Control of a 3d flexible catilever beam

The primary focus is to develop a dynamic model of a flexible beam and reduce the residual vibration of the flexible beam produced by the hub’s motion.. The dynamic model is, however, a

ĐẠI HỌC QUỐC GIA TP.HCM

TRƯỜNG ĐẠI HỌC BÁCH KHOA

-

VŨ NGUYỄN TRÍ GIANG

ĐIỀU KHIỂN VỊ TRÍ VÀ DAO ĐỘNG CHO CƠ CẤU THANH

MỀM CÔNG XÔN TRONG KHÔNG GIAN CONTROL OF A 3D FLEXIBLE CANTILEVER BEAM

Chuyên ngành : KỸ THUẬT CƠ ĐIỆN TỬ

Mã số: 8520114

LUẬN VĂN THẠC SĨ

TP HỒ CHÍ MINH, tháng 09 năm 2020

Công trình được hoàn thành tại:Trường Đại học Bách Khoa – ĐHQG-HCM

Cán bộ hướng dẫn khoa học : PGS TS Nguyễn Quốc Chí

(Ghi rõ họ, tên, học hàm, học vị và chữ ký) Cán bộ chấm nhận xét 1: PGS TS Trương Đình Nhơn

(Ghi rõ họ, tên, học hàm, học vị và chữ ký) Cán bộ chấm nhận xét 2: PGS TS Lê Mỹ Hà

(Ghi rõ họ, tên, học hàm, học vị và chữ ký) Luận văn thạc sĩ được bảo vệ tại Trường Đại học Bách Khoa, ĐHQG Tp HCM

Ngày 03 tháng 09 năm 2020

Thành phần Hội đồng đánh giá luận văn thạc sĩ gồm:

(Ghi rõ họ, tên, học hàm, học vị của Hội đồng chấm bảo vệ luận văn thạc sĩ)

ĐẠI HỌC QUỐC GIA TP.HCM

NHIỆM VỤ LUẬN VĂN THẠC SĨ

I TÊN ĐỀ TÀI:

Điều khiển vị trí và dao động cho cơ cấu thanh mềm công xôn

trong không gian Control of a 3D flexible cantilever beam

II NHIỆM VỤ VÀ NỘI DUNG:

- Phân tích động lực học cho cơ cấu thanh dầm mềm có giảm chấn trong không gian một chiều (1D)

- Thiết kế bộ điều khiển khử dao dộng cho cơ cấu thanh dầm mềm 1D

- Kiểm chứng mô hình toán 1D và bộ điều khiển bằng mô phỏng và thực nghiệm

- Phân tích động lực học cho cơ cấu thanh dầm có giảm chấn trong không gian

ba chiều (3D)

- Kiểm chứng mô hình toán của thanh dầm 3D bằng mô phỏng

III NGÀY GIAO NHIỆM VỤ: 19/08/2019

IV NGÀY HOÀN THÀNH NHIỆM VỤ: 07/06/2020

Khoa Cơ Khí

Tp HCM, ngày tháng 09 năm 2020

TRƯỞNG KHOA

(Họ tên và chữ ký)

PGS TS Nguyễn Hữu Lộc

i

ACKNOWLEDGEMENT

First and foremost, I would like to praise and thank God, the almighty, who has granted countless blessings, knowledge, and opportunity to the writer, so that I have been finally able to accomplish the thesis

I would first like to thank my thesis advisor Assoc Prof Nguyen Quoc Chi of the Department of Mechatronics at Ho Chi Minh City University of Technology for the continuous support of my master study and research, for his patience, motivation, enthusiasm, and immense knowledge His guidance helped me in all the time of research and writing of this thesis

I thank my colleagues in Think Alpha Co., Ltd for the stimulating discussions, for theoretical and experimental, and for all the fun we have had

Finally, I must express my very profound gratitude to my parents and to my sister for providing me with unfailing support and continuous encouragement throughout my years of study and through the process of researching and writing this thesis This accomplishment would not have been possible without them Thank you

ii

ABSTRACT

A study of a flexible beam fixed on a translating hub is presented in this thesis The primary focus is to develop a dynamic model of a flexible beam and reduce the residual vibration of the flexible beam produced by the hub’s motion The dynamic model is, however, a discrete model using the Galerkin discretization method to predict the behaviors of the system in the experimental process A damping mechanism is introduced to apply to the undamped discrete model by the Galerkin method First, a 1D flexible beam will be investigated

