a comparative study of vibration analysis of piezoelectrically …rx... · 2019. 2. 13. · a...

A Comparative Study of Vibration Analysis of

Piezoelectrically-Actuated Cantilever Beam Systems

under Different Modeling Frameworks

A Thesis Presented

By

Payman Zolmajd

To

The Department of Mechanical and Industrial Engineering

in partial fulfillment of the requirements

for the degree of

Master of Science

in the field of

Mechanical Engineering

Northeastern University

Boston, Massachusetts

August 2015

1

Contents

Abstract ………………………………………………………………………………...... 3

1 Piezoelectric Materials ……………………………………………………………….. 3

1.1 Overall Concept …………………………………………………………............. 3

1.2 History ………………………………………………………………………….... 3

1.3 General Categories ………………………………………………………………. 4

1.4 Materials …………………………………………………………………………. 4

1.5 How it works …………………………………………………………………….. 4

1.6 Practical applications ……………………………………………………………. 5

1.6.1 Tennis Racquet Case Study ………………………………………………. 6

1.6.2 Wind Power Generator …………………………………………………… 6

1.6.3 Knock Sensors ……………………………………………………………. 7

1.6.4 Tuned Mass Damper ……………………………………………………… 7

1.6.5 Nano-Mechanical Cantilever (NMC) probes …………………………….. 8

2 Energy Method for Piezoelectric Materials ………………………………………….. 9

2.1 Electrical Potential Energy ……………………………………………………… 9

2.2 Definition of Material Constants ………………………………………………... 12

2.3 Piezoelectric Constants …………………………………………………………. 13

3 Piezoelectric-Based System Modeling ………………………………………………. 15

3.1 Modeling Assumptions and Preliminaries ………………………………………. 15

3.2 Modeling Piezoelectric Actuators in Transverse Configuration ………………… 15

3.3 Piezoelectric-Based Cantilever Beam Modeling – Euler Bernoulli Theory ……… 18

3.4 Piezoelectric-Based Cantilever Beam Modeling – Rayleigh Theor ……………… 30

3.5 Piezoelectric-Based Cantilever Beam Modeling – Timoshenko Theory ……........ 43

4 Numerical Results ……………………………………………………………………. 68

4.1 Calculation of and values ……………………………………………………. 68

4.2 Calculation of eigenfunction coefficients ………………………………………… 71

4.3 Mode shapes ……………………………………………………………………… 73

4.4 Time and frequency domain ……………………………………………………... 76

5 Conclusions …………………………………………………………………………… 85

List of Symbols …………………………………………………………………………. 86

Appendix A1- Finding equations (3.15) and (3.17) ………..……………………………. 88

Appendix A2- Finding equations (3.37a-l) …….………………………………..………. 94

Appendix A3- Finding equations (3.69) and (3.71) ……………………………..………. 96

Appendix A4- Solution of equation (3.82) ………………………………………………. 102

Appendix A5- Finding equations (3.88a-l) ….………………………………...…………. 103

2

Appendix A6- Finding equation (3.89) …………………………………………………. 106

Appendix A7- Finding equations (3.123), (3.125) and (3.128a) …………..……………. 108

Appendix A8- Solution of equation (3.141) ….…………………………………………. 115

Appendix A9- Finding equations (3.150a-l) .……………………………………………. 117

Appendix A10- Finding equation (3.151) ..……………………………………………… 120

Appendix A11- Solution of equation (3.155) ….………………………………………... 122

Appendix A12- Finding equations (3.164a-l) ….………………………………………... 124

Appendix B- MATLAB codes ..………………………………………………………..... 128

References ………………………………………………………………………………. 173

3

Abstract

In this research a comprehensive modeling framework for a piezoelectrically-actuated

cantilever beam is developed and a detailed model and vibration analyses is performed. To

achieve this goal, the governing dynamics for the system as well as boundary conditions are

derived using the extended Hamilton’s principle. The equations of motion of cantilever beam are

derived according to the Euler-Bernoulli, Rayleigh and Timoshenko theories separately. The

Euler-Bernoulli theory neglects the effects of rotary inertia and shear deformation and is only

applicable to analysis of thin beams. The Rayleigh theory considers the effect of rotary inertia,

while the Timoshenko theory considers the effects of both rotary inertia and shear deformation

for thick beams. It is evident from the nature of discontinuous geometry of system, equation of

stress-strain relationship are modified as shown in theory subsection meanwhile the natural

surface in the composite (beam-piezoelectric layer) portion of the cantilever beam must be

considered in this stage of calculations. Then the first five natural frequencies of this composite

system are obtained by those three different theories and the results are compared. Relevant

mode shapes are also drawn and effects of including rotary inertia and shear deformation are

discussed for slender and stocky beams. Then, the forced vibration problem is solved and the

cantilever tip deflection is obtained in which applied voltage to the piezoelectric layer is

considered to be a unit-step input. The results are compared again for slender and stocky beams.

1. Piezoelectric Materials

1.1 Overall Concept

“Piezoelectricity, also called the piezoelectric effect, is the ability of certain materials to

generate an AC (alternating current) voltage when subjected to mechanical stress or vibration, or

to vibrate when subjected to an AC voltage, or both”,[26]. “The piezoelectric effect is

understood as the linear electromechanical interaction between the mechanical and the electrical

state in crystalline materials with no inversion symmetry. The piezoelectric effect is a reversible

process in that materials exhibiting the direct piezoelectric effect (the internal generation of

electrical charge resulting from an applied mechanical force) also exhibit the reverse

piezoelectric effect (the internal generation of a mechanical strain resulting from an applied

electrical field)”,[25]. In other words, Piezoelectricity refers to an electromechanical

phenomenon in particular solid-state materials that demonstrate a coupling between their

electrical, mechanical, and thermal states generated by applying mechanical stress to dielectric

crystals. The word piezoelectricity means electricity resulting from pressure. “It is derived from

the Greek piezo or piezein, which means to squeeze or press, and electric or electron, which

means amber, an ancient source of electric charge”,[25].

1.2 History

Piezoelectricity was discovered in 1880 by French physicist Jacques and Pierre Curie. They

combined what they knew about pyroelectricity and about structures of crystals to demonstrate

the effect with tourmaline, quartz, topaz, cane sugar and Rochelle salt. The converse effect

however was discovered later by Gabriel Lippmann in 1881 through the mathematical aspect of

http://whatis.techtarget.com/definition/alternating-current-AC

http://searchcio-midmarket.techtarget.com/definition/voltage

http://en.wikipedia.org/wiki/Piezoelectricity#Mechanism

http://en.wikipedia.org/wiki/Centrosymmetry

http://en.wikipedia.org/wiki/Reversible_process_(thermodynamics)

http://en.wikipedia.org/wiki/Reversible_process_(thermodynamics)

http://en.wikipedia.org/wiki/Force

http://en.wikipedia.org/wiki/Greek_language

http://en.wikipedia.org/wiki/Amber

http://en.wikipedia.org/wiki/Pierre_Curie

4

theory. These behaviors were labeled the piezoelectric effect and the inverse piezoelectric effect

respectively.

1.3 General Categories

“Piezoelectric devices fit into four general categories, depending of what type of physical

effect is used: generators, sensors, actuators, and transducers. Generators and sensors make use

of the direct piezoelectric effect, meaning that mechanical energy is transformed into a dielectric

displacement. This, in turn, is measurable as a charge or voltage signal between the metallized

surfaces of the piezoelectric material. Actuators work vice-versa when transforming electrical

energy into mechanical by means of the inverse piezoelectric effect. Finally, in transducers both

effects are used within one and the same device. For all of these basic functionalities, different

designs are available”,[28].

1.4 Materials

The most commonly known piezoelectric material is quartz. But piezoelectric materials are

numerous, the most used are:

Aluminium nitride

Apatite

Barium titanate

Bimorph

Gallium phosphate

Lanthanum gallium silicate

Lead scandium tantalate

Lead zirconate titanate

Lithium tantalate

Piezoelectric accelerometer

Polyvinylidene fluoride

Potassium sodium tartrate

Quartz

Unimorph

1.5 How it works

1) “Normally, the charges in a piezoelectric crystal are exactly balanced, even if they're not

symmetrically arranged.

2) The effects of the charges exactly cancel out, leaving no net charge on the crystal faces.

(More specifically, the electric dipole moments—vector lines separating opposite

charges—exactly cancel one another out.)

3) If you squeeze the crystal, you force the charges out of balance.

http://en.wikipedia.org/wiki/Aluminium_nitride

http://en.wikipedia.org/wiki/Apatite

http://en.wikipedia.org/wiki/Barium_titanate

http://en.wikipedia.org/wiki/Bimorph

http://en.wikipedia.org/wiki/Gallium_phosphate

http://en.wikipedia.org/wiki/Lanthanum_gallium_silicate

http://en.wikipedia.org/wiki/Lead_scandium_tantalate

http://en.wikipedia.org/wiki/Lead_zirconate_titanate

http://en.wikipedia.org/wiki/Lithium_tantalate

http://en.wikipedia.org/wiki/Piezoelectric_accelerometer

http://en.wikipedia.org/wiki/Polyvinylidene_fluoride

http://en.wikipedia.org/wiki/Potassium_sodium_tartrate

http://en.wikipedia.org/wiki/Quartz

http://en.wikipedia.org/wiki/Unimorph

5

4) Now the effects of the charges (their dipole moments) no longer cancel one another out

and net positive and negative charges appear on opposite crystal faces. By squeezing the

crystal, you've produced a voltage across its opposite faces—and that's piezoelectric

effect”,[27]. (Fig. 1.1)

“Fig. 1.1 Piezoelectric effect in quartz”,[24] with permission

1.6 Practical applications

“One of the first applications of the piezoelectric effect was an ultrasonic submarine detector

developed during the First World War. A mosaic of thin quartz crystals glued between two steel

plates acted as a transducer that resonated at 50MHz. By submerging the device and applying a

voltage they succeeded in emitting a high frequency 'chirp' underwater, which enabled them to

measure the depth by timing the return echo. This was the basis for sonar and the development

encouraged other applications using piezoelectric devices both resonating and non-resonating

such as microphones, signal filters and ultrasonic transducers. However many devices were not

commercially viable due to the limited performance of the materials at the time”,[20].

“The continued development of piezoelectric materials has led to a huge market of products

ranging from those for everyday use to more specialized devices. Some typical applications can

be seen below:

Automotive: Air bag sensor, air flow sensor, audible alarms, fuel atomiser,

keyless door entry, seat belt buzzers, knock sensors.

Computer: Disc drives, inkjet printers.

Consumer: Cigarette lighters, depth finders, fish finders, humidifiers,

jewellery cleaners, musical instruments, speakers, telephones.

Medical: Disposable patient monitors, foetal heart monitors, ultrasonic imaging.

Military: Depth sounders, guidance systems, hydrophones, sonar”,[20].

6

1.6.1 Tennis Racquet Case Study

“A more recent innovation using piezoelectric technology is in the sports industry. Tennis

manufacturers, Head, were requested by players to design racquets with comfort as well as

power. Previously, racquets had been designed to be stiff so that they return maximum energy to

the ball when it is hit but this means that the racquet transmits shock vibration to the players

arm.

In an attempt to reduce vibration, piezoelectric fibers have been embedded around the racquet

throat and a computer chip embedded inside the handle (Fig. 1.2). The frame deflects slightly

when the ball is hit so that the piezoelectric fibers bend and generate a charge (by the direct

effect) which is collected by the patterned electrode surrounding the fibers. The charge and

associated current is carried to an embedded silicon chip via a flexible circuit containing

inductors capacitors and resistors, which boost the current and send it back to the fibers out of

phase in an attempt to reduce the vibration by destructive interference.

The fibers then bend (by the converse effect) to counter the motion of the racket and reduce

vibration. The current generated is said to be only a couple of hundred micro amps generating

600 to 800 volts in only 2 to 3 milliseconds.

The manufacturers claim 50% reduction in vibration compared with conventional rackets and

the International Tennis Federation have approved them for tournament play”,[20].

“Fig. 1.2 Piezoelectric fibers on tennis racquet”,[20] with permission

1.6.2 Wind Power Generator

“This low cast solid state wind power generator turns the flexing of an omnidirectional shaft

directly into electricity, using piezoelectric materials (Fig. 1.3).

7

“Fig. 1.3 Wind power generator”,[21] with permission

A tall flexible stalk is surrounded with many embedded piezoelectric discs that are alternately

sandwiched in-between rigid backup plates. These piezoelectric structures (toroids) compress

and stretch when flexed in any direction, converting any motion directly into electricity with no

intermediary mechanical generators, transmissions or propellers. A weighted wind-capturing tip

can sustain the energy output from a single gust of wind by the continuing oscillation of this

inverted pendulum after the gust fades. In light winds the power extraction would be maximized

while remaining robust in high winds”,[21].

1.6.3 Knock Sensors

“Knock sensors are placed near the engine in order to detect irregular combustions. The

measurement principle is the one also used in accelerometers. The piezoelectric material is

placed between the vibrating structure and a seismic mass introducing the vibration forces into

the piezo element. The piezo element itself converts the vibrations into an electric charge

proportional to the applied force. Usually, piezoelectric ceramics (PZT) with specially tailored

properties are used. The material has to withstand high temperatures (up to 200°C) as well as

rapid temperature changes. Also, the piezoelectric coefficient of the material must be almost

independent of the temperature and remain stable over the vehicle’s lifetime. Only recently, first

attempts were made to replace PZT by thin PVDF foil sensors”,[28].

1.6.4 Tuned Mass Damper

A tuned mass damper (TDM) is composed of a mass, a spring and a damper and is supposed

to reduce the dynamic response of the structure to which it is attached. The values of the mass,

spring and damper of TDM are calculated so that the TDM will resonate out of phase with the

structure when exited by an external loading. It is basically a damping system that minimizes the

displacement of the main mass with a combination of both its spring and its viscous damping.

8

1.6.5 Nano-Mechanical Cantilever (NMC) probes

“NMC probes have recently attracted widespread attention in variety of applications including

atomic force and friction microscopy, biomass sensing, thermal scanning microscopy and

MEMS switches. For instance, in the Atomic Force Microscopy (AFM), the NMC oscillates at or

near its resonant frequency (Fig. 1.4). The shift in the natural frequency due to the tip-sample

interaction is used to quantitatively characterize the topography of the surface. In the biosensing

applications, the NMC surface is functionalized to adsorb desired biological species which

induce surface stress on the NMC. In this application, the added mass of species is estimated

from the shift in the resonant frequency of the system away from that of the original NMC.

An Active Probe is typically covered by a piezoelectric layer (e.g., ZnO) on the top surface.

To develop an accurate dynamic model for NMCs with jump discontinuities in cross-section, a

comprehensive framework has been recently developed. It has been shown that the effects of

added mass and stiffness on the beam mode shapes and natural frequencies are significant. Also,

results from forced vibration analysis indicate that the system frequency response is affected by

geometrical discontinuities of the structure. It is experimentally shown that assuming uniform

geometry and configuration for the dynamic analysis of the current NMC Active Probes is not a

valid assumption since it oversimplifies the problem and creates significant error in

measurements”,[1].

“Fig. 1.4 Nano-Mechanical Cantilever (NMC) probes”,[1] with permission

9

2 Energy Method for Piezoelectric

2.1 Electrical Potential Energy

“The total electrical potential energy of the electrostatic filed є (with electrical potential ),

while neglecting losses, is equal to the work needed to move total charge in this field. This

relationship in variational form can be simply given by:

(2.1)

where is the total electric potential energy.

The total accumulated charge for a continuum of volume V is defined as:

∫

(2.2)

where the total charge is measured in Coulombs (C) and q is the charge density in C/ . The

current intensity, I , is defined as the rate of change of total charge as:

(2.3)

Since Maxwell equation is defined as:

(2.4)

Insertion of Maxwell equation (2.4) for charge density q into the definition (2.2) for total

charge and substitution of resultant expression in (2.1) yields:

∫

(2.5)

Using the following relationship for divergence:

φ (2.6)

expression (2.5) can be rewritten as:

(∫( )

∫( )

)

(2.7)

The application of divergence theorem on the first term in (2.7) results in”,[1]:

*The materials and procedure in this section come directly from: Jalili N (2010) Piezoelectric-Based Vibration Control: From Macro to

Micro/Nano Scale Systems, Springer, New York

10

( ∮

∫( )

)

(2.8)

“Assuming either high-frequency applications or taking into account the fact that potential

energy decreases at least with 1/r (where r is the distance) while also dielectric displacement D

decreases at least with 1/ , the first term in expression (2.7) can be safely ignored.

Since The electric field ϵ with unit V/m, the analog of a conservative force field in mechanics,

can be related to electric potential φ, similar to the relationship between conservative potential

function and force in mechanics is defined as:

φ (2.9)

Considering this fact and the definition of electric field ϵ in (2.9), the electrical energy (2.8)

reduces to:

∫ ∫

(2.10)

The electrical potential energy (2.8) can be recast in indicial notation form as:

∫

(2.11)

Since the total potential energy can be rewritten in its compressed notation as:

∫

(2.12)

This electrical potential energy can now be augmented with the developed strain energy (2.12) to

form the total potential energy as:

∫( )

(2.13)

It is clear that the total energy (2.13) does not take into account other coupled and interacting

fields such as magnetomechanical, electromagnetic, thermoelectric, thermomagnetic, and

thermomechanical couplings. Since our primary objective here is to derive the constitutive

relationships for standard piezoelectric materials in which the magnetic effects can be safely

assumed negligible. It is also assumed that the thermal effects may be neglected; that is, either

the heat exchange with the environment is assumed to be negligible (an adiabatic process) or the

temperature is constant (an isothermal process). Although this is not a good assumption as most

piezoelectric materials are virtually pyroelectric, this is a common practice and could save a lot

of undue complications”,[1].

11

“Representing the total energy density (energy per volume V ) as , one can write (2.13) in

the following density form:

(2.14)

which, in comparison with (2.13), implies that total variation can be described as:

(

)

(

)

(2.15)

where the subscripts “ ” and “ ” imply that those values are measured at constant stress ( )

or constant electrical field ( 0). By comparing (2.14) and (2.15), one can relate the conjugated

and dependent variables and as functions of independent variables and that is:

(

)

(

)

(2.16a)

(

)

(

)

(2.16b)

i,j = 1,2,3 and p,q = 1,2,…,6

Alternatively, the conjugated and dependent variables and can be related to independent

variables and as:

(

)

(

)

(2.17a)

(

)

(

)

(2.17b)

i,j = 1,2,3 and p,q = 1,2,…,6

Equations (2.16) and (2.17) are called linear constitutive equations. These constitutive

relationships can be recast in the following more useful form:

+

(2.18)

where the indices i,j = 1,2,3 and p,q = 1,2,…,6 refer to different directions within the material

coordinate systems.

