Structural Dynamics, St. Louis, Missouri, 2006

28

AfFM

FV

Jknrn

tvn

Vn

Voltage Signal, V(t) Force Signal, F(t)

Black Box (Non-linear

System)

k

F (t) =k∑

n=1

(r2n−1 sin(2πnft) + r2n cos(2πnft)

)f

⎡⎢⎢⎢⎣

sin(2πft0) cos(2πft0) · · · cos(2πkft0)sin(2πft1) cos(2πft1) · · · cos(2πkft1)

sin(2πftT ) cos(2πftT ) · · · cos(2πkftT )

⎤⎥⎥⎥⎦⎧⎪⎪⎪⎨⎪⎪⎪⎩

r1

r2

r2k

⎫⎪⎪⎪⎬⎪⎪⎪⎭ =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

F (t0)F (t2)

F (tT )

⎫⎪⎪⎪⎬⎪⎪⎪⎭{

r1 r2 r3 . . . r2k

}t0 tT

Z0 =√

r21 + r2

2

ψ0 = tan−1(r2/r1)

{FM

}=

{Z0, ψ0, r3, r4, . . . , r2k

}

{FD

}=

{Zd, ψd, 0, 0, 0 . . .

}{FM

} {FD

}

V (t) =k∑

n=1

(v2n−1 sin(2πnft) + v2n cos(2πnft)

){VN

}=

{v1, v2, v3, . . . , v2k

}{VN

}k

2k

{VN

} {FM

}

n x n ⎧⎪⎪⎪⎨⎪⎪⎪⎩

g1(x1, x2, . . . , xN )g2(x1, x2, . . . , xN )

gN(x1, x2, . . . , xN )

⎫⎪⎪⎪⎬⎪⎪⎪⎭ =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

00

0

⎫⎪⎪⎪⎬⎪⎪⎪⎭

{g1, g2, . . . , gN

}

{FV

}=

{FM

}− {FD

}

{FV (v1, v2, . . . , vN )

}=

{0}

{VN

}n {VN

}n+1

{FV

}n+1<

{FV

}n

{FV

}{VN

}⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

∆F 1V

∆F 2V

∆F 3V

∆F 4V

∆F 2kV

⎫⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎭

=

⎡⎢⎢⎢⎢⎢⎢⎣

j11 j12 j13 j14 · j1(2k)

j21 j22 j23 j24 · j2(2k)

j31 j32 j33 j34 · j3(2k)

j41 j42 j43 j44 · j4(2k)

· · · · · j5(2k)

j(2k)1 j(2k)2 j(2k)3 j(2k)4 · j6(2k)

⎤⎥⎥⎥⎥⎥⎥⎦

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

∆v1

∆v2

∆v3

∆v4

·∆v2k

⎫⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎭

∆{VN

}

jnm

[Jnm

]=

∂FnV

∂vm

[Jnm

]=

12h

(∆Fn

V (vm + h) − ∆FnV (vm − h)

)h

h m

m [J ]k

2k 4k

{VN

}n+1 ={VN

}n − [J]−1 {

FV

}{VN

} {VN

}n+1

Fcrit =2k∑

n=1

{FV

}n

Fcrit

{VN

}

k 4k

g′(x) =g(xn) − g(xn−1)

xn − xn−1

g′(x)(xn − xn−1) ≈ g(xn) − g(xn−1)

J({

VN

}n)({VN

}n − {VN

}n−1) ≈ FV

({VN

}n)− FV

({VN

}n−1)

A({

VN

}n )≈ J({

VN

}n)[A]

{yk

}=

{FV

}n − {FV

}n−1

{sk

}=

{VN

}n − {VN

}n−1

An = An−1 +yk − An−1sk

‖ sk ‖22

(sk)T

‖ . ‖2

{VN

}1

{VI

}n+1 ={VI

}n − [An

]−1 {FV

}

{VI

}n+1 ={VI

}n − u[An

]−1 {FV

}u

GV =12FT

V FV

{0.1, . . . , 1

}

{VN

}

m c k P

m =0.024 kg c = 1.5 Ns/m k = 1.6 · 106 N/m P = 1.6 · 108 N/m3

11.93 11.94 11.95 11.96 11.97 11.98 11.99 12

-8

-6

-4

-2

0

2

4

6

8

Time [Sec]