The input shaping control method is used to suppress the vibration of the system where various input shapers including simple and advanced shapers are applied The simulation and experiment are conducted to show the qualitative agreement between the numerical and experimental results

From the understanding obtained from previous works, a 3D flexible beam – vibrations in 3D coordinates is modeled with decoupled relationships among these 3D vibrations is presented

TÓM TẮT

Luận văn này trình bày nghiên cứu về thanh dầm công xôn mềm được cố định trên bệ gá tịnh tiến Trọng tâm là phát triển mô hình động lực học và giảm độ rung của thanh dầm do chuyển động của bệ gá gây ra Mô hình động lực học rời rạc được xây dựng bằng phương pháp Galerkin để dự đoán các chuyển động của hệ thống trong quá trình thực nghiệm Mô hình giảm chấn được áp dụng cho mô hình rời rạc không giảm chấn Đầu tiên, thanh dầm công xôn mềm một chiều sẽ được khảo sát

Phương pháp điều khiển input shaping được sử dụng để triệt tiêu rung động của

hệ thống với việc áp dụng các input shaper khác nhau từ đơn giản tới nâng cao Mô phỏng và thực nghiệm được thực hiện để cho thấy sự thống nhất giữa kết quả số và thực nghiệm

Từ sự hiểu biết thu được, thanh dầm công mềm ba chiều – với các dao động trong tọa độ ba chiều được mô hình hóa với các mối quan hệ được tách rời giữa các rung động 3D này được trình bày

iii

LỜI CAM ĐOAN

Tôi xin cam đoan tất cả các số liệu và nội dung trình bày trong luận văn này là trung thực và không sao chép các công trình nghiên cứu của bất kì cá nhân hay tổ chức nào Tôi xin đảm bảo thực hiện nghiêm túc việc trích dẫn các tài liệu tham khảo được sử dụng trong luận văn

TP Hồ Chí Minh, ngày 25 tháng 09 năm 2020

Vũ Nguyễn Trí Giang

iv

CONTENTS

ACKNOWLEDGEMENT i

ABSTRACT ii

LỜI CAM ĐOAN iii

CONTENTS iv

LIST OF FIGURE vi

LIST OF TABLE viii

CHAPTER 1: INTRODUCTION 1

1.1 Previous work 2

1.2 Research objectives 4

1.3 Thesis outline 5

CHAPTER 2: ONE-DIMENSIONAL DYNAMIC MODEL 6

2.1 Problem formulation 6

2.1.1 Kinetic energy 7

2.1.2 Potential energy 7

2.1.3 Work done 7

2.2 The continuous model 7

2.2.1 The variation with respect to variable h(t) 8

1.1.2 The variations with respect to w z t( , ) 8

2.3 The discrete model 10

CHAPTER 3: BASIC INPUT SHAPING THEORY 14

3.1 Basic theory 14

3.2 Multi-mode input shaping 17

v

CHAPTER 4: SIMULATION AND EXPERIMENT OF THE ONE –

DIMENSIONAL SYSTEM 19

4.1 Experimental testbed 19

4.2 Parameter determination 20

4.2.1 Direct method 20

4.2.2 Experimental modal analysis 24

4.3 Simulation and experiment 28

4.3.1 Model verification 28

4.3.2 Input shaping verification 30

CHAPTER 5: THREE DIMENSIONAL DYNAMIC MODEL 36

5.1 Modeling 36

5.1.1 Inextensional beam 37

5.1.2 Three-dimensional beam theory 38

5.1.3 Coordinate projection 38

5.1.4 The damped system 41

5.2 Simulation 42

CHAPTER 6: CONCLUSION 47

6.1 Conclusions 47

6.2 Future work 48

REFERENCES 49

vi

LIST OF FIGURE

Figure 1 a) Industrial robot arm; b) Satellite; c) MEMS sensor 1

Figure 2 Flexible cantilever beam system with a moving hub 6

Figure 3 Block diagram of input shaping 14

Figure 4 Basic concept of input shaping 14

Figure 5 Experimental testbed 19

Figure 6 The cross-section of the experimental beam 20

Figure 7 The excitation signal [0 10] Hz in: a) Time-domain; b) Frequency domain 21

Figure 8 The tip response of the beam corresponding to [0 20] Hz excitation in: a) Time-domain; b) Frequency domain 22

Figure 9 The excitation signal [0 8] Hz in: a) Time-domain; b) Frequency domain 23

Figure 10 The tip response of the beam corresponding to [0 8] Hz excitation in: a) Time-domain; b) Frequency domain 23