It must be noted that in constitutive relationships (2.18) or subsequent configurations, the

differentials in (2.16) or (2.17) have been replaced by the variables themselves. To justify this

action, we have assumed that the nominal values of the variables used in either (2.16) or (2.17)

are zero. Hence, the differentials are defined as the comparison between the variables themselves

to these zero-value states”,[1].

12

“In matrix form of (2.18), S is the strain vector, is the stress vector,

is the electrical field vector measured in V/m, and D is the displacement vector

measured in C/ .

The first relationship in (2.18) refers to converse piezoelectric effect (actuation), while the

second equation describes the direct piezoelectric effect (sensing). These equations can be

alternatively rewritten in the following form, which is mainly used for sensing applications:

+

(2.19)

where (in a similar manner to

) is the compliance coefficients matrix under constant

dielectric displacement (D = 0). Similar to constitutive relationships (2.18), the first equation in

(2.19) refers to converse effect (i.e., actuation mechanism) while the second equation denotes

the direct effect (i.e., sensing mechanism).

Alternatively, (2.19) can be manipulated to arrive at the following more suitable form for

actuation applications:

+

(2.20a)

+

(2.20b)

where is the elasticity coefficients matrix under constant dielectric displacement (D = 0), and

and are the piezoelectric constants matrices (the superscript S in refers to constant or zero

strain condition for the impermittivity constants matrix). It is worthy to note that a set of

relationships between material constants defined in (2.18) and (2.19) can be obtained by simple

cross-insertion of these equations in each other”,[1].

2.2 Definition of Material Constants

“Table 2.1 Deffinition of material constant”,[1] with permission

Material Constant Notation Units

(

)

Compliance coefficients matrix

(inverse of elastic coefficient matrix)

under constant electric field

(

)

(

)

Matrix of piezoelectric strain

constants relating electric

displacement (measured in

C/ ) to stress under zero electric

field (short-circuited electrodes)

m/V or

C/N

(

)

Dielectric or permittivity constants

matrix under constant stress

F/m

(Farad,

F=C/V)

13

(

)

(

)

Matrix of piezoelectric voltage

constants relating strain to

electric filed under zero stress

Vm/N

or

(

)

Impermittivity constants matrix under

constant stress

m/F

(

)

(

)

Matrix of piezoelectric constants V/m

(

)

Elastic stiffness coefficients matrix

under constant dielectric

displacement

(

)

(

)

Matrix of piezoelectric constants

2.3 Piezoelectric Constants

“To better visualize the material constants defined in the preceding subsection, the

piezoelectric constitutive relations (2.18) can be written in matrix form as:

{

}

(

)

{

}

(

)

{

}

(2.21a)

{

} (

)

{

}

(

) {

}

(2.21b)

The matrix forms (2.21) are in the most general form; however, when the material’s elastic

properties are invariant with respect to rotation of any angle about a given axis, the total number

of compliance coefficients reduces to 5. These materials are referred to as transversely isotropic.

Piezoceramics belong to this class of materials. It is commonly assumed that the third axis or

direction 3 is along the polarization direction which also coincides with the axis of transverse

isotropy. Hence, (2.21) for these materials (piezoceramics) reduces to”,[1]:

14

{

}

(

)

{

}

(

)

{

}

(2.22a)

{

} (

)

{

}

(

) {

}

(2.22b)

“Equations (2.22) imply that for transversely isotropic piezoceramics, there are five elastic

constants, three iezoelectric strain constants, and two dielectric or permittivity constants.

One can clearly see the transversely isotropic assumption for piezoceramics where an electric

field applied in direction of polarization vector ( for instance) will result in same strains in 1

and 2 directions (see (2.22a) where ). This assumption, however, is not valid for

nonisotropic piezoelectric materials such as PVDF where their piezoelectric strain constants

matrix takes the form:

(

)

(2.23)

Equation (2.23) clearly demonstrates that the application of an electric field in the polarization

direction for these piezoelectric materials results in different strains in directions 1 and 2 since

. As a matter of fact, PVDF films are highly anisotropic with . Also, the

dielectric strength of PVDF polymers is about 20 times higher than that of PZT, and hence, can

endure much higher electric field compared to PZT materials. For both PZT and PVDF

materials, piezoelectric strain constant implies that the application of electric field

(normal to the polarization direction 3) produces a shear deformation or

( ). Since typically has the largest values among all piezoelectric constants, this

property can be utilized to design effective shear actuators and sensors.

As defined in previous section, the piezoelectric strain (or sometime referred to as charge)

constant is the ratio of the induced electric polarization per unit applied mechanical stress.

Alternatively, it is defined as produced mechanical strain per unit applied electric field.

Therefore, representing this definition using the indicial notation, the piezoelectric strain

constant can be defined as the generated strain along j -axis due to a unit electric field applied

along i -axis, provided that all external stresses are kept constant. For example, is the

15

induced strain in direction 1 due to a unit electric field in direction 3 (polarization direction),

while the system is kept under a stress-free field”,[1].

3 Piezoelectric-Based System

3.1 Modeling Assumptions and Preliminaries

“As discussed in previous section, the total potential energy of a linear piezoelectric material

is expressed as:

∫( )

(3.1)

By substituting the piezoelectric constitutive relationships (2.18), (2.19), or (2.20), the

potential energy (3.1) reduces to appropriate relationships depending on the nature of the

problem at hand. For instance, for actuator applications, substituting constitutive equation

(2.20a) into energy (3.1) results in:

∫(

)

(3.2)

Equation (3.2) can be further simplified to:

∫(

)

(3.3)

Clearly, (3.3) can be separated into three parts: a purely mechanical term (elastic energy), a

purely electrical term (dielectric energy), and a combined term (coupled energy).

It must be noted that the electrical kinetic energy is still ignored in the calculations. However,

the electrical virtual work due to application of electrical voltage in piezoelectric material will be

considered in Hamilton’s formulation as discussed in the next section (section 3.2)”,[1].

3.2 Modeling Piezoelectric Actuators in Transverse Configuration

“Many structural vibration-control systems utilize piezoceramic materials that are typically

implemented in the form of monolithic wafers. The term “monolithic” refers to a piezoceramic

material which is free from added materials or augmenting structural components. While the

axial configuration is mainly used for positioning applications, the laminar configuration is

typically utilized in structural vibration control and sensing applications. This mode of actuation

or sensing relies on in-plane actuation and sensing, i.e., induced stresses and strains parallel to

the structure’s surfaces (see Fig. 3.1). As a result, the piezoceramic wafers operate in mode.

Using this approach, in-plane strains can be readily measured with an attached piezoceramic.

Note that the values of are typically lower than those of ”,[1].

*The procedure in this section comes to some extend from: Jalili N (2010) Piezoelectric-Based Vibration Control: From Macro to Micro/Nano

Scale Systems, Springer, New York

16

“Fig. 3.1 Piezoelectric Patch Actuator”,[1] with permission

“For the purpose of model development and undue complications, a uniform flexible beam

with piezoelectric patch actuator bonded on its top surface is considered. As shown in Fig. 3.2,

the beam has total thickness , and length L, while the piezoelectric film possesses thickness

and length and , respectively. It is assumed that beam has width and piezoelectric

has width . It is also assumed that the piezoelectric actuator is perfectly bonded on the beam at

distance measured from the beam support and the input voltage (t) applied to the

piezoelectric actuator is considered to be independent of spatial coordinate x and serves as the

only external effect.

“Fig. 3.2 Coordinate System and Detailed Descriptions of the Attachment”,[1] with permission

To establish a coordinate system for the beam, the x-axis is taken in the longitudinal direction

and the z-axis is specified in the transverse direction of the beam with mid-plane of the beam to

be z = 0 as shown in Figure 3.2.

The simplified version of the constitutive equation (2.20a) for this configuration can be

expressed as”,[1]:

17

(3.4)

“Notice that for this configuration, the strain-displacement relationship is utilized as:

(3.5)

where is the transverse displacement of the neutral axis. Before utilizing (3.4) and (3.5)

into the potential energy (3.3), care must be taken for the multi-material and non-uniform nature

of the system. For this, the original form of the potential energy (3.1) is a better choice when

dealing with this variable geometry structure, i.e., this energy is written for three parts: the

section before piezoelectric patch starts (0 to ), the zone where patch is affixed ( ), and

the zone after patch ( to L).

It must also be noted that in the segment of the beam where the piezoelectric patch is

attached, the material properties change along the height (or z-axis); hence, both strain equation

(3.5) and potential energy (3.1) need to be modified. That is, wherever the piezoelectric patch is

not attached on the beam (i.e., ), the neutral surface is the geometric center of

the beam (z = 0) and strain equation (3.5) holds. For the portions where the piezoelectric patch is

attached (i.e., ), the strain equation (3.5) is modified to:

(3.6)

where is the neutral surface (see Figure 3.2). This new neutral surface can be calculated by

setting the sum of all forces in x-direction over the entire cross-section zero as:

∫ ∫

(3.7)

where and

are referred to as stresses induced in beam and piezoelectricmaterial segments,

respectively. Utilizing Hook’s law (

) for each segment, while substituting strain

relationship (3.6), yields :

∫ ∫

(3.8)

where and

are the respective Young’s moduli of elasticity for beam and piezoelectric

materials. Upon implifying (3.8), the neutral axis can be readily obtained as”,[1]:

(3.9)

18

3.3 Piezoelectric-Based Cantilever Beam Modeling – Euler Bernoulli Theory

“In order to deal with the material dissimilarity and geometrical non-uniformity, the integral

for the potential energy (3.1) for this configuration is also broken into several integrals, based on

the location of the piezoelectric actuator. Hence, (3.1) is recast in the following form:

∫ ∫

∫ ∫

∫ ∫ (( )

)

∫ ∫

(3.10)

Note that in (3.10), strain from (3.5) is used in the first two and last integrals, while strain

from (3.6) is used in the third integral. Similar to potential energy, the kinetic energy

associated with this non-uniform configuration can be expressed as (notice that the electrical

kinetic energy is neglected):

{∫ (

)

∫ ( ) (

)

∫ (

)

}

∫

(

)

∫

(

)

(3.11)

where

(3.12)

and H(x) is the Heaviside function, and are the respective beam and piezoelectric

volumetric densities.

Considering both viscous and structural damping mechanisms for beam material, the total

mechanical virtual work can be given by”,[1]:

∫ (

)

∫ (

)

(3.13)

19

“where and are the viscous and structural damping coefficients, respectively. The electrical

virtual work due to input voltage to piezoelectric patch is given by:

∫

(3.14)

Notice that for generalization, we again assume that the input voltage to piezoelectric actuator

is a function of both spatial and temporal coordinates as presented in (3.14). At this stage, all the

intermediate steps in deriving different expressions for use in the extended Hamilton’s principle

(∫

) have been completed. By insertion of (3.5) and (3.6) into energy

equation (3.10), and inserting the results along with kinetic energy (3.11) and total virtual works

(3.13) and (3.14) into ∫

, and after some manipulations, we get (see

Appendix A1) :

∫ [∫

(

)

∫ {

(

)

(

)}

∫

∫ (

)

∫ (

)

]

(3.15)

where:

[

(

)

(

)]

⁄

(3.16)

After some manipulations, one can simplify (3.15) as follows (see Appendix A1)”,[1]:

20

∫ [∫ {(

(

)

)

(

) }

(

) (

)|

(

(

)

) |

]

(3.17)

“For (3.17) to vanish regardless of independent variations and , the integrant

must vanish, and for the integrant to vanish we must have:

For :

(

)

(3.18a)

For

(3.18b)

along with the boundary conditions :

(

) (

)|

(

(

)

) |

(3.18c)

Equation (3.18a) represents the distributed-parameters equation of beam coupled with the

dielectric displacement, (3.18b) indicates a static coupling between piezoelectric actuator and

structure and finally (3.18c) presents the boundary conditions that need to be satisfied.

Substituting the dielectric displacement from (3.18b) into both (3.18a) and boundary

conditions (3.18c), one can obtain the PDE governing this type of actuator in response to input

voltage as”,[1]:

21

((

)

)

(3.19a)

[(

)

] (

)|

(3.19b)

[

((

)

)

] |

(3.19c)

“Due to geometrical non-uniformity arising from piezoelectric patch attachment, the coupling

term ⁄ appearing in (3.19) cannot be further simplified at this stage, but for some

special or simple arrangements this expression can be further simplified. However, expression

⁄ in (3.19) can be simplified to:

( )

( )

( )

( )

(3.20)

where is the equivalent stiffness of the piezoelectric actuator in laminar configuration defined

as:

(3.21)

A widely accepted assumption for laminar piezoelectric actuator is to assume a uniform input

voltage wherever the actuator is attached on the beam, and naturally zero input voltage

elsewhere. To mathematically describe this voltage profile, can be expressed as:

(3.22)

where was defined earlier in (3.12) and is the input voltage to the actuator.

Inserting the input voltage profile (3.22) and property (3.20) into (3.19), while noticing that

in boundary equations (3.19b and 3.19c) yields”,[1]:

(

)

(3.23)

22

(

) (

)|

(3.24a)

[

(

)] |

(3.24b)

where:

( )

(3.25)

“Equation (3.23) and boundary conditions (3.24a) and (3.24b) represent the governing

equations describing piezoelectric laminar actuators. They form the fundamental ground from

which many vibration-control systems can be designed for these types of actuators.

Now Consider a piezoelectric-based cantilever beam system which the beam has total

thickness , and length L, while the piezoelectric film possesses thickness and length and

, respectively. It is assumed that beam has width and piezoelectric has width . It is

also assumed that the piezoelectric actuator is perfectly bonded on the beam at distance

measured from the cantilevered end of the beam and the input voltage (t) applied to the

piezoelectric actuator is considered to be independent of spatial coordinate x and serves as the

only external effect. For this configuration, the variable mass per unit length, stiffness, and

moment of inertia are given as:

{

(3.26)

where:

∫

(3.27a)

∫

∫

23

{

( )

( ( )

)} ( )

(3.27b)

∫

(3.27b)

and:

{

(3.28)

where is the neutral axis of the beam on the composite portion, and are the densities of

the beam and piezoelectric layer, respectively, and the rest of the parameters were defined

before. Moreover, the distribution of damping can be safely assumed to be uniform in the entire

length of the cantilever.

So free and undamped conditions associated with the transverse vibration of the beam are

given by:

(

)

(3.29)

Assuming that the solution of (3.29) is separable in the form of (3.29) can

be rewritten in the form of:

(

)

(3.30)

where is the natural frequency of the system. In order to obtain an analytical solution for

(3.30), the entire length of the beam is divided into three uniform segments with two sets of

continuity conditions at stepped points. Therefore, (3.30) can be divided into three equations

given by:

(3.31)

where are mode shapes, flexural stiffness, and mass per unit length of

beam at the nth segment, respectively”,[1].

24

“The general solution for (3.31) can be written as:

(3.32)

where ⁄ and are the constants of integration to be

obtained by solving the characteristics equation of the system. So the three eigenfunctions for the

three-section cantilever can be written as:

(3.33)

For this purpose, the boundary conditions for the beam as well as the continuity conditions at

the stepped points must be applied. The clamped-free boundary conditions of the beam require:

(3.34a)

(3.34b)

and the respective conditions for the continuity of deflection, slope of the deflection, bending

moment, and shear force of the beam at the nth stepped point, where are given by:

(3.35a)

(3.35b)

(3.35c)

(3.35d)

Applying 12 boundary conditions (6 geometric and 6 natural) (3.33) and (3.34) into (3.32),

the characteristics matrix equation of system can be written as:

[ ]

[ ] [ ]

(3.36)

where the components of matrix can be obtained from boundary conditions as follows (see

Appendix A2)”,[1]:

25

(3.37a)

(3.37b)

(3.37c)

(3.37d)

(3.37e)

(3.37f)

(3.37g)

26

(3.37h)

(3.37i)

(3.37j)

(3.37k)

(3.37l)

27

where according to following formulation can be utilized as functions of to make as a function of :

⁄

(

)

(

)

[( )

]

⁄

[

]

⁄

(3.38)

“Setting the determinant of the characteristics matrix to zero leads to finding the system

natural frequencies.

The mode shape coefficients at each natural frequency can be obtained by solving the

characteristics equation and using a normalization condition with respect to mass as follows:

∫

∫ ( )

(3.39)

where is the Kronecker delta, and and are rth and sth mode shapes

corresponding to the rth and sth natural frequency of beam. For instance, is expressed

as:

{

(

) (

)

(

) (

)

( )

( )

(

) (

)

( )

( )

(

) (

)

(3.40)

The obtained natural frequencies and mode shapes are utilized to derive the governing

equations of motion for the forced vibration of the system. According to the eigenfunctions

expansion method, the response of system can be expressed in the form of”,[1]:

∑

(3.41)

28

“where and are the eigenfunction and generalized time-dependent coordinates

for the rth mode for each section. Substituting (3.41) into (3.23) and carrying out the forced

vibration analysis, the equation of motion of the system can be expressed in the following form:

∑

∑

∑

∑

(3.42)

or in indicial form:

(3.43)

On the other hand, are the eigenfunctions and satisfy the free and undamped vibration

problem:

(3.44)

Now, substituting (3.44) into (3.43), premultiply the resulting expression by eigenfunction

and integrating over the domain while utilizing the orthogonality conditions between

eigenfunction and yields:

∑(∫

)

∑ (∫

)

∑(∫

)

∑(∫

)

( ∫

)

(3.45)

In this stage we assume which means no structural damping, and also using a

normalization condition with respect to mass (3.39) we obtain”,[1]:

29

∑

∑

∑(∫

)

∫

(3.46)

“If we get and then change we obtain:

∑{ }

(3.47)

∫

{∫

∫

∫

}

(3.48)

∫

(

) ∫ ( )

(3.49)

For the second distributional derivative of the Heaviside function used in (3.49), we can write:

∫

∫

( )|

∫

∫

( )|

(3.50)

where represents the Dirac delta function. In this simplification, we have utilized the

following property of Dirac delta function:

∫

(3.51)

Now substituting (3.50) into (3.49) yields”,[1]:

30

( ) ( )

(3.52)

“The truncated p-mode description of the beam model of (3.47) can now be presented in the

following matrix form:

(3.53)

where:

[ ] [ ]

[ ]

[ ]

(3.54)

Consequently, the state-space representation of (3.53) can be expressed as:

(3.55)

Where”,[1]:

[

]

[

]

{ }

(3.56)

3.4 Piezoelectric-Based Cantilever Beam Modeling – Rayleigh Theory

In this case similar to Euler Bernoulli’s Theory, the total potential energy of a linear piezoelectric

material can be expressed as:

∫( )

(3.57)

or

∫(

)

(3.58)

31

In order to deal with the material dissimilarity and geometrical non-uniformity, the integral for

the potential energy (3.58) for this configuration is also broken into several integrals, based on

the location of the piezoelectric actuator. Hence, (3.58) is recast in the following form :

∫ ∫

∫ ∫

∫ ∫ (( )

)

∫ ∫

(3.59)

Note that in (3.59), strain from (3.5) is used in the first two and last integrals, while strain

from (3.6) is used in the third integral. Note that the effect of rotary inertia was ignored so far in

the derivation. That means, the kinetic energy due to rotation of the beam was ignored. If this

can’t be ignored (Rayleigh’s Beam Theory), then the kinetic energy must be modified as:

∫

(

)

∫

(

)

(3.60)

or:

∫

(

) ∫

(

)

(3.61)

where:

(3.62)

is the mass moment of inertia for unit length and is the radius of gyration, both about

the neutral axis (bending), and for small vibration:

(3.63)

So similar to potential energy, the kinetic energy associated with this non-uniform configuration

can be expressed as (notice that the electric kinetic energy is neglected):

32

{∫ (

)

∫ (

)

∫ ( ) (

)

∫ ( ) (

)

∫ (

)

∫ (

)

}

∫ (

)

∫

(

)

(3.64)

So we can say:

∫

(

) ∫

(

)

(3.65)

where:

( ) ( )

(3.66)

and H(x) is the Heaviside function, and are the respective beam and piezoelectric

volumetric densities.


mechanical virtual work can be given by:

∫ (

)

∫ (

)

(3.67)

where C and B are the viscous and structural damping coefficients, respectively. The electrical


∫

(3.68)


is a function of both spatial and temporal coordinates as presented in (3.68). At this stage, all the

intermediate steps in deriving different expressions for use in the extended Hamilton’s principle

33

(∫

) have been completed. By insertion of (3.5) and (3.6) into energy

equation (3.59), and inserting the results along with kinetic energy (3.64) and total virtual works

(3.67) and (3.68) into ∫

, and after some manipulations, we get (see

Appendix A3) :

∫ [∫

(

)

∫

(

)

∫ {

(

)

(

)}

∫

∫ (

)

∫ (

)

]

(3.69)

where:

[

(

)

(

)]

⁄

(3.70)

After some manipulations, one can simplify (3.69) as follows (see Appendix A3):

∫ [∫ {(

(

)

(

)

)

(

) }

(

) (

)|

(

(

)

) |

]

(3.71)

34

For (3.71) to vanish regardless of independent variations and , the integrant

must vanish, and for the integrant to vanish we must have:

For

(

)

(

)

(3.72a)

For

(3.72b)


(

) (

)|

(

(

)

) |

(3.72c)

Equation (3.72a) represents the distributed-parameters equation of Rayleigh beam coupled

with the dielectric displacement, (3.72b) indicates a static coupling between piezoelectric

actuator and structure and finally (3.72c) presents the boundary conditions that need to be

satisfied.

