The Pennsylvania State University

The Graduate School

Department of Mechanical and Nuclear Engineering

DESIGN AND MODELING OF A MOTION AMPLIFIER USING AN AXIALLY-

DRIVEN BUCKLING BEAM

A Thesis in

Mechanical Engineering

by

Jie Jiang

2004 Jie Jiang

Submitted in Partial Fulfillment of the Requirements

for the Degree of

Doctor of Philosophy

August 2004

The thesis of Jie Jiang was reviewed and approved* by the following:

Gary H. Koopmann Distinguished Professor of Mechanical & Nuclear Engineering Thesis Co-Advisor Co-Chair of Committee

Eric M. Mockensturm Assistant Professor of Mechanical & Nuclear Engineering Thesis Co-Advisor Co-Chair of Committee

George A. Lesieutre Professor of Aerospace Engineering

Richard C. Benson Professor /Department Head of Mechanical & Nuclear Engineering

Richard C. Benson Professor of Mechanical Engineering Head of the Department of Mechanical & Nuclear Engineering

*Signatures are on file in the Graduate School

iii

ABSTRACT

For active materials such as piezoelectric stacks, which produce large forces and

small displacements, motion amplification mechanisms are often necessary -- not simply

to trade force for displacement, but to increase the output work transferred through a

compliant structure. Here, a motion amplifier for obtaining large rotations from small

linear displacements produced by a piezoelectric stack is built and tested. The concept for

this motion amplifier uses elastic (buckling) and dynamic instabilities of an axially driven

buckling beam. The optimal design of the buckling beam end conditions was determined

from a static analysis of the system using Euler’s elastica theory. This analysis was

verified experimentally. A stack-driven, buckling beam prototype actuator consisting of a

pre-compressed PZT stack (140 mm long, 10 mm diameter) and a thin steel beam (60

mm x 12 mm x 0.508 mm) was constructed. The buckling beam served as the motion

amplifier, while the PZT stack provided the input actuation. The experimental setup,

measuring instrumentation and method, the beam preloading condition, and the excitation

are fully described. The frequency response of the system for three preloading levels and

three stack driving amplitudes was obtained. A maximum 16° peak-to-peak rotation was

measured when the stack was driven at amplitude of 325 V and frequency of 39 Hz. The

experiments on the details of period-n motions and the effects of beam preload were also

conducted.

Since the amplifier is driving a large mass at the pinned end, for simplicity, the

mass of the buckling beam is neglected and the system is modeled as a single-degree-of-

freedom, non-linear system. The beam simply behaves a non-linear rotational spring

iv

having a prescribed displacement on the input end and a moment produced by the inertial

mass acting on the output end. The moment applied to the mass is then a function of the

beam end displacement and the mass rotation. The system is then modeled simply as a

base-excited spring-mass oscillator.

Results of the response for an ideal beam using the SDOF model agree with the

experimental data to a high degree. Loading and geometric imperfections are also studied

to determine the sensitivity of the actuator. The behavior with slight imperfection is

similar to the response for the ideal beam and the experimental results; the response is not

particularly sensitive to imperfection.

Parameter studies for the ideal buckling beam amplifier were conducted using the

validated spring-beam model; these can be used as guidance for improving the design of

the motion amplifier and finding the optimal operational conditions for different

applications.

v

TABLE OF CONTENTS

LIST OF FIGURES .....................................................................................................vii

LIST OF TABLES.......................................................................................................x

ACKNOWLEDGEMENTS.........................................................................................xi

Chapter 1 Introduction ................................................................................................1

1.1 Motivation and Objectives..............................................................................1 1.2 Background.....................................................................................................3

1.2.1 Cumulative motion amplification mechanism......................................3 1.2.1.1 Frictional clamps by normal forces ............................................4 1.2.1.2 ER (Electro-Rheological) fluid clutches ....................................5 1.2.1.3 Hydraulic check valves ..............................................................6 1.2.1.4 Self-locking wedge clamp..........................................................6

1.2.2 Direct motion amplification mechanism ..............................................9 1.3 Summary of Study ..........................................................................................11

Chapter 2 Static Structure Design for Axially Driven Buckling Beam......................15

2.1 Structural Design ............................................................................................15 2.1.1 End support design ...............................................................................16 2.1.2 Static test stand .....................................................................................17

2.2 Static Analysis of Buckling Beam..................................................................17 2.2.1 Effect of external load ..........................................................................17

2.2.1.1 Spring load .................................................................................18 2.2.1.2 Constant end load .......................................................................18

2.2.2 Beam imperfection ...............................................................................19 2.2.3 Energy flow ..........................................................................................19

Chapter 3 Experimental Performance of an Axially Excited Post-Buckling Beam ...27

3.1 Experimental Setup.........................................................................................27 3.2 Experimental Procedure..................................................................................28

3.2.1 Quasi-static experiments ......................................................................28 3.2.2 Dynamic experiments...........................................................................28

3.3 Discussion.......................................................................................................29 3.3.1 Quasi-static testing ...............................................................................29 3.3.2 Frequency response ..............................................................................30 3.3.3 Sub-harmonic regions...........................................................................34 3.3.4 Preloading effects .................................................................................36

3.4 Summary of Dynamic Experiments................................................................37

vi

Chapter 4 Modeling of Axially Driven Post-Buckled Beam......................................52

4.1 Literature Review on Dynamic Buckling Beam Analysis..............................52 4.2 Modal Analysis of an Inextensible Post-Buckled Beam ................................57

4.2.1 Linear vibration of post-buckled beam.................................................58 4.3 Beam-Spring Model........................................................................................62

4.3.1 Non-unique spring constant kt(φe, u) ....................................................63 4.3.2 Construction of a non-linear rotational spring......................................64

4.4 Model Validation ............................................................................................66 4.4.1 Model validation for an ideal beam......................................................67 4.4.2 Model validation for an ideal beam with loading imperfection ...........68 4.4.3 Model validation for a curved beam.....................................................69 4.4.4 Discussion.............................................................................................71

4.5 Parameter Studies ...........................................................................................72 4.5.1 Finding period-1 output motion............................................................73

4.5.1.1 Effects of preloading level .........................................................73 4.5.1.2 Effects of driving amplitude.......................................................73 4.5.1.3 Effects of damping .....................................................................74

4.5.2 Optimal period-1 rotation angle ...........................................................75

Chapter 5 Summary and Future Work ........................................................................106

5.1 Summary.........................................................................................................106 5.2 Future Work....................................................................................................108

Bibliography ................................................................................................................110

Appendix Programming of Beam-Spring Model in Mathematica..............................117

A.1 Single Static Solution for End Displacement u=0.29 (Example)..................117 A.2 3-D Surface Construction ..............................................................................120

A.2.1 Data processing....................................................................................120 A.2.2 Data interpolation ................................................................................122 A.2.3 Moment function .................................................................................123

A.2.3.1 Ideal beam .................................................................................123 A.2.3.2 Loading imperfection ................................................................123

A.3 Dynamic Modeling ........................................................................................124

vii

LIST OF FIGURES

Figure 1-1: Comparison of the work done with and without motion amplification ....12

Figure 1-2: A self-locking taper [20, 29] .....................................................................13

Figure 1-3: Roller clutch’s operating principle............................................................13

Figure 1-4: L-L amplification mechanism (a) schematic (b) linkage representation [31]........................................................................................................................14

Figure 2-1: End conditions...........................................................................................21

Figure 2-2: Buckling amplification..............................................................................21

Figure 2-3: Concept of buckling beam motion amplifier ............................................22

Figure 2-4: Static experimental setup ..........................................................................23

Figure 2-5: Effect of end spring stiffness ....................................................................24

Figure 2-6: (a) Stall torque and (b) effect of constant shaft torque .............................24

Figure 2-7: The effect of initial shape imperfection and experimental results ............25

Figure 2-8: The energy transferred to the shaft with spring (a) and constant (b) loads......................................................................................................................26

Figure 3-1: Photograph of (a) the physical setup of the experiment and schematic of (b) the instrumentation .....................................................................................39

Figure 3-2: Quasi-static experimental results ..............................................................40

Figure 3-3: Driving conditions for data set A..............................................................41

Figure 3-4: Frequency responses for test described in Table 3-1. The dots and pluses represent decreasing and increasing frequency sweeps, respectively. The upper (red) and lower (green) profiles are the maximum and minimum rotation for each driving period, respectively.......................................................45

Figure 3-5: Phase plots and frequency spectra of rotation angle and slider signals for three driving frequencies highlighted in Figure 3-3 (a) ..................................46

Figure 3-6: Phase plots and frequency spectra of rotation angle and slider signals for four driving frequencies highlighted in Figure 3-3 (d) ...................................47

viii

Figure 3-7: Phase plots and frequency spectra of rotation angle and slider signals for three driving frequencies highlighted in Figure 3-3 (g) ..................................48

Figure 3-8: Time domain signal of period-n motion ...................................................49

Figure 3-9: Period-n motion in frequency domain ......................................................50

Figure 3-10: The effects of preloading level................................................................51

Figure 4-1: Natural frequencies are plotted as functions of end-displacement, uo, for an ideal beam ..................................................................................................77

Figure 4-2: First four mode shapes of end-displacement uo=0.3% for an ideal beam......................................................................................................................78

Figure 4-3: Beam-spring model ...................................................................................79

Figure 4-4: Solutions for Equation 4.16 for u=0.03%................................................80

Figure 4-5: The relationship of moment M versus end rotation φe at a certain displacement u=1% for a buckled beam ..............................................................81

Figure 4-6: Moment function of non-linear rotational spring for an ideal post-buckled beam in 3D configuration .......................................................................82

Figure 4-7: Model validation of data sets A and B for an ideal beam.........................85

Figure 4-8: Simulated time responses graphs (left) and phase-plane portraits (right) for test B1 of driving frequencies 34 Hz ~ 42 Hz .....................................88

Figure 4-9: Model validation of data sets A and B for an ideal beam with loading imperfection..........................................................................................................91

Figure 4-10: Simulated time responses graph (left) and phase-plane portrait (right) for test B1 of driving frequencies 37 Hz ..............................................................92

Figure 4-11: Flow chart for find the free stress configuration of a curved beam which has initial pinned-end angle of φo ..............................................................92

Figure 4-12: Relations of (ψi, xi, and yi) and (ψinit, xinit, and yinit) ................................93

Figure 4-13: The relationships of moment M versus φe for a curved beam at the end displacement u=0.03% ..................................................................................94

Figure 4-14: Frequency response of test A1 for different loading imperfection offset e ..................................................................................................................95

ix

Figure 4-15: The effects of loading imperfection offset e on resonance frequency and maximum peak-to-peak output for test A1 ....................................................96

Figure 4-16: The effects of preloading level for stack driving amplitude 0.1255% (peak-to-peak) and damping 42.0ˆ =c ..................................................................97

Figure 4-17: Period-n motion driving condition for damping 18.0ˆ =c ......................100

Figure 4-18: Effects of damping on the output motion................................................101

Figure 4-19: Maximum period-1 motion rotation angles for different driving conditions..............................................................................................................104

Figure 4-20: Maximum period-1 motion driving frequencies for different driving conditions..............................................................................................................105

x

LIST OF TABLES

Table 3-1: List of dynamic testing conditions .............................................................31

Table 3-2: Hinged end frequency response summary..................................................33

Table 4-1: Parameters for modeling validation ...........................................................66

Table 4-2: Experimental driving conditions summary ................................................67

Table 4-3: Parameter ranges and step sizes used in numerical tests............................72

xi

ACKNOWLEDGEMENTS

I would like to thank my advisors, Dr. Eric M. Mockensturm who provided me

with an extraordinary amount of patience, encouragement, and enthusiasm during the

development of the study, and Dr. Gary H. Koopmann for his support, guidance and

wisdom throughout my Ph.D. education. I would also like to thank my committee

members Dr. George A. Lesieutre and Dr. Richard C. Benson for their time and

invaluable assistance. Special thanks go to my lab-mates: Jacob Loverich, Arash

Mahdavi, Dr. Michael Grissom, Dongjai Lee and lots more for your help and suggestions

for my research.

Finally and most significantly, I would like to give my whole-hearted thanks to

my husband, Zhengyu Pang, for the time we went through together and for his forever

encouragement. I also want to thank my parents, Wenda Jiang and Caifeng Fei, who

always encourage me to pursue the best I can do.

Chapter 1

Introduction

1.1 Motivation and Objectives

With their excellent dynamic performance and high energy density, active

materials have been successfully applied in actuators and motors for nearly 40 years.

Piezoelectric ceramics produce strains of approximately 0.1%. While little displacement

can be extracted from these raw active materials, a great deal of force can be produced

and they can operate at high frequency. For many applications, designers developing

mechanical actuators with active materials often need to trade decreased actuation force

for increased displacement output. Efficient, tunable motion amplification mechanisms

are then necessary.

Raw active materials are seldom used in isolation. In most applications a structure

surrounds the material to protect it and better transfer work to an applied load. This

surrounding structure will always have some compliance which will diminish the energy

passing through the structure to the load; some of the work done by the active material is

stored as strain energy in the structure. In the typical case when the active material is

surrounded by a compliant structure, motion amplification mechanisms can be used to

maximize the output or pass-through work of the actuator.

This idea is illustrated qualitatively in Figure 1-1 where Fb and δmax are the

blocked force and the free strain of the active material, respectively. The dashed line

2

represents the compliance of the surrounding structure. Given this structural compliance

one can determine the maximum amount of work that can be done against any possible

load. As the force generated by the active material is increased from zero, the work done

is first transferred into strain energy; the spring in the inset of Figure 1-1 is compressed.

Once the force transmitted through the structure is great enough to overcome the load,

work is transferred to it; the mass in the inset is lifted off the support. The work done on

the load will depend on the magnitude of the load and will be maximized for a particular

load level. If the active material has stiffness ka = Fb/δmax and the structure has stiffness ks

= α ka, then the ratio of the blocked force to the load P at which maximum work is done

is P/Fb = α/2(1+α). This ratio increases monotonically from zero when alpha is zero to

one half as alpha goes to infinity (a rigid structure). The maximum work done on the load

is (αδmax)(Fb/4)/(1+α) which approaches δmaxFb/4 for a rigid structure. If the motion is

amplified by β then the maximum work done on the load by the motion-amplified system

is magnified by β2(1+α)/(αβ2+1). The example case in which β = 2 and α = 1/4 is

illustrated in Figure 1-1 . In this case, with no motion amplification, the work done on the

load is 1/5 that done if the structure were rigid; shown in Figure 1-1 as the dark gray box.