Figure 11 Free vibration of the tip in: a) Time-domain; b) Frequency domain 24

Figure 12 The modal parameter estimation procedure 26

Figure 13 FRF of the tip of the beam with Lin1 excitation 26

Figure 14 FRF of the tip of the beam with Exp5 excitation 27

Figure 15 FRF of the tip of the beam with Exp10 excitation 27

Figure 16 FRF of the tip of the beam with Exp15 excitation 27

Figure 17 An S-curve displacement 29

Figure 18 Simulation verification of log-decrement and EMA methods 29

Figure 19 Comparison of the results of EMA and log-decrement methods 31

vii

Figure 20 ZV shaper for the first mode 32

Figure 21 ZVD and ZVDD shapers experiment for the first mode 33

Figure 22 Two-mode ZV shaper experiment 33

Figure 23 Two-mode ZVD and ZVDD shapers experiments 34

Figure 24: A schematic of a vertical cantilever beam with axial, lateral deformations 36

Figure 25: Deformation of a beam element 37

Figure 26 The hub motion in a planar in simulation 1 42

Figure 27 The transverse vibration along with X-axis in simulation 5.1 43

Figure 28 The transverse vibration along with Y-axis in simulation 5.1 43

Figure 29.The longitudinal vibration in the simulation 5.1 44

Figure 30 A beam with a circular cross-section 44

Figure 31 The hub motion in a planar in simulation 5.2 45

Figure 32 The transverse vibration along with X-axis in simulation 5.2 46

Figure 33 The transverse vibration along with Y-axis in simulation 5.2 46

Figure 34 The longitudinal vibration in simulation 5.2 46

viii

LIST OF TABLE

Table 1 System parameters 20

Table 2 Excitation singal types 26

Table 3 Estimated modal parameters 28

Table 4 Rayleigh damping constants 29

Table 5 Comparison of input shapers 34

Table 6 The parameters for the simulation 5.2 45

1

CHAPTER 1: INTRODUCTION

1.1 Motivation and background

A cantilever beam is a structural member whose one end is fixed and the other end is free This structure is a basic component of engineering structures with applications in various fields such as mechanical [1], civil, aerospace [2], and electronic engineering [3] (see Figure 1) Basically, every cantilever beam produces vibration under applied forces In order to overcome this unwanted phenomenon, we can increase the moment of inertia by enlarging dimensions or increase the modulus

of elasticity Those methods usually result in large, heavy structures Regardless, the flexible beams are lightweight, small in size but they are not stable as the rigid ones Therefore, vibration suppression of a flexible beam is the study object in this thesis

2

1.1 Previous work

From the engineering point of view, beam models can be used to descript the flexible cantilevers A number of researches have been investigated the dynamics of the flexible cantilever dynamics, which can be classified into two cases: (i) Stationary case [4 - 6] (ii) Moving case including rotating beams [7 - 9] and translating beams [10 - 13] For Case (i), the beams are clamped at one end, while it is free at the other end In Case (ii), the beams are fixed on moving hubs, which generates rotational or translational motions These motions not only drive the hubs but also govern the vibrations of the beams Therefore, it is noticeable that the dynamics of the moving hubs should be coupled with the dynamics of the beam systems, in which the work

on this issue is rare

A dynamic model of the hub-beam system is necessary to analyze the behavior of the system where Euler-Bernoulli and Timoshenko beam theories have been widely applied to develop a mathematic model In the Euler-Bernoulli beam theory [14], the beam is subjected to bending moment only, which provides an essential solution for common engineering problems, especially for beam systems made of isotropic materials Timoshenko [15 - 16] considered the effects of shear deformation and cross-section rotation to the Euler-Bernoulli beam Therefore, the Timoshenko beam model is advanced but yields complications for dynamic analysis Han et al [17] found that if the slenderness ratio is large enough, the differences between the Euler-Bernoulli model and the others are not significant Therefore, the Euler-Bernoulli beam model is still valid in the dynamic analysis of the beam structures The Euler-Bernoulli beam systems are usually represented by partial differential equations (PDEs) Since PDEs are infinite-order systems, they result in the complexity of obtaining the solution and consequently, the difficulty of the dynamic analysis Therefore, approximation methods such as the Galerkin decomposition method [3] and the finite element method [4] are used to tackle the issues Instead of the PDE models, these approximations convert PDEs into a set of ordinary differential