Substituting the dielectric displacement from (3.72b) into both (3.72a) and boundary

conditions (3.72c), one can obtain the PDE governing this type of actuator in response to input

voltage as:

(

)

((

)

)

(3.73a)

[(

)

] (

)|

(3.73b)

[

((

)

)

] |

(3.73c)

Due to geometrical non-uniformity arising from piezoelectric patch attachment, the coupling


35



( )

( )

( )

( )

(3.74)


as:

(3.75)




(3.76)


Inserting the input voltage profile (3.76) and property (3.74) into (3.73), while noticing that

in boundary equations (3.73b and 3.73c) yields:

(

)

(

)

(3.77)

(

) (

)|

(3.78a)

[

(

)

] |

(3.78b)

where:

( )

(3.79)

36

Equation (3.77) and boundary conditions (3.78a) and (3.78b) represent the governing

equations describing piezoelectric laminar actuators. They form the fundamental ground from

which many vibration-control systems can be designed for these types of actuators.

Now for free-undamped case we can say:

(

)

(

)

(3.80)

Assuming that the solution of (3.80) is separable in the form of (3.80) can

be rewritten in the form of:

(

)

(

)

(3.81)




given by:

(3.82)

where are mode shapes, flexural stiffness, mass moment of

inertia per unit length and mass per unit length of beam at the nth segment, respectively.

The general solution for (3.82) can be written as (see Appendix A4):

(3.83)

where have been presented in App. A4 and are the constants of

integration to be obtained by solving the characteristics equation of the system. So the three

eigenfunctions for the three-section cantilever can be written as:

(3.84)

For this purpose, the boundary conditions for the beam as well as the continuity conditions at the

stepped points must be applied. The clamped-free boundary conditions of the beam require:

(3.85a)

37

(3.85b)

(

)

(3.85c)

and the respective conditions for the continuity of deflection, slope of the deflection, bending


(3.86a)

(3.86b)

(3.86c)

(3.86d)

Applying 12 boundary conditions (6 geometric and 6 natural) (3.85) and (3.86) into (3.84), the

characteristics matrix equation of system can be written as:

[ ]

[ ] [ ]

(3.87)


Appendix A5):

(3.88a)

(3.88b)

38

(3.88c)

(3.88d)

(3.88e)

(3.88f)

(3.88g)

(3.88h)

39

(3.88i)

(3.88j)

(3.88k)

(3.88l)

Setting the determinant of the characteristics matrix to zero leads to finding the system natural

frequencies.


characteristics equation and using a normalization condition with respect to mass as follows (see

Appendix A6):

40

∫

∫

∫ ( )

∫ (

)

(3.89)

where is the Kronecker delta, and and are rth and sth mode shapes

corresponding to the rth and sth natural frequency of beam. For instance, is expressed

as:

{

(3.90)


equations of motion for the forced vibration of the system.

According to the eigenfunctions expansion method, the response of system can be expressed in

the form of:

∑

(3.91)

where and are the eigenfunction and generalized time-dependent coordinates

for the rth mode for each section. Substituting (3.91) into (3.77) and carrying out the forced

vibration analysis, the equation of motion of the system can be expressed in the following form:

∑

∑

∑

∑

∑

(3.92)

On the other hand, are the eigenfunctions and satisfy the free and undamped vibration

problem:

41

(

)

(3.93)

Now, substituting (3.93) into (3.92), premultiply the resulting expression by eigenfunction

and integrating over the domain while utilizing the orthogonality conditions between

eigenfunction and yields:

∑(∫

∫

)

∑

|

∑ (∫

∫

)

∑ (

|

)

∑(∫

)

∑(∫

)

( ∫

)

(3.94)

In this stage we assume which means no structural damping, and also using a

normalization condition with respect to mass (3.89) we obtain:

∑

∑

∑(∫

)

∫

(3.95)

If we get and then change we obtain:

∑{ }

(3.96)

42

∫

{∫

∫

∫

}

(3.97)

∫

(

) ∫ ( )

(3.98)

For the second distributional derivative of the Heaviside function used in (3.98), we can write:

∫

∫

( )|

∫

∫

( )|

(3.99)



∫

(3.100)

Now substituting (3.99) into (3.98) yields:

( ) ( )

(3.101)

The truncated p-mode description of the beam model of (3.96) can now be presented in the


(3.102)

where:

[ ] [ ]

43

[ ]

[ ]

(3.103)


(3.104)

where:

[

]

[

]

{ }

(3.105)

3.5 Piezoelectric-Based Cantilever Beam Modeling – Timoshenko Theory

In this case is the angle for which internal moment is acting. The slope of the deflection

curve is now

which is expressed as:

(3.106)

where is rotation of cross-section from vertical axis and is shear distortion of the

cross-section (Fig. 3.3).

Fig.3.3 A clear description of equation (3.106)

So in this special case, equations (3.5) and (3.6) will be modified to:

44

(3.107)

(3.108)

So the total potential energy equation (3.1) will be changed due to effects of rotation and

shear distortion of cross section to:

∫( ) ∫

(3.109)

or:

∫(

) ∫

(3.110)

Now the shear force could be related to by multiplying shear stress by area. To account

for the fact that shear is parabolically distributed on a cross-section, a constant shape-dependent

variable ‘s’ is defined which depends on the shape of the cross-section, hence:

(

)

(3.111)

is also called reduced section and is computed from classical beam theory. For

example, for a plane rectangular cross-section, and for a plane circular section,

.

So we can say:

∫

∫

∫ ( )

∫

∫

(3.112)

where:

45

( )

(3.113)

So the total potential energy (3.110) will be:

∫ ∫

∫ ∫

∫ ∫ (( )

)

∫ ∫

∫

(3.114)

Note that in (3.114), strain from (3.107) is used in the first two and fourth integrals, while

strain from (3.108) is used in the third integral.

In this case similar to previous beam theory (Rayleigh’s beam), we should consider the effect

of rotary inertia, so the kinetic energy must be written as:

∫

(

)

∫

(

)

(3.115)

or:

∫

(

) ∫

(

)

(3.116)

where:

(3.117)

As we mentioned before, is the mass moment of inertia for unit length and is the

radius of gyration, both about the neutral axis (bending).

So similar to potential energy, the kinetic energy associated with this non-uniform configuration

can be expressed as (notice that the electric kinetic energy is neglected):

46

{∫ (

)

∫ (

)

∫ ( ) (

)

∫ ( ) (

)

∫ (

)

∫ (

)

}

∫ (

)

∫

(

)

(3.118)

So we can say:

∫

(

) ∫

(

)

(3.119)

where:

( ) ( )

(3.120)


mechanical virtual work can be given by:

∫ (

)

∫ (

)

(3.121)

where C and B are the viscous and structural damping coefficients, respectively. The electrical


∫

(3.122)


is a function of both spatial and temporal coordinates as presented in (3.122). At this stage, all

the intermediate steps in deriving different expressions for use in the extended Hamilton’s

principle (∫

) have been completed. By insertion of (3.107) and (3.108)

into energy equation (3.114), and inserting the results along with kinetic energy (3.118) and total

47

virtual works (3.121) and (3.122) into ∫

, and after some manipulations,

we get (see Appendix A7):

∫ [∫

(

)

∫

(

)

∫ {

(

)

(

)}

∫ (

)

(

)

∫

∫ (

)

∫ (

)

]

(3.123)

where:

[

(

)

(

)]

⁄

(3.124)

After some manipulations, one can simplify (3.123) as follows (see Appendix A7):

48

∫ [∫ {(

( (

))

)

(

(

)

( )

(

))

(

) }

(

) |

( (

)) |

]

(3.125)

For (3.125) to vanish regardless of independent variations , and ,

the integrant must vanish, and for the integrant to vanish we must have:

For :

( (

))

(3.126a)

For :

(

)

( )

(

)

(3.126b)

For :

(3.126c)

49


(

) |

( (

)) |

(3.126d)

Substituting the dielectric displacement from (3.126c) into both (3.126b) and also

boundary conditions (3.126d), one can obtain the PDE governing this type of actuator in

response to input voltage as:

( (

))

(3.127a)

((

)

) (

)

(

)

(3.127b)

((

)

) |

(3.127c)

( (

)) |

(3.127d)

Now if we assume constants values for after some

manipulations one can simplify (3.127) as follows (see Appendix A7):

(

)

(

)

[

(

)]

(

)

(

)

(

)

(

)

(

)

(3.128a)

50

((

)

) |

(3.128b)

( (

)) |

(3.128c)

Due to geometrical non-uniformity arising from piezoelectric patch attachment, the coupling




( )

( )

( )

( )

(3.129)


as:

(3.130)




(3.131)


Inserting the input voltage profile (3.131) and property (3.129) into (3.127), while noticing

that in boundary equation (3.127c) yields:

( (

))

(3.132)

(

) (

)

(3.133)

51

(

) |

(3.134)

( (

)) |

(3.135)

where:

( )

(3.136)

Equations (3.132) and (3.133) and boundary conditions (3.134) and (3.135) represent the

governing equations describing piezoelectric laminar actuators. They form the fundamental

ground from which many vibration-control systems can be designed for these types of actuators.

Now for free-undamped case we can say:

( (

))

(3.137)

(

) (

)

(3.138)

Now by mixing equations (3.137) and (3.138) we obtain:

(

)

(3.139)

Assuming that the solution of (3.139) is separable in the form of (3.139)

can be rewritten in the form of:

(

)

(3.140)



52


given by:

(

)

(

)

(3.141)

where are mode shapes, flexural stiffness, mass

moment of inertia per unit length, mass per unit length and shear correction factor times shear

modulus times area of beam at the nth segment, respectively.


(3.142)




(3.143)


stepped points must be applied. So we should assume:

(3.144)

So if we put equations (3.144) in (3.137) and (3.138) we will obtain:

(3.145)

Now by eliminating the function from equations (3.145) we will obtain:

(

)

(3.146)

so we can use boundary conditions (3.134) and (3.135) just in terms of !

53

The clamped-free boundary conditions of the beam require:

(3.147a)

(

)

(3.147b)

(

)

(3.147c)

(

)

(

)

{

(

)}

(

)

(3.147d)

and the respective conditions for the continuity of deflection, bending angle, bending


(3.148a)

(

)

(

)

(3.148b)

(

)

(

)

(3.148c)

54

(

) (

)

(

)

(

)

(3.148d)

Applying 12 boundary conditions (6 geometric and 6 natural) (3.147) and (3.148) into

(3.143), the characteristics matrix equation of system can be written as:

[ ]

[ ] [ ]

(3.149)


Appendix A9):

(3.150a)

(3.150b)

(3.150c)

(3.150d)

55

(3.150e)

(3.150f)

(3.150g)

56

(3.150h)

(

) (

)

(3.150i)

(

) (

)

57

(3.150j)

(3.150k)

(

)

(3.150l)

Setting the determinant of the characteristics matrix to zero leads to finding the system natural

frequencies.


characteristics equation and using a normalization condition with respect to mass as follows (see

Appendix A10):

∫

∫

∫ ( )

∫ ( )

(3.151)

where is the Kronecker delta, and pairs of { } and { } are

rth and sth mode shapes corresponding to the rth and sth natural frequency of beam. For

instance, is expressed as:

58

{

(3.152)

in this stage similar to previous procedure we should find the function so by mixing

equations (3.137) and (3.138) we will obtain:

(

)

(3.153)

which is exactly same with equation (3.139), so assuming that the solution of (3.153) is

separable in the form of (3.153) can be rewritten in the form of:

(

)

(3.154)




given by:

(

)

(

)

(3.155)

where are mode shapes, flexural stiffness, mass

moment of inertia per unit length, mass per unit length and shear correction factor times shear

modulus times area of beam at the nth segment, respectively.


(3.156)




59

(3.157)


stepped points must be applied. So we should assume:

(3.158)

so if we put equation (3.158) in (3.137) and (3.138) we will obtain:

(3.159)

now by eliminating the function from equation (3.159) we will obtain:

(

)

(3.160)

so we can use boundary conditions (3.134) and (3.135) just in terms of ! The clamped-free boundary conditions of the beam require:

(

)

(3.161a)

(3.161b)

(3.161c)

(

)

(

)

{

(

) }

60

(

)

(3.161d)

and the respective conditions for the continuity of deflection, bending angle, bending moment,

and shear force of the beam at the nth stepped point, where are given by:

(

)

(

)

(3.162a)

(3.162b)

(3.162c)

(

) (

)

(

)

(

)

(3.162d)

Applying 12 boundary conditions (6 geometric and 6 natural) (3.161) and (3.162) into

(3.157), the characteristics matrix equation of system can be written as:

[ ]

[ ] [ ]

(3.163)


Appendix A12):

(3.164a)

61

(3.164b)

(3.164c)

(3.164d)

62

(3.164e)

(3.164f)

(3.164g)

(3.164h)

(

) (

)

63

(3.164i)

(

) (

)

(3.164j)

(3.164k)

(

)

(3.164l)

So similar to equation (3.152), is expressed as:

64

{

(3.165)


equations of motion for the forced vibration of the system.

According to the eigenfunctions expansion method, the response of system can be expressed

in the form of:

∑

∑

(3.166)

where are the eigenfunctions and is generalized time-dependent

coordinates for the rth mode for each section. Substituting (3.166) into (3.132) and (3.133) and

carrying out the forced vibration analysis, the equations of motion of the system can be

expressed in the following form:

∑

(∑

∑

)

∑

∑

(3.167)

∑

∑

(∑

∑

)

(3.168)

on the other hand, are the eigenfunctions and satisfy the free and

undamped vibration problem:

65

(

)

(3.169)

(

)

(3.170)

So equations (3.167) and (3.168) convert to:

∑

∑

∑

∑

(3.171)

∑

∑

(3.172)

Now, premultiply equation (3.171) by eigenfunction and equation (3.172) by

eigenfunction and integrating over the domain while utilizing the orthogonality

conditions between eigenfunction and and also and

and finally

adding two equations to each other yields:

∑(∫

∫

)

∑ (∫

∫

)

∑(∫

)

∑(∫

)

∫

(3.173)

Now regarding to orthogonalithy condition and assuming yields:

66

∑

∑

∑(∫

)

∫

(3.174)

If we get and then change we obtain:

∑{ }

(3.175)

∫

{∫

∫

∫

}

(3.176)

∫

(

) ∫ ( )

(3.177)

for the first distributional derivative of the Heaviside function used in (3.177), we can write:

∫

∫

|

∫

∫

|

(3.178)



∫

(3.179)

now substituting (3.178) into (3.177) yields:

( ) ( )

(3.180)

67

The truncated p-mode description of the beam model of (3.184) can now be presented in the


(3.181)

where:

[ ] [ ]

[ ]

[ ]

(3.182)


(3.183)

where:

[

]

[

]

{ }

(3.184)

68

4 Numerical Results

The material properties for the discontinuous cantilever system are listed in table 4.1. The

MATLAB code develops to perform the simulations described in this research is shown in

Appendix B.

Table 4.1 Variables descriptions, definitions and values

Symbol Definition Value Unit

Length of 1st section of cantilever 20 mm

Total length of 1st and 2nd section of cantilever 80 mm

Total length of cantilever 250 mm

Beam young’s modulus 210 Gpa

Piezoelectric young’s modulus 67 Gpa

Beam shear modulus 82 Gpa

Piezoelectric shear modulus 26 Gpa

Volumetric mass density of beam 7850 kg/m3

Volumetric mass density of piezoelectric 7910 kg/m3

Width of beam 25 mm

Width of piezoelectric 15 mm

Thickness of beam 0.5-24 mm

Thickness of piezoelectric 0.5 mm

Piezoelectric strain constant 183 pC/N

Beam constant shape-dependent variable 0.833 ---

Piezoelectric constant shape-dependent variable 0.833 ---

4.1 Calculation of and values

Calculating values for Euler-Bernoulli beam and values for Rayleigh and Timoshenko

beam, ( ) and ( ) are calculated as varies from 0 to 120,000 rad/s (for example

for varies from 0 to 3000 and for varies from 0 to 120,000

rad/s). The relationship between is shown in equation (4.1):

(

)

(4.1)

So variation of is dependent on variation of . Equation (3.38) is utilized for to

make only a function of .

| ( )| is plotted versus yielding Fig. 4.1a-d for 4 different value of thickness of beam

(0.5, 4, 16 and 24 mm) which shows five first natural frequencies of beam.

69

Fig. 4.1.a-b values for (left) and (right)

Fig. 4.1.c-d values for (left) and (right)

Percentage of difference between five first natural frequencies of Euler-Bernoulli, Rayleigh

and Timoshenko beam are shown in table 4.2.a-g. As we see, for slender beam the difference

values are negligible, while with increasing of thickness, percentage of difference increases.