Res

pons

e

Reference Sine

Output From Black BoxAfter 1 iteration

After 2 iterations

11.91 11.92 11.93 11.94 11.95 11.96 11.97 11.98 11.99 12-40

-30

-20

-10

0

10

20

30

40

Time [Sec]

Inpu

t S

igna

l

Initial Input Signal (Sine)

Input Signal After 3 iterations

11.96 11.965 11.97 11.975 11.98 11.985 11.99 11.995 12

-8

-6

-4

-2

0

2

4

6

8

Time [Sec]

Res

pons

e

Reference Sine

Output From Black Box

After 1 iterationAfter 2 iterations

After 3 iterations

Paper B

On Nonlinear ParameterEstimation

41

Paper B is published as:

Magnevall, M., Josefsson, A. and Ahlin, K. On Nonlinear Parameter Es-timation, ISMA 2006, Leuven, Belgium, 2006

42

H

H{cx

}= cH

{x}

H{x1 + x2

}= H

{x1

}+ H

{x2

}

27.5 28 28.5 29 29.5 30 30.510

−7

10−6

10−5

10−4

Frequency [Hz]

Rec

epta

nce

[m/N

]

Hardening Spring

F = 0.01NF = 0.2NF = 0.7N

28.5 29 29.5 30 30.5 31 31.5 32 32.510

−7

10−6

10−5

10−4

Softening Spring

Frequency [Hz]

Rec

epta

nce

[m/N

]

F = 0.05NF = 0.2NF = 1.0N

27.5 28 28.5 29 29.5 30 30.510

−7

10−6

10−5

10−4

Frequency [Hz]

Rec

epta

nce

[m/N

]

Nonlinear Damping

F = 0.01NF = 0.2NF = 1.0N

10 20 30 40 50 6010

−8

10−7

10−6

Hysteresis Damping

Frequency [Hz]

Rec

epta

nce

[m/N

]

F = 0.01NF = 1NF = 10N

H22

H11

10 20 30 40 50 60 70

10-5

Frequency [Hz]

Rec

epta

nce

[m/N

]

10 20 30 40 50 60 70

10-5

Frequency [Hz]

Rec

epta

nce

[m/N

]

Low Force Amplitude

High Force Amplitude

Low Force Amplitude

High Force Amplitude

H11 H22

H22

N M

{ }F { }X

= XF−1FF

XF (N x N) FF

(M x N)

p q

γ2pq =

| XpFq |2FqFq XpXp

γ2m = XF

−1FF

HXF

XX

XF (1 x N) XX

M

mx + cx + kx + px3 = f

BX + pF (x3) = F

B F (.)p

H = B−1

( )3x ( )3x

( )3x

{x}

+{x}

+{x}

={f}− N

[{x}

,{x}]

N x x

N[{

x}

,{x}]

=NEL∑m=1

pm

{wm

}gm

({x}

,{x})

pm{wm

}gm

gm =({

wm

}T {x})3 (Cubic hardening spring)

gm =({

wm

}T {x})2 (Quadratic damping)

{X

}=

{F}−

NEL∑i=1

pm

{wm

}F

(gm(

{x}

,{x}))

{R}

{R}

=

⎧⎪⎪⎪⎨⎪⎪⎪⎩

{F}

r1

rm

⎫⎪⎪⎪⎬⎪⎪⎪⎭

rm

rm = −F(gm(

{x}

,{x}))

H1

M = XR−1RR

{ }XINPUT OUTPUT

( ) 1×+ ELNN 1×M( ) MNNsize EL ×+:

MH{ }R

M{HL

}

pi =1{

wi

}T {HL

} (r,i+1)M i ∈ {

1, 2, . . . , NEL

}(r,i+1)M M r i+1

RR

{U}

= Φ{R}

[GUU

]= Φ

{R} (

Φ{R})H = Φ RR ΦH

Φ UU

Φ

XU = XR ΦH

UU XU

N + NEL

32 xP ⋅

vvP ⋅⋅1

P1 = 2e5 P2 = 6e13

10 15 20 25 30 35 40

10-6

10-5

Frequency [Hz]