3

equations (ODEs), which is a finite-order system It should be noted that the accuracy

of the approximation models depends on the choice of order of the ODEs

A number of studies have been investigated by the dynamics of the dimensional beam Khadem et al [18] developed dynamic equations of motion for a manipulator link where prismatic and revolute joints were considered Kane et al [19] established an analysis of a cantilever beam with a generic cross-section built into a moving base undergoing general motions Zhao et al [20] investigated the coupling equations of motion of a rotating three-dimensional cantilever beam

three-Vibration control of a flexible beam can be categorized into two types: open-loop and closed-loop controls In the field of feedback control, positive position feedback (PPF) was applied on the Euler-Bernoulli beam by using a piezoelectric actuator [21], while strain rate feedback (SRF) controller was used in [22] Both PPF and SRF control algorithms are essentially second-order compensators to vary the damping ratio and stiffness of the system called active damping and active stiffness, respectively A robust control scheme was presented in [23], where a modification of system transfer function allowed integral feedback generating improvement of performance In [24], multi positive feedback (MPF), which included feedback signals of position and velocity to suppress vibrations Feedback control requires the installation of extra actuators and/or sensors, which may not be implemented in practice (for example, in machining tools)

In contrast to feedback control, feedforward control that does not require feedback signals also attracts many researchers An S-curve profile was proposed associated with time-optimal constraint by Bai et al in the work [25] Yoon et al applied a trapezoidal velocity profile to reduce the vibration of an object attached to commercial robots Singer et al [26] developed a new feedforward technique, i.e., preshaped command, based on the characteristics of control objects, called ZV shaper However, ZV shaper is insufficient for most systems due to its insensitiveness

of modeling error Additional constraint equations were used to overcome this problem, which created ZVD shaper and ZVDD shaper based on added derivative

4

constraints [27] An alternative robustness shaper was known as Extra-Insensitive (EI) shaper, which was used by Singhose et al [28] to put residual vibration into acceptable tolerance while the derivative of residual vibration converged to zero Nguyen et al [29] designed the input shaping control to minimize the vibration of a rotational beam Sadat-Hoseini et al [30] derived an optimal-integral feedforward control scheme to control vibrations of aircraft wings enabling the auto-landing of aircraft In [31], an adaptive feedforward control scheme was proposed to suppress vibrations of a cantilever beam with unknown multiple frequencies Sahinkaya [32] developed a method, which is based on inverse dynamics, to generate shaped inputs for a lightly damped system Physik Instrumente company introduced a vibration suppression solution including hardware (a controller is compatible with linear motors) and software that is based on input shaping method [33]

1.2 Research objectives

This thesis represents a continuation of the previous work by Pham Phuong Tung [34] where a discrete mathematic model of a flexible beam attached on a moving hub (1-D) was developed with a single-mode input shaping controller was applied Anisotropic flexible beam with asymmetric cross-section is adopted in this thesis with the following contributions:

 Implement a damped mathematic model of a flexible beam fixed on a 1-D moving hub based on the existing undamped model This work includes consideration of

an appropriate damping model and verification of the obtained model

 Applying multi-mode input shaping controller to determine the effectiveness of the controller

 Verifying the 1-D model and the controller by simulation and experiment

 Develop a mathematic model for a flexible beam fixed on a 3-D moving hub Then numerical simulation is conducted by Matlab ®

5

1.3 Thesis outline

This thesis is structured as follows:

Chapter 1 – Introduction

Chapter 2 – One-dimensional dynamic model

Chapter 3 – Basic input shaping theory

Chapter 4 – Simulation and experiment of the one –dimensional system

Chapter 5 – Three-dimensional dynamic model

Chapter 6 – Conclusion

6

CHAPTER 2: ONE-DIMENSIONAL DYNAMIC MODEL 2.1 Problem formulation

Figure 2 Flexible cantilever beam system with a moving hub

Figure 2 illustrates a flexible cantilever beam system, where the solid line represents the deformed beam, and the dashed line indicates the neutral axis of the beam The deflections of the beam result from the motions of the hub, which moves

on the X-axis of the world coordinate OXYZ The beam transverse displacement with

the neutral axis is denoted by w(z, t) while b(z, t) and h(t) indicate the positions of the beam and the hub, respectively The beam parameters are the length l, the mass per

unit length , Young’s modulus E and the inertial moment I The external force f(t) exerts on the hub, where m is the mass of the hub

The following assumptions, which are known as the Euler-Bernoulli assumptions for beams, are adopted as