Raising of difference between eigenfrequency values are considerable in stocky beam and

specially in last modes.

0 500 1000 1500 2000 2500 300010

-15

10-10

10-5

100

105

1010

1015

1020

1025

1030

1035

Omega (rad/s)

|Det(

J)|

Omega values for first five eigenfrequencies

Discontinuous Euler-Bernoulli beam

Discontinuous Rayleigh beam

Discontinuous Timoshenko beam

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 104

10-10

100

1010

1020

1030

1040

1050

Omega (rad/s)

|Det(

J)|





0 1 2 3 4 5 6 7 8

x 104

100

1010

1020

1030

1040

1050

1060

Omega (rad/s)

|Det(

J)|





0 2 4 6 8 10 12

x 104

1010

1020

1030

1040

1050

1060

1070

Omega (rad/s)

|Det(

J)|





70

Table 4.2.a-g First five eigenfrequency values

a - Piezoelectric-Based Cantilever Beam System (tb=0.5 mm)

Natural Frequencies of Beam (rad/s)

1st 2nd 3rd 4th 5th

Theory

Euler-Bernoulli 51.90 263.60 745.50 1,537.80 2,531.20

Rayleigh 51.90 263.60 745.50 1,537.70 2,531.00

Timoshenko 51.90 263.60 745.40 1,537.60 2,530.60

Percentage of Difference between

EB & R 0.00 0.00 0.00 0.01 0.01

R & T 0.00 0.00 0.01 0.01 0.02

EB & T 0.00 0.00 0.01 0.01 0.02

b - Piezoelectric-Based Cantilever Beam System (tb=1 mm)


1st 2nd 3rd 4th 5th

Theory

Euler-Bernoulli 93.60 527.20 1,471.60 2,925.20 4,830.20

Rayleigh 93.60 527.20 1,471.50 2,924.90 4,829.20

Timoshenko 93.60 527.20 1,471.20 2,924.00 4,826.70


EB & R 0.00 0.00 0.01 0.01 0.02

R & T 0.00 0.00 0.02 0.03 0.05

EB & T 0.00 0.00 0.03 0.04 0.07

c - Piezoelectric-Based Cantilever Beam System (tb=2 mm)


1st 2nd 3rd 4th 5th

Theory

Euler-Bernoulli 176.20 1,052.60 2,932.50 5,769.20 9,546.50

Rayleigh 176.20 1,052.50 2,931.90 5,766.90 9,540.10

Timoshenko 176.20 1,052.20 2,929.90 5,759.80 9,521.70


EB & R 0.00 0.01 0.02 0.04 0.07

R & T 0.00 0.03 0.07 0.12 0.19

EB & T 0.00 0.04 0.09 0.16 0.26

d - Piezoelectric-Based Cantilever Beam System (tb=4 mm)


1st 2nd 3rd 4th 5th

Theory

Euler-Bernoulli 343.00 2,105.00 5,872.00 11,521.00 19,063.00

Rayleigh 343.00 2,104.00 5,867.00 11,503.00 19,015.00

Timoshenko 343.00 2,102.00 5,852.00 11,449.00 18,874.00


EB & R 0.00 0.05 0.09 0.16 0.25

R & T 0.00 0.10 0.26 0.47 0.74

EB & T 0.00 0.14 0.34 0.62 0.99

71

e - Piezoelectric-Based Cantilever Beam System (tb=8 mm)


1st 2nd 3rd 4th 5th

Theory

Euler-Bernoulli 679.00 4,210.00 11,763.00 23,061.00 38,144.00

Rayleigh 679.00 4,204.00 11,724.00 22,921.00 37,772.00

Timoshenko 678.00 4,186.00 11,607.00 22,508.00 36,718.00


EB & R 0.00 0.14 0.33 0.61 0.98

R & T 0.15 0.43 1.00 1.80 2.79

EB & T 0.15 0.57 1.33 2.40 3.74

f - Piezoelectric-Based Cantilever Beam System (tb=16 mm)


1st 2nd 3rd 4th 5th

Theory

Euler-Bernoulli 1,350.00 8,420.00 23,552.00 46,161.00 76,332.00

Rayleigh 1,349.00 8,374.00 23,247.00 45,072.00 73,501.00

Timoshenko 1,346.00 8,237.00 22,388.00 42,224.00 66,754.00


EB & R 0.07 0.55 1.30 2.36 3.71

R & T 0.22 1.64 3.70 6.32 9.18

EB & T 0.30 2.17 4.94 8.53 12.55

g - Piezoelectric-Based Cantilever Beam System (tb=24 mm)


1st 2nd 3rd 4th 5th

Theory

Euler-Bernoulli 2,022.00 12,631.00 35,343.00 69,265.00 114,530.00

Rayleigh 2,019.00 12,476.00 34,337.00 65,745.00 105,610.00

Timoshenko 2,008.00 12,037.00 31,779.00 57,928.00 88,582.00


EB & R 0.15 1.23 2.85 5.08 7.79

R & T 0.54 3.52 7.45 11.89 16.12

EB & T 0.69 4.70 10.08 16.37 22.66

4.2 Calculation of eigenfunction coefficients

To solve for the 12 eigenfunction coefficients for the three section discontinuous beam and

orthonormilize the eigenfunctions, the [ ] matrixes are converted to reduced row echelon

form and all coefficients are written in terms of the first coefficient of the first eigenfunction.

The reduced row echelon form of [ ] can be written as:

[ ] [[

] [

]

]

(4.2)

72

The orthonormalization of discontinuous cantilever beam is performed by varying from 0

to maximum 10 (dependent on thickness of beam) and performing orthonormalization

calculation with respect to the mass distribution for the mode which are shown in (4.3), (4.4)

and (4.5) for Euler-Bernoulli, Rayleigh and Timoshenko beam respectively:

∫

∫ ( )

(4.3)

∫

∫

∫ ( )

∫ (

)

(4.4)

∫

∫

∫ ( )

∫ ( )

(4.5)

A sample plot showing appropriate value yielding orthonormilized eigenfunctions of three

theory is shown in Fig. 4.2. The eigenfunctions are orthonormalized when their respective plot of

equations (4.3), (4.4) and (4.50) crosses the x-axis.

Fig. 4.2 coefficient estimate to yield orthonormalized eigenfunctions

0 0.2 0.4 0.6 0.8 1 1.2 1.4-1

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4A1 coefficient estimate to yield orthonormalized eigenfunctions

A1 Coefficient

Zero

Cro

ssin

g

73

4.3 Mode shapes

Using and eigenfunction coefficients calculated in previous section, the first five mode

shapes are plotted and are shown in Fig. 4.3.a-e and Fig. 4.4.a-e.

Fig. 4.3.a-e The first five mode shapes for

0 0.05 0.1 0.15 0.2 0.250

2

4

6

8

10

12

14

Cantilever length (m)

0 0.05 0.1 0.15 0.2 0.25-15

-10

-5

0

5

10


0 0.05 0.1 0.15 0.2 0.25-10

-5

0

5

10

15


0 0.05 0.1 0.15 0.2 0.25-15

-10

-5

0

5

10


0 0.05 0.1 0.15 0.2 0.25-10

-5

0

5

10

15


74

Fig. 4.4.a-e The first five mode shapes for

The first group demonstrates the first five mode shapes of three kind of theory for slender

beam with and second group demonstrates those for stocky beam with

. As the plots indicate, in slender beam, Euler-Bernoulli, Rayleigh and Timoshenko

curves cover each other, while in stocky beam differences between them are considerable and

0 0.05 0.1 0.15 0.2 0.25-0.5

0

0.5

1

1.5

2


0 0.05 0.1 0.15 0.2 0.25-2

-1.5

-1

-0.5

0

0.5

1

1.5


0 0.05 0.1 0.15 0.2 0.25-1.5

-1

-0.5

0

0.5

1

1.5

2


0 0.05 0.1 0.15 0.2 0.25-2

-1.5

-1

-0.5

0

0.5

1

1.5


0 0.05 0.1 0.15 0.2 0.25-1.5

-1

-0.5

0

0.5

1

1.5

2


75

visible. Table 4.3.a-e show difference between normalized mode shapes (MAC value) of three

theory for 7 values of thickness of beam.” In general it measures the degree of proportion

between two modal vectors in the form of a correlation coefficient. MAC is defined in equation

(4.6). If { } and { } are estimates from the same physical mode shape the MAC value should

be close to un1ty. If { } and { } are estimates of different physical mode shape the MAC

value should be low”,[13]. MAC value 1 means there is no difference, while factor 0.95 indicates

%5 difference between normalized vector of mode shapes. As we see, for

and for all five modes, MAC factor equals 1 while with increasing thickness, this factor

decreases. The difference for higher thickness and higher eigenfrequencies are considerable. In

other words for stocky beam with and in fifth mode, MAC Value is 0.95 which

means %5 difference between modal amplitude of Rayleigh and Timoshenko or between Euler-

Bernoulli and Timoshenko beam.

Table 4.3.a-e Difference between normalized mode shapes

a - Piezoelectric-Based Cantilever Beam System (tb=0.5, 1 and 2 mm)

Natural Frequencies of Beam (rad/s) 1st 2nd 3rd 4th 5th

Difference between Normalized Mode Shape

Vectors (MAC value)

EB & R 1.0000 1.0000 1.0000 1.0000 1.0000

R & T 1.0000 1.0000 1.0000 1.0000 1.0000

EB & T 1.0000 1.0000 1.0000 1.0000 1.0000

b - Piezoelectric-Based Cantilever Beam System (tb=4 mm)


Difference between Normalized Mode Shapes

(MAC value)

EB & R 1.0000 1.0000 1.0000 1.0000 1.0000

R & T 1.0000 1.0000 1.0000 1.0000 0.9999

EB & T 1.0000 1.0000 1.0000 1.0000 0.9999

c - Piezoelectric-Based Cantilever Beam System (tb=8 mm)



(MAC value)

EB & R 1.0000 1.0000 1.0000 1.0000 0.9999

R & T 1.0000 1.0000 0.9999 0.9996 0.9989

EB & T 1.0000 1.0000 0.9999 0.9997 0.9992

d - Piezoelectric-Based Cantilever Beam System (tb=16 mm)



(MAC value)

EB & R 1.0000 1.0000 0.9999 0.9997 0.9992

R & T 1.0000 0.9998 0.9998 0.9954 0.9867

EB & T 1.0000 0.9998 0.9991 0.9965 0.9903

76

e - Piezoelectric-Based Cantilever Beam System (tb=24 mm)



(MAC value)

EB & R 1.0000 1.0000 0.9997 0.9987 0.9961

R & T 1.0000 0.9991 0.9952 0.9820 0.9535

EB & T 1.0000 0.9992 0.9961 0.9868 0.9683

{ } { } |{ }

{ } |

{ } { } { }

{ }

(4.6)

4.4 Time and frequency domain

All system models are cast in state space for further time/frequency domain analysis. No

damping is added to the system models. The excitation force is applied by the piezo layer and

can be modeled as a concentrated moment at the locations where the piezo layers starts and ends.

The moment applied by the piezo layers due to the applied voltage can be written as

equations (4.7) for Euller-Bernoulli and Rayleigh and (4.8) for Timoshenko beam.

( ) ( )

(4.7)

( ) ( )

(4.8)

The output C vector is composed of the superposition of the mode shape values at .

Bode plots of three theory are shown in Fig. 4.5.a-g for different thickness of beam. This

discontinuous system exhibits different behavior for different theory and different thickness. For

example for , the first natural frequency of Euler-Bernoulli and Timoshenko beam

dominates the oscillatory content of the tip displacement while in Rayleigh beam fifth natural

frequency dominates, and for , the fifth natural frequency of Euler-Bernoulli,

Rayleigh and Timoshenko beam dominates the oscillatory content of the tip displacement. In

other words, for stocky beam higher frequencies are dominant to achieve the tip displacement.

Supporting the Bode plots, the time response of Timoshenko theory due to a unit step input are

shown in Fig. 4.6.a-f.

77

Fig. 4.5.a Bode plot of tip displacement for

-300

-200

-100

0

100

200

Mag

nitud

e (d

B)

101

102

103

104

105

-180

0

180

360

540

720

900

Phas

e (d

eg)

Bode Diagram

Frequency (rad/s)


-300

-200

-100

0

100

200

Mag

nitud

e (d

B)

101

102

103

104

105

-180

0

180

360

540

720

900

Phas

e (d

eg)

Bode Diagram

Frequency (rad/s)


-300

-200

-100

0

100

Mag

nitud

e (d

B)

101

102

103

104

105

0

180

360

540

720

Phas

e (d

eg)

Bode Diagram

Frequency (rad/s)


78

Fig. 4.5.b Bode plot of tip displacement for

-300

-200

-100

0

100

200

Mag

nitud

e (d

B)

101

102

103

104

105

0

180

360

540

720

900

Phas

e (d

eg)

Bode Diagram

Frequency (rad/s)


-300

-200

-100

0

100

200

Mag

nitud

e (d

B)

101

102

103

104

105

-360

0

360

720

1080

Phas

e (d

eg)

Bode Diagram

Frequency (rad/s)


-300

-200

-100

0

100

200

Mag

nitud

e (d

B)

101

102

103

104

105

0

180

360

540

720

900

Phas

e (d

eg)

Bode Diagram

Frequency (rad/s)


79

Fig. 4.5.c Bode plot of tip displacement for

-300

-200

-100

0

100

200

Mag

nitud

e (d

B)

101

102

103

104

105

-360

0

360

720

1080

Phas

e (d

eg)

Bode Diagram

Frequency (rad/s)


-300

-200

-100

0

100

200

Mag

nitud

e (d

B)

101

102

103

104

105

-360

0

360

720

1080

Phas

e (d

eg)

Bode Diagram

Frequency (rad/s)


-300

-200

-100

0

100

200

Mag

nitud

e (d

B)

101

102

103

104

105

0

180

360

540

720

900

Phas

e (d

eg)

Bode Diagram

Frequency (rad/s)


80

Fig. 4.5.d Bode plot of tip displacement for

-300

-200

-100

0

100

200

Mag

nitud

e (d

B)

101

102

103

104

105

106

0

360

720

1080

Phas

e (d

eg)

Bode Diagram

Frequency (rad/s)


-300

-200

-100

0

100

200

Mag

nitud

e (d

B)

101

102

103

104

105

106

0

180

360

540

720

Phas

e (d

eg)

Bode Diagram

Frequency (rad/s)


-300

-200

-100

0

100

Mag

nitud

e (d

B)

101

102

103

104

105

106

0

180

360

540

720

Phas

e (d

eg)

Bode Diagram

Frequency (rad/s)


81

Fig. 4.5.e Bode plot of tip displacement for

-400

-300

-200

-100

0

100

200

Mag

nitud

e (d

B)

102

103

104

105

106

-360

0

360

720

1080

Phas

e (d

eg)

Bode Diagram

Frequency (rad/s)


-400

-300

-200

-100

0

100

200

Mag

nitud

e (d

B)

102

103

104

105

106

0

360

720

1080

Phas

e (d

eg)

Bode Diagram

Frequency (rad/s)


-400

-300

-200

-100

0

100

200

Mag

nitud

e (d

B)

102

103

104

105

106

-360

0

360

720

1080

Phas

e (d

eg)

Bode Diagram

Frequency (rad/s)


82

Fig. 4.5.f Bode plot of tip displacement for

-400

-300

-200

-100

0

100

Mag

nitud

e (d

B)

102

103

104

105

106

0

180

360

540

720

Phas

e (d

eg)

Bode Diagram

Frequency (rad/s)


-400

-300

-200

-100

0

100

200

Mag

nitud

e (d

B)

102

103

104

105

106

0

180

360

540

720

Phas

e (d

eg)

Bode Diagram

Frequency (rad/s)


-400

-300

-200

-100

0

100

200

Mag

nitud

e (d

B)

102

103

104

105

106

-360

0

360

720

1080

1440

Phas

e (d

eg)

Bode Diagram

Frequency (rad/s)


83

Fig. 4.5.g Bode plot of tip displacement for

-400

-300

-200

-100

0

100

Mag

nitud

e (d

B)

102

103

104

105

106

-360

0

360

720

1080

Phas

e (d

eg)

Bode Diagram

Frequency (rad/s)


-400

-300

-200

-100

0

100

Mag

nitud

e (d

B)

102

103

104

105

106

-180

0

180

360

540

720

900

Phas

e (d

eg)

Bode Diagram

Frequency (rad/s)


-400

-300

-200

-100

0

100

Mag

nitud

e (d

B)

102

103

104

105

106

0

180

360

540

720

900

Phas

e (d

eg)

Bode Diagram

Frequency (rad/s)


84

Fig. 4.6.a-f Step response of tip displacement for

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-2

0

2

4

6

8

10

12x 10

-6

Step Response

Time (seconds)

Am

plit

ude


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1

0

1

2

3

4

5x 10

-6

Step Response

Time (seconds)

Am

plit

ude


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-2

0

2

4

6

8

10

12

14x 10

-7

Step Response

Time (seconds)

Am

plit

ude


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-0.5

0

0.5

1

1.5

2

2.5

3

3.5x 10

-7

Step Response

Time (seconds)

Am

plit

ude


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-1

0

1

2

3

4

5

6

7

8

9x 10

-8

Step Response

Time (seconds)

Am

plit

ude


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5-2

0

2

4

6

8

10x 10

-9

Step Response

Time (seconds)

Am

plit

ude


85

5 Conclusions

In this research, equations of motion and boundary conditions of a discontinuous three section

piezoelectric based cantilever based system were developed. To achieve the goal, three different

theory were utilized, Euler-Bernoulli, Rayleigh and Timoshenko. As mentioned before, Euler-

Bernoulli theory neglects the effects of rotary inertia and shear deformation and is applicable to

analyze slender beams while Rayleigh theory considers the effect of rotary inertia and

Timoshenko theory consider the effects of both rotary inertia and shear deformation and can be

used for thick beams.

In next step, natural frequencies of system for seven different thickness were obtained and

compared to each other. The results showed for stocky beams difference between which

calculated by Euler-Bernoulli or Rayleigh and Timoshenko theory is considerable while those

values for slender beams are negligible and for low thickness are precisely zero! Difference

values for higher natural frequencies (higher mode shapes) are major.

In next step, eigenfunction coefficients were calculated and relevant eigenfunctions were

obtained and plotted. Similar to previous results, in slender beams, mode shapes cover each other

while for higher eigenfrequencies, difference between orthonormilized eigenfunction vectors

which were obtained by Timoshenko theory and two other ones are visible and considerable.

Next, reaction of system to excitation force was analyzed. The force was applied by the piezo

layer and was modeled as a concentrated moment at the locations where the piezo layer starts

and ends. The moment was due to the applied voltage to piezo layer. Although there was not

obtained a general role for system’s behavior, usually for slender beams the first natural

frequency of Euler-Bernoulli, Rayleigh and Timoshenko beams dominates the oscillatory content

of the tip displacement while for stocky beams, higher eigenfrequencies of all three method,

particularly Timoshenko, dominates the oscillatory content of the tip displacement. At the end of

research, system’s behavior to unit input step function was presented.