Rec

epta

nce

[m/N

]

0.5 N rms

3.7 N rms

9.8 N rms

20 N rms

37 N rms

62 N rms

{R}

={F −x2 −x3 −x sgn(x) −v|v| −v3 −v5

}T

w1,2 ={1}

r1 = −F((x1)3

)r2 = −F

((v1|v1|)3

)

+/−

10 15 20 25 30 35 40

10-6

10-5

Frequency [Hz]

Rec

epta

nce

[m/N

]

True Linear FRF

Estimated Linear FRFRaw FRF

10 15 20 25 30 35 4010

2

104

106

108

1010

1012

1014

Frequency [Hz]

Non

linea

r C

oeff

icie

nts

Cubic Spring Element

Quadratic Damping Element

( )3311 xxP −⋅

( ) ( )31312 xxxxP −⋅−⋅

P1 = 4e16 P2 = 6e10

w1 ={0 −1 1

}T, w2 =

{−1 0 1}T

r1 = −F(({

w1

}T {x} )3

)r2 = −F

({w2

}T {x} |{w2

}T {x} |

)

15 20 25 30 3510

-9

10-8

10-7

10-6

10-5

10-4

Frequency [Hz]

Rec

epta

nce

[m/N

]

Estimated H11Estimated H21

Estimated H31

Theoretical H11

Theoretical H21Theoretical H31

15 20 25 30 3510

5

1010

1015

Frequency [Hz]

Non

linea

r C

oeff

icie

nts

P1

P2

15 20 25 30 350

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency [Hz]

Mul

tiple

Coh

eren

ce

X1, Nonlinear MIMO-Model

X1, Linear Model

15 20 25 30 350

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency [Hz]

Mul

tiple

Coh

eren

ce


X2, Linear Model

15 20 25 30 350

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency [Hz]

Mul

tiple

Coh

eren

ce


X3, Linear Model

28.5 29 29.5 30

10-6

10-5

Frequency [Hz]

Rec

epta

nce

[m/N

]

Forward Sweep

Backward Sweep

Harmonic BalanceMulti-valued

region

mx + cx + kx = F − Fnlf

Fnlf

F (t) = F0 ejωt

x(t) =∞∑

k=1

xk ejkωt

x(t) =∞∑

k=1

jkωxk ejkωt

x(t) = −∞∑

k=1

k2ω2xk ejkωt

Fnlf (x) =∞∑

k=1

Fnlk ejkωt

[1 3 5]

−mx1ω2 ejωt +jcx1ω

2 ejωt +kx1 ejωt −F0 ejωt +Fnl1 ejωt = 0

−9mx3ω2 ej3ωt +3jcx3ω

2 ej3ωt +kx3 ej3ωt +Fnl3 ej3ωt = 0

−25mx5ω2 ej5ωt +5jcx5ω

2 ej5ωt +kx5 ej5ωt +Fnl5 ej5ωt = 0

(−mω2 + jcω + k)x1 − F0 + Fnl1 = 0(−9mω2 + 3jcω + k

)x3 + Fnl3 = 0(−25mω2 + 5jcω + k)x5 + Fnl5 = 0

Z1x1 − F0 + Fnl1 = 0Z3x3 + Fnl3 = 0Z5x5 + Fnl5 = 0

ε = ‖Hmeasurednonlinear − Hanalyticnonlinear‖2

kd

Fd

0 10 20 30 40 50 6010

−8

10−7

10−6

Frequency [Hz]

Rec

epta

nce

[m/N

]

F = 0.01*Fd

F = 0.05*Fd

F = 0.1*Fd

F = 0.2*Fd

F = 0.5*Fd

F = 1*Fd

F = 2*Fd

F = 5*Fd

F = 12*Fd

0.01*Fd (Pure Stick)

1*Fd (Stick−Slip)

12*Fd (Pure Slip)

F =xkd(

1 +(

xkd

Fd

)N)1/N

kd

Fd

N

N

[1 3 5 7]

Fd kd N246.048 N 2.7e7 N/m242.05 N 2.33e7 N/m 22.55

kd Fd N

−5 −4 −3 −2 −1 0 1 2 3 4 5

x 10−4

−250

−200

−150

−100

−50

0

50

100

150

200

250

Displacement [m]