Assumption 1: The cross-section is infinitely rigid in its own plane

Assumption 2: The cross-section of a beam remains plane after deformation

Assumption 3: The cross-section remains normal to the deformed axis of the

beam

In this study, the overdot  ˙ stands for  / t and the prime  ´ denotes

/ l

 

1.1.1.3 The total kinetic energy of the system

The total kinetic energy of the system is a sum of the kinetic energy of the beam and the kinetic energy of the hub as follows:

2 2

1

( , ) 2

l zz

2.2 The continuous model

The extended Hamilton’s principle, that is used to obtain the equations of motion, is given by

8

2.2.1 The variation with respect to variable h(t)

The variation with respect to h(t) is expressed as:

1.1.2 The variations with respect to w z t( , )

The variation with respect to w z t( , ) is expressed as

2 1 1 2

0 0

0

l

l l

2.3 The discrete model

To develop the discrete model, the approximation of the dynamic model Eqs (2.10)and (2.15) is carried out by using the Galerkin method, where the transverse displacement of the beam can be represented by Galerkin decomposition:

1

n

i i r

11

1 2 3 4

sincos

.sinhcosh

( )0

0

n

f t q

a set of single-degree-of-freedom (SDOF) systems An actual system always consists

of damping To apply the modal analysis of undamped systems to damped systems, well-known damping called Rayleigh damping or proportional damping [4] is usually employed This damping model is a linear combination of the mass and stiffness matrices as:

where and are the two Rayleigh damping coefficients The formula

between them and the N modal damping factors is described as:

0 1

1

.2

r

a a

Figure 3 Block diagram of input shaping

Figure 4 Basic concept of input shaping

15

The method of input shaping was attributed to Singer [36], where he considered

a second-order vibratory system to develop a linear system represented as a series of

a second-order system with decaying sinusoidal response as [37]:

0

0 2

finite series of n impulses:

To eliminate the total vibration caused by a series of impulses, the value B amp

must equal to zero This condition leads to both terms inside the square root in Eq (3.4) equal to zero

A





The condition in Eq (3.7) minimizes the time delay of the shaper The condition

in Eq (3.8) normalizes the shaper to guarantee that there will not be a gain of the

For a complete derivation of these equations see [36]

3.2 Multi-mode input shaping

Singer stated the solution for the systems consists of more than one natural frequency Shapers can be convolved together due to their linear operation In this way, many frequencies can be canceled simultaneously but the delay time is increased

because the impulses, N, are exponential with the number of frequencies are considered, m

m

Another disadvantage of this method is that the total time of the shaper equals

to the sum of the convolved impulses Hype [38] showed a new method called direct –solution sequences for calculating sequences for multiple-mode systems From the basic equations from (3.5) ~ (3.8), (3.11), he repeats the simple and derivative constraints for each mode:

There are 4m+2 equations, where m is the total number of modes to cancel

Therefore, Hype’s sequences will have

19

CHAPTER 4: SIMULATION AND EXPERIMENT OF THE ONE –

DIMENSIONAL SYSTEM 4.1 Experimental testbed

The experiment set-up (as shown in Figure 5) is used to verify the investigation cantilever beam A stainless steel beam (see Figure 6) with the parameters as shown

in Table 1 is clamped on a linear motor Yokogawa, which generates the external force The linear motor system is controlled by a motion controller UMAC The motion program controlling the whole linear motor system is generated from accompanying software with the UMAC controller (PeWin32Pro) The measured signals are collected via a PC with an NI PCI motion card and processed in the LabVIEW program The input signal is encoder feedback pulse from the linear motor drive and the beam vibration’s data in pixels is produced by a Keyence® high-speed camera, then transferred to the computer through an Ethernet cable

Figure 5 Experimental testbed

20

Figure 6 The cross-section of the experimental beam

Table 1 System parameters

The dynamic model of the one-dimensional hub-beam system (see Figure 2) is

implemented using the parameters shown in Table 1 It can be seen in Eq (2.39) that

the damping matrix is unknown In order to complete the dynamic model, the damping ratios must be obtained There are many methods to extract the modal parameters of a system In this thesis, direct method employing Fast Fourier transform and logarithmic decrement to obtain natural frequencies and damping ratio; and fitting method using experimental modal analysis are used to extract the parameters and a discussion will be made

4.2.1 Direct method

a) Fast Fourier transform

Firstly, the system is excited by an input signal to collect input and output of interest point in the time domain In this method, the input signal is a sinusoidal sweep sinusoid whose frequency varies with time and the interest point is the tip of the beam