With motion amplification, the work done is just 1/2 that done for a rigid structure; the

work output is thus magnified by 5/2 as shown by the light gray box in Figure 1-1 . If let

Fb = 1 kN and αδmax = 0.1%, the maximum work done is (αδmax)(Fb/4)/(1+α) = 0.05 N-m

and (αδmax)(Fb/4)β2/(αβ2+1) = 0.125 N-m for without and with motion amplification,

respectively.

3

The objectives of the present study are:

• To design a motion amplification mechanism using elastic (buckling) and

dynamic instabilities for high performance piezoelectric actuators,

• To derive a non-linear beam model that can be used to predict the

response of post-buckled structures driving large inertial loads, and

• To study how design parameters and operational conditions alter the

motion amplification.

1.2 Background

Various creative motion amplification mechanisms have been developed to

increase piezoelectric actuator stroke, doing so at the expense of decreased force. Motion

amplifiers in piezo-actuators and motors have evolved into two primary classes:

cumulative motion amplifiers and direct motion amplifiers. Cumulative motion

amplifiers, such as overrunning clutches, take the small periodic motions of an actuator

and accumulate them into larger steady-state motions. Devices that amplify input motion

from an active material by a certain, finite factor, such as levers and flextensionals, are

referred to as direct motion amplifiers.

1.2.1 Cumulative motion amplification mechanism

Designers have created a variety of cumulative motion amplifiers for

piezoelectronic motors. They can be categorized as (1) frictional clamps using normal

4

forces, (2) ER fluid clutches, (3) hydraulic check valves and (4) self-locking wedge

clamps. Friction clamps and ER fluid clutches are active amplifier while hydraulic check

valves and self-locking wedge clamps are passive amplifiers.

1.2.1.1 Frictional clamps by normal forces

One of the first cumulative motion amplifiers to use clamps utilized the friction

force generated by actively pressing two objects together was the so-called inchworm. In

this design a quasi-static clamping force was provided by active materials. One active

actuator was oscillated and the other two active clamps were actuated alternately to

rectify the oscillatory motion of the active actuator. The friction force was produced by

the normal force acting between the active clamp and the output element. While the basic

idea is similar, the inchworm designs vary substantially.

A number of patents have been awarded, mostly in achieving nanometer

resolution rather than high forces [1-9]. In 1964, Sibitz & Steele [10] were among the

first to patent an inchworm-type actuator as a low-force, high precision (micro-inch

range) positioner. Ling, et. al. [11] in 1998, present a device that made use of dry friction

and impulsive inertial forces caused by rapid oscillatory motion of a piezoelectric stack.

The clamping force was the friction force produced by gravity. This actuator could

produce a motion resolution of several nanometers with unlimited range.

Although the piezo-ceramics themselves are capable of producing large forces,

actuators made from them are typically not. In recent years, focus has turned to

improving the force actuators are capable of producing. Miesner and Teter [12] designed

5

a piezoelectric/magnetostrictive motor in 1994 which achieved 115 N output force and 25

mm/s free speed. At the same year, Zhang and Zhen [13] achieved an output force of 200

N and a positioning resolution of 5 nm by careful grinding and optimum design of flexure

frame to avoid shear forces.

To achieve a more accurate and precise device, a caterpillar design was presented

by Pandell and Garcia in 1996 [14]. Three piezoelectric clamps instead of two were used.

While the phase controls of the four input signals were complicated, the caterpillar device

decreased the amount of slip compared to the double clamping mechanism. In 1999,

Henderson and Fasick [15] developed a NGST (Next Generation Space Telescope)

inchworm design involving only one active clamp. This clamp used a PZT stack mounted

in a flexure that had very high stiffness in the travel direction and low stiffness in the

clamping direction. This one-clamp design simplified the control circuitry and it was

possible to achieve the desired position.

1.2.1.2 ER (Electro-Rheological) fluid clutches

ER fluid clutches use a similar concept as the inchworm-type motor, where the

clamping force is produced by the change in viscosity of ER (electro-rheological) fluids

rather than a friction force.

Dong, Li et al. [16] reported a new type of linear piezo-stepper motor using ER

clamps that avoided the clamping impulse vibration and had advantages of no noise, no

wear, and low power consumption. While the velocity was only about 1.5 µm/s with 0.25

6

kgf push force, it showed the potential to be an alternative to conventional friction

clamps.

1.2.1.3 Hydraulic check valves

Hydraulic check valves are also used to rectify the oscillatory motion of an active

material actuator to create, in this instance, quasi-steady fluid motion.

Since Aiba, et. al. in 1985 [17] and Kuwana, et. al. in 1992 [18] designed low

discharge rate and pressure piezo-pumps using mechanical check valves, many different

designs using check valves were presented to achieve higher flow rate and discharge

pressure. Mauck and Lynch [19] developed a hybrid piezo-hydraulic pump that produced

a working pressure of 6.9 Mpa and a flow rate of 45 ccm. The basic design used a stack

actuator to oscillate a piston in a hydraulic pump. The piston pulled fluid through an inlet

valve on its back stroke and pumped fluid out of an outlet valve on its forward stroke.

1.2.1.4 Self-locking wedge clamp

A “self-locking taper” was exploited in a wedgeworm stepper designed by Frank,

Koopmann et al. [20, 21]. Figure 1-2 illustrates the concept. Under a load F, no force C is

required for clamping at some combinations of wedge angle θ, and friction coefficients

µ1, µ2, and µ3. In place of the active clamps, two self-locking wedges were used to

accumulate the oscillatory motion of the driving stack. Only one stack was necessary for

forward motion; this simplified the electronics significantly. An actuator using self-

7

locking wedges was developed. It could deliver a maximum of 250 N dynamic force and

a maximum free-running speed of 10 mm/sec when the stacks were driven at 150 Vpp

and 200 Hz. Further modifications were made to improve the function of the self-locking

wedges such as using line contacts rather than surface contacts to get the wedges to seat

properly against the taper.

In the late 1940s [22], one-way clutches were first utilized in multi-phase, multi-

element torque converters. Since then, three major applications for OWCs have

developed [23, 24]. The first is backstopping in which the clutch is used to prevent

rotation in one direction. Clutches are used in this way in conveyors, lifts, and speed

reducers. The second OWC application is over-running in which the clutch discriminates

between the rotary speeds of the races and disengages one race from the other when the

speed of the first race becomes less than that of the second race. It is used in applications

where the driven member must separate from the driver. The third application is indexing

in which the OWC is used to convert reciprocating motion to one-way rotary motion. The

first and third applications have been recently exploited in piezoelectric motors [21, 25-

27].

Fanella [22] defined a roller one-way clutch as a clutch that has an outer and inner

race (one of which contains the cam profile), rollers, springs which load the rollers, and a

means of positioning the springs. The operating principle for a roller clutch is similar to a

ratchet mechanism, oscillatory motion driving the clutch element is rectified to rotation in

one direction. Figure 1-3 shows two rectifying designs for a wedge roller clutch: (a)

vibrating the outer race and produces counter-clockwise output rotation, (b) vibrating the

8

inner race and produces clock-wise output rotation. Both types are currently used in

piezoelectric actuators.

To obtain continuous rotation efficiently, two roller clutches are required. These

two roller clutches could be identical or of similar size. One acts as a “driving clutch”, the

other acts as a “grounding clutch”. For roller clutch piezomotors, roller clutches are

critical elements making an accurate clutch model vital for design and analysis. However,

the action of one-way roller clutches introduces a strong non-linearity that greatly adds to

the complexity of the analysis of the system.

In 1996, King and Xu [25] were the first to utilize commercialized, inexpensive

roller clutches as accumulative amplification mechanisms. The linear oscillatory motion

of a stack was amplified by a lever and rectified by the roller clutches to provide

unlimited rotary motion of output shaft. Zhang Q.M. [27] also used roller clutches

attached to a piezoelectric torsion tube which could produce angular vibration. In 1999, a

high performance actuator using roller clutches was developed by Frank, Koopmann et

al. [21]. Twelve bimorphs were evenly spaced around a central hub that was press fit

around the outer ring of a OWC. A mass was attached to the end of each bimorph to

create a resonance near 1000 Hz. The output shaft also acted as the inner race of the

clutch and met another grounding clutch to prevent it from rotating backwards. The

actuator had a free speed of 600 rpm, a stall torque of 0.5 N-m, and a peak mechanical

power output of nearly 4 watts.

9

1.2.2 Direct motion amplification mechanism

A variety of direct motion amplification mechanisms have been developed. In the

class of flexure-hinged displacement amplifiers, there are three basic amplifying

mechanisms: simple lever displacement amplifiers, bridge displacement amplifiers, and

four-bar displacement amplifiers [25, 26, 28]. The hinged, L-shaped lever displacement

amplifiers use single L-shaped levers, or multiple levers. The roller clutch piezomotor

developed by Frank et al. in [20, 21, 29] adopted one rigid L-shaped lever arm to amplify

the small motion of the stack. An L-shaped arm was also applied in the trailing edge flap

of a wing model by Chandra and Chopra in 1997 [30]. Lee and Chopra [31] developed a

“L-L” two stage lever-fulcrum amplifier for a piezostack-driven trailing-edge flap

actuator (Figure 1-4). This L-L amplification mechanism combined two lever-fulcrum

systems with an elastic linkage. Compared with the single lever design, this design had

higher amplification but was more complex. Figure 1-4 shows a schematic diagram of the

L-L amplifier: the small stroke of the piezostack was amplified by an inner lever with a

low amplification ( 6≤ ) and amplified again by the outer lever. King and Xu [25, 26]

studied the characteristics of flexure-hinged simple lever displacement amplifiers using

the finite element methods. Three limiting cases were examined: a right-angle hinge

profile, a right-circular profile, and in between elliptical or corner-filleted high profiles.

Displacement amplifiers for piezomotors were designed based on the performances of the

studied cases. Lau, Du et al. [32] proposed a systematic methodology to design

displacement amplifiers based on topology optimization. This methodology was applied

successfully to optimize Yano et al.’s multiple lever system for a printer head that was

10

referenced by Uchino in 1986 [33]. The multiple lever magnification mechanism had two

flexural levers driven by a piezostack that was similar to the L-L design of Lee and

Chopra. The rear lever and front lever were driven clockwise and counter clockwise

respectively by the motion of the stack. Three objective functions for both static and

dynamic operations were examined, namely maximum output stroke, magnification

factor, and mechanical efficiency.

Giurgiutiu and Rogers [34] proposed a solid-state axial-to-rotary converter-

amplifier for obtaining large rotation from small linear displacements generated by piezo-

stacks. The concept used the twist-warping coupling in thin-wall open tubes. A proof-of-

concept was built and tested with a 28 mm diameter, 1.2 m long, 0.8 mm wall-thickness

steel open tube. A maximum rotary displacement of 8o was measured in experiments.

Prechtl and Hall [35, 36] incorporated two stack actuators into two criss-crossed frames,

which they called an “X-frame” actuator. Trailing edge rotor blade flap of ± 5.9o

deflections were predicted for an 11.5% of span and 20% of chord slotted flap at the

hover operating point by actuator bench tests and a simple design code.

11

1.3 Summary of Study

Since the direct motion amplifier can overcome the backlash of cumulative

motion amplifiers and maximize the pass through work of actuators. A motion

amplification mechanism is created using elastic (buckling) and dynamic instabilities for

high performance piezoelectric actuators. A proof-of-concept prototype is built based on

buckling beam static analysis. Static and dynamic experimental results are presented and

validate the theoretical predictions of large motion amplification. A single-degree-of-

freedom non-linear beam-spring model is developed and verified with the experimental

results. Parameter studies for the ideal buckling beam amplifier are conducted using this

beam-spring model; these can be used as guidance for improving the design of the motion

amplifier and finding the optimal operational conditions for different applications.

12

Figure 1-1: Comparison of the work done with and without motion amplification

13

Figure 1-2: A self-locking taper [20, 29]

Output

Output

Figure 1-3: Roller clutch’s operating principle

14

Figure 1-4: L-L amplification mechanism (a) schematic (b) linkage representation [31]

Chapter 2

Static Structure Design for Axially Driven Buckling Beam

For active materials such as piezoelectric stacks, which produce large force and

small displacements, motion amplification mechanisms are often necessary – not simply

to trade force for displacement, but to increase the active materials’ output work against a

compliant load. In this chapter, a new concept for obtaining large rotations from the small

linear displacement produced by a piezoelectric stack is presented and analyzed statically.

The concept uses elastic (buckling) and dynamic instabilities of an axially driven

buckling beam.

2.1 Structural Design

The buckling beam’s static amplification is estimated using non-linear,

inextensible beam (elastica) theory to determine the rotation angle of an output shaft for a

given amount of longitudinal end deflection. The governing equations are (Love [37]):

where EI is the beam bending stiffness, P is the constant compressive load acting in the x

direction defined by a line connecting the end supports, V is the constant load acting

perpendicular to P, ψ(s) is the angle between the vectors tangent to the deformed and

)](sin[)(

)](cos[)(

0)](cos[)](sin[)(

ssy

ssx

sVsPsEI

ψ

ψ

ψψψ

=′

=′

=−+′′

(2.1)

16

undeformed curves, and X(s) and Y(s) are the coordinates of particles identified by the

convected arc-length coordinate s. Primes denote differentiation with respect to s.

Equation 2.1 is solved using a shooting method in Mathematica 4.2.

2.1.1 End support design

Two types of end conditions, shown in Figure 2-1 , are considered for the device.

In both cases, one end of the beam, called the output end, is clamped along a radial line

of a shaft. As the shaft radius, R, approaches zero, the shaft end condition becomes a

pinned end. The other end of beam, called the input end, is clamped to the stack input for

case (A), and pinned to the stack input for case (B). In the following, L is the beam

length, δ is the longitudinal end displacement generated by the active material (about

0.1% of the length of a PZT stack), and θ is the shaft rotation angle.