86

List of symbols

When introducing the beam theories several constants/functions are used. Here is a summary

of these:

Volumetric mass density of beam

Volumetric mass density of piezoelectric

Electric potential

ϵ Electric field

Stress

Piezoelectric constant

Eigenfrequency

Mode shape or eigenfunction

Rotation of cross section from vertical axis

Eigenfunction

Kronecker delta

Shear distortion of cross section

Structural damping coefficient

Elasticity coefficients matrix under constant dielectric displacement

Viscous damping coefficient

Piezoelectric strain constant

D Dielectric displacement


Beam young’s modulus

Piezoelectric young’s modulus

Flexural stiffness

Beam shear modulus

Piezoelectric shear modulus


H(x) Heaviside function

Current intensity

Mass moment of inertia

Radius of gyration

Length of 1st section of cantilever

Total length of 1st and 2nd section of cantilever

Total length of cantilever

Total length of cantilever

Mass per unit length

Charge density

generalized time-dependent coordinates

87

Total accumulated charge

Beam constant shape dependent variable

Piezoelectric constant shape-dependent variable

Strain

Thickness of beam

Thickness of piezoelectric

Kinetic energy

Total energy density per unit volume

Potential energy

Total electrical potential energy

Volume

Applied voltage to piezoelectric

Width of beam

Width of piezoelectric

Transverse displacement of the neutral axis

Mechanical virtual work

Electrical virtual work

Neutral surface

88

Appendix A1- Finding equations (3.15) and (3.17):

∫

∫ [∫

(

)

∫ ∫

∫ ∫

∫ ∫ (( )

)

∫ ∫

∫ (

)

∫ (

)

∫

]

(A1.1)

or we can say:

∫

(A1.2)

where:

∫

(

)

(A1.3)

89

∫ ∫

∫ ∫ (

) (

)

∫ ∫

(

)

∫ (

)|

(

)

∫ (

)

(

)

(A1.4)

∫ ∫

∫ ∫ (

) (

)

∫ ∫

(

)

∫ (

)|

(

)

∫ (

)

(

)

(A1.5)

90

∫ ∫ (( )

)

∫ ∫ (( (

) ) (

)

( (

)

) )

∫ ∫

(

)

∫ ∫

(

)

∫ ∫

∫ ∫

∫ (

)|

(

)

∫ (

)|

(

)

∫ (

)|

∫ |

(A1.6)

91

∫ ∫

∫ ∫ (

) (

)

∫ ∫

(

)

∫ (

)|

(

)

∫ (

)

(

)

(A1.7)

∫ (

)

(A1.8)

∫ (

)

(A1.9)

∫

(A1.10)

so:

∫

∫ [∫ {

(

)}

∫ {

(

)

(

)}

∫

∫ (

)

∫ (

)

]

(A1.11

where:

92

[

(

)

(

)]

⁄

(A1.12)

Now using integration by parts theorem we can say:

∫ ∫

(

) ∫ ∫

(A1.13)

∫ ∫

(

)

∫ {

(

)|

(

) |

∫

(

)

}

(A1.14)

∫ ∫ (

)

∫ { (

)|

|

∫

}

(A1.15)

so after some manipulations equation (3.15) leads to (3.17):

93

∫

∫ [∫ {(

(

)

)

(

) }

(

) (

)|

(

(

)

) |

]

(A1.16)

94

Appendix A2- Finding equations (3.37a-l)

(A2.1)

(A2.2)

(A2.3)

(A2.4)

(A2.5)

(A2.6)

(A2.7)

(A2.8)

95

(A2.9)

(A2.10)

(A2.11)

(A2.12)

96

Appendix A3- Finding equations (3.69) and (3.71)

∫

∫ [∫

(

)

∫

(

)

∫ ∫

∫ ∫

∫ ∫ (( )

)

∫ ∫

∫ (

)

∫ (

)

∫

]

(A3.1)

or we can say:

∫

(A3.2)

where:

∫

(

)

(A3.3)

∫

(

)

(A3.4)

97

∫ ∫

∫ ∫ (

) (

)

∫ ∫

(

)

∫ (

)|

(

)

∫ (

)

(

)

(A3.5)

∫ ∫

∫ ∫ (

) (

)

∫ ∫

(

)

∫ (

)|

(

)

∫ (

)

(

)

(A3.6)

98

∫ ∫ (( )

)

∫ ∫ (( (

) ) (

)

( (

)

) )

∫ ∫

(

)

∫ ∫

(

)

∫ ∫

∫ ∫

∫ (

)|

(

)

∫ (

)|

(

)

∫ (

)|

∫ |

(A3.7)

99

∫ ∫

∫ ∫ (

) (

)

∫ ∫

(

)

∫ (

)|

(

)

∫ (

)

(

)

(A3.8)

∫ (

)

(A3.9)

∫ (

)

(A3.10)

∫

(A3.11)

so:

∫

∫ [∫

(

)

∫

(

)

∫ {

(

)

(

)}

∫

∫ (

)

∫ (

)

]

(A3.12)

100

where:

[

(

)

(

)]

⁄

(A3.13)


∫ ∫

(

) ∫ ∫

(A3.14)

∫ ∫

(

)

∫ {

|

∫

(

)

}

(A3.15)

∫ ∫

(

)

∫ {

(

)|

(

) |

∫

(

)

}

(A3.16)

∫ ∫ (

)

∫ { (

)|

|

∫

}

(A3.17)

so after some manipulations equation (3.69) leads to (3.71):

101

∫

∫ [∫ {(

(

)

(

)

)

(

) }

(

) (

)|

(

(

)

) |

]

(A3.18)

102

Appendix A4- Solution of equation (3.82)

(A4.1)

so if we assume we will obtain:

(A4.2)

so the roots of first equation will be:

(

⁄ (

⁄ )

⁄

)

⁄

(

⁄ (

⁄ )

⁄

)

⁄

(

⁄ (

⁄ )

⁄

)

⁄

(

⁄ (

⁄ )

⁄

)

⁄

(

⁄ (

⁄ )

⁄

)

⁄

(

⁄ (

⁄ )

⁄

)

⁄

(A4.3)

103


(A5.1)

(A5.2)

(A5.3)

(A5.4)

(A5.5)

(A5.6)

(A5.7)

104

(

)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

(A5.8)

(A5.9)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

(

)

(A5.10)

(A5.11)

105

(

)

(

)

(

)

(

)

(A5.12)

106

Appendix A6- Finding equation (3.89)

(

)

(

)

(A6.1)

now if we assume we will obtain:

(

)

(

)

(A6.2)

or we can say:

(

)

(

)

(

)

(

)

(A6.3)

Multiplying the first equation by and the second one by and integrate them

over the domain and then subtract them we will obtain:

∫

∫

(

)

∫

(

)

∫

(

)

∫

(

)

(A6.4)

Now using intergration by part (twice) the last equation will hanged into:

(∫

∫

)

|

(

)

|

|

(

)

|

(A6.5)

107

If we compare second and third terms of above equation with equation (3.78b) we see that

both of them due to boundary conditions are zero, so finally we will obtain:

∫

∫

∫ ( )

∫ (

)

(A6.6)

which is orthogonality condition of Rayleigh’s beam with respect to mass.

108

Appendix A7- Finding equations (3.123), (3.125) and (3.128a)

∫

∫ [∫

(

)

∫

(

)

∫ ∫

∫ ∫

∫ ∫ (( )

)

∫ ∫

∫

∫ (

)

∫ (

)

∫

]

(A7.1)

or we can say:

∫

(A7.2)

where:

∫

(

)

(A7.3)

∫

(

)

(A7.4)

109

∫ ∫

∫ ∫ (

) (

)

∫ ∫

(

)

∫ (

)|

(

)

∫ (

)

(

)

(A7.5)

∫ ∫

∫ ∫ (

) (

)

∫ ∫

(

)

∫ (

)|

(

)

∫ (

)

(

)

(A7.6)

110

∫ ∫ (( )

)

∫ ∫ (( (

) ) (

)

( (

)

) )

∫ ∫

(

)

∫ ∫

(

)

∫ ∫

∫ ∫

∫ (

)|

(

)

∫ (

)|

(

)

∫ (

)|

∫ |

(A7.7)

111

∫ ∫

∫ ∫ (

) (

)

∫ ∫

(

)

∫ (

)|

(

)

∫ (

)

(

)

(A7.8)

∫

∫ (

)

(

)

∫ (

)

(

)

∫ (

)

(A7.9)

∫ (

)

(A7.10)

∫ (

)

(A7.11)

∫

(A7.12)

So:

112

∫

∫ [∫

(

)

∫

(

)

∫ {

(

)

(

)}

∫ (

)

(

)

∫

∫ (

)

∫ (

)

]

(A7.13)

where:

[

(

)

(

)]

⁄

(A7.14)


∫ ∫

(

) ∫ ∫

(A7.15)

∫ ∫

(

) ∫ ∫

(A7.16)

∫ ∫

(

)

∫ {

|

∫

(

)

}

(A7.17)

113

∫ ∫ (

)

∫ { | ∫

( )

}

(A7.18)

∫ ∫ (

) (

)

∫ { (

) |

∫

( (

))

}

(A7.19)

So after some manipulations equation (3.123) leads to (3.125):

∫

∫ [∫ {(

( (

))

)

(

(

)

( )

(

))

(

) }

(

) |

( (

)) |

]

(A7.20)

Now from equation (3.127a) we can say:

114

(

)

(

)

(

)

(A7.21)

(

)

(

)

(

)

(A7.22)

(

)

(

)

(

)

(A7.23)

And similarly from equation (3.127b) we can say:

(

)

(

)

(A7.24)

(

)

(

)

(A7.25)

Now put (A7.21), (A7.22) and (A7.23) into (A7.25) gives:

(

)

(

)

[

(

)]

(

)

(

)

(

)

(

)

(

)

(A7.26)

115


(

)

(

)

(A8.1)

so if we assume we will obtain:

(

)

(

)

(A8.2)

so for √

the roots of first equation will be:

(

⁄ (

⁄ )

⁄

)

⁄

(

⁄ (

⁄ )

⁄

)

⁄

(

⁄ (

⁄ )

⁄

)

⁄

(

⁄ (

⁄ )

⁄

)

⁄

(

⁄ (

⁄ )

⁄

)

⁄

(

⁄ (

⁄ )

⁄

)

⁄

(A8.3)

and for √


(

⁄ (

⁄ )

⁄

)

⁄

(

⁄ (

⁄ )

⁄

)

⁄

116

(

⁄ (

⁄ )

⁄

)

⁄

(

⁄ (

⁄ )

⁄

)

⁄

(

⁄ (

⁄ )

⁄

)

⁄

(

⁄ (

⁄ )

⁄

)

⁄

(

⁄ (

⁄ )

⁄

)

⁄

(

⁄ (

⁄ )

⁄

)

⁄

(A8.4)

It is easy to observe that (

⁄ ) for all And also because solution of first case is

more comprehensive, we consider it.

117


(A9-1)

(

)

(

)

(A9-2)

(A9-3)

(A9-4)

(

)

(

)

{

[

]}

{

[

]}

(A9-5)

(

)

(

)

{

[

]}

{

[

]}

(A9-6)

118

(

) (

)

{

[ ]} {

[ ]}

(A9-7)

(

) (

)

{

[ ]} {

[ ]}

(A9-8)

(

) (

)

(

)

(

)

{

[

]}

{

[ ]}

(A9-9)

(

) (

)

119

(

)

(

)

{

[

]}

{

[ ]}

(A9-10)

(

)

{

[ ]}

(A9-11)

(

)

(

)

{

[ ]}

(A9-12)

120

Appendix A10- Finding equation (3.151)

{ (

)}

(A10.1)

(

) (

)

(A10.2)

now we assume:

(A10.3)

we will obtain:

{ (

)}

(A10.4)

(

) (

)

(A10.5)

or we can say:

{ (

)}

{ (

)}

(A10.6)

(

) (

)

(

) (

)

(A10.7)

Multiplying the first equation 0f (A10-6) by and the second one by and also

multiplying the first equation 0f (A10-7) by and the second one by

and integrate

them over the domain and then subtract them we will obtain:

121

∫

∫ (

)

(A10.8)

∫

∫ (

)

(A10.9)

by adding equations (A10-8) and (A10-9) we will obtain:

∫

∫

∫ ( )

∫ ( )

(A10.10)

which is orthogonality condition of Yimoshenko’s beam with respect to mass.

122


(

)

(

)

(A11-1)

So if we assume we will obtain:

(

)

(

)

(A11-2)

so for √


(

⁄ (

⁄ )

⁄

)

⁄

(

⁄ (

⁄ )

⁄

)

⁄

(

⁄ (

⁄ )

⁄

)

⁄

(

⁄ (

⁄ )

⁄

)

⁄

(

⁄ (

⁄ )

⁄

)

⁄

(

⁄ (

⁄ )

⁄

)

⁄

(A11-3)

and for √


(

⁄ (

⁄ )

⁄

)

⁄

(

⁄ (

⁄ )

⁄

)

⁄

123

(

⁄ (

⁄ )

⁄

)

⁄

(

⁄ (

⁄ )

⁄

)

⁄

(

⁄ (

⁄ )

⁄

)

⁄

(

⁄ (

⁄ )

⁄

)

⁄

(

⁄ (

⁄ )

⁄

)

⁄

(

⁄ (

⁄ )

⁄

)

⁄

(A11-4)

It is easy to observe that (

⁄ ) for all And also because solution of first case is

more comprehensive, we consider it.

124


(

)

(

)

(A12.1)

(A12.2)

(

)

(

)

{

}

{

}

(A12.3)

(

)

(

)

{

}

{

}

(A12.4)

125

(A12.5)

(A12.6)

(A12.7)

(A12.8)

(

) (

)

(

)

(

)

{

[ ]}

126

{

[ ]}

(A12.9)

(

) (

)

(

)

(

)

{

[ ]}

{

[ ]}

(A12.10)

(A12.11)

(

)

(

)

127

{

[ ]}

(A12.12)

128

Appendix B- MATLAB codes

% Payman Zolmajd % Thesis 7990, summer 2015

clear all close all clc

% Material properties

l1=20E-3; l2=80E-3; l3=250E-3; L=l3;

Eb=210E9; Ep=67E9;

Gb=82E9; Gp=26E9;

rhob=7850; rhop=7910;

wb=25E-3; wp=15E-3;

tb=0.5E-3; %tb=1E-3; %tb=2E-3; %tb=4E-3; %tb=8E-3; %tb=16E-3; %tb=24E-3; tp=0.5E-3;

d31=-183*10^-12;

Sb=5/6; Sp=Sb;

% Mass properties

Ab=tb*wb; mb=rhob*Ab;

Ap=tp*wp; mp=rhop*Ap;

mequ1=mb; mequ2=mb+mp;

129

mequ3=mb; mequ=mequ2;

% Nutral axis

zn=wp*Ep*tp*(tp+tb)/(2*(wp*Ep*tp+wb*Eb*tb));

% Equivalent young's modulus times moment of inertia

Ib1=(wb*(tb)^3)/12; EIequ1=Eb*Ib1;

Ib2=(wb*(tb)^3)/12+wb*tb*(zn^2); Ip=(zn^2)*(wb*tb+(Ep/Eb)*wp*tp)-(Ep/Eb)*wp*(zn*(tb*tp+tp^2)-(tp^3)/3-tb*... (tp^2)/2-tp*(tb^2)/4); EIequ2=Eb*(Ib1+Ip);


Ipnet=wp*tp*(tb^2)/4-wp*tp*tb*zn+wp*(tp^3)/3+wp*(tp^2)*tb/2-wp*(tp^2)*... zn+wp*tp*(zn^2); %EIequ2test=Eb*Ib2+Ep*Ipnet;

% Equivalent shear correction factor times shear modulus times area (sagequ)

sagequ1=Sb*Ab*Gb; sagequ2=Sb*(Ab*Gb+Ap*Gp); sagequ3=Sb*Ab*Gb;

% Equivalent polar moment of inertia of cross section

Jequ1=(mb*Ib1)/Ab; Jequ2=(mb*Ib2)/Ab+(mp*Ipnet)/Ap; Jequ3=(mb*Ib3)/Ab;

% Solve for natural frequencies and eigenfunction coefficients (3-part beam)

n=80001; %beta1=linspace(0,90,n); omega=linspace(0,3000,n);%tb=0.5mm %omega=linspace(0,5000,n);%tb=1mm %omega=linspace(0,10000,n);%tb=2mm %omega=linspace(0,20000,n);%tb=4mm %omega=linspace(0,40000,n);%tb=8mm %omega=linspace(0,80000,n);%tb=16mm %omega=linspace(0,120000,n);%tb=24mm

beta1=((omega.^2)*mequ1/EIequ1).^(0.25);

%betatest1=((2500^2)*mequ1/EIequ1).^(0.25) %betatest2=((3000^2)*mequ1/EIequ1).^(0.25) %betatest3=((15000^2)*mequ1/EIequ1).^(0.25)

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%betatest4=((18000^2)*mequ1/EIequ1).^(0.25) %betatest5=((39000^2)*mequ1/EIequ1).^(0.25) %betatest6=((48000^2)*mequ1/EIequ1).^(0.25) %betatest7=((65000^2)*mequ1/EIequ1).^(0.25) %betatest8=((95000^2)*mequ1/EIequ1).^(0.25) %betatest9=((100000^2)*mequ1/EIequ1).^(0.25) %betatest10=((155000^2)*mequ1/EIequ1).^(0.25)

for i=1:n

% Three-section solution - Euler Bernouli's beam J=BETA(beta1(i)); Jdet(i)=det(J);

% Three-section solution - Rayleigh's beam JR=omega_r(omega(i)); Jdet_r(i)=det(JR);

% Three-section solution - Timoshenko's beam1 JT1=omega_t1(omega(i)); Jdet_t1(i)=det(JT1);

% Three-section solution - Timoshenko's beam2 JT2=omega_t2(omega(i)); Jdet_t2(i)=det(JT2);

end

% 3-part beta lims - Euler Bernouli's beam beta_lims=[5 10;15 25;28 35;40 50;52 60];%tb=0.5mm,5modes %beta_lims=[5 10;15 25;28 35;40 50;52 60];%tb=1mm,5modes %beta_lims=[5 12;15 20;30 35;40 45;55 60];%tb=2mm,5modes %beta_lims=[7 9;18 20;30 32;43 45;55 57];%tb=4mm,5modes %beta_lims=[7 9;18 20;30 32;43 45;55 57];%tb=8mm,5modes %beta_lims=[7 9;18 20;30 32;43 45;55 58];%tb=16mm,5modes %beta_lims=[7 9;17 20;28 32;36 45;45 58];%tb=24mm,5modes

% 3-part omega lims - Rayleigh and Timoshenko's beam omega_lims=[40 60;260 270;740 760;1500 1560;2520 2540];%tb=0.5mm,5modes %omega_lims=[60 120;400 600;1400 1600;2800 3000;4700 5000];%tb=1mm,5modes %omega_lims=[100 300;1000 1100;2800 3000;5600 5800;9400 9600];... %tb=2mm,5modes %omega_lims=[300 400;2000 2200;5800 6000;11400 11600;18800 19200];... %tb=4mm,5modes %omega_lims=[600 700;4000 4400;11500 12000;22200 23200;36500 38500];... %tb=8mm,5modes %omega_lims=[1000 1500;8000 8500;22000 24000;40000 48000;66000 80000];... %tb=16mm,5modes %omega_lims=[1000 3000;11500 13000;31000 36000;57000 70000;85000 115000]; %tb=24mm,5modes

figure(1) semilogy(omega,abs(Jdet),'g',omega,abs(Jdet_r),'r',omega,abs(Jdet_t1),'b') grid; legend('Discontinuous Euler-Bernoulli beam',... 'Discontinuous Rayleigh beam',...