For

ce [N

]

ReferenceEstimated

10 20 30 40 50 6010

−8

10−7

10−6

Frequency [Hz]

Rec

epta

nce

[m/N

]

ReferenceEstimated

Paper C

On Parameter Estimation andSimulation of Zero Memory

Nonlinear Systems

61

Paper C is published as:

Magnevall, M., Josefsson, A. and Ahlin, K. On Parameter Estimationand Simulation of Zero Memory Nonlinear Systems, IMAC-XXVI Con-ference & Exposition on Structural Dynamics, Orlando, Florida, 2008

62

Hg(x, x)P

Rλ

A, BF

¨ + ˙ + + ( ˙ ) =

nl˙

=NEL∑

m

Pm m gm( ˙ )

Pm gm m

Pm gm m

gm = Tm | T

m | Square hardening spring

gm = ( Tm ˙ )2 Quadratic damping

=

(−

NEL∑m

Pm m F(gm( ˙ )

))

H1

N

i = i F i i i =[1 2 . . . N

]i

i =[Xi r1 r2 . . . rNEL

]T

rm gm

rm = F(gm( ˙ )

)i i

i i+1 Xi

i

i Pm

Pm =1InpInp

N∑i=1

m(i)1iInp

m(i) i

m Inp iInp

iInp =

1iInp

Pm

x(t) =∫ t

0

h(t − τ)f(τ)dτ

Rr λr

H(s) =N∑

i=1

Rr

s − λr+

R∗r

s − λ∗r

s N

r

HD

HD(s) =Rr

s − λr

HD(s)

hd(t) = Rr eλrt

xr(nT +T ) Tx

f

xr(nT + T ) =∫ nT+T

0

Rr eλr(nT+T−τ) f(τ)dτ

= Rr eλrT xr(nT ) + Rr eλrT

∫ T

0

e−λru f(u + nT )du

xr(nT +T )xr(nT ) [f(nT ) f(nT +T )]

f(t)

f(nT )

xr(nT + T ) = Rr eλrT xr(nT ) + f(nT )Rr

λr

(eλrT −1

)

HD(z) =z−1 Rr

λr

(eλrT −1

)1 − z−1Rr eλrT

NH

H(z) =N∑

r=1

[z−1 Rr

λr

(eλrT −1

)1 − z−1Rr eλrT

+z−1 R∗

r

λ∗r

(eλ∗

rT −1)

1 − z−1 eλ∗rT

]

A B

N D

Nr =[0 Rr

λr

(eλrT −1

)]Dr =

[1 − Rr eλrT

]Br = 2 Re(Nr) ∗ Re(Dr) + 2 Im(Nr) ∗ Im(Dr)

Ar = Re(Dr ∗ D∗r)

x(n) =N∑

r=1

B0rf(n) + B1

rf(n − 1) + . . . + Bmr f(n − m)

−A1rxr(n − 1) − . . . − Ap

rxr(n − p)

x(n)

Z(z)X(z) + NL(z) = F (z)

Z(z) NL(z)

X(z) = H(z)(F (z) − NL(z)

)

x(n) =N∑

r=1

B0r

(f(n) − nl(n)

)+ B1

r

(f(n − 1) − nl(n − 1)

)+ . . .

+Bmr

(f(n − m) − nl(n − m)

)− A1rxr(n − 1) − . . . − Ap

rxr(n − p)

x(n)f(n) − nl(n) nl(n) x(n) f(n)

( )32 cb xxP −3

1 axP

H

1 2

[1 0 0 0

][0 1 −1 0

]⎡⎢⎢⎣

Haa Hab Hac Had

Hba Hbb Hbc Hbd

Hca Hcb Hcc Hcd

Hda Hdb Hdc Hdd

⎤⎥⎥⎦⎡⎢⎢⎣

−P1F (x3a)

−P2F((xb − xc)3

)P2F

((xb − xc)3

)F

⎤⎥⎥⎦ =

⎡⎢⎢⎣Xa

Xb

Xc

Xd

⎤⎥⎥⎦

Xd

FHdd − P1HdaF (x3a) − P2(Hdb − Hdc)F

((xb − xc)3

)= Xd

F

[1

HddP1

Hda

HddP2

(Hdb−Hdc)Hdd

]︸︷︷︸

d= d F−1

d d

⎡⎣ Xd

F (x3a)