As shown in Figure 2-2, relatively large rotations can be obtained for small,

dimensionless longitudinal end displacements, δ /L. Note that the clamped condition at

the stack input end produces higher rotations than the pinned condition does and that as

R/L increases, the amount of rotation for a given end displacement decreases. Note also

that δ /L is the ratio of the input displacement δ over the beam length L, not the active

material strain. As δ is likely fixed by the active material that generates it, increasing the

beam length will decrease δ /L and the shaft rotation.

For optimal amplification, design (A) (clamped at the input end) and a small shaft

radius is desirable. It is possible to make the effective shaft radius essentially zero by

machining away half the shaft over the length where the beam is attached.

17

2.1.2 Static test stand

A test stand, illustrated schematically in Figure 2-3, was constructed to confirm

the static theoretical predictions made using elastica theory. Figure 2-4 A shows the

entire experimental setup consisting of a 140 mm long PZT stack, a preload set screw, a

thin steel beam of dimensions 130 mm x 12 mm x 0.508 mm, and a pivoted shaft

supported by two pillow blocks. The stack is supported by two blocks riding on a linear

slider that will bear any shear or bending load on the stack.

Micrometers were used to measure the displacement of the clamped beam end and

the compliance of the fixture. The shaft was machined flat such that the effective radius is

nearly zero. For measurements with a constant torque applied to the shaft, a lever arm

was attached to the shaft and weights hung from it. The shaft rotation was measured by

an arm attached to the shaft as shown in Figure 2-4 B.

2.2 Static Analysis of Buckling Beam

2.2.1 Effect of external load

All the theoretical results to follow are evaluated with the output end pinned (R =

0) and input end clamped. Although buckling beams appear to be excellent mechanisms

for amplifying active material motion, it is important to understand how applied loads

reduce the amount of rotation.

18

2.2.1.1 Spring load

A torsional spring is theoretically attached to the output end to simulate, for

example, aerodynamic loads on a flap. Figure 2-5 illustrates the effect of this torsional

spring load on the shaft rotation. For a given displacement δ and beam length L, the

amount of shaft rotation decreases with the increasing dimensionless torsional stiffness

Κτ = kτL/EI. Thus, by making the beam stiffer compared to the spring, greater rotations

can be obtained for a given end displacement.

2.2.1.2 Constant end load

For use in other applications, such as a motion amplifier in an active material

motor, an understanding of how rotation is reduced by a constant applied moment is

needed. The dimensionless stall torque, the torque at which the shaft rotation is zero, is

shown in Figure 2-6(a) as a function of δ /L. The stall torque increases with δ /L as

expected. Figure 2-6(b) demonstrates how the shaft rotation angle decreases with

increasing dimensionless torque for various end displacements. The buckled beam acts

like a softening torsional spring. The dots in Figure 2-6(b) represent the experimental

data collected with initial shaft rotations of 10°, 8°, 6°, 4°, and 2°. Considering the

imperfection of the beam and the lack of precision in the experiment, the experimental

and theoretical results agree well.

19

2.2.2 Beam imperfection

The presence of imperfections is inevitable in real systems. Imperfections

affecting a buckling beam can be considered geometric or loading. The elastica model

assumes a perfectly flat beam loaded in pure axial compression. Geometric imperfections

can be included by assuming the beam has some initial curvature. Loading imperfections

are included by addition of a moment at the beam end. Since these slight imperfections

will always exist in any actual device, it is necessary to determine how sensitive the

amplification is to these imperfections.

An initial shape imperfection is illustrated as the dashed curve in the inset of

Figure 2-7 . As shown in this figure, an increasing initial shape imperfection, as measured

by the unloaded rotation of the shaft, reduces the rotational amplification, but not

dramatically. The black dots in Figure 2-7 are experimental results which, for the crude

beam used, are quite encouraging.

The theoretical effect of a loading imperfection is similar to that of the geometric

imperfection.

2.2.3 Energy flow

An understanding of how energy flows through the amplifier is important for

maximizing the efficiency of passing work from the active material to the load. Part of

the energy from the active material is stored as strain energy in the buckled beam, and

since no dissipative elements are modeled, the remaining energy is transferred to the load

20

acting on the shaft. The strain energy stored in an inextensible beam is ∫BdsEI 2/2κ . For

a spring or constant load acting on the output shaft, the energy transferred to the load is

kτθ 2/2 or Mθ, respectively. When the amplifier returns to its natural state, the energy

stored in the beam will flow back to the active material.

The percentage of energy delivered by the active material which is transferred

through the amplifier to the shaft is shown in Figure 2-8 . For spring loads (Figure 2-8 a),

the energy efficiency increases with increasing spring stiffness, and is almost constant

under the investigated rotation range. There is a saturation level at approximately 19%.

When a constant load acts on the shaft (Figure 2-8 b) there is an optimal rotation

amplitude that delivers energy most efficiently. The optimal rotation increases as shaft

torque EIMLM /ˆ = increases and the peak percentage power transfer is approximately

31% for various constant torques.

21

Case (A) Clamped end case Case (B) Pinned end case

Figure 2-1: End conditions

Figure 2-2: Buckling amplification

22

Figure 2-3: Concept of buckling beam motion amplifier

23

Figure 2-4: Static experimental setup

24

Figure 2-5: Effect of end spring stiffness

Figure 2-6: (a) Stall torque and (b) effect of constant shaft torque

25

Figure 2-7: The effect of initial shape imperfection and experimental results

26

(a)

(b)

Figure 2-8: The energy transferred to the shaft with spring (a) and constant (b) loads

Chapter 3

Experimental Performance of an Axially Excited Post-Buckling Beam

Based on the static structural design of a buckling beam motion amplifier, a

prototype stack-driven buckling beam was built. In this test stand, the post-buckling beam

drove a large inertial mass at the output end. Quasi-static experiments were conducted

first and followed by dynamic experiments.

3.1 Experimental Setup

The static test fixture was modified slightly to accept more sensors for dynamic

data collection; a photograph and schematic are shown in Figure 3-1 . A shorter steel

beam (2) with length 60 mm and the same rectangular cross section was attached to

supports at both ends. One end was attached to a steel shaft that is supported by two

bearings (1); the other end was rigidly clamped to a slider block (3). A pre-compressed

piezoelectric stack (4) encased in a brass cylinder has one end fixed to a base block (5)

and the other end fixed to the slider block (3). Between the slider block and the stack is a

dynamic load cell (9). The beam can be statically preloaded by a set screw (6) and there

is a static load cell (11) installed between the set screw and the base block. The slider

block and base block translate on a linear, low-friction slider (12). The shaft rotation was

measured by an optical encoder (7) that provides 10000 pulses per revolution. The

28

displacement of the slider and base blocks were measured by Philtec optical displacement

sensors D125 (8) and RC100 (10).

3.2 Experimental Procedure

3.2.1 Quasi-static experiments

Since the dynamic setup is somewhat different than the static setup, a quasi-static

experiment was first conducted to study the quasi-static relationships between the

preload, the beam end displacement, and the shaft rotation angle. This helped to

determine the buckling level of the beam used in the dynamic tests. The displacement of

the slider block, the static load between the set screw and base block, and the shaft

rotation angle were recorded as the beam preload was slowly cycled manually to five

different buckling levels corresponding to 2°, 4°, 6°, 8°, and 10° of shaft rotations.

3.2.2 Dynamic experiments

The operating conditions could be any combinations of the preload level, and the

stack driving amplitude and frequency. Thus, the frequency response for three different

preload levels and three different stack driving amplitudes using slow frequency sweeps

were conducted. For certain stack driving amplitudes and preload levels, interesting

frequency regions were studied in more detail. Finally, experiments were performed

holding the stack driving frequency and amplitude constant, and slowly varying the

preload level.

29

If the period of output rotation is an integer, n, times the stack driving period, it is

called period-n motion. For example, response that has twice the period as the excitation

is period-2 response; linear systems always have period-1 response.

The data were colleted with a Data Acquisition System dSPACE ds1102 board

using a sampling frequency of 2.5 kHz. For each test point, data were collected for 13

seconds after steady state was reached. Once the time traces over the desired range of the

controlled parameters were captured, the time domain signals were converted to the

frequency domain and the amplitudes of the signals were extracted using MATLAB.

3.3 Discussion

3.3.1 Quasi-static testing

The relationships between preloading force and output rotation angle, and

dimensionless end displacement and shaft rotation angle are illustrated in Figure 3-2 (a)

and Figure 3-2 (b), respectively. It is clear that the loading profile is different from the

unloading profile, indicating this setup has a fairly large friction force and significant

hysteretic damping. When the preload is increasing, the friction force opposes the shaft

rotation and the shaft rotation is less than expected. When the preload is decreasing, the

friction force is reversed and resists the shaft returning back to its original position. After

each loop, the output rotation is displaced from the origin slightly (< 0.7°). The first

buckling load, Pcr1, was found to be approximately 130 N, 11.6% less than the theoretical

value of 147N for an ideal beam.

30

3.3.2 Frequency response

In the following, the data presented were obtained by methodically varying the

excitation amplitude and frequency to gain insight into the dynamics at a given level of

preload. The lines (A), (B), and (C) in Figure 3-2 (b) show the three preload levels at

which data sets were acquired. The beam was first preloaded by the set screw and then

the stack was energized. Since the stack requires a DC bias of 360 volts, the beam was

further preloaded. Data sets corresponding to (A), (B) and (C) in Figure 3-2 (b) represent

three regions along the beam buckling curve: fully buckled, transitioning from pre-

buckled to buckled, and pre-buckled. At each preloading level, the stack was provided

sinusoidal voltages with three different amplitudes from the source of the HP analyzer.

For each driving amplitude, the driving frequency was swept from 1 Hz to 70 Hz and

then from 70 Hz to 1 Hz using a step size of 1 Hz. This frequency range was based on the

calculated value for the first natural frequency of a hinged-clamped beam with an inertial

mass at the hinged end (44.7 Hz).

Figure 3-3 gives the data set A’s driving conditions. One can see there are certain

interactions between the stack’s input driving voltages and the beam’s end input

displacement for different driving frequencies because of the non-linearity of the

buckling beam. The amplitude of the beam’s end displacement is higher for frequencies

below resonance than for frequencies above resonance. The response of the beam and the

input driving signal are likely out-of-phase above the resonance frequency causing the

reduced driving amplitude in this frequency region. The experimental end input

displacement driving conditions are summarized in Table 3-1 .

31

Table 3-1: List of dynamic testing conditions

Driving amplitude of end input peak-to peak Test # Condition Preloading

level (δ/L %)Stack input voltage (V)

(1-70 Hz) (1-20 Hz) (60-70 Hz)

A1 0.1940 325 0.1289 0.1421 0.1076

A2 0.1872 272 0.1138 0.1209 0.0971

A3

Buckled

0.1827 227 0.0896 0.0952 0.0785

B1 0.1298 325 0.1221 0.1415 0.0959

B2 0.1233 272 0.0995 0.1165 0.0827

B3

Transition

0.1213 227 0.0832 0.0968 0.0699

C1 0.0726 325 0.1013 0.1189 0.0806

C2 0.0592 272 0.0836 0.0970 0.0703

C3

Pre-buckled

-- 227 -- -- --

To characterize the dynamics in each data set, the data is presented in two ways.

First the frequency response for each driving amplitude is shown in Figure 3-4 . These

graphs were obtained by first finding the maximum and minimum rotation angles

recorded for the entire test. From these points, samples were taken at each driving period

both forward and backward in time. The dots and pluses represent the maximum (red)

and minimum (green) values for increasing and decreasing frequency sweeps,

respectively. The data obtained when the frequency is increased follows that obtained

when the frequency is decreased. Displaying the data in this way allows one to quickly

determine if the response is period-1 or not.

32

For frequencies highlighted in the Poincaré plots, corresponding phase plots and

frequency spectra of the response data are shown in Figure 3-5 , Figure 3-6 and Figure 3-

7 . The frequency spectra of the input motions are also displayed. The phase plots show

the shaft rotation angle plotted against the shaft angular velocity. Zero-padding and a

low-pass filter were applied to smooth the phase plots. The cut-off frequency of the low-

pass filter is four times the driving frequency. Data was also collected from sensors

measuring the motion of the base block and the force acting between the stack and the

slider. However, because the base block motion is prevented in one direction by the set

screw, these data appear very noisy as contact between the screw and the block was

apparently lost for short intervals. Impacts that occur when contact is reestablished

appear to be the source of the noise. Thus, this data is not presented. The data collected

from the slider’s motion signal is, however, quite clean, which can be observed from the

spectrum graphs. Thus, only the spectra of the shaft rotation and slider translation are

presented here. The power spectral densities were computed with the signal processing

toolbox in MATLAB.

Table 3-2 summarizes the dynamic response for the output shaft for the three data

sets.

33

The output rotation angles are similar when the stack driving frequency is

increased and decreased except for the regions that do not have period-1 response.

In test A1, a jump phenomenon is observed in the ‘resonant’ region from 37 Hz to

38 Hz. The system acts like one with a softening spring in this region. The phase plot in

Figure 3-5 shows the output is period-1 for the driving frequencies where multiple steady

motions occur.

Table 3-2: Hinged end frequency response summary

Largest periodic rotation Data set Test # Periods

Freq. (Hz) Sweep up (o)

Sweep down (o)

Figure 3-4

37 9.76 17.4 1 1 (multiple

solutions 37-38 Hz)(39) (15.8) (16.1)

a

2 1 40 11.2 10.6 B A

3 1 40 5.15 5.36 c

30 9.14 10.2 1 4-7 (31-37 Hz)

38 13.9 14.5 d

31 8.93 9.14 2 6-9 (32-34 Hz)

35 13.2 13.5 e

B

3 1 36 9.61 9.54 f

1 1 29 11.1 11.2 g

2 1 32 8.64 8.42 h C

3 The shaft does not rotate.

34

For each data set, as the stack driving amplitude increases, the shaft rotation in the

region of the ‘resonant’ frequency increases. The largest rotation angle obtained was

17.3° (peak to peak) in test A1 at a driving frequency of 37 Hz. The largest unique steady

state response was approximately 16° at 39 Hz.

In data set B, period-n response appears in test 1 from 31 Hz to 37 Hz (Figure 3-4

d), and in test 2 from 32 Hz to 34 Hz (Figure 3-4 e). For 32 Hz and 34 Hz in test 1

(Figure 3-6 ), one can see that there are multiple loops in the phase plots and many sub-

harmonic peaks appear in the frequency spectra of the response.