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'Discontinuous Timoshenko beam','Location','southeast') title('Omega values for first five eigenfrequencies') xlabel('Omega (rad/s)') ylabel('|Det(J)|') h=fig(1,'units','inches','width',6,'height',6,'font','Times New Roman',... 'fontsize',8);

for i=1:size(beta_lims,1) %************* Euler Bernouli *************%

% 3-part, Euler Bernouli's beam Jdet_num=@(zi) det(BETA(zi)); beta1_num(i)=fzero(Jdet_num,beta_lims(i,:)); omega_num_eb(i)=(((beta1_num(i)).^4)*EIequ1/mequ1).^(0.5);

end

beta2_num=beta1_num*((mb+mp)*EIequ1/(mb*EIequ2))^(1/4); beta3_num=beta1_num*(EIequ1/EIequ3)^(1/4); omega_num_eb=((beta1_num.^4*(EIequ1/mequ1)).^0.5);

for i=1:size(omega_lims,1) %******* Rayleigh and Timoshenko ********%

% 3-part, Rayleigh's beam Jdet_num_r=@(zi) det(omega_r(zi)); omega_num_r(i)=fzero(Jdet_num_r,omega_lims(i,:));

% 3-part, Timoshenko's beam1 Jdet_num_t1=@(zi) det(omega_t1(zi)); omega_num_t1(i)=fzero(Jdet_num_t1,omega_lims(i,:));

% 3-part, Timoshenko's beam2 Jdet_num_t2=@(zi) det(omega_t2(zi)); omega_num_t2(i)=fzero(Jdet_num_t2,omega_lims(i,:));

end

% Solve for eigenfunction coefficients

x1=linspace(0,l1,101); x2=linspace(l1,l2,101); x3=linspace(l2,l3,101); C=linspace(0,10,501); % tb=0.5 mm %C=linspace(0,8,501); % tb=1 mm %C=linspace(0,5,501); % tb=2 mm %C=linspace(0,3.5,501); % tb=4 mm %C=linspace(0,3,501); % tb=8 mm %C=linspace(0,2,501); % tb=16 mm %C=linspace(0,1.4,501); % tb=24 mm

%$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$% Clims=[5 5.5;7.5 8;7 7.5;7.4 8;8 8.5]; % tb=0.5 mm %Clims=[3.5 3.6;4.98 5;4.8 4.9;4.96 4.99;4.96 4.99];% tb=1 mm %Clims=[2.35 2.45;3.37 3.40;3.33 3.36;3.33 3.36;3.31 3.34]; %tb=2 mm %Clims=[1.6 1.8;2.32 2.37;2.29 2.33;2.29 2.33;2.27 2.31]; %tb=4 mm

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%Clims=[1.15 1.2;1.63 1.65;1.56 1.62;1.56 1.62;1.52 1.62]; %tb=8 mm %Clims=[0.8 0.86;1.1 1.18;1.0 1.14;1.0 1.14;0.9 1.14]; % tb=16 mm %Clims=[0.66 0.70;0.88 0.95;0.80 0.93;0.72 0.93;0.65 0.93]; % tb=24 mm

colorfinder=['k';'b';'r';'g';'m']; %$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$%

color1=['b-';'r-';'g-']; color2=['b:';'r:';'g:']; color3=['b--';'r--';'g--'];

% Initialize output vectors for state space

C_disc_eb=zeros(1,10); C_disc_r=zeros(1,10); C_disc_t=zeros(1,10);

for i=1:size(beta_lims,1)

%************* 3-part solution, Euler Bernouli's beam **************** Jrre=COEFF(beta1_num(i)); a1=1; a2=Jrre(2,12)/Jrre(1,12); a3=Jrre(3,12)/Jrre(1,12); a4=Jrre(4,12)/Jrre(1,12); a5=Jrre(5,12)/Jrre(1,12); a6=Jrre(6,12)/Jrre(1,12); a7=Jrre(7,12)/Jrre(1,12); a8=Jrre(8,12)/Jrre(1,12); a9=Jrre(9,12)/Jrre(1,12); a10=Jrre(10,12)/Jrre(1,12); a11=Jrre(11,12)/Jrre(1,12); a12=-1/Jrre(1,12);

phi1_2_num=@(x1_num) (a1*sin(beta1_num(i)*x1_num)+a2*... cos(beta1_num(i)*x1_num)+a3*sinh(beta1_num(i)*x1_num)+... a4*cosh(beta1_num(i)*x1_num)).^2; phi2_2_num=@(x2_num) (a5*sin(beta2_num(i)*x2_num)+a6*... cos(beta2_num(i)*x2_num)+a7*sinh(beta2_num(i)*x2_num)+... a8*cosh(beta2_num(i)*x2_num)).^2; phi3_2_num=@(x3_num) (a9*sin(beta3_num(i)*x3_num)+a10*... cos(beta3_num(i)*x3_num)+a11*sinh(beta3_num(i)*x3_num)+... a12*cosh(beta3_num(i)*x3_num)).^2;

C_zero_eb(i,:)=C.^2*((mequ1)*integral(phi1_2_num,0,l1)+(mequ2)*... integral(phi2_2_num,l1,l2)+(mequ3)*... integral(phi3_2_num,l2,l3))-1;

C1(i)=fzero(@(Cnum) (Cnum).^2*((mequ1)*integral(phi1_2_num,0,l1)... +(mequ2)*integral(phi2_2_num,l1,l2)+(mequ3)*... integral(phi3_2_num,l2,l3))-1,Clims(i));

D1(i)=a2*C1(i); E1(i)=a3*C1(i);

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F1(i)=a4*C1(i);

C2(i)=a5*C1(i); D2(i)=a6*C1(i); E2(i)=a7*C1(i); F2(i)=a8*C1(i);

C3(i)=a9*C1(i); D3(i)=a10*C1(i); E3(i)=a11*C1(i); F3(i)=a12*C1(i);

phi1(i,:)=C1(i)*sin(beta1_num(i)*x1)+D1(i)*cos(beta1_num(i)*x1)... +E1(i)*sinh(beta1_num(i)*x1)+F1(i)*cosh(beta1_num(i)*x1); phi2(i,:)=C2(i)*sin(beta2_num(i)*x2)+D2(i)*cos(beta2_num(i)*x2)... +E2(i)*sinh(beta2_num(i)*x2)+F2(i)*cosh(beta2_num(i)*x2); phi3(i,:)=C3(i)*sin(beta3_num(i)*x3)+D3(i)*cos(beta3_num(i)*x3)... +E3(i)*sinh(beta3_num(i)*x3)+F3(i)*cosh(beta3_num(i)*x3);

dphi2(i,:)=C2(i)*beta2_num(i)*cos(beta2_num(i)*x2)-D2(i)*beta2_num(i)*... sin(beta2_num(i)*x2)+E2(i)*beta2_num(i)*... cosh(beta2_num(i)*x2)+F2(i)*beta2_num(i)*... sinh(beta2_num(i)*x2);

dphi2l1(i,:)=C2(i)*beta2_num(i)*cos(beta2_num(i)*l1)-D2(i)*... beta2_num(i)*sin(beta2_num(i)*l1)+E2(i)*beta2_num(i)*... cosh(beta2_num(i)*l1)+F2(i)*beta2_num(i)*... sinh(beta2_num(i)*l1); dphi2l2(i,:)=C2(i)*beta2_num(i)*cos(beta2_num(i)*l2)-D2(i)*... beta2_num(i)*sin(beta2_num(i)*l2)+E2(i)*beta2_num(i)*... cosh(beta2_num(i)*l2)+F2(i)*beta2_num(i)*... sinh(beta2_num(i)*l2);

fdisc_eb(i,1)=0.5*(dphi2l1(i,:)-dphi2l2(i,:))*wp*Ep*d31*(tb+tp-2*zn);

C_disc_eb(1,i)=C3(i)*sin(beta3_num(i)*l3)+D3(i)*cos(beta3_num(i)*l3)+... E3(i)*sinh(beta3_num(i)*l3)+F3(i)*cosh(beta3_num(i)*l3);

%*************** 3-part solution, Rayleigh's beam ********************% Jrre_r=coeff_r(omega_num_r(i)); a1r=1; a2r=Jrre_r(2,12)/Jrre_r(1,12); a3r=Jrre_r(3,12)/Jrre_r(1,12); a4r=Jrre_r(4,12)/Jrre_r(1,12); a5r=Jrre_r(5,12)/Jrre_r(1,12); a6r=Jrre_r(6,12)/Jrre_r(1,12); a7r=Jrre_r(7,12)/Jrre_r(1,12); a8r=Jrre_r(8,12)/Jrre_r(1,12); a9r=Jrre_r(9,12)/Jrre_r(1,12); a10r=Jrre_r(10,12)/Jrre_r(1,12); a11r=Jrre_r(11,12)/Jrre_r(1,12); a12r=-1/Jrre_r(1,12);

alpha1(i)=(omega_num_r(i).^2)*Jequ1/EIequ1; alpha2(i)=(omega_num_r(i).^2)*Jequ2/EIequ2;

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alpha3(i)=(omega_num_r(i).^2)*Jequ3/EIequ3;

zeta1(i)=mequ1*(omega_num_r(i).^2)/(EIequ1); zeta2(i)=mequ2*(omega_num_r(i).^2)/(EIequ2); zeta3(i)=mequ3*(omega_num_r(i).^2)/(EIequ3);

s11(i)=((-alpha1(i)/2)+((alpha1(i).^2)/4+zeta1(i)).^(1/2)).^(1/2); s21(i)=((-alpha2(i)/2)+((alpha2(i).^2)/4+zeta2(i)).^(1/2)).^(1/2); s31(i)=((-alpha3(i)/2)+((alpha3(i).^2)/4+zeta3(i)).^(1/2)).^(1/2);

s12(i)=s11(i); s22(i)=s21(i); s32(i)=s31(i);

s13(i)=((alpha1(i)/2)+((alpha1(i).^2)/4+zeta1(i)).^(1/2)).^(1/2); s23(i)=((alpha2(i)/2)+((alpha2(i).^2)/4+zeta2(i)).^(1/2)).^(1/2); s33(i)=((alpha3(i)/2)+((alpha3(i).^2)/4+zeta3(i)).^(1/2)).^(1/2);

s14(i)=s13(i); s24(i)=s23(i); s34(i)=s33(i);

phi1_2_num_r=@(zi1) (a1r*sinh(s11(i)*zi1)+a2r*cosh(s12(i)*zi1)+... a3r*sin(s13(i)*zi1)+a4r*cos(s14(i)*zi1)).^2; phi2_2_num_r=@(zi2) (a5r*sinh(s21(i)*zi2)+a6r*cosh(s22(i)*zi2)+... a7r*sin(s23(i)*zi2)+a8r*cos(s24(i)*zi2)).^2; phi3_2_num_r=@(zi3) (a9r*sinh(s31(i)*zi3)+a10r*cosh(s32(i)*zi3)+... a11r*sin(s33(i)*zi3)+a12r*cos(s34(i)*zi3)).^2;

dphi1_2_num_r=@(zi1) (a1r*s11(i)*cosh(s11(i)*zi1)+a2r*s12(i)*... sinh(s12(i)*zi1)+a3r*s13(i)*cos(s13(i)*zi1)-a4r*s14(i)*... sin(s14(i)*zi1)).^2; dphi2_2_num_r=@(zi2) (a5r*s21(i)*cosh(s21(i)*zi2)+a6r*s22(i)*... sinh(s22(i)*zi2)+a7r*s23(i)*cos(s23(i)*zi2)-a8r*s24(i)*... sin(s24(i)*zi2)).^2; dphi3_2_num_r=@(zi3) (a9r*s31(i)*cosh(s31(i)*zi3)+a10r*s32(i)*... sinh(s32(i)*zi3)+a11r*s33(i)*cos(s33(i)*zi3)-a12r*... s34(i)*sin(s34(i)*zi3)).^2;

C_zero_r(i,:)=C.^2*((mequ1)*integral(phi1_2_num_r,0,l1)+(mequ2)*... integral(phi2_2_num_r,l1,l2)+(mequ3)*... integral(phi3_2_num_r,l2,l3)+Jequ1*... integral(dphi1_2_num_r,0,l1)+Jequ2*... integral(dphi2_2_num_r,l1,l2)+Jequ3*... integral(dphi3_2_num_r,l2,l3))-1;

C1_r(i)=fzero(@(Cnum) (Cnum).^2*((mequ1)*integral(phi1_2_num_r,0,l1)+... (mequ2)*integral(phi2_2_num_r,l1,l2)+(mequ3)*... integral(phi3_2_num_r,l2,l3)+Jequ1*... integral(dphi1_2_num_r,0,l1)+Jequ2*... integral(dphi2_2_num_r,l1,l2)+Jequ3*... integral(dphi3_2_num_r,l2,l3))-1,Clims(i));

C1_r(i)=(-1)*C1_r(i);

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D1_r(i)=a2r*C1_r(i); E1_r(i)=a3r*C1_r(i); F1_r(i)=a4r*C1_r(i);

C2_r(i)=a5r*C1_r(i); D2_r(i)=a6r*C1_r(i); E2_r(i)=a7r*C1_r(i); F2_r(i)=a8r*C1_r(i);

C3_r(i)=a9r*C1_r(i); D3_r(i)=a10r*C1_r(i); E3_r(i)=a11r*C1_r(i); F3_r(i)=a12r*C1_r(i);

phi1_r(i,:)=C1_r(i)*sinh(s11(i)*x1)+D1_r(i)*cosh(s12(i)*x1)... +E1_r(i)*sin(s13(i)*x1)+F1_r(i)*cos(s14(i)*x1); phi2_r(i,:)=C2_r(i)*sinh(s21(i)*x2)+D2_r(i)*cosh(s22(i)*x2)... +E2_r(i)*sin(s23(i)*x2)+F2_r(i)*cos(s24(i)*x2); phi3_r(i,:)=C3_r(i)*sinh(s31(i)*x3)+D3_r(i)*cosh(s32(i)*x3)... +E3_r(i)*sin(s33(i)*x3)+F3_r(i)*cos(s34(i)*x3);

C_disc_r(1,i)=C3_r(i)*sinh(s31(i)*l3)+D3_r(i)*cosh(s32(i)*l3)... +E3_r(i)*sin(s33(i)*l3)+F3_r(i)*cos(s34(i)*l3);

dphi2_r_l1(i,:)=C2_r(i)*s21(i)*cosh(s21(i)*l1)+D2_r(i)*s22(i)*... sinh(s22(i)*l1)+E2_r(i)*s23(i)*cos(s23(i)*l1)-F2_r(i)*... s24(i)*sin(s24(i)*l1); dphi2_r_l2(i,:)=C2_r(i)*s21(i)*cosh(s21(i)*l2)+D2_r(i)*s22(i)*... sinh(s22(i)*l2)+E2_r(i)*s23(i)*cos(s23(i)*l2)-F2_r(i)*... s24(i)*sin(s24(i)*l2);

fdisc_r(i,1)=0.5*(dphi2_r_l1(i,:)-dphi2_r_l2(i,:))*wp*Ep*d31*... (tb+tp-2*zn);

C_disc(1,i)=C3(i)*sin(beta3_num(i)*l3)+D3(i)*cos(beta3_num(i)*l3)+... E3(i)*sinh(beta3_num(i)*l3)+F3(i)*cosh(beta3_num(i)*l3);

%************** 3-part solution, Timoshenko's beam1 *****************% Jrre_t1=coeff_t1(omega_num_t1(i)); a1t1=1; a2t1=Jrre_t1(2,12)/Jrre_t1(1,12); a3t1=Jrre_t1(3,12)/Jrre_t1(1,12); a4t1=Jrre_t1(4,12)/Jrre_t1(1,12); a5t1=Jrre_t1(5,12)/Jrre_t1(1,12); a6t1=Jrre_t1(6,12)/Jrre_t1(1,12); a7t1=Jrre_t1(7,12)/Jrre_t1(1,12); a8t1=Jrre_t1(8,12)/Jrre_t1(1,12); a9t1=Jrre_t1(9,12)/Jrre_t1(1,12); a10t1=Jrre_t1(10,12)/Jrre_t1(1,12); a11t1=Jrre_t1(11,12)/Jrre_t1(1,12); a12t1=-1/Jrre_t1(1,12);

alpha1t1(i)=((omega_num_t1(i).^2)/EIequ1)*(Jequ1+(mequ1*EIequ1)/sagequ1);