F((xb − xc)3

)⎤⎦

︸︷︷︸d

= F

Bd

Hdd

Bc Bb Ba

221

217

Fs

P1

P2

100 200 300 400 500 600 70010

−10

10−8

10−6

10−4

10−2

Frequency [Hz]

Rec

epta

nce

[m/N

]


10 20 30 40 50 60 70 80

10−5

10−4

10−3

10−2

Frequency [Hz]

Rec

epta

nce

[m/N

]


100 200 300 400 500 600 7000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency [Hz]

Coh

eren

ce

Raw−CoherenceMultiple Coherence

100 200 300 400 500 600 700 800

107

108

109

Frequency [Hz]

Rea

l Par

t of N

onlin

ear

Coe

ffici

ents

P1

P2

P1 P2

P1 P2

14.3 N

|x2.5872| sgn(x)2604.2

215 50 120

20 30 40 50 60 70 80 90 100 110 120 1300

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency [Hz]

Coh

eren

ce

Reverse−Path

Raw CoherenceMultiple Coherence

20 30 40 50 60 70 80 90 100 110 120 1300

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency [Hz]

Coh

eren

ce

NIFO

Raw CoherenceMultiple Coherence

20 40 60 80 100 12010

−6

10−5

10−4

10−3

10−2

Frequency [Hz]

Rec

epta

nce

[m/N

]Raw−FRFReverse PathNIFO

15 20 25 30 35 40 45 50 55

10−4

10−3

Frequency [Hz]

Rec

epta

nce

[m/N

]

Raw−FRFReverse PathNIFO

[60 70]55 90

[15 55]

20 30 40 50 60 70 80 90 100 110 120 13010

7

108

109

Frequency [Hz]

Non

linea

r C

oeffi

cien

t

Reverse−Path

|x2.5827|*sgn(x)

20 30 40 50 60 70 80 90 100 110 120 13010

7

108

109

Frequency [Hz]

Non

linea

r C

oeffi

cien

t

NIFO

|x2.5827|*sgn(x)

−3 −2 −1 0 1 2 3

x 10−3

−100

−80

−60

−40

−20

0

20

40

60

80

100

Displacement [m]

For

ce [N

]

Estimated FunctionMeasured Values

−3 −2 −1 0 1 2 3

x 10−3

−40

−30

−20

−10

0

10

20

30

40

Displacement [m]

For

ce [N

]

Reverse PathStatic Measurement

1.2224 · 108 1.2376 · 108 1.2740 · 108

0 50 100 150 200 250 300 350−4

−3

−2

−1

0

1

2

3

4x 10−3

Time [s]

Dis

plac

emen

t [m

]

Node 2

Simulated ResponseMeasured Response

20 40 60 80 100 120

10−12

10−10

10−8

10−6

Frequency [Hz]

PS

D [m

2 /Hz]

Node 2

Simulated ResponseMeasured Response

Paper D

Simulation andCharacterization of a

Nonlinear Hysteretic Damper

79

Paper D is submitted for publication as:

Magnevall, M., Josefsson, A., Ahlin, K. and Broman, G. Simulation andCharacterization of a Nonlinear Hysteretic Damper, Submitted for pub-lication, September 2008

80

m c knlf(t)

f(t)

mx(t) + cx(t) + kx(t) + nlf(t)

= f(t)

nlf(t)

nlf(t) = knlαx(t) + (1 − α)knlz(t)

z(t) = x(t)[κ − ∣∣z(t)

∣∣n(γ + β sgn(x(t)) sgn(z(t)))]

zx

α0 < α < 1 knl

κ, β, n γ

β γ κ = n = 1

−0.05 0 0.05−0.05

0

0.05(a)

x [m]

FN

L [N

]

−0.05 0 0.05−0.04

−0.02

0

0.02

0.04(b)

x [m]

FN

L [N

]

−0.05 0 0.05−0.04

−0.02

0

0.02

0.04(c)

x [m]