From the phase plots, it is observed that even period-1 motions do not make

elliptic traces; the motion is not pure harmonic. Many super harmonic peaks appear in the

frequency spectra for such output motion.

For those tests with period-1 output, the frequency spectra of the slider’s motion

shows one very dominant peak; the motion is nearly harmonic. For those tests with

output which is not period-1 motion, the slider motion is not as harmonic and the

corresponding frequency spectra show many super- and sub-harmonic peaks.

In data set C, the testing condition for test 3 is the lowest preload level with

lowest driving amplitude. No output motion was observed. The preloading level and

driving amplitude are not provided in Table 3-1.

3.3.3 Sub-harmonic regions

A more detailed study of the regions that exhibit responses with periods greater

than the driving period was conducted. In these regions finer steps in driving frequency

35

were taken. Using a similar driving condition as test B1, the driving frequency was swept

up from 28.87 Hz to 38.17 Hz and down from 37.92 Hz to 29.67 Hz with a variable step

size. At the frequencies when the output signal jumps from period-1 motion to period-n

motion, a frequency step size of 0.125 Hz was used. Outside of this area, the frequency

step size was 0.25 Hz. Figure 3-8 shows signals in the time domain as the driving

frequency is swept down from 30.04 Hz to 30.17 Hz. Sections between the red lines are

for the particular driving frequency listed. The upper frame of the figure shows the output

rotation signal in the time domain. The numbers in the upper frame indicate the

multiplicity of the output signal compared to the driving period. One can see that for

some fixed-frequency driving conditions the response periods are changing and the

motions are chaotic. For example, when the driving frequency is 30.54 Hz, the output

signal includes period-13, -14, -16, and -17 motions. The order in which these motions

occur is random.

The time signal in the narrow shaded frame is the slider motion. For driving

frequencies of 30.04 and 36.17 Hz, corresponding to period-1 response, the slider motion

is nearly harmonic. For frequencies with period-n response, whenever there is a jump in

the output signal in time domain there is a jump in the slider signal. The ratio of the

response period to the driving period is shown as a function of the frequency in Figure 3-

9. When the driving frequency is increasing, this ratio starts from one (period-1 motion)

and changes to five (period-5 motion) at 35.919 Hz. The period ratio n increases as the

driving frequency decreases. Below 30.545 Hz, the shaft’s motion returns to period-1

motion. For increasing driving frequency, there is a wider frequency range for which the

shaft motion is not period-1, from 29.920 Hz to 37.669 Hz.

36

3.3.4 Preloading effects

Experiments were also conducted to further study how the preload affects the

output motion. For these experiments the stack was driven at a constant frequency and

amplitude (325 V). The response was recorded as the steady axial load acting on the

beam was manually and slowly increased. The steady load was increased through the

range in which the response period was greater than the input period and then slowly

decreased back to zero. This cycling of the steady axial beam load was repeated for a

variety of driving frequencies. The steady load level was estimated using the average

beam end displacement and is shown in Figure 3-10. The upper and lower graphs

correspond to the steady load level increasing and decreasing, respectively. When the

steady load level is swept up, there is a period-n band for the driving frequencies from 28

Hz to 35 Hz; this is shown as the shaded area in the graph. Outside this region, the

response is period-1 motion only. The situation is similar when the steady load level is

swept down. However, for this case the period-n region is wider and longer, covering

more steady loading levels and frequencies from 28 Hz to 36 Hz. The steady loading

levels of data sets A, B and C are also shown in the plots. One can see that the data set A

is beyond the shaded area in which the response to driving period ratio is greater than

one. This confirms the results obtained during prior experiments. Data set B was

collected using a steady load that crosses through the period-n region and again confirms

prior results. Data set C appears to be on the boundary of the period-n region. Again,

these results are consistent with the previous experimental results that show that test B1

has a period-n region, while tests A1 and C1 do not.

37

3.4 Summary of Dynamic Experiments

A motion amplification concept has been proposed for obtaining large rotary

amplification from the small linear displacements generated by piezo-ceramic stacks. The

concept utilizes the elastic (buckling) and dynamic instabilities of a thin beam. Static

analysis and experimental results are given. The static experimental results match

theoretical analyses closely.

A prototype piezoelectric-stack-driven buckling beam actuator was constructed.

The actuator consisted of a 140 mm long pre-compressed PZT stack and a 60 mm x 12

mm x 0.508 mm thin steel beam. The beam served as a motion amplifier, while the PZT

stack provided the actuation.

Frequency responses of the system for three different preloading and three

different stack driving amplitudes were obtained. A maximum 16° peak-to-peak rotation

was measured when the stack was driven at the amplitude of 325 V and frequency of 39

Hz. The details of sub-harmonic regions and the preloading effects were also studied.

When the beam is preloaded to the transition region, the period-n motions appear at the

resonance. For the driving frequencies at the resonance region, the output motions start

from period-1 motion, then meet period-n motion, finally return back to period-1 motion

as the beam’s preloading level increases. Through the experimental tests, this

investigation shows the proposed stack-driven buckling beam actuator is an easily

constructed, feasible motion amplification mechanism. It should be also noted that using

a buckling beam as a motion amplifier is not limited to stack driven actuation.

Electroactive polymers (EAP) or shape memory alloy (SMA) wires could also be used to

38

buckle the beam. The maximum stable rotary motion can be optimized to other values to

meet the operation requirements of specific applications. The following chapters

investigate the theoretical dynamic response of an axially driven buckling beam in order

to develop a complete design guide for this stack-driven buckling beam actuator.

39

(a)

dSPACEA/D

HP source

Stack input signal monitor

Base block displacement

Dynamic load cell

Slider block displacement

Optical encoder (rotation)

PC data acquisition

Stack power amplifier

Oscilloscope

HP analyzer Stack power input

(b)

Figure 3-1: Photograph of (a) the physical setup of the experiment and schematic of (b) the instrumentation

40

(a) (b)

(c)

Figure 3-2: Quasi-static experimental results

41

Figure 3-3: Driving conditions for data set A

42

(a) Test A1

(b) Test A2

43

(c) Test A3

(d) Test B1

44

(e) Test B2

(f) Test B3

45

(g) Test C1

(h) Test C2

Figure 3-4: Frequency responses for test described in Table 3-1. The dots and pluses represent decreasing and increasing frequency sweeps, respectively. The upper (red) and lower (green) profiles are the maximum and minimum rotation for each driving period,

respectively

46

Figure 3-5: Phase plots and frequency spectra of rotation angle and slider signals for three driving frequencies highlighted in Figure 3-3 (a)

47

Figure 3-6: Phase plots and frequency spectra of rotation angle and slider signals for four

driving frequencies highlighted in Figure 3-3 (d)

48

Figure 3-7: Phase plots and frequency spectra of rotation angle and slider signals for three driving frequencies highlighted in Figure 3-3 (g)

49

Figure 3-8: Time domain signal of period-n motion

50

30 32 34 36 380

5

10

15

# of

driv

ing

perio

d Driving frequency sweep down

30 32 34 36 380

5

10

15

Driving frequency (HZ)

# of

driv

ing

perio

d Driving frequency sweep up

(35.91949 HZ)

(30.54488 HZ)

(29.91992 HZ) (37.66936 HZ)

(32.41974 HZ)

Figure 3-9: Period-n motion in frequency domain

51

Figure 3-10: The effects of preloading level

Chapter 4

Modeling of Axially Driven Post-Buckled Beam

For the stack-driven buckling beam amplifier, because the piezoelectric stack’s

internal frequency is approximately 8 kHz and the driving frequency is less than 100 Hz,

the dynamics of the stack can be neglected and the amplifier can be modeled as a hinged-

clamped post-buckled beam driving large inertial loads. In this chapter, free vibration of

an inextensible post-buckled hinged-clamped beam is first studied to understand how the

free resonance changes with preloading levels. Then a single-degree-of-freedom

nonlinear dynamic model for an axially driven post-buckling beam with large end inertial

mass is developed and verified against experimental results presented in Chapter 3.

Lastly, parameters studies are conducted to provide design guidelines for obtain optimal

design and operational conditions.

4.1 Literature Review on Dynamic Buckling Beam Analysis

The dynamic buckling beam has drawn researchers’ attention over the past 50

years. Early works began in 1951, when Burgreen [38] investigated the free vibrations of

a simply supported beam that was given an initial end displacement. Eisley [39, 40]

considered free and forced vibration of simply supported and clamped beams for which

the initial end displacement was also prescribed. Both used a single-degree-of-freedom

representation of the equations of motion. Results were obtained in the post-buckled

53

region as well as the pre-buckled region. Burgreen found that the natural frequencies of

the buckled beam depend on the initial amplitude of oscillation. Experimental results

validated his theory. Burgreen also mentioned that when the axial load is greater than the

Euler load, snap through may occur depending upon the initial amplitude of deflection;

this was observed in experiments.

In the 1970s, Tseng and Dugundji [41] used a linear combination of the first two

linear buckled modes for a clamped-clamped beam to study the non-linear vibration of a

buckled beam under transverse harmonic excitation. They also concluded that the second

asymmetric mode does not contribute to the response unless it is parametrically excited

by the first mode through an internal resonance. Away from the region where the first and

second modes are close to each other, the result of a single-mode approximation is close

to that using a two-mode approximation. In a similar approach, Min and Eisley [42],

Yamaki and Mori [43], and Afaneh and Ibrahim [44] considered three modes with an

assumption that the modes of a buckled beam could be expressed in terms of the linear

modes of a straight beam with corresponding boundary conditions. Abou-rayan, Nayfeh

et al. [45] analyzed a nonlinear response of a simply-supported buckled beam to a

harmonic axial load using a single mode approximation. He found complicated dynamic

behaviors including period-multiplying and period-demultiplying bifurcations, period-

three and period-six motions, jump phenomena, and chaos.

To provide an exact solution to the linear vibration about a slightly buckled

configuration, Nayfeh, Kreider et al. [46] investigated the linear modes of vibration of

buckled beams experimentally and analytically by studying weakly nonlinear beam

equations. An exact solution was obtained by assuming a static buckled shape

54

corresponding to the nth buckling mode. The associated natural frequencies of post

buckled beams are found for fixed-fixed, fixed-hinged, and hinged-hinged boundary

conditions. The first natural frequencies of all three types of boundary conditions increase

as the maximum static buckled deflection increases. Experimental data were obtained

only for fixed-fixed boundary conditions and were in agreement with results obtained

analytically. Lestari and Hanagud [47] presented some exact solutions for the nonlinear

vibration of buckled beams subjected simultaneously to axial and lateral loads with

elastic end restraints and axial stretch due to immovable ends. Exact solutions were

obtained by using modes from the linear theory, which readily satisfy the boundary

conditions, and Jacobi elliptic functions. They concluded that the nonlinear natural

frequency of a beam increases with the amplitude of vibration with a constant axial load

or a constant end separation.

Kreider, Nayfeh et al. [48, 49] investigated experimentally and analytically the

nonlinear single-mode responses of a fixed-fixed, buckled beam under the case of

uniform, transverse harmonic excitation. Lacarbonara, Nayfeh et al. [50] also studied

experimentally and analytically the frequency response of the case of primary resonance

of the nth mode of the beam without activating the internal resonances in this mode. He

used a single-mode Galerkin method and studied directly the governing integral-partial-

differential equation and associated boundary conditions. Ji and Hansen [51]

experimentally investigated the non-linear response of a clamped-sliding post buckled

beam subjected to a harmonic axial load. Several non-linear phenomena including period-

doubling, sequence bifurcation, period-three, and chaotic motion were observed.

55

A theoretical model was proposed by Perkins [52] that solved the planar response

of an elastica rod about a generally curved, axially pre-stressed equilibrium. He derived

the governing equations of motion for the rod from Hamilton’s principle using

Kirchhoff’s assumptions for rod deformation (Love [37]). The case of free linear

vibration about the elastica equilibrium was then specialized and numerically solved to

determine the natural frequencies and mode shapes of a simply supported rod that

buckles under a large, steady end-load and moment. For the studied boundary condition,

the natural frequencies of the elastica arch decrease monotonically as the end load

exceeds the Euler buckling load, which was different from the results of Nayfeh, Kreider

et al. [46]’s free resonance of slightly buckled simply-supported case. When the two ends

of the rod meet, the fundamental natural frequency vanishes and further increases in the

end-load lead to a divergence instability. When a light load eccentricity is introduced, the

rod becomes significantly stiffened by the developing curvature and the first natural

frequency increases correspondingly. Results from an experimental test of a simply

supported rod provided support for the model. Levitas and Weller [53] also examined the

dynamic global post-buckling behavior of an axially loaded inextensible simply

supported beam.

Chin, Nayfeh et al. [54] and Nayfeh, Lacarbonara et al. [55] examined three-to-

one, two-to-one and one-to-one internal resonances using multiple scales method to

construct the non-linear normal modes for parametrically excited buckled beams. They

identified “rich” nonlinear behaviors. Afaneh and Ibrahim [44] also investigated the

nonlinear response of an initially buckled beam in the neighborhood of one-to-one

internal resonance via three different approaches: multiple scales (analytical) method,

56

numerical simulation, and experimental testing. They found that energy was transferred

from the first mode, which is externally excited, to the second mode. The analytical

results were qualitatively compared with those obtained by numerical simulation and

experimental measurements.

To obtain accurate quantitative as well as qualitative dynamic results, different

numerical approaches were studied and compared by many researches. Abhyankar, Hall

et al. [56] provided a general solution technique applicable to problems in chaotic

dynamics. In the paper, they determined the nonlinear vibration response of a simply

supported buckled beam under lateral harmonic excitation using a stable, explicit, finite-

difference method for both space and time and compared it with a single-mode Galerkin

discretization approach. He predicted a series of period-doubling bifurcations leading to

chaos in both methods. He also demonstrated that the finite difference method is more

powerful in that it may be applied for those problems difficult for the Galerkin

approximations. Emam and Nayfeh [57] used a Galerkin approximation to discretize the

nonlinear partial-differential weekly nonlinear equation governing the motion of the

beam about one of the buckled configurations by extending the work of Nayfeh, Kreider

et al. [46] to a clamped-clamped buckled beam. Single- and multi-mode Galerkin

methods are compared. The results show that a single–mode discretization yields

quantitative and qualitative errors in static and dynamic results for relatively high

buckling levels. A four-mode discretization provided good agreement with the

experimental results of Kreider and Nayfeh [49].