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alpha2t1(i)=((omega_num_t1(i).^2)/EIequ2)*(Jequ2+(mequ2*EIequ2)/sagequ2); alpha3t1(i)=((omega_num_t1(i).^2)/EIequ3)*(Jequ3+(mequ3*EIequ3)/sagequ3);

zeta1t1(i)=(mequ1*(omega_num_t1(i).^2)/(EIequ1))*... (Jequ1*(omega_num_t1(i).^2)/sagequ1-1); zeta2t1(i)=(mequ2*(omega_num_t1(i).^2)/(EIequ2))*... (Jequ2*(omega_num_t1(i).^2)/sagequ2-1); zeta3t1(i)=(mequ3*(omega_num_t1(i).^2)/(EIequ3))*... (Jequ3*(omega_num_t1(i).^2)/sagequ3-1);

s11t1(i)=((-alpha1t1(i)/2)+... ((alpha1t1(i).^2)/4-zeta1t1(i)).^(1/2)).^(1/2); s21t1(i)=((-alpha2t1(i)/2)+... ((alpha2t1(i).^2)/4-zeta2t1(i)).^(1/2)).^(1/2); s31t1(i)=((-alpha3t1(i)/2)+... ((alpha3t1(i).^2)/4-zeta3t1(i)).^(1/2)).^(1/2);

s12t1(i)=s11t1(i); s22t1(i)=s21t1(i); s32t1(i)=s31t1(i);

s13t1(i)=((alpha1t1(i)/2)+((alpha1t1(i).^2)/4-zeta1t1(i)).^(1/2)).^(1/2); s23t1(i)=((alpha2t1(i)/2)+((alpha2t1(i).^2)/4-zeta2t1(i)).^(1/2)).^(1/2); s33t1(i)=((alpha3t1(i)/2)+((alpha3t1(i).^2)/4-zeta3t1(i)).^(1/2)).^(1/2);


phi1_2_num_t1=@(zi1t1) (a1t1*sinh(s11t1(i)*zi1t1)+a2t1*... cosh(s12t1(i)*zi1t1)+a3t1*sin(s13t1(i)*zi1t1)+a4t1*... cos(s14t1(i)*zi1t1)).^2; phi2_2_num_t1=@(zi2t1) (a5t1*sinh(s21t1(i)*zi2t1)+a6t1*... cosh(s22t1(i)*zi2t1)+a7t1*sin(s23t1(i)*zi2t1)+a8t1*... cos(s24t1(i)*zi2t1)).^2; phi3_2_num_t1=@(zi3t1) (a9t1*sinh(s31t1(i)*zi3t1)+a10t1*... cosh(s32t1(i)*zi3t1)+a11t1*sin(s33t1(i)*zi3t1)+a12t1*... cos(s34t1(i)*zi3t1)).^2;

dphi1_2_num_t1=@(zi1t1) (a1t1*s11t1(i)*cosh(s11t1(i)*zi1t1)+a2t1*... s12t1(i)*sinh(s12t1(i)*zi1t1)+a3t1*s13t1(i)*... cos(s13t1(i)*zi1t1)-a4t1*s14t1(i)*sin(s14t1(i)*zi1t1)).^2; dphi2_2_num_t1=@(zi1t1) (a5t1*s21t1(i)*cosh(s21t1(i)*zi1t1)+... a6t1*s22t1(i)*sinh(s22t1(i)*zi1t1)+a7t1*s23t1(i)*... cos(s23t1(i)*zi1t1)-a8t1*s24t1(i)*sin(s24t1(i)*zi1t1)).^2; dphi3_2_num_t1=@(zi1t1) (a9t1*s31t1(i)*cosh(s31t1(i)*zi1t1)+a10t1*... s32t1(i)*sinh(s32t1(i)*zi1t1)+a11t1*s33t1(i)*... cos(s33t1(i)*zi1t1)-a12t1*s34t1(i)*sin(s34t1(i)*zi1t1)).^2;

%************** 3-part solution, Timoshenko's beam2 *****************% Jrre_t2=coeff_t2(omega_num_t2(i)); a1t2=1; a2t2=Jrre_t2(2,12)/Jrre_t2(1,12); a3t2=Jrre_t2(3,12)/Jrre_t2(1,12); a4t2=Jrre_t2(4,12)/Jrre_t2(1,12);

137

a5t2=Jrre_t2(5,12)/Jrre_t2(1,12); a6t2=Jrre_t2(6,12)/Jrre_t2(1,12); a7t2=Jrre_t2(7,12)/Jrre_t2(1,12); a8t2=Jrre_t2(8,12)/Jrre_t2(1,12); a9t2=Jrre_t2(9,12)/Jrre_t2(1,12); a10t2=Jrre_t2(10,12)/Jrre_t2(1,12); a11t2=Jrre_t2(11,12)/Jrre_t2(1,12); a12t2=-1/Jrre_t2(1,12);

alpha1t2(i)=((omega_num_t2(i).^2)/EIequ1)*(Jequ1+(mequ1*EIequ1)/sagequ1); alpha2t2(i)=((omega_num_t2(i).^2)/EIequ2)*(Jequ2+(mequ2*EIequ2)/sagequ2); alpha3t2(i)=((omega_num_t2(i).^2)/EIequ3)*(Jequ3+(mequ3*EIequ3)/sagequ3);

zeta1t2(i)=(mequ1*(omega_num_t2(i).^2)/(EIequ1))*... (Jequ1*(omega_num_t2(i).^2)/sagequ1-1); zeta2t2(i)=(mequ2*(omega_num_t2(i).^2)/(EIequ2))*... (Jequ2*(omega_num_t2(i).^2)/sagequ2-1); zeta3t2(i)=(mequ3*(omega_num_t2(i).^2)/(EIequ3))*... (Jequ3*(omega_num_t2(i).^2)/sagequ3-1);

s11t2(i)=((-alpha1t2(i)/2)+... ((alpha1t2(i).^2)/4-zeta1t2(i)).^(1/2)).^(1/2); s21t2(i)=((-alpha2t2(i)/2)+... ((alpha2t2(i).^2)/4-zeta2t2(i)).^(1/2)).^(1/2); s31t2(i)=((-alpha3t2(i)/2)+... ((alpha3t2(i).^2)/4-zeta3t2(i)).^(1/2)).^(1/2);


s13t2(i)=((alpha1t2(i)/2)+((alpha1t2(i).^2)/4-zeta1t2(i)).^(1/2)).^(1/2); s23t2(i)=((alpha2t2(i)/2)+((alpha2t2(i).^2)/4-zeta2t2(i)).^(1/2)).^(1/2); s33t2(i)=((alpha3t2(i)/2)+((alpha3t2(i).^2)/4-zeta3t2(i)).^(1/2)).^(1/2);


teta1_2_num_t2=@(zi1t2) (a1t2*sinh(s11t2(i)*zi1t2)+a2t2*... cosh(s12t2(i)*zi1t2)+a3t2*sin(s13t2(i)*zi1t2)+a4t2*... cos(s14t2(i)*zi1t2)).^2; teta2_2_num_t2=@(zi2t2) (a5t2*sinh(s21t2(i)*zi2t2)+a6t2*... cosh(s22t2(i)*zi2t2)+a7t2*sin(s23t2(i)*zi2t2)+a8t2*... cos(s24t2(i)*zi2t2)).^2; teta3_2_num_t2=@(zi3t2) (a9t2*sinh(s31t2(i)*zi3t2)+a10t2*... cosh(s32t2(i)*zi3t2)+a11t2*sin(s33t2(i)*zi3t2)+a12t2*... cos(s34t2(i)*zi3t2)).^2;

%**************************** Timoshenko ****************************% C_zero_t(i,:)=C.^2*((mequ1)*integral(phi1_2_num_t1,0,l1)+(mequ2)*... integral(phi2_2_num_t1,l1,l2)+(mequ3)*... integral(phi3_2_num_t1,l2,l3)+Jequ1*... integral(teta1_2_num_t2,0,l1)+Jequ2*... integral(teta2_2_num_t2,l1,l2)+Jequ3*...

138

integral(teta3_2_num_t2,l2,l3))-1;

C1_t(i)=fzero(@(Cnum) (Cnum).^2*((mequ1)*integral(phi1_2_num_t1,0,l1)+... (mequ2)*integral(phi2_2_num_t1,l1,l2)+(mequ3)*... integral(phi3_2_num_t1,l2,l3)+Jequ1*... integral(teta1_2_num_t2,0,l1)+Jequ2*... integral(teta2_2_num_t2,l1,l2)+Jequ3*... integral(teta3_2_num_t2,l2,l3))-1,Clims(i));

%C_zero_t(i,:)=C.^2*((mequ1)*integral(phi1_2_num_t1,0,l1)+(mequ2)*... %integral(phi2_2_num_t1,l1,l2)+(mequ3)*... %integral(phi3_2_num_t1,l2,l3)+Jequ1*... %integral(dphi1_2_num_t1,0,l1)+Jequ2*... %integral(dphi2_2_num_t1,l1,l2)+Jequ3*... %integral(dphi3_2_num_t1,l2,l3))-1;

%C1_t(i)=fzero(@(Cnum) (Cnum).^2*((mequ1)*... %integral(phi1_2_num_t1,0,l1)+(mequ2)*... %integral(phi2_2_num_t1,l1,l2)+(mequ3)*... %integral(phi3_2_num_t1,l2,l3)+Jequ1*... %integral(dphi1_2_num_t1,0,l1)+Jequ2*... %integral(dphi2_2_num_t1,l1,l2)+Jequ3*... %integral(dphi3_2_num_t1,l2,l3))-1,Clims(i));

C1_t(i)=(-1)*C1_t(i);

D1_t1(i)=a2t1*C1_t(i); E1_t1(i)=a3t1*C1_t(i); F1_t1(i)=a4t1*C1_t(i);

C2_t1(i)=a5t1*C1_t(i); D2_t1(i)=a6t1*C1_t(i); E2_t1(i)=a7t1*C1_t(i); F2_t1(i)=a8t1*C1_t(i);


D1_t2(i)=a2t2*C1_t(i); E1_t2(i)=a3t2*C1_t(i); F1_t2(i)=a4t2*C1_t(i);



139

phi1_t1(i,:)=C1_t(i)*sinh(s11t1(i)*x1)+D1_t1(i)*cosh(s12t1(i)*x1)... +E1_t1(i)*sin(s13t1(i)*x1)+F1_t1(i)*cos(s14t1(i)*x1); phi2_t1(i,:)=C2_t1(i)*sinh(s21t1(i)*x2)+D2_t1(i)*cosh(s22t1(i)*x2)... +E2_t1(i)*sin(s23t1(i)*x2)+F2_t1(i)*cos(s24t1(i)*x2); phi3_t1(i,:)=C3_t1(i)*sinh(s31t1(i)*x3)+D3_t1(i)*cosh(s32t1(i)*x3)... +E3_t1(i)*sin(s33t1(i)*x3)+F3_t1(i)*cos(s34t1(i)*x3);

dphi1_t1(i,:)=C1_t(i)*s11t1(i)*cosh(s11t1(i)*x1)+D1_t1(i)*s12t1(i)*... sinh(s12t1(i)*x1)+E1_t1(i)*s13t1(i)*cos(s13t1(i)*x1)-... F1_t1(i)*s14t1(i)*sin(s14t1(i)*x1); dphi2_t1(i,:)=C2_t1(i)*s21t1(i)*cosh(s21t1(i)*x2)+D2_t1(i)*s22t1(i)*... sinh(s22t1(i)*x2)+E2_t1(i)*s23t1(i)*cos(s23t1(i)*x2)-... F2_t1(i)*s24t1(i)*sin(s24t1(i)*x2); dphi3_t1(i,:)=C3_t1(i)*s31t1(i)*cosh(s31t1(i)*x3)+D3_t1(i)*s32t1(i)*... sinh(s32t1(i)*x3)+E3_t1(i)*s33t1(i)*cos(s33t1(i)*x3)-... F3_t1(i)*s34t1(i)*sin(s34t1(i)*x3);

phi2_t1_l1(i,:)=C2_t1(i)*sinh(s21t1(i)*l1)+D2_t1(i)*cosh(s22t1(i)*l1)... +E2_t1(i)*sin(s23t1(i)*l1)+F2_t1(i)*cos(s24t1(i)*l1); phi2_t1_l2(i,:)=C2_t1(i)*sinh(s21t1(i)*l2)+D2_t1(i)*cosh(s22t1(i)*l2)... +E2_t1(i)*sin(s23t1(i)*l2)+F2_t1(i)*cos(s24t1(i)*l2);

dphi2_t1_l1(i,:)=C2_t1(i)*s21t1(i)*cosh(s21t1(i)*l1)+D2_t1(i)*... s22t1(i)*sinh(s22t1(i)*l1)+E2_t1(i)*s23t1(i)*... cos(s23t1(i)*l1)-F2_t1(i)*s24t1(i)*sin(s24t1(i)*l1); dphi2_t1_l2(i,:)=C2_t1(i)*s21t1(i)*cosh(s21t1(i)*l2)+D2_t1(i)*... s22t1(i)*sinh(s22t1(i)*l2)+E2_t1(i)*s23t1(i)*... cos(s23t1(i)*l2)-F2_t1(i)*s24t1(i)*sin(s24t1(i)*l2);

teta1_t2(i,:)=C1_t(i)*sinh(s11t2(i)*x1)+D1_t2(i)*cosh(s12t2(i)*x1)... +E1_t2(i)*sin(s13t2(i)*x1)+F1_t2(i)*cos(s14t2(i)*x1); teta2_t2(i,:)=C2_t2(i)*sinh(s21t2(i)*x2)+D2_t2(i)*cosh(s22t2(i)*x2)... +E2_t2(i)*sin(s23t2(i)*x2)+F2_t2(i)*cos(s24t2(i)*x2); teta3_t2(i,:)=C3_t2(i)*sinh(s31t2(i)*x3)+D3_t2(i)*cosh(s32t2(i)*x3)... +E3_t2(i)*sin(s33t2(i)*x3)+F3_t2(i)*cos(s34t2(i)*x3);

teta2_t2_l1(i,:)=C2_t2(i)*sinh(s21t2(i)*l1)+D2_t2(i)*cosh(s22t2(i)*l1)... +E2_t2(i)*sin(s23t2(i)*l1)+F2_t2(i)*cos(s24t2(i)*l1); teta2_t2_l2(i,:)=C2_t2(i)*sinh(s21t2(i)*l2)+D2_t2(i)*cosh(s22t2(i)*l2)... +E2_t2(i)*sin(s23t2(i)*l2)+F2_t2(i)*cos(s24t2(i)*l2);

fdisc_t(i,1)=0.5*(teta2_t2_l1(i,:)-teta2_t2_l2(i,:))*wp*Ep*d31*... (tb+tp-2*zn); %fdisc_t(i,1)=0.5*(dphi2_t1_l1(i,:)-dphi2_t1_l2(i,:))*wp*Ep*d31*... %(tb+tp-2*zn);

C_disc_t(1,i)=C3_t1(i)*sinh(s31t1(i)*l3)+D3_t1(i)*cosh(s32t1(i)*l3)... +E3_t1(i)*sin(s33t1(i)*l3)+F3_t1(i)*cos(s34t1(i)*l3);

figure(2) plot(C,C_zero_eb(i,:),colorfinder(i,:),C,C_zero_r(i,:),... colorfinder(i,:),C,C_zero_t(i,:),colorfinder(i,:)); hold on grid title('A1 coefficient estimate to yield orthonormalized eigenfunctions') xlabel('A1 Coefficient')

140

ylabel('Zero Crossing')

figure(i+2) plot(x1,phi1(i,:),'g',x2,phi2(i,:),'g',x3,phi3(i,:),'g'); hold on plot(x1,phi1_r(i,:),'r',x2,phi2_r(i,:),'r',x3,phi3_r(i,:),'r'); hold on plot(x1,phi1_t1(i,:),'b',x2,phi2_t1(i,:),'b',x3,phi3_t1(i,:),'b');... hold on xlabel('Cantilever length (m)') grid

phi_total_eb(i,:)=[phi1(i,:) phi2(i,:) phi3(i,:)]; phi_total_r(i,:)=[phi1_r(i,:) phi2_r(i,:) phi3_r(i,:)]; phi_total_t1(i,:)=[phi1_t1(i,:) phi2_t1(i,:) phi3_t1(i,:)]; mac_eb_r=(phi_total_eb(i,:)*(phi_total_r(i,:))').^2/... ((phi_total_eb(i,:)*(phi_total_eb(i,:))')*(phi_total_r(i,:)*... (phi_total_r(i,:))')); mac_r_t1=(phi_total_r(i,:)*(phi_total_t1(i,:))').^2/... ((phi_total_r(i,:)*(phi_total_r(i,:))')*(phi_total_t1(i,:)*... (phi_total_t1(i,:))')); mac_eb_t1=(phi_total_eb(i,:)*(phi_total_t1(i,:))').^2/... ((phi_total_eb(i,:)*(phi_total_eb(i,:))')*... (phi_total_t1(i,:)*(phi_total_t1(i,:))'));

end

% Cast all systems into state space

% Discontinuous, three-part model, Euler Bernouli's beam Mdisc_eb=eye(5); Cdisc_eb=zeros(5); Kdisc_eb=diag(omega_num_eb.^2); [Adisc_eb,Bdisc_eb]=state_space(Mdisc_eb,Cdisc_eb,Kdisc_eb,fdisc_eb); Ddisc_eb=0; sys_disc_eb=ss(Adisc_eb,Bdisc_eb,C_disc_eb,Ddisc_eb);

% Discontinuous, three-part model, Rayleigh's beam Mdisc_r=eye(5); Cdisc_r=zeros(5); Kdisc_r=diag(omega_num_r.^2); [Adisc_r,Bdisc_r]=state_space(Mdisc_r,Cdisc_r,Kdisc_r,fdisc_r); Ddisc_r=0; sys_disc_r=ss(Adisc_r,Bdisc_r,C_disc_r,Ddisc_r);

% Discontinuous, three-part model, Timosheko's beam Mdisc_t=eye(5); Cdisc_t=zeros(5); Kdisc_t=diag(omega_num_t1.^2); [Adisc_t,Bdisc_t]=state_space(Mdisc_t,Cdisc_t,Kdisc_t,fdisc_t); Ddisc_t=0; sys_disc_t=ss(Adisc_t,Bdisc_t,C_disc_t,Ddisc_t);

figure(8) bode(sys_disc_eb,'g-') grid legend('Discontinuous Euler-Bernoulli beam')

141

figure(9) bode(sys_disc_r,'r-') grid legend('Discontinuous Rayleigh beam') figure(10) bode(sys_disc_t,'b-') grid legend('Discontinuous Timoshenko beam')

figure(11) step(sys_disc_eb,0.5,'g-'); hold on grid legend('Discontinuous Euler-Bernoulli beam') figure(12) step(sys_disc_r,0.5,'r-'); hold on grid legend('Discontinuous Rayleigh beam') figure(13) step(sys_disc_t,0.5,'b-'); hold on grid legend('Discontinuous Timoshenko beam')

142

function J=BETA(beta1)



l1=20E-3; l2=80E-3; l3=250E-3; L=l3;

Eb=210E9; Ep=67E9;

Gb=82E9; Gp=26E9;

rhob=7850; rhop=7910;

wb=25E-3; wp=15E-3;


d31=-183*10^-12;

Sb=5/6; Sp=Sb;

% Mass properties



mequ1=mb; mequ2=mb+mp; mequ3=mb; mequ=mequ2;

% Nutral axis

143











% beta2 and beta3 as functions of beta1

beta2=beta1*((mb+mp)*EIequ1/(mb*EIequ2))^(1/4); beta3=beta1*(EIequ1/EIequ3)^(1/4);

% Boundary conditions

G1=[0,1,0,1,0,0,0,0,0,0,0,0];

G2=[beta1,0,beta1,0,0,0,0,0,0,0,0,0];

G3=[sin(beta1*l1),cos(beta1*l1),sinh(beta1*l1),cosh(beta1*l1),... -sin(beta2*l1),-cos(beta2*l1),-sinh(beta2*l1),-cosh(beta2*l1),0,0,0,0];

G4=[beta1*cos(beta1*l1),-beta1*sin(beta1*l1),beta1*cosh(beta1*l1),... beta1*sinh(beta1*l1),-beta2*cos(beta2*l1),beta2*sin(beta2*l1),... -beta2*cosh(beta2*l1),-beta2*sinh(beta2*l1),0,0,0,0];

G5=[0,0,0,0,sin(beta2*l2),cos(beta2*l2),sinh(beta2*l2),cosh(beta2*l2),... -sin(beta3*l2),-cos(beta3*l2),-sinh(beta3*l2),-cosh(beta3*l2)];

G6=[0,0,0,0,beta2*cos(beta2*l2),-beta2*sin(beta2*l2),beta2*...