FN

L [N

]

−0.05 0 0.05−0.04

−0.02

0

0.02

0.04(d)

x [m]

FN

L [N

]

−0.05 0 0.05−0.04

−0.02

0

0.02

0.04(e)

x [m]

FN

L [N

]γ β

m = 0.3 c = 0.01 k = 2knl = 1 α = 0.3 κ = n = 1 γ − β < γ + β γ + β = 0 γ − β < 0γ + β > γ − β > 0 γ + β > 0 γ − β = 0 γ + β > 0 γ − β < 0

m

mm2

[Haa Hab

Hba Hbb

] [F − NLF (Xa − Xb)

NLF (Xa − Xb)

]=

[Xa

Xb

]

F (f) NLF (f)

x(t) =∫ t

0

h(t − τ)f(τ)dτ

R λ

H(s) =N∑

r=1

rR

s − rλ+

rR∗

s − rλ∗

s N

HD(s) =R

s − λ

HD(s)

hd(t) = R eλt

x(nT + T )x f T

x(nT + T ) =∫ nT+T

0

R eλ(nT+T−τ) f(τ)dτ

= eλT x(nT ) + R eλT

∫ T

0

e−λu f(u + nT )du

x(nT + T )x(nT ) [nT, nT + T ]

f(nT )

x(nT + T ) = eλT x(nT ) + f(nT )R

λ

(eλT −1

)z

HD(z) =z−1 R

λ

(eλT −1

)1 − z−1 eλT

N

H(z) =N∑

r=1

[z−1 rR

rλ

(erλT −1

)1 − z−1 e rλT

+z−1 rR∗

rλ∗(erλ∗T −1

)1 − z−1 e rλ∗T

]

A Bz

z

rNrD r

rN =[0,

rRrλ

(erλT −1

)]rD =

[1, − e

rλT]

rB = 2 Re(rN) ∗ Re(rD) + 2 Im(rN) ∗ Im(rD)

rA = Re(rD ∗ rD∗)

xa xb

xa(n) =N∑

r=1

[−nlf(n)

(rB0

aa − rB0ab

)+

2∑m=0

f(n − m) rBmaa

−2∑

m=1

nlf(n− m)(

rBmaa − rBm

ab

)− 2∑m=1

rxma (n − m)

(rAm

aa + rAmab

)]

xb(n) =N∑

r=1

[−nlf(n)

(rB0

ba − rB0bb

)+

2∑m=0

f(n − m) rBmba

−2∑

m=1

nlf(n− m)(

rBmba − rBm

bb

)− 2∑m=1

rxmb (n − m)

(rAm

ba + rAmbb

)]

xa(n) xb(n)

∆x

∆x = xa − xb

Cx(n)nlf(n) Ex

∆x(n) + nlf(n)Ex = Cx(n)

∆x(n)

∆x(n)

∆x(n) + nlf(n)Ev = Cv(n)

∆x(n)Ev Cv(n) Ex

Cx(n)∆x(n) nlf(n)

z(n)

z(n) =Cx(n) − ∆x(n)(1 + αknlEx)

(1 − α)knlEx

∆x(n) z(n)z(n)

∆x(n) z(n) z(n)

z(n) = z(n − 1) e−fcutT +z(n)fcutT + e−fcutT −1

f2cutT

+z(n − 1)1 − e−fcutT (fcutT + 1)

f2cutT

fcut z(n)

z(n)(1 − α)knlEx + ∆x(n)(1 + αknlEx) − Cx(n) = 0

c = 1 k = 1000

κ = 10 knl = 5000 n = 2 α = 0.3 β = 4 · 105 γ = −5 · 105

−0.5 0 0.5−20

−10

0

10

20(b)

Displacement [mm]

Non

linea

r F

eedb

ack

[N]

0 500 1000 1500 2000 2500 3000 3500 4000

10−10

100

(c)

Frequency [Hz]

Am

plitu

de [m

]

0 5 10−4

−2

0

2

4x 10

−4 (a)

Time [s]

Dis

plac

emen

t [m

]

mxb = (1 − α)knlz + c∆x + knlα∆x + k∆x

n knl αz

κ β γ

z =[∆x −∆x|z|n sgn(∆x) sgn(z) −∆x|z|n]