In addition, Nayfeh and Lacarbonara [58] compared discretization and direct

treatment for a general distributed-parameter system with quadratic and cubic non-

57

linearities. They showed that the discretization approach failed to predict the correct

dynamics of the original system. In the case of primary resonance, the even mode results

disagree in the two approaches, while the odd mode results with both approaches agree

with great accuracy. They further demonstrated that in most common nonlinear

distributed-parameter systems, the discretization procedure, such as the Galerkin

procedure, under certain conditions is not able to capture the spatial dependence of the

motion. A blind application of the discretization method might yield incorrect results.

4.2 Modal Analysis of an Inextensible Post-Buckled Beam

The nonlinear equations governing the transverse planar vibrations of a hinged-

clamped beam subject to an axial static load and a transverse harmonic load is given by a

set of five partial differential equations in space s and time t for a set of five dependent

variables (P, V, ψ, x, y).

subject to the boundary conditions

where the meanings of system EI and L parameters, and the dependent variables (Ψ, P, V,

x, y) are explained in Section 2.1; ρ is the mass per unit length; ψi is the angle between

)],(sin[),(')5()],(cos[),(')4(0),(),(')3(0),(),(')2(

)('')],(cos[),()],(sin[),(),('')1(

tstsytstsx

tsytsVtsxtsP

sEItstsVtstsPtsEI i

ψψ

ρρ

ψψψψ

==

=+=+

=++

&&

&&

(4.1)

0),(0),0('

==

tLt

ψψ

, 0),(0),0(

==

tLyty

, and 0),('

0),0(=

=tLx

tx (4.2)

58

the vectors tangent to the preloaded static beam curve and straight beam centerline;

primes denote differentiation with respect to arc length s and dots denote differentiation

with respect to time t.

Let us use the following non-dimensional variables

As a result, we can rewrite Equation 4.1 and 4.2 as

and

where the primes now indicate differentiation with respect to s and the dots now indicate

differentiation with respect to t .

4.2.1 Linear vibration of post-buckled beam

Assume that the beam response is the sum of the static buckled configuration

resulting from a static end displacement, and a time-dependent perturbation; that is

Lss /ˆ = , Lxx /ˆ = , Lyy /ˆ = ,

EIPLp /ˆ 2= , EIVLv /ˆ 2= , 4/ˆ LEItt ρ= (4.3)

)]ˆ,ˆ(sin[)ˆ,ˆ('ˆ)5()]ˆ,ˆ(cos[)ˆ,ˆ('ˆ)4(

0)ˆ,ˆ(ˆ)ˆ,ˆ('ˆ)3(

0)ˆ,ˆ(ˆ)ˆ,ˆ('ˆ)2(

)ˆ('')]ˆ,ˆ(cos[)ˆ,ˆ(ˆ)]ˆ,ˆ(sin[)ˆ,ˆ(ˆ)ˆ,ˆ('')1(

tstsytstsx

tsytsv

tsxtsp

ststsvtstspts i

ψψ

ψψψψ

=

=

=+

=+

=++

&&

&&

(4.4)

0)ˆ,1(0)ˆ,0('

=

=

tt

ψψ

, 0)ˆ,1(ˆ0)ˆ,0(ˆ

=

=

tyty

, and 0)ˆ,1('ˆ0)ˆ,0(ˆ

=

=

txtx

(4.5)

59

To find the natural frequencies and mode shapes, we employ separation of variables by

assuming a time-harmonic solution. Thus we let

where ω is the dimensionless natural frequency. Further more, to capture the linear

vibrations, we assume that the time-dependent perturbations are small relative to the

static configuration. As a result, the linear vibration of a beam around its post-buckled

equilibrium configuration is governed by the following equations

where λ = -ω2; variables φ, x and y constitute the mode shape for any solution that also

satisfies the boundary conditions. Note that Equations 4.8 are coupled and contains non-

constant ( )ˆ(ssψ ) and constant (ps and vs) coefficients, which are the solutions to the

equilibrium equations

)ˆ,ˆ()ˆ()ˆ,ˆ(ˆ tssts s ψψψ += , )ˆ,ˆ()ˆ,ˆ(ˆ tspptsp s += , )ˆ,ˆ()ˆ,ˆ(ˆ tsvvtsv s += , )ˆ,ˆ()ˆ()ˆ,ˆ(ˆ tsxsxtsx s += , and )ˆ,ˆ()ˆ()ˆ,ˆ(ˆ tsysytsy s +=

(4.6)

tiests ˆ)ˆ()ˆ,ˆ( ωφψ = , tiesptsp ˆ)ˆ()ˆ,ˆ( ω= , tiesvtsv ˆ)ˆ()ˆ,ˆ( ω= , tiesxtsx ˆ)ˆ()ˆ,ˆ( ω= , and tiesytsy ˆ)ˆ()ˆ,ˆ( ω=

(4.7)

)ˆ()]ˆ(cos[)ˆ(')5()ˆ()]ˆ(sin[)ˆ(')4(

0)ˆ()ˆ(')3(0)ˆ()ˆ(')2(

0)ˆ()])ˆ(sin[)]ˆ(cos[()]ˆ(cos[)ˆ()]ˆ(sin[)ˆ()ˆ('')1(

sssysssx

sysvsxsp

ssvspssvssps

s

s

ssss

ss

φψφψ

λλ

φψψψψφ

=−=

=+=+

=++−+

(4.8)

)ˆ('')]ˆ(cos[)]ˆ(sin[)ˆ('' ssvsps isssss ψψψψ =−+ 0',0' == ss vp , 0)1(',0)0(' == ss ψψ ,

0=sy at 1,0ˆ =s , and oss uxx −== 1)1(,0)0( (4.9)

60

where uo is the static end displacement at the clamped end. The last four equations in

Equation 4.8 can be rewritten as

In this formulation, the end load p is not perturbed p(0)=0 and the end )1(x is free to

move. Substituting Equation 4.10 into the first equation of Equation 4.8 , we obtain

Approximate eigen-solutions (φ, λ) are found from the admissible solution series:

Substituting Equation 4.12 into 0)1( =iφ and 0)1( =iy gives one βi and bi. Note that the

first two terms are the mode shapse for the linear vibration of a straight hinged-clamped

beam and always equals zero at 1ˆ =s .

Substituting βi, bi, and Equation 4.12 into 4.11, and integrating by parts, the

eigen-value problem becomes

The matrix K is given by

ζξξφξψλ

ζξξφξψλ

ξξφξψ

ξξφξψ

ζ

ζ

ddsv

ddsp

dsy

dsx

s

s

s

s

s

s

s

s

)()]([cos)ˆ()4(

)()]([sin)ˆ()3(

)()](cos[)ˆ()2(

)()](sin[)ˆ()1(

ˆ

0 0

ˆ

0 0

ˆ

0

ˆ

0

∫ ∫∫ ∫

∫∫

−=

=

−=

−=

(4.10)

0)()]([cos)]ˆ(cos[

)()]([sin)]ˆ(sin[

)ˆ()])ˆ(sin[)]ˆ(cos[()ˆ(''

ˆ

0 0

ˆ

0 0

=+

+

++

∫ ∫∫ ∫

ζξξφξψψλ

ζξξφξψψλ

φψψφ

ζ

ζ

dds

dds

ssvsps

s

s

s

s

s

s

ssss

(4.11)

)]ˆ1(sin[]sin[]ˆcos[

1)ˆ( sbs

s iiii

ii −+−= ββ

ββ

φ (4.12)

02 =− MK ω (4.13)

61

and the matrix M is given by

It can be shown that the linearized post-buckling beam is a self-adjoint system, so that the

matrices K and M are symmetric. The natural frequencies ω and mode shapes (φ, x, y) for

the inextensible hinged-clamped beam using static end displacement uo are determined

from the numerical solutions of the symmetric eigenvalue problem (Equation 4.13 ).

From Equation 4.3, the non-dimensional frequency parameter ω is related to the

dimensional frequency Ω (rad/sec) through ω=Ω(ρL4/EI)1/2.

Figure 4-1 illustrates the dependence of an ideal beam’s four natural frequencies,

using seven modes, on the end-displacement uo. Note that the first natural frequency

increases monotonically as the end-displacement increases. As shown in Table 3-2, the

frequencies of largest periodic rotation increase as the preloading levels increase from

data set C to A, which validates the modal analysis of the post-buckled inextensible

buckling beam. The arch equilibrium (dashed cures) is shown in Figure 4-2 at an end

displacement of uo=0.3%. The first four mode shapes of the elastica for that equilibrium

are also illustrated in that figure (solid curves).

∫∫ −+=1

0

1

0ˆ)ˆ()ˆ(ˆ)ˆ()ˆ()])ˆ(sin[)]ˆ(cos[( sdsssdsssvspK jijissssij φφφφψψ (4.14)

∫ ∫∫∫ ∫∫

+

=1

0

ˆ

0

ˆ

0

1

0

ˆ

0

ˆ

0

ˆ))](cos[)(())](cos[)((

ˆ))](sin[)(())](sin[)((

sddd

sdddMs

sj

s

si

s

sj

s

siij

ξξψξφξξψξφ

ξξψξφξξψξφ (4.15)

62

4.3 Beam-Spring Model

To create a design tool for predicting the response of post-buckled structures

driving large inertial loads requires a different analysis model from the analysis

approaches reviewed in Section 4.1. Direct numerical integration of the fully dynamic

elastic theory (non-linear PDEs) can be done but is an extremely time-consuming

procedure for investigating the steady state dynamic behavior of the system. Also, the

discretization method using the Galerkin approach is not accurate for a complex

nonlinear dynamic system as noted in several references [56-59]. Since the amplifier is

driving a large mass at the pinned end (output end), for simplicity, the mass of the

buckled beam is neglected and the system is modeled as a single degree of freedom non-

linear system. The beam simply behaves as a non-linear rotational spring having a

prescribed displacement on the input end and a moment produced by the inertial mass

acting on the output end (see Figure 4-3). The moment applied to the mass is then a

function of the beam end displacement u and the mass rotation φe.

To obtain the non-linear torsional spring function, say kt(φe, u), the elastica

equations

are solved quasi-statically to find the output rotation for typical ranges of input

displacement. M is the end moment acting on the pinned end of the beam. The spring

,1)1(,1)1(,0)0(,0)0(0)1(,ˆ)0(',)0(

)]ˆ(cos[)ˆ(')],ˆ(sin[)ˆ('0)]ˆ(cos[)]ˆ(sin[)ˆ(''

=−======

===−+

yuxyxM

ssxssysvsps

e φφφφ

φφφφφ

(4.16)

63

constant kt(φe, u) can be calculated from the moment function ),(ˆ uM eφ , namely

e

eet

uMuk

φφ

φ∂

∂=

),(ˆ),( .

4.3.1 Non-unique spring constant kt(φe, u)

For each displacement and rotation, the moment acting between the beam and the

mass can be determined. The loads p and v can also be calculated to determine if p

remains well below the blocked force of the stack. Figure 4-4 shows the relationship of p,

v, and M versus the end rotation for a displacement of u = 0.03%. One can see the same

loading condition can give different rotation values.

Figure 4-5 gives more details to illustrate the relationship of the moment M and

the mass rotation φe for a specific end displacement (u=1%). Again, one finds that M is

not a true function of φe and u; for given values of φe and u, there may be multiple values

of M . This can be readily understood by imagining a buckled pinned-clamped beam.

The buckled configuration is not unique. If the end rotation is initially, say, positive

(point A), a positive moment can be applied to reduce the rotation to zero as in a

clamped-clamped beam (point C).

Conversely, if the end rotation is initially negative (point a), a negative moment

will be required to bring the angle back to zero (point c). Thus for the set (u, 0) at least

two values of M exist. In fact, many other values of M could occur for (u, 0) as the

beam can theoretically buckled in higher modes (see Figure 4-5 ). Points E/e show the 2nd

64

mode buckled shapes. However, only two values are feasible in the actual system. For an

ideal beam, A moment interpolation function can then be composed of two surfaces

which are symmetric with respect to the axis u. If the system is moment-controlled, the

angle will snap at a critical moment, from point C/c to f/F. If the system is angle-

controlled, the moment will jump at a critical angle, from point D/d to b/B. Standard

structures that have snap-through behavior (arches, spherical caps, etc.) do not behave in

this way. Typically for every displacement there is a unique load; in a displacement-

controlled experiment no jump in load is observed.

The experimental results show and physics requires that the output rotational

angle is continuous. Thus for the amplifier driving a large inertial mass, the dynamic

structure is an angle-controlled system and a moment jump will happen for certain end

displacement and rotational angles.

4.3.2 Construction of a non-linear rotational spring

To construct a non-linear rotational spring, let the input end displacement u range

from 0.01% to 0.45% with a step size of 0.01%. For each u, the moment M is

determined numerically for a range of rotation angles using Equation 4.16. The

dependence of the driving moment on the beam’s end displacement and output rotational

angle is constructed by interpolating the 3-D surface, ),(ˆ uM eφ , with 5th order splines in

Mathematica 4.0. Figure 4-6 shows the non-linear rotational spring’s moment function

M with respect to the rotation φe and end displacement u for an ideal beam. The two

surfaces represent the two states of the moment for the same rotation and end

65

displacement. Since the present structure is an angle controlled system, a third 'toggle'

parameter is necessary to define the state of the system. If the previous dynamic state of

the beam is on one of the surfaces and the present rotation value is calculated beyond the

moment jump threshold, the dynamic state of system would jump to the other surface.

Using the interpolated moment surfaces of the non-linear rotational spring, the

equations of motion for the one-degree-of-freedom system can be written as

where Jh is the inertial mass at the pinned-end of the beam; c is the damping; φe is the

end rotation (rad); M is the dimensional moment acting on the inertial mass; Uo and Ua

are the static and time-varying components of the end displacement input, respectively; Ω

is the end displacement driving frequency; and dots represent a differentiation with

respect to time t. We non-dimensionalize the variables such that

Note that t in Equation 4.18 is defined differently than in Equation 4.3. Substituting

Equations 4.18 into 4.17, we obtain the non-dimensional equations of motion

])sin[),(()()( tUUtMtctJh aoeee Ω+=+ φφφ &&& (4.17)

JhLIEtt =ˆ ,

IEJhL

Ω=Ω , JhIE

Lcc =ˆ ,

LUu o

o = , L

Uu aa = , and

IELMM =ˆ

(4.18)

])ˆˆsin[),ˆ((ˆ)ˆ(ˆ)ˆ( tuutMtct aoeee Ω+=+ φφφ &&& (4.19)

66

By numerically integrating the equation of motion with certain initial end rotation )0(eφ

and angular velocity )0(eφ& , one can find the dynamic response of the output end of the

beam.