It is noticeable that the frequency of the input signal must cover the frequency of interest and the system is settled when it is measured A sinusoidal sweep with the frequency range of [0, 10] Hz with the slope   is applied to the system (see 1,Figure 7) Then, a pair of input and output data is measured every 0.01 seconds

It can be seen in Figure 8 that the beam becomes unstable when the frequency reaches 9 Hz Thus, the frequency of interest is chosen at [0, 8] Hz to ensure the stability of the beam and the presence of the two first modes From the FFT spectrum

Figure 8 The tip response of the beam corresponding to [0 20] Hz excitation in:

a) Time-domain; b) Frequency domain

23

Figure 9 The excitation signal [0 8] Hz in: a) Time-domain; b) Frequency

domain

Figure 10 The tip response of the beam corresponding to [0 8] Hz excitation in:

a) Time-domain; b) Frequency domain

Logarithmic decrement

The logarithmic decrement assumes a single-degree-of-freedom (SDOF) harmonic oscillator and brings the best result for underdamped exponentially decaying data( Figure 11) In this thesis, the damping ratios of the two first modes are assumed to be equal The damping ratio evaluated by logarithmic decrement will represent both modes

The logarithmic decrement is defined as the natural log of the ratio of the amplitudes of any two peaks:

where x t( ) is the amplitude at time t and x t( nT) is the overshoot of the peak n

periods away, where n is any integer number of successive, positive peaks

24

The damping ratio is then found from the logarithmic decrement:

Figure 11 Free vibration of the tip in: a) Time-domain; b) Frequency domain

2

121

4.2.2 Experimental modal analysis

In the previous section, a damping ratio estimation method of a SDOF is applied

to a MDOF system with an assumption of equalization among damping ratios of different modes Therefore, it necessary to conduct an estimation method for MDOF method to a MDOF system

In this thesis, a brief introduction of an experimental modal analysis is

25

presented, more detailed discussions can be found in [5, 6, 7, 8] Experimental modal analysis (EMA) is used as an identification method by the measurements of the frequency response functions (FRFs) with fast Fourier transform (FFT) analyzer Then, modal parameters are extracted by a set of FRFs using curve fitting (see Figure 12)

Eq.(2.39) can be represented with proportional damping and harmonic excitation as:

1

,

T N

i t

r r ss

where ω is the forcing frequency in rad/s, which covers the frequency range of

interest in the modal test The receptance FRF, which deals with displacement response at due to excitation, has the form [43, Eq (18 3)]:

r r ij

,

N

ijr ijr ij

26

Firstly, the system is excited by a test signal to collect input and output of interest point in the time domain The excitation signal is the decisive factor to expose the characteristics of the system The test signal in the previous section is adopted ( Figure 9 and Figure 10)

The frequency response functions of the tip are obtained by using a MATLAB® function – modalfrf provided in Signal Processing Toolbox™ - MATLAB® in Figure

13 To determine the parameters, the frequency response functions are processed by

a MATLAB® function – modalfit to estimate the natural frequencies and damping ratios with the least-squares rational function estimation method [44] in Table 2

Figure 12 The modal parameter estimation procedure

Table 2 Excitation singal types

Sine sweep types Frequency

growth rate

Sine amplitude (mm)

27

Figure 14 FRF of the tip of the beam with Exp5 excitation

Figure 15 FRF of the tip of the beam with Exp10 excitation

Figure 16 FRF of the tip of the beam with Exp15 excitation

28

Table 3 Estimated modal parameters

Excitation signal Natural frequency (Hz) Damping ratio

According to Table 3, the estimated parameters form Lin1 is wrong This could

be explained in Figure 13 that the magnitude of FRF for the first mode is much smaller than the second mode Thus, the fitting algorithm cannot work properly Roy

et al [45] demonstrated that an exponential sine sweep excitation should be employed when the first mode is close to the lower limit frequency A variation of exponential sins sweeps are presented in Table 2 with the exponential growth rate is defined as:

exp

,ln(2)

T

where f0 and f1are the initial and final frequencies, T is the total exciting time

There is only a bit of difference among natural frequencies while damping ratios vary vastly in different excitation signals

4.3 Simulation and experiment

Table 4 and parameter in Table 1 into Eq (2.39) the discrete dynamic model of

the system is determined An S-curve displacement is applied to verify the consistency between simulation and experiment in terms of free vibration and forced vibration

29

Figure 17 An S-curve displacement

Figure 18 Simulation verification of log-decrement and EMA methods

Table 4 Rayleigh damping constants