⎡⎣κ

βγ

⎤⎦

zz

z z∆x

error =1K

K∑k=1

|fnlm(k) − fnls(k)|

fnlm fnls

n knl αz

z z

κ β γ

+/−

n 1.6 < n < 2.4knl 20 < knl < 5 · 104

α 0 < α < 1

nknl · 10−3

ακ

β · 10−5

γ · 10−5

nknl · 10−3

ακ

β · 10−5

γ · 10−5

z α knl

zz

0.5 0.51 0.52 0.53 0.54 0.55 0.56−20

−15

−10

−5

0

5

10

15

20

25

Time [s]

Non

linea

r F

eedb

ack

For

ce [N

]

ReferenceEstimated

(a)

0.5 0.51 0.52 0.53 0.54 0.55 0.56−20

−15

−10

−5

0

5

10

15

20

25

Time [s]

Non

linea

r F

eedb

ack

For

ce [N

]

ReferenceEstimated

(b)

0 5 10 15 20 25 305.8

6

6.2

6.4

6.6

6.8

7

7.2

7.4x 10

−3

Generations

Err

or

(a)

0 20 40 60 80 1001

2

3

4

5

6

7

8x 10

−3

Generations

Err

or

(b)

m = 0.0926k = 500

c = 0.1

−8 −6 −4 −2 0 2 4 6 8

x 10−5

−15

−10

−5

0

5

10

15

Relative displacement [m]

Non

linea

r F

eedb

ack

For

ce [N

]

80 Hz90 Hz100 Hz

κ = +/− +/−

4 4.01 4.02 4.03 4.04 4.05 4.06−20

−15

−10

−5

0

5

10

15

20

25

Time [s]

Non

linea

r F

eedb

ack

For

ce [N

]

ReferenceEstimated

(a)

2.5 2.51 2.52 2.53 2.54 2.55 2.56−20

−15

−10

−5

0

5

10

15

20

25

Time [s]

Non

linea

r F

eedb

ack

For

ce [N

]

ReferenceEstimated

(b)

κ

Blekinge Institute of TechnologyLicentiate Dissertation Series No. 2008:13

School of Engineering

Methods for siMulation and CharaCterization of nonlinear MeChaniCal struCtures

Martin Magnevall

Trial and error and the use of highly time-consu-ming methods are often necessary for modeling, simulating and characterizing nonlinear dynamical systems. However, for the rather common special case when a nonlinear system has linear relations between many of its degrees of freedom there are particularly interesting opportunities for more efficient approaches. The aim of this thesis is to develop and validate new efficient methods for the theoretical and experimental study of mecha-nical systems that include significant zero-memory or hysteretic nonlinearities related to only small parts of the whole system.

The basic idea is to take advantage of the fact that most of the system is linear and to use much of the linear theories behind forced response simula-tions. This is made possible by modeling the nonli-nearities as external forces acting on the underly-ing linear system. The result is very fast simulation routines where the model is based on the residues and poles of the underlying linear system. These residues and poles can be obtained analytically, from finite element models or from experimental

measurements, making these forced response rou-tines very versatile. Using this approach, a com-plete nonlinear model contains both linear and nonlinear parts. Thus, it is also important to have robust and accurate methods for estimating both the linear and nonlinear system parameters from experimental data.

The results of this work include robust and user-friendly routines based on sinusoidal and random noise excitation signals for characterization and description of nonlinearities from experimental measurements. These routines are used to create models of the studied systems. When combined with efficient simulation routines, complete tools are created which are both versatile and compu-tationally inexpensive.

The developed methods have been tested both by simulations and with experimental test rigs with promising results. This indicates that they are use-ful in practice and can provide a basis for future research and development of methods capable of handling more complex nonlinear systems.

aBstraCt

ISSN 1650-2140

ISBN 978-91-7295-156-32008:13

Me

th

od

s f

or

siM

ul

at

ion

an

d C

ha

ra

Ct

er

iza

tio

n

of

no

nl

ine

ar

Me

Ch

an

iCa

l s

tr

uC

tu

re

sM

artin Magnevall

2008:13

methods for simulation and hani characterization of nonlinear mechanical...

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