4.4 Model Validation

To validate the one-degree-of-freedom non-linear rotational spring model for a

buckling beam driving a large inertial mass, Equation 4.19 is solved by adopting the

parameters measured in experiments and the results are compared with the experimental

results. The parameters for the amplifier are listed in Table 4-1. The damping c is

determined empirically.

Table 4-1: Parameters for modeling validation

Parameters (Units) Value

Beam width b (mm) 12

Beam thickness h (mm) 0.508

Beam length L (mm) 60

Beam Young’s modulus E (GN/m2) 200

End inertial mass Jh (kg-m2) 5.386E-5

Damping c (m-N-s/rad) 2.154E-2

67

Note that data set C did not give good amplification and the driving amplitudes

are even smaller than the preloading level for some frequencies; this causes vibration

shocks acting back on the stack. Thus, in validating the model, only data sets A and B are

considered. The average values of preloading levels and end displacement amplitudes for

data sets A and B ( Table 4-2 ) are adopted for use in the model.

4.4.1 Model validation for an ideal beam

For an ideal beam, the one-degree-of-freedom beam-spring system is solved by

using the experimental parameters and driving conditions for data sets A and B listed in

Table 4-1 and 4-2. The simulated frequency responses for data sets A and B are shown in

Figure 4-7 (a) to (f), which are qualitatively and quantitatively close to the experimental

data shown in Figure 3-6 (a) to (f).

Table 4-2: Experimental driving conditions summary

Test # Preloading (uo %) (1-70 Hz)

Driving amplitude of end input peak-to-peak (2 x ua %) (1-70 Hz)

A1 0.194 0.1289

A2 0.1872 0.1138

A3 0.1827 0.0896

B1 0.1298 0.1221

B2 0.1233 0.0995

B3 0.1213 0.0832

68

The simulated results confirmed the softening spring behavior seen in test A1 and

also captured the period-n behavior seen in tests B1 and B2. To further study the period-n

region in test B1, the time responses for driving frequencies from 34 Hz to 42 Hz are

shown in Figure 4-8. The graphs on the left are the time responses and on the right are the

phase-plane portraits. For driving frequencies less than 34 Hz, the response of the end

rotation is period-1 motion. At driving frequency 35 Hz, the output response oscillates

about the positive equilibrium for a while, then oscillates about the negative equilibrium

for a while; this cycle repeats. From driving frequencies 36 to 41 Hz, period-n motions

are obtained in simulations. Although the moment function of a perfect beam is

symmetric with respect to axis u, the time response and phase-plane plots are generally

not because of preloading. The experimental time response and phase plots in Figure 3-10

and 3-8, respectively, also show asymmetric characteristics.

4.4.2 Model validation for an ideal beam with loading imperfection

Loading and geometric imperfections are studied to determine if the motion

amplifier is sensitive to the presence of imperfection. In the case of a loading

imperfection, one assumes the load acts at an offset e from the beam neutral axis; this

produces an external moment on the beam proportional to the load. For the ideal case, the

force is described by two symmetric surfaces (Figure 4-6). To calculate the driving

moment caused by the offset, one can add the moment produced by the perfect beam to

the axial force function times the small offset e. Figure 4-9 shows the simulated

69

frequency response results for an offset of 50 µm (0.083%). The other parameters and

driving conditions are exactly the same as those used in the last section.

The simulated results for an ideal beam with this loading imperfection are similar

to those for the ideal case. The frequency range for period-n motion response is slightly

narrower, but the trend of the output response over the resonance region is the same.

Figure 4-10 gives the time response and phase-plane for test B1 with a driving frequency

of 37 Hz; if compared with the same driving conditions for an ideal beam in Figure 4-8

(d), the system is more biased to oscillate about the positive (or negative) equilibrium

when a loading imperfection is present.

4.4.3 Model validation for a curved beam

In the case of geometric imperfection, the buckling beam is assumed to be a

shallow beam ( 0)ˆ( ≠siψ ) originally. The non-linear moment relations must be

recalculated for every different geometric imperfection.

The initial shape of the curved beam, which has an initial angle φo at the pinned

end, can be found by shooting methods illustrated in Figure 4-11. For simplicity, assume

)ˆ(sinitψ to be a three-quarter cosine function (Equation 4.20). The configuration of this

curved beam could be calculated by integrating

]ˆ2/3cos[)ˆ( ss initinit πφψ = (4.20)

)]ˆ(cos[)ˆ(')],ˆ(sin[)ˆ(' ssxssy initinitinitinit ψψ == 0)0(,0)0( == initinit xy

(4.21)

70

First, guess a φinit to get the function of ψinit using Equation 4.20; then, derive xinit

and yinit by integrating Equation 4.21 and find ls where the slope for the curve of xinit and

yinit is the same as the angle of ψinit(ls); finally, check whether φinit- ψinit(ls) is the specific

initial angle φo; if not, go back to the first step and guess another φinit, or if yes, the final

configuration (ψi, xi, and yi) is defined by

where ls is the length of the beam. The graphical relationship is shown in Figure 4-12.

Going through this procedure to find the initial shape function ψi one obtains also the

length of the beam ls. The curved beam length is scaled by ls, so that it has an initial unit

length.

After defining the curved beam, the first equation of Equation 4.16 is modified as

which gives the relations for p, v, M and φe. The relationship of M and φe for a curved

beam with φo=3o and the input end displacement u = 0.03% is shown Figure 4-13. For an

ideal beam, the relations between M and φe is anti-symmetric as shown in the last plot of

Figure 4-4 . However, for a curved beam, the anti-symmetric relation between M and φe

is broken; the curves are slightly shifted and tilted.

The numerical results for the curved beam are very close to those for an ideal

beam. However, the period-n motions in test B1 and B2 do not appear for the same

damping conditions ( 444.0ˆ =c ) as used in the ideal beam case, but appear for less

damping ( 276.0ˆ ≤c ).

)ˆ()](sin[)ˆ()](cos[)ˆ()ˆ()](sin[)ˆ()](cos[)ˆ(

sxlssylssysylssxlssx

initinitinitiniti

initinitinitiniti

ψψψψ

−=+=

(4.22)

)ˆ()]ˆ(cos[)]ˆ(sin[)ˆ('' ssvsps iψφφφ =−+ (4.23)

71

4.4.4 Discussion

To understand when imperfection causes a significant change from the ideal case,

a further study on preloading offset e is conducted. Figure 4-14 gives the frequency

response using the test A1 conditions for 10 different loading offsets from 50 µm to 500

µm with step size of 50 µm. The dependence of resonance frequency and maximum

peak-to-peak rotational angle on loading offset are illustrated in Figure 4-15. As the

offset increases, the resonance frequencies and maximum output increase as well. This is

reasonable if the static buckling behavior is considered; the beam buckles more when the

imperfection is present. The period-n motion also continues to appear at the resonance

region using test B1 conditions for all offsets studied between 50 µm and 500 µm.

The geometric loading imperfection was only studied using an initial shape with

an angle of 3o at the pinned end, since for each different initial shape, the moment

function needs to be recalculated. Since an initial shape with 3o slope at the pinned end is

relatively large, studies on different initial shape were not conducted.

In summary, the behavior with slight loading or geometric imperfection is similar

to the response for the ideal beam and the experimental results; thus the response is not

particularly sensitive to slight imperfection.

72

4.5 Parameter Studies

In this section, parameter studies on the dynamic behavior of the ideal buckling

beam amplifier are conducted using the experimentally-validated one-degree-of-freedom

beam-spring model. Four dimensionless parameters are selected for further study: stack

driving amplitude ua, end displacement preloading level uo, stack driving frequency Ω ,

and damping c . The beam’s physical parameters can be derived from those four

dimensionless parameters given the beam’s stiffness and length. Since the experimental

results for data set C are obviously not the driving conditions one wants for a motion

amplifier, a larger operational driving range is defined based on the driving conditions for

the experimental data sets A and B. Table 4-3 gives the ranges and step sizes of the

parameters in the study. For comparison with the experiments, the values given in the

parentheses for driving frequency Ω are the dimensional frequency with units of Hz. In

all, there were 148716 (27x18x6x51) combinations investigated.

Table 4-3: Parameter ranges and step sizes used in numerical tests

Parameter Start End Step size Test points number

Preloading level: uo (%) 0.1 0.36 0.01 27 Peak-to-peak driving amplitude: 2ua (%) 0.054 0.1475 0.0055 18

Damping: c 0.18 0.78 0.12 6

Driving frequency: Ω (f: Hz) 0.754 (10) 4.525 (60) 0.075 (1) 51

73

4.5.1 Finding period-1 output motion

For any buckling beam applied as a motion amplifier, different stack driving

conditions and damping values (bearing lubrication, friction, etc.) affect the output of the

amplifier. To operate the amplifier in period-1 motion is vital for most applications. Thus,

the operational region for any beam design to obtain period-1 motion response is to be

found first.

4.5.1.1 Effects of preloading level

A set of numerical tests with driving conditions close to the tests of largest driving

amplitudes in the experiments, are selected to show how the preloading effects the output

motion. For the same peak-to-peak driving amplitude (0.1255%) and damping (0.42), the

frequency response of the output with respect to preloading levels is illustrated in

Figure 4-16. The shaded area represents the period-n motion at the corresponding driving

frequency range. Beyond the shaded area, the response is period-1 motion. As the

preloading level increases, the frequency region for period-n motion decreases until it

disappears. The effect of preloading levels obtained numerically agrees with the

experimental results shown in Figure 3-9.

4.5.1.2 Effects of driving amplitude

To illustrate how the stack driving amplitude affects the output motion, Figure 4-

17 shows the type of output motion at different preloading level versus driving frequency

74

for different driving amplitudes. The damping in this case is 18.0ˆ =c . Each graph

represents one driving amplitude, which is given at the left top of each plot. One can see

that as the driving amplitude increases, the minimum preloading level needed to obtain

period-1 motion increases for the entire driving frequency range of interest; and for the

same preloading level, the higher the driving amplitude, the wider the period-n motion

frequency region gets.

4.5.1.3 Effects of damping

The effect of damping on the output motion is illustrated in Figure 4-18. For each

damping level, similar data as shown in Figure 4-17 are collected to find the minimum

preloading level so that the response is period-1 motion over the frequency range of

interest. Each curve in Figure 4-18 represents one damping level and divides the driving

condition space of preloading level and driving amplitude into two areas: the area above

the curve gives the period-1 motion response; and for some frequencies, period-n motions

appear for parameter combinations below the curve. As the damping increases, the area

for period-1 motion becomes larger. For example, Figure 4-16 and Figure 4-17 (14) gives

the effects of preloading level for stack peak-to-peak driving amplitude of 0.1255% at

damping levels of 0.42 and 0.18, respectively; for the same preloading level, the driving

frequency region of period-n response increases as the damping decreases.

75

4.5.2 Optimal period-1 rotation angle

Once the parameter space for period-1 output motion is defined, the optimal

period-1 output rotation angle of the buckling beam can be obtained. First, the period-1

motion driving amplitude and preloading levels are searched for different damping

conditions; then, the maximum rotation angle and corresponding driving frequency are

found.

The six graphs in Figure 4-19 show the maximum period-1 motion rotation angles

for six different damping estimates listed at the bottom of the graph. Beyond the dashed

black line, the parameter space gives period-n motions in the resonance region. The last

plot in Figure 4-19 is for the smallest damping 0.18. Because the damping is small, while

the frequency step size is relatively big, the resonance frequency might not be found

exactly. Thus, the curve of maximum rotation angle is not smooth; a smoother curve

would be obtained if a smaller frequency step size was used. Up to a static end

displacement of 0.36%, there always exist period-n motions for the parameter space when

the damping is 0.18 and the driving peak-to-peak amplitude is 0.1475%. As the damping

decreases, the output rotational angle increases. For each damping level, if a small

enough frequency step size is taken, the output might increase monotonically as the

preloading level decreases for the same driving amplitude. And for the same preloading

level, the output increases when the driving amplitude increases.

For any driving amplitude, one wants the lowest preloading level so that the

response is still period-1 motion. In other words, if the driving amplitude is fixed by the

76

stack or the beam, one needs to adjust the preloading level to be as small as possible until

period-n motion appears.

The optimal driving frequencies for damping 42.0ˆ =c are illustrated in Figure 4-

20. As the preloading level increases, the optimal driving frequency increases for the

same driving amplitude, this was predicted analytically in section 4.2.1 and validated in

experiments. Furthermore, for the same preloading level, when the driving amplitude

increases, the optimal driving frequency decreases.

77

Figure 4-1: Natural frequencies are plotted as functions of end-displacement, uo, for an ideal beam

78

1st mode

2nd mode

3rd mode

4th mode

Figure 4-2: First four mode shapes of end-displacement uo=0.3% for an ideal beam

79

Figure 4-3: Beam-spring model

80

Figure 4-4: Solutions for Equation 4.16 for u=0.03%

81

Figure 4-5: The relationship of moment M versus end rotation φe at a certain displacement u=1% for a buckled beam

Figure 4-6: Moment function of non-linear rotational spring for an ideal post-buckled beam in 3D configuration 82

82

83

(b) Test A2

(a) Test A1

84

(c) Test A3

(d) Test B1

85

(e) Test B2

(f) Test B3

Figure 4-7: Model validation of data sets A and B for an ideal beam

86

(a) Driving frequency f = 34 Hz

(b) Driving frequency f = 35 Hz

(c) Driving frequency f = 36 Hz

87

(d) Driving frequency f = 37 Hz

(e) Driving frequency f = 38 Hz

(f) Driving frequency f = 39 Hz

88

(g) Driving frequency f = 40 Hz

(h) Driving frequency f = 41 Hz

(i) Driving frequency f = 42 Hz

Figure 4-8: Simulated time responses graphs (left) and phase-plane portraits (right) for test B1 of driving frequencies 34 Hz ~ 42 Hz

89

(a) Test A1

(b) Test A2

90

(c) Test A3

(d) Test B1

91

(e) Test B2

(f) Test B3

Figure 4-9: Model validation of data sets A and B for an ideal beam with loading imperfection

92

Figure 4-10: Simulated time responses graph (left) and phase-plane portrait (right) for test

B1 of driving frequencies 37 Hz

Set φinit and assume initial shape as Equation 4.20

Solve Equation 4.21 and find ls which satisfies

)(])()(

[ lslsxlsy

ArcTan initinit

init ψ=

φo =φinit-ψinit(ls) ?