144

cosh(beta2*l2),beta2*sinh(beta2*l2),-beta3*cos(beta3*l2),beta3*... sin(beta3*l2),-beta3*cosh(beta3*l2),-beta3*sinh(beta3*l2)];

N1=[-EIequ1*(beta1^2)*sin(beta1*l1),-EIequ1*(beta1^2)*cos(beta1*l1),... EIequ1*(beta1^2)*sinh(beta1*l1),EIequ1*(beta1^2)*cosh(beta1*l1),... EIequ2*(beta2^2)*sin(beta2*l1),EIequ2*(beta2^2)*cos(beta2*l1),... -EIequ2*(beta2^2)*sinh(beta2*l1),-EIequ2*(beta2^2)*cosh(beta2*l1),... 0,0,0,0];

N2=[-EIequ1*(beta1^3)*cos(beta1*l1),EIequ1*(beta1^3)*sin(beta1*l1),... EIequ1*(beta1^3)*cosh(beta1*l1),EIequ1*(beta1^3)*sinh(beta1*l1),... EIequ2*(beta2^3)*cos(beta2*l1),-EIequ2*(beta2^3)*sin(beta2*l1),... -EIequ2*(beta2^3)*cosh(beta2*l1),-EIequ2*(beta2^3)*sinh(beta2*l1),... 0,0,0,0];

N3=[0,0,0,0,-EIequ2*(beta2^2)*sin(beta2*l2),-EIequ2*(beta2^2)*... cos(beta2*l2),EIequ2*(beta2^2)*sinh(beta2*l2),EIequ2*(beta2^2)*... cosh(beta2*l2),EIequ3*(beta3^2)*sin(beta3*l2),EIequ3*(beta3^2)*... cos(beta3*l2),-EIequ3*(beta3^2)*sinh(beta3*l2),-EIequ3*(beta3^2)*... cosh(beta3*l2)];

N4=[0,0,0,0,-EIequ2*(beta2^3)*cos(beta2*l2),EIequ2*(beta2^3)*... sin(beta2*l2),EIequ2*(beta2^3)*cosh(beta2*l2),EIequ2*(beta2^3)*... sinh(beta2*l2),EIequ3*(beta3^3)*cos(beta3*l2),-EIequ3*(beta3^3)*... sin(beta3*l2),-EIequ3*(beta3^3)*cosh(beta3*l2),-EIequ3*(beta3^3)*... sinh(beta3*l2)];

N5=[0,0,0,0,0,0,0,0,-EIequ3*(beta3^2)*sin(beta3*l3),-EIequ3*(beta3^2)*... cos(beta3*l3),EIequ3*(beta3^2)*sinh(beta3*l3),EIequ3*(beta3^2)*... cosh(beta3*l3)];

N6=[0,0,0,0,0,0,0,0,-EIequ3*(beta3^3)*cos(beta3*l3),EIequ3*(beta3^3)*... sin(beta3*l3),EIequ3*(beta3^3)*cosh(beta3*l3),EIequ3*(beta3^3)*... sinh(beta3*l3)];

J=[G1; G2; G3; G4; G5; G6; N1; N2; N3; N4; N5; N6];

145

function Jrre=COEFF(beta1)



l1=20E-3; l2=80E-3; l3=250E-3; L=l3;

Eb=210E9; Ep=67E9;

Gb=82E9; Gp=26E9;

rhob=7850; rhop=7910;

wb=25E-3; wp=15E-3;


d31=-183*10^-12;

Sb=5/6; Sp=Sb;

% Mass properties




% Nutral axis

146











% beta2 and beta3 as functions of beta1

beta2=beta1*((mb+mp)*EIequ1/(mb*EIequ2))^(1/4); beta3=beta1*(EIequ1/EIequ3)^(1/4);


G1=[0,1,0,1,0,0,0,0,0,0,0,0];

G2=[beta1,0,beta1,0,0,0,0,0,0,0,0,0];

G3=[sin(beta1*l1),cos(beta1*l1),sinh(beta1*l1),cosh(beta1*l1),... -sin(beta2*l1),-cos(beta2*l1),-sinh(beta2*l1),-cosh(beta2*l1),0,0,0,0];

G4=[beta1*cos(beta1*l1),-beta1*sin(beta1*l1),beta1*cosh(beta1*l1),... beta1*sinh(beta1*l1),-beta2*cos(beta2*l1),beta2*sin(beta2*l1),... -beta2*cosh(beta2*l1),-beta2*sinh(beta2*l1),0,0,0,0];

G5=[0,0,0,0,sin(beta2*l2),cos(beta2*l2),sinh(beta2*l2),cosh(beta2*l2),... -sin(beta3*l2),-cos(beta3*l2),-sinh(beta3*l2),-cosh(beta3*l2)];

G6=[0,0,0,0,beta2*cos(beta2*l2),-beta2*sin(beta2*l2),beta2*...

147

cosh(beta2*l2),beta2*sinh(beta2*l2),-beta3*cos(beta3*l2),beta3*... sin(beta3*l2),-beta3*cosh(beta3*l2),-beta3*sinh(beta3*l2)];

N1=[-EIequ1*(beta1^2)*sin(beta1*l1),-EIequ1*(beta1^2)*cos(beta1*l1),... EIequ1*(beta1^2)*sinh(beta1*l1),EIequ1*(beta1^2)*cosh(beta1*l1),... EIequ2*(beta2^2)*sin(beta2*l1),EIequ2*(beta2^2)*cos(beta2*l1),... -EIequ2*(beta2^2)*sinh(beta2*l1),-EIequ2*(beta2^2)*cosh(beta2*l1),... 0,0,0,0];

N2=[-EIequ1*(beta1^3)*cos(beta1*l1),EIequ1*(beta1^3)*sin(beta1*l1),... EIequ1*(beta1^3)*cosh(beta1*l1),EIequ1*(beta1^3)*sinh(beta1*l1),... EIequ2*(beta2^3)*cos(beta2*l1),-EIequ2*(beta2^3)*sin(beta2*l1),... -EIequ2*(beta2^3)*cosh(beta2*l1),-EIequ2*(beta2^3)*sinh(beta2*l1),... 0,0,0,0];

N3=[0,0,0,0,-EIequ2*(beta2^2)*sin(beta2*l2),-EIequ2*(beta2^2)*... cos(beta2*l2),EIequ2*(beta2^2)*sinh(beta2*l2),EIequ2*(beta2^2)*... cosh(beta2*l2),EIequ3*(beta3^2)*sin(beta3*l2),EIequ3*(beta3^2)*... cos(beta3*l2),-EIequ3*(beta3^2)*sinh(beta3*l2),-EIequ3*(beta3^2)*... cosh(beta3*l2)];

N4=[0,0,0,0,-EIequ2*(beta2^3)*cos(beta2*l2),EIequ2*(beta2^3)*... sin(beta2*l2),EIequ2*(beta2^3)*cosh(beta2*l2),EIequ2*(beta2^3)*... sinh(beta2*l2),EIequ3*(beta3^3)*cos(beta3*l2),-EIequ3*(beta3^3)*... sin(beta3*l2),-EIequ3*(beta3^3)*cosh(beta3*l2),-EIequ3*(beta3^3)*... sinh(beta3*l2)];

N5=[0,0,0,0,0,0,0,0,-EIequ3*(beta3^2)*sin(beta3*l3),-EIequ3*(beta3^2)*... cos(beta3*l3),EIequ3*(beta3^2)*sinh(beta3*l3),EIequ3*(beta3^2)*... cosh(beta3*l3)];

N6=[0,0,0,0,0,0,0,0,-EIequ3*(beta3^3)*cos(beta3*l3),EIequ3*(beta3^3)*... sin(beta3*l3),EIequ3*(beta3^3)*cosh(beta3*l3),EIequ3*(beta3^3)*... sinh(beta3*l3)];

J=[G1; G2; G3; G4; G5; G6; N1; N2; N3; N4; N5; N6]; Jrre=rref(J);

148

function JR=omega_r(omega)



l1=20E-3; l2=80E-3; l3=250E-3; L=l3;

Eb=210E9; Ep=67E9;

Gb=82E9; Gp=26E9;

rhob=7850; rhop=7910;

wb=25E-3; wp=15E-3;


d31=-183*10^-12;

Sb=5/6; Sp=Sb;

% Mass properties




% Nutral axis

149











% Roots of Characteristic Equation

alpha1=(omega.^2)*Jequ1/EIequ1; alpha2=(omega.^2)*Jequ2/EIequ2; alpha3=(omega.^2)*Jequ3/EIequ3;

zeta1=mequ1*(omega.^2)/(EIequ1); zeta2=mequ2*(omega.^2)/(EIequ2); zeta3=mequ3*(omega.^2)/(EIequ3);

s11=((-alpha1/2)+((alpha1.^2)/4+zeta1).^(1/2)).^(1/2); s21=((-alpha2/2)+((alpha2.^2)/4+zeta2).^(1/2)).^(1/2); s31=((-alpha3/2)+((alpha3.^2)/4+zeta3).^(1/2)).^(1/2);

s12=s11; s22=s21; s32=s31;

s13=((alpha1/2)+((alpha1.^2)/4+zeta1).^(1/2)).^(1/2); s23=((alpha2/2)+((alpha2.^2)/4+zeta2).^(1/2)).^(1/2); s33=((alpha3/2)+((alpha3.^2)/4+zeta3).^(1/2)).^(1/2);

150

s14=s13; s24=s23; s34=s33;


G1=[0,1,0,1,0,0,0,0,0,0,0,0];

G2=[s11,0,s13,0,0,0,0,0,0,0,0,0];

G3=[sinh(s11*l1),cosh(s12*l1),sin(s13*l1),cos(s14*l1),... -sinh(s21*l1),-cosh(s22*l1),-sin(s23*l1),-cos(s24*l1),0,0,0,0];

G4=[s11*cosh(s11*l1),s12*sinh(s12*l1),s13*cos(s13*l1),-s14*sin(s14*l1),... -s21*cosh(s21*l1),-s22*sinh(s22*l1),-s23*cos(s23*l1),s24*sin(s24*l1)... ,0,0,0,0];

G5=[0,0,0,0,sinh(s21*l2),cosh(s22*l2),sin(s23*l2),cos(s24*l2),... -sinh(s31*l2),-cosh(s32*l2),-sin(s33*l2),-cos(s34*l2)];

G6=[0,0,0,0,s21*cosh(s21*l2),s22*sinh(s22*l2),s23*cos(s23*l2),-s24*... sin(s24*l2),-s31*cosh(s31*l2),-s32*sinh(s32*l2),-s33*cos(s33*l2),... s34*sin(s34*l2)];

N1=[EIequ1*(s11^2)*sinh(s11*l1),EIequ1*(s12^2)*cosh(s12*l1),-EIequ1*... (s13^2)*sin(s13*l1),-EIequ1*(s14^2)*cos(s14*l1),-EIequ2*(s21^2)*... sinh(s21*l1),-EIequ2*(s22^2)*cosh(s22*l1),EIequ2*(s23^2)*sin(s23*l1),... EIequ2*(s24^2)*cos(s24*l1),0,0,0,0];

N2=[EIequ1*(s11^3)*cosh(s11*l1)+Jequ1*(omega^2)*s11*cosh(s11*l1),... EIequ1*(s12^3)*sinh(s12*l1)+Jequ1*(omega^2)*s12*sinh(s12*l1),... -EIequ1*(s13^3)*cos(s13*l1)+Jequ1*(omega^2)*s13*cos(s13*l1),... EIequ1*(s14^3)*sin(s14*l1)-Jequ1*(omega^2)*s14*sin(s14*l1),... -EIequ2*(s21^3)*cosh(s21*l1)-Jequ2*(omega^2)*s21*cosh(s21*l1),... -EIequ2*(s22^3)*sinh(s22*l1)-Jequ2*(omega^2)*s22*sinh(s22*l1),... EIequ2*(s23^3)*cos(s23*l1)-Jequ2*(omega^2)*s23*cos(s23*l1),... -EIequ2*(s24^3)*sin(s24*l1)+Jequ2*(omega^2)*s24*sin(s24*l1),... 0,0,0,0];

N3=[0,0,0,0,EIequ2*(s21^2)*sinh(s21*l2),EIequ2*(s22^2)*cosh(s22*l2),... -EIequ2*(s23^2)*sin(s23*l2),-EIequ2*(s24^2)*cos(s24*l2),-EIequ3*... (s31^2)*sinh(s31*l2),-EIequ3*(s32^2)*cosh(s32*l2),EIequ3*(s33^2)*... sin(s33*l2),EIequ3*(s34^2)*cos(s34*l2)];

N4=[0,0,0,0,... EIequ2*(s21^3)*cosh(s21*l2)+Jequ2*(omega^2)*s21*cosh(s21*l2),... EIequ2*(s22^3)*sinh(s22*l2)+Jequ2*(omega^2)*s22*sinh(s22*l2),... -EIequ2*(s23^3)*cos(s23*l2)+Jequ2*(omega^2)*s23*cos(s23*l2),... EIequ2*(s24^3)*sin(s24*l2)-Jequ2*(omega^2)*s24*sin(s24*l2),... -EIequ3*(s31^3)*cosh(s31*l2)-Jequ3*(omega^2)*s31*cosh(s31*l2),... -EIequ3*(s32^3)*sinh(s32*l2)-Jequ3*(omega^2)*s32*sinh(s32*l2),... EIequ3*(s33^3)*cos(s33*l2)-Jequ3*(omega^2)*s33*cos(s33*l2),... -EIequ3*(s34^3)*sin(s34*l2)+Jequ3*(omega^2)*s34*sin(s34*l2)];

151

N5=[0,0,0,0,0,0,0,0,... EIequ3*(s31^2)*sinh(s31*l3),EIequ3*(s32^2)*cosh(s32*l3),... -EIequ3*(s33^2)*sin(s33*l3),-EIequ3*(s34^2)*cos(s34*l3)];

N6=[0,0,0,0,0,0,0,0,... EIequ3*(s31^3)*cosh(s31*l3)+Jequ3*(omega^2)*s31*cosh(s31*l3),... EIequ3*(s32^3)*sinh(s32*l3)+Jequ3*(omega^2)*s32*sinh(s32*l3),... -EIequ3*(s33^3)*cos(s33*l3)+Jequ3*(omega^2)*s33*cos(s33*l3),... EIequ3*(s34^3)*sin(s34*l3)-Jequ3*(omega^2)*s34*sin(s34*l3)];

JR=[G1; G2; G3; G4; G5; G6; N1; N2; N3; N4; N5; N6];

152

function Jrre_r=coeff_r(omega)



l1=20E-3; l2=80E-3; l3=250E-3; L=l3;

Eb=210E9; Ep=67E9;

Gb=82E9; Gp=26E9;

rhob=7850; rhop=7910;

wb=25E-3; wp=15E-3;


d31=-183*10^-12;

Sb=5/6; Sp=Sb;

% Mass properties




% Nutral axis

153












alpha1=(omega.^2)*Jequ1/EIequ1; alpha2=(omega.^2)*Jequ2/EIequ2; alpha3=(omega.^2)*Jequ3/EIequ3;

zeta1=mequ1*(omega.^2)/(EIequ1); zeta2=mequ2*(omega.^2)/(EIequ2); zeta3=mequ3*(omega.^2)/(EIequ3);

s11=((-alpha1/2)+((alpha1.^2)/4+zeta1).^(1/2)).^(1/2); s21=((-alpha2/2)+((alpha2.^2)/4+zeta2).^(1/2)).^(1/2); s31=((-alpha3/2)+((alpha3.^2)/4+zeta3).^(1/2)).^(1/2);

s12=s11; s22=s21; s32=s31;

s13=((alpha1/2)+((alpha1.^2)/4+zeta1).^(1/2)).^(1/2); s23=((alpha2/2)+((alpha2.^2)/4+zeta2).^(1/2)).^(1/2); s33=((alpha3/2)+((alpha3.^2)/4+zeta3).^(1/2)).^(1/2);

154

s14=s13; s24=s23; s34=s33;


G1=[0,1,0,1,0,0,0,0,0,0,0,0];

G2=[s11,0,s13,0,0,0,0,0,0,0,0,0];

G3=[sinh(s11*l1),cosh(s12*l1),sin(s13*l1),cos(s14*l1),... -sinh(s21*l1),-cosh(s22*l1),-sin(s23*l1),-cos(s24*l1),0,0,0,0];

G4=[s11*cosh(s11*l1),s12*sinh(s12*l1),s13*cos(s13*l1),-s14*sin(s14*l1),... -s21*cosh(s21*l1),-s22*sinh(s22*l1),-s23*cos(s23*l1),s24*sin(s24*l1),... 0,0,0,0];

G5=[0,0,0,0,sinh(s21*l2),cosh(s22*l2),sin(s23*l2),cos(s24*l2),... -sinh(s31*l2),-cosh(s32*l2),-sin(s33*l2),-cos(s34*l2)];

G6=[0,0,0,0,s21*cosh(s21*l2),s22*sinh(s22*l2),s23*cos(s23*l2),-s24*... sin(s24*l2),-s31*cosh(s31*l2),-s32*sinh(s32*l2),-s33*cos(s33*l2),... s34*sin(s34*l2)];

N1=[EIequ1*(s11^2)*sinh(s11*l1),EIequ1*(s12^2)*cosh(s12*l1),-EIequ1*... (s13^2)*sin(s13*l1),-EIequ1*(s14^2)*cos(s14*l1),-EIequ2*(s21^2)*... sinh(s21*l1),-EIequ2*(s22^2)*cosh(s22*l1),EIequ2*(s23^2)*sin(s23*l1),... EIequ2*(s24^2)*cos(s24*l1),0,0,0,0];

N2=[EIequ1*(s11^3)*cosh(s11*l1)+Jequ1*(omega^2)*s11*cosh(s11*l1),... EIequ1*(s12^3)*sinh(s12*l1)+Jequ1*(omega^2)*s12*sinh(s12*l1),... -EIequ1*(s13^3)*cos(s13*l1)+Jequ1*(omega^2)*s13*cos(s13*l1),... EIequ1*(s14^3)*sin(s14*l1)-Jequ1*(omega^2)*s14*sin(s14*l1),... -EIequ2*(s21^3)*cosh(s21*l1)-Jequ2*(omega^2)*s21*cosh(s21*l1),... -EIequ2*(s22^3)*sinh(s22*l1)-Jequ2*(omega^2)*s22*sinh(s22*l1),... EIequ2*(s23^3)*cos(s23*l1)-Jequ