Objective ψi(0)=φo

Rotate the original axisψinit(ls) and get the new ψi, xi, and yi

Yes

No

Figure 4-11: Flow chart for find the free stress configuration of a curved beam which

has initial pinned-end angle of φo

93

Figure 4-12: Relations of (ψi, xi, and yi) and (ψinit, xinit, and yinit)

94

Figure 4-13: The relationships of moment M versus φe for a curved beam at the end displacement u=0.03%

95

Figure 4-14: Frequency response of test A1 for different loading imperfection offset e

96

Figure 4-15: The effects of loading imperfection offset e on resonance frequency and

maximum peak-to-peak output for test A1

97

Figure 4-16: The effects of preloading level for stack driving amplitude 0.1255% (peak-

to-peak) and damping 42.0ˆ =c

98

(1) (2)

(3) (4)

(5) (6)

99

(7) (8)

(9) (10)

(11) (12)

100

(13) (14)

(15) (16)

(17) (18)

Figure 4-17: Period-n motion driving condition for damping 18.0ˆ =c

101

Figure 4-18: Effects of damping on the output motion

102

103

104

Figure 4-19: Maximum period-1 motion rotation angles for different driving conditions

105

Figure 4-20: Maximum period-1 motion driving frequencies for different driving

conditions

Chapter 5

Summary and Future Work

5.1 Summary

A motion amplification concept has been proposed for obtaining large rotary

amplification from the small linear displacements generated by a stack. This motion

amplifier based on the response of a buckling beam is an easily-constructed, feasible

dynamic actuator. A prototype piezoelectric-stack-driven buckling beam actuator was

constructed. The actuator consisted of a 140 mm long pre-compressed PZT stack and a

60 mm x 12 mm x 0.508 mm thin steel beam. The beam served as a motion amplifier,

while the PZT stack provided the input actuation.

The amplifier using an axially-driven buckling beam has not only large motion

amplification in quasi-static operation, but much larger and stable motion amplification in

dynamic operation. Frequency responses of the system for three different preloading

levels (post-buckled, transition, and pre-buckled) and three different stack driving

amplitudes were experimentally obtained. If the period of output rotation is the same as

the driving period, the response is called period-1 motion, otherwise period-n motion or

sub-harmonic motion. When the beam was preloaded to the transition level, sub-

harmonic responses were observed at resonance of transition preloading level. A

maximum 16° peak-to-peak rotation of the period-1 motion output was achieved when

the stack was driven at an amplitude of 325 V and frequency of 39 Hz.

107

Most applications of motion amplifiers require an output motion with the same

period as the excitation; thus, it is vital to find operational conditions to avoid the sub-

harmonic motions. A simple one-degree-of-freedom nonlinear beam-spring model of the

actuator structure agrees with the experimental results to a high degree, thus validating its

fidelity. In this analytical model formulated in this thesis for the post-buckled beam, the

moment associated with the torsional beam-spring exhibits multi-value response

characteristics. Parameter studies using this beam-spring model for the ideal buckling

beam amplifier indicate the following behaviors:

• If period-1 motion is required over the entire driving frequency range of

interest, higher driving amplitudes require higher preloading levels for the

same damping;

• Increasing damping widens the regions for which period-1 motions occur,

but decreases the maximum period-1 motion output;

• To maximize period-1 motion, the preloading level must be as small as

possible for a fixed driving amplitude; the resonance frequency decreases

when the preloading level decreases for the same driving amplitude;

• Finally, increasing the driving amplitude decreases the resonance

frequency for the same preloading level.

The beam-spring model and the results of parameter studies can be used as

guidance for improving the design of the motion amplifier and finding the optimal

operational conditions for different applications. In other words, the beam dimensions

and profile can be tailored to meet the operating requirements of many applications by

using this beam-spring model.

108

5.2 Future Work

In this study, the single-degree-of-freedom beam-spring model neglects the

dynamics of the beam because the buckling beam is driving a large rotational mass.

Further studies need to address this assumption to determine how large the end mass must

be to be compared to the beam mass for the beam mass to be safely neglected. This must

be clarified both experimentally and numerically.

The loading imperfection offset study using the beam-spring model shows that as

the loading offset e increases, the resonance frequency and the maximum rotation angle

increase as well. Experiments on loading imperfection would help to verify the model

and obtain larger motion amplification.

In the modeling verification of geometric imperfection, only 3o of end rotation

angle for the initial shape of the beam is considered. Thus, more issues need to be studied

further, such as how much geometric imperfection causes a significant change so that the

model is no longer applicable, how the geometric profile of the beam affects the output,

or is there an optimal initial shape of the beam to get the maximum output.

Because of simplicity and large rotation amplification of the motion amplifier, it

can be applied to applications such as active flap rotor blades and active material motors.

In the future, the development of applications for the buckling beam motion amplifier is

needed.

It should be also noted that using a buckling beam as motion amplifier is not

limited to stack driven actuation. Electroactive polymers (EAP) or shape memory alloy

(SMA) wires could also be used to buckle the beam. Developing buckling beam motion

109

amplifiers using other actuation mechanisms other than piezo-electric stacks should be

tested simultaneously with the exploring of applications.

110

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Appendix

Programming of Beam-Spring Model in Mathematica

A.1 Single Static Solution for End Displacement u=0.29 (Example)

Off@General::"spell1"D;SetDirectory@"C:\Documents and Settings\Jie Jiang\My Documents\Beam Spring\IdealCase"D;

<<Graphics Graphics3D<<Graphics ParametricPlot3DNeeds@"Graphics Legend "D;H∗Needs@"Enhancements RootSearch "D;∗L

lengL@arg1_D:=Hp= arg1;ψsol= NDSolve@8ψ''@sD + pSin@ψ@sDD 0, y'@sD Sin@ψ@sDD, x'@sD Cos@ψ@sDD,

ψ@0D ψ0, ψ'@0D κ0,x@0D RCos@ψ0D, y@0D RSin@ψ0D<, 8ψ, x, y<, 8s, 0, 2<D;L= sê. FindRoot@HArcTan@y@sDêx@sDD ê. ψsolL@@1DD Hψ@sD ê. ψsolL@@1DD,8s,0.75, 1.25<DL

endstrain@arg2_, arg3_D:= Hψ0= arg2 π ê180; R =arg3;load= xxê. FindRoot@lengL@xxD 1, 8xx, 20,21<D;θ = Hψ@LD ê. ψsolL@@1DD;xb@s_D = Cos@θDx@sD+Sin@θDy@sD;yb@s_D = Cos@θDy@sD−Sin@θDx@sD;8strain= 100 HL+ R− Hxb@LD ê. ψsolL@@1DDLêL, Hψ0− θL180êπ,loadCos@θD,HHψ'@LD − κ0Lêxb@LDL ê. ψsol@@1DD<L

ψsolfind@varg_?NumericQ, κ0arg_?NumericQ, parg_?NumericQ, ψ0arg_, uarg_D:=Jp= parg;v =varg; κ0= κ0arg;ψ0 =

ψ0arg π

180;

ψsol= NDSolve@8ψ''@sD −vCos@ψ@sDD+ pSin@ψ@sDD 0,y'@sD Sin@ψ@sDD,x'@sD Cos@ψ@sDD, ψ@0D ψ0, ψ'@0D κ0,x@0D 0, y@0D 0<, 8ψ,x, y<, 8s,0,1<D;8Hψ'@1D − κ0−vx@1DL,100 H1−x@1DL −uarg, ψ@1D< ê. ψsol@@1DDN;H∗ The solution should satisfy moment balance, defined end displacement,

and defined rotation angle ∗L

118

fixedufcntry1@ψ0arg_, uarg_D:= Hsol= FindRoot@ψsolfind@xx, yy, zz, ψ0arg, uargD 80,0, 0<,8xx, vprev−vrange, vprev+vrange<, 8yy, κ0prev− κ0range, κ0prev+ κ0range<,8zz, pprev−prange, pprev+ prange<, MaxIterations→ 200D;H∗Print@ψ0arg," ",solD;∗Lvprev= xxê.sol;κ0prev =yy ê.sol; pprev= zzê. sol;8xx,yy,zz< ê. solL

fixedufcntry5@ψ0arg_,uarg_D:=Hvrange= Abs@Hvprev−voldprevLD; κ0range= Abs@Hκ0prev− κ0oldprevLD;prange= Abs@Hpprev− poldprevLD;voldprev= vprev;κ0oldprev = κ0prev; poldprev= pprev;aa= 8xx,vprev,vprev+vrange<;bb= 8yy, κ0prev−κ0range, κ0prev<;cc= 8zz, pprev−prange, pprev<;sol= FindRoot@ψsolfind@xx, yy, zz, ψ0arg,uargD 80, 0, 0<,8xx,vprev,vprev+vrange<, 8yy, κ0prev− κ0range, κ0prev<,8zz, pprev−prange, pprev<, MaxIterations→ 200D;H∗If@Mod@Hii−startpoint@@2DDLê0.5,1D 0,Print@iiDD;∗Lvprev= xxê. sol;κ0prev =yy ê.sol; pprev= zzê. sol;8xx,yy,zz< ê.solL

myu= 0.29;κ0= 0;xx =xx ê.FindRoot@endstrain@xx,0D@@1DD myu, 8xx, 1,13<D;startpoint= endstrain@xx,0D80.29, 7.70004, 20.2126, 0.485559<

tab0= 88startpoint@@2DD, 8startpoint@@4DD, κ0,startpoint@@3DD<<<;Clear@κ0D;vrange= 0.1;κ0range =0.1; prange= 1;vprev= startpoint@@4DD; κ0prev =0; pprev= startpoint@@3DD;tab1= Table@8ii, fixedufcntry1@ii, myuD<,8ii, startpoint@@2DD −0.1, startpoint@@2DD−0.2, −0.1<Dvoldprev= tab1@@1, 2DD@@1DD;κ0oldprev =tab1@@1, 2DD@@2DD;poldprev= tab1@@1, 2DD@@3DD;887.60004, 80.472635, 0.0149927, 20.5604<<, 87.50004, 80.459915, 0.0298008, 20.904<<<

ForAcutix= Dimensions@tab2D@@1DD−20, cutix ≤ Dimensions@tab2D@@1DD,IfAtab2@@cutix,2DD@@2DD < 0 »» ik tab2@@cutix, 2DD

tab2@@cutix−1, 2DD y@@2DD >1, Break@DE;cutix++E;tab3= Join@tab0,tab1, Take@tab2,cutix−1DD;

119 plot1v= ListPlot@Table@8tab9@@ii,1DD,tab9@@ii,2DD@@1DD<,8ii,1, Dimensions@tab9D@@1DD<D,PlotJoined→ True,PlotStyle→ RGBColor@1,0, 0D,

DisplayFunction→ Identity,AxesLabel−> "v",PlotRange→ AllD;...plot2p= ListPlot@Table@8−tab9@@ii,1DD, tab9@@ii,2DD@@3DD<,8ii,1, Dimensions@tab9D@@1DD<D,PlotJoined→ True,PlotStyle→ RGBColor@0,0, 1D,

DisplayFunction→ Identity,AxesLabel−> "p",PlotRange→ AllD;Show@GraphicsArray@8Show@plot1v, plot2vD,Show@plot1k, plot2kD,

Show@plot1p, plot2pD<DD;

-6 -4 -2 2 4 6

-1

-0.5

0.5

1

v

-6 -4 -2 2 4 6

-1

-0.5

0.5

1

k

-6 -4 -2 2 4 6

-80-60-40-20

204060p

120

A.2 3-D Surface Construction

A.2.1 Data processing

nd1= 23ê24;nd2 =11ê12; m1 =100; m2= 100;m3 =100; m4= 200;

<<tabmyu1.mxmyv= Table@8tab9@@ii, 1DD, tab9@@ii, 2DD@@1DD<, 8ii,1, Dimensions@tab9D@@1DD<D;myκ = Table@8tab9@@ii,1DD,tab9@@ii,2DD@@2DD<, 8ii,1, Dimensions@tab9D@@1DD<D;myp= Table@8tab9@@ii, 1DD, tab9@@ii, 2DD@@3DD<, 8ii,1, Dimensions@tab9D@@1DD<D;myvfun= Interpolation@myvD;myκfun= Interpolation@myκD;mypfun= Interpolation@mypD;maxψ0= Max@Table@tab9@@ii,1DD, 8ii,1, Dimensions@tab9D@@1DD<DD;minψ0= Min@Table@tab9@@ii,1DD, 8ii,1, Dimensions@tab9D@@1DD<DD;d1=

nd1 minψ0− minψ0m1

; d2=nd2minψ0−nd1 minψ0

m2;d3 =

0−nd2 minψ0m3

; d4=maxψ0m4

;

newtab1=

Join@Table@8myu, 8ii, myvfun@iiD, myκfun@iiD, mypfun@iiD<<, 8ii, minψ0,nd1minψ0,d1<D,Table@8myu, 8ii, myvfun@iiD, myκfun@iiD, mypfun@iiD<<, 8ii, nd1minψ0+d2,nd2minψ0, d2<D,88myu, 8nd2minψ0, myvfun@nd2minψ0D, myκfun@nd2 minψ0D, mypfun@nd2 minψ0D<<<,Table@8myu, 8ii, myvfun@iiD, myκfun@iiD, mypfun@iiD<<, 8ii, nd2minψ0+d3,0, d3<D,Table@8myu, 8ii, myvfun@iiD, myκfun@iiD, mypfun@iiD<<, 8ii, d4, maxψ0,d4<DD;

DumpSave@"newtab1.mx",newtab1D;

……

vecv1= Table@8newtab1@@ii, 1DD,newtab1@@ii, 2DD@@1DD, newtab1@@ii, 2DD@@2DD<,8ii,1, Dimensions@newtab1D@@1DD<D;vecκ1= Table@8newtab1@@ii, 1DD,newtab1@@ii, 2DD@@1DD, newtab1@@ii, 2DD@@3DD<,8ii,1, Dimensions@newtab1D@@1DD<D;vecp1= Table@8newtab1@@ii, 1DD,newtab1@@ii, 2DD@@1DD, newtab1@@ii, 2DD@@4DD<,8ii,1, Dimensions@newtab1D@@1DD<D;

……

121 matrixv= 8vecv1,vecv2,vecv3,vecv4,vecv5,vecv6, vecv7,vecv8,vecv9,vecv10,

vecv11,vecv12,vecv13,vecv14,vecv15,vecv16, vecv17,vecv18,vecv19,vecv20,vecv21,vecv22,vecv23,vecv24,vecv25,vecv26, vecv27,vecv28,vecv29,vecv30,vecv31,vecv32,vecv33,vecv34,vecv35,vecv36, vecv37,vecv38,vecv39,vecv40,vecv41,vecv42,vecv43,vecv44,vecv45<;

matrixκ = 8vecκ1,vecκ2, vecκ3,vecκ4,vecκ5, vecκ6, vecκ7, vecκ8, vecκ9,vecκ10,vecκ11, vecκ12,vecκ13,vecκ14,vecκ15,vecκ16, vecκ17, vecκ18, vecκ19,vecκ20,vecκ21, vecκ22,vecκ23,vecκ24,vecκ25,vecκ26, vecκ27, vecκ28, vecκ29,vecκ30,vecκ31, vecκ32,vecκ33,vecκ34,vecκ35,vecκ36, vecκ37, vecκ38, vecκ39,vecκ40,vecκ41, vecκ42,vecκ43,vecκ44,vecκ45<;

matrixp= 8vecp1,vecp2,vecp3,vecp4,vecp5,vecp6, vecp7,vecp8,vecp9,vecp10,vecp11,vecp12,vecp13,vecp14,vecp15,vecp16, vecp17,vecp18,vecp19,vecp20,vecp21,vecp22,vecp23,vecp24,vecp25,vecp26, vecp27,vecp28,vecp29,vecp30,vecp31,vecp32,vecp33,vecp34,vecp35,vecp36, vecp37,vecp38,vecp39,vecp40,vecp41,vecp42,vecp43,vecp44,vecp45<;

obvecv1= Table@8newtab1@@ii,1DD, newtab1@@ii, 2DD@@1DD, newtab1@@ii, 2DD@@2DD<,8ii, 1, Dimensions@newtab1D@@1DD<D;obvecκ1= Table@8newtab1@@ii,1DD, −newtab1@@ii, 2DD@@1DD, −newtab1@@ii, 2DD@@3DD<,8ii, 1, Dimensions@newtab1D@@1DD<D;obvecp1= Table@8newtab1@@ii,1DD, −newtab1@@ii, 2DD@@1DD, newtab1@@ii, 2DD@@4DD<,8ii, 1, Dimensions@newtab1D@@1DD<D;

……

obmatrixv= 8obvecv1,obvecv2, obvecv3, obvecv4,obvecv5, obvecv6, obvecv7, obvecv8,obvecv9, obvecv10, obvecv11, obvecv12,obvecv13, obvecv14, obvecv15, obvecv16,obvecv17,obvecv18, obvecv19, obvecv20,obvecv21, obvecv22, obvecv23, obvecv24,obvecv25,obvecv26, obvecv27, obvecv28,obvecv29, obvecv30, obvecv31, obvecv32,obvecv33,obvecv34, obvecv35, obvecv36,obvecv37, obvecv38, obvecv39, obvecv40,obvecv41,obvecv42, obvecv43, obvecv44,obvecv45<;

obmatrixκ = 8obvecκ1,obvecκ2, obvecκ3, obvecκ4,obvecκ5,obvecκ6, obvecκ7, obvecκ8,obvecκ9, obvecκ10, obvecκ11, obvecκ12,obvecκ13, obvecκ14, obvecκ15, obvecκ16,obvecκ17,obvecκ18, obvecκ19, obvecκ20,obvecκ21, obvecκ22, obvecκ23, obvecκ24,obvecκ25,obvecκ26, obvecκ27, obvecκ28,obvecκ29, obvecκ30, obvecκ31, obvecκ32,obvecκ33,obvecκ34, obvecκ35, obvecκ36,obvecκ37, obvecκ38, obvecκ39, obvecκ40,obvecκ41,obvecκ42, obvecκ43, obvecκ44,obvecκ45<;

obmatrixp= 8obvecp1,obvecp2, obvecp3, obvecp4,obvecp5, obvecp6, obvecp7, obvecp8,obvecp9, obvecp10, obvecp11, obvecp12,obvecp13, obvecp14, obvecp15, obvecp16,obvecp17,obvecp18, obvecp19, obvecp20,obvecp21, obvecp22, obvecp23, obvecp24,obvecp25,obvecp26, obvecp27, obvecp28,obvecp29, obvecp30, obvecp31, obvecp32,obvecp33,obvecp34, obvecp35, obvecp36,obvecp37, obvecp38, obvecp39, obvecp40,obvecp41,obvecp42, obvecp43, obvecp44,obvecp45<;

122

A.2.2 Data interpolation

tabθs1= Table@8jj,ii, matrixκ@@jj,ii,2DD<, 8ii, 1, Dimensions@matrixκD@@2DD<,8jj,1, Dimensions@matrixκD@@1DD<D;tabus1= Table@8jj,ii, matrixκ@@jj,ii,1DD<, 8ii, 1, Dimensions@matrixκD@@2DD<,8jj,1, Dimensions@matrixκD@@1DD<D;tabκs1= Table@8jj,ii, matrixκ@@jj,ii,3DD<, 8ii, 1, Dimensions@matrixκD@@2DD<,8jj,1, Dimensions@matrixκD@@1DD<D;tabps1= Table@8jj,ii, matrixp@@jj,ii,3DD<, 8ii, 1, Dimensions@matrixκD@@2DD<,8jj,1, Dimensions@matrixκD@@1DD<D;tabvs1= Table@8jj,ii, matrixv@@jj,ii,3DD<, 8ii, 1, Dimensions@matrixκD@@2DD<,8jj,1, Dimensions@matrixκD@@1DD<D;obtabθs1= Table@8jj,ii,obmatrixκ@@jj,ii,2DD<, 8ii, 1, Dimensions@obmatrixκD@@2DD<,8jj,1, Dimensions@matrixκD@@1DD<D;obtabus1= Table@8jj,ii,obmatrixκ@@jj,ii,1DD<, 8ii, 1, Dimensions@obmatrixκD@@2DD<,8jj,1, Dimensions@matrixκD@@1DD<D;obtabκs1= Table@8jj,ii,obmatrixκ@@jj,ii,3DD<, 8ii, 1, Dimensions@obmatrixκD@@2DD<,8jj,1, Dimensions@matrixκD@@1DD<D;obtabps1= Table@8jj,ii,obmatrixp@@jj,ii,3DD<, 8ii, 1, Dimensions@obmatrixκD@@2DD<,8jj,1, Dimensions@matrixκD@@1DD<D;obtabvs1= Table@8jj,ii,obmatrixv@@jj,ii,3DD<, 8ii, 1, Dimensions@obmatrixκD@@2DD<,8jj,1, Dimensions@matrixκD@@1DD<D;snapθ1= Table@8matrixκ@@ii,1,1DD, matrixκ@@ii, 1, 2DD<,8ii,1, Dimensions@matrixκD@@1DD<D;

intθs1= Interpolation@Flatten@tabθs1, 1D,InterpolationOrder→ 5D;intus1= Interpolation@Flatten@tabus1, 1D,InterpolationOrder→ 5D;intκs1= Interpolation@Flatten@tabκs1, 1D,InterpolationOrder→ 5D;intps1= Interpolation@Flatten@tabps1, 1D,InterpolationOrder→ 5D;intvs1= Interpolation@Flatten@tabvs1, 1D,InterpolationOrder→ 5D;obintθs1= Interpolation@Flatten@obtabθs1,1D, InterpolationOrder→ 5D;obintus1= Interpolation@Flatten@obtabus1,1D, InterpolationOrder→ 5D;obintκs1= Interpolation@Flatten@obtabκs1,1D, InterpolationOrder→ 5D;obintps1= Interpolation@Flatten@obtabps1,1D, InterpolationOrder→ 5D;obintvs1= Interpolation@Flatten@obtabvs1,1D, InterpolationOrder→ 5D;intsnapθ1= Interpolation@snapθ1,InterpolationOrder→ 5D

123

A.2.3 Moment function

A.2.3.1 Ideal beam

moment@θarg_?NumericQ,uarg_?NumericQD :=Hii= Huarg−0.01Lê0.01+1;If@flag 1 &&θarg <intsnapθ1@uargD,flag =2D;If@flag 2 &&θarg > −intsnapθ1@uargD,flag =1D;H∗Print@"flag = ",flagD;∗LIf@flag== 1,8jsol= FindRoot@intθs1@ii,jjD θarg, 8jj, 81, Dimensions@tabκs1D@@1DD<<D;myκ = intκs1@ii,jjê.jsolD<,8jsol= FindRoot@obintθs1@ii,jjD θarg, 8jj, 81, Dimensions@tabκs1D@@1DD<<D;myκ = obintκs1@ii,jjê.jsolD<D;H∗Print@"jj = ",jjê.jsolD;∗L

myκL

A.2.3.2 Loading imperfection

momentimp@θarg_?NumericQ,uarg_?NumericQD :=Jii=

uarg−0.010.01

+1;

If@flag 1 &&θarg <intsnapθ1@uargD, flag =2D;If@flag 2 &&θarg > −intsnapθ1@uargD, flag =1D;H∗Print@"flag = ",flagD;∗LIf@flag== 1,8jsol= FindRoot@intθs1@ii,jjD θarg, 8jj, 81, Dimensions@tabκs1D@@1DD<<D;myκ = intκs1@ii, jjê.jsolD +eintps1@ii, jjê.jsolD<,8jsol= FindRoot@obintθs1@ii,jjD θarg, 8jj, 81, Dimensions@tabκs1D@@1DD<<D;myκ = obintκs1@ii,jjê. jsolD +eobintps1@ii, jjê.jsolD<D;H∗Print@"jj = ",jjê.jsolD;∗L

myκN

124

A.3 Dynamic Modeling

freqrespond@uinit_, upp_, freq_,damping_,ncycle_D :=ik

time1 = TimeUsed@D;ii=

uinit−0.010.01

+1;

jsol= FindRoot@intκs1@ii, jjD 0, 8jj, 81, Dimensions@tabκs1D@@1DD<<D;θinit= intθs1@ii, jjê. jsolD πê180;flag= 1;

tend1=tratiofreq

ncycle;

sol1= NDSolveA9θs''@tD +dampingtratio

θs'@tD cc momentAθs@tD180ê π, uinit+uppê2 SinA2 freq π

tratio tEE,

θs@0D θinit, θs'@0D 0=, θs, 8t, 0, tend1<,Compiled→ False, MaxSteps→ 5000000E;tstablestart= tend1−

tratiofreq

3;

nn= 4000;

temp= TableA180π

θs@ttD ê.sol1@@1DD, 9tt,tstablestart, tend1, tend1−tstablestartnn

=E;PtoP= Max@tempD− Min@tempD;time2 = TimeUsed@D;Print@"freq = ", freq, ", PtoP = ",PtoP, ", Time used = ", time2−time1D;8freq, Max@tempD, Min@tempD,PtoP<y

ifreq = 36;freqrespond@0.1298,0.1221,ifreq, 40,80Dfreq = 36, PtoP = 9.48254, Time used = 28.063836, 0.311274, −9.17126, 9.48254<

PlotA180π

θs@ttD ê.sol1@@1DD, 9tt, tratioifreq

40,tratioifreq

80=,PlotPoints→ 10000,

PlotStyle→ RGBColor@1, 0,0DE;

120 140 160 180 200

-10

-5

5

10

125

ParametricPlotA9180π

θs@ttD ê. sol1@@1DD, θs'@ttD ê.sol1@@1DD=, 9tt, tratioifreq

40,tratioifreq

80=,PlotStyle→ RGBColor@1, 0, 0D, PlotRange→ AllE;

-10 -5 5 10

-0.3

-0.2

-0.1

0.1

0.2

0.3

VITA

Jie Jiang

Education The Pennsylvania State University, University Park, PA 16803 Ph.D. in Mechanical Engineering, 2004 East China University of Science and Technology, Shanghai, China M.S. in Mechanical Engineering, 1996 Nanjing University of Chemical Technology, Nanjing, China B.S. in Mechanical Engineering, 1993

Experience The Pennsylvania State University, Aug. 1999-May 2004 Center for Acoustics and Vibration, Aug. 1999-Aug. 2002 Dynamic & Structural Stability Laboratory, Sept. 2002 – May 2004 Department of Mechanical and Nuclear Engineering Graduate Research Assistant East China University of Science and Technology, Sept. 1993 – July 1996 Department of Mechanical Engineering Research Assistant

Publication Jiang, J.; Mockensturm, EM, “Non-linear dynamics of buckling beam actuators”, 45th AIAA/ASME/ASCE/AHS Structures, Structural Dynamics, and Materials Conference, Palm Springs, California, 2004, AIAA-2004-1866 Jiang, J.; Mockensturm, EM, “A novel motion amplifier using axially driven buckling beam”, Proceedings of IMECE’03, 2003 ASME International Mechanical Engineering Congress & Exposition, Washington, D.C., November 16-21, 2003, IMECE2003-42317

Mockensturm, EM; Jiang, J., “Active Rotors and Motors Using Buckling Beam Amplifiers”, 2002 United States National Congress of Theoretical and Applied Mechanics Chen, W.; Jiang, J.; Zhang, D., “The development of HL fault detecting system based on neural networks”, Process equipment technology, 1996, n3 Chen, W.; Jiang, J.; Zhang, D., “HL fault detecting system based on fuzzy neural networks”, Process equipment design, 1996, v33, n2, p35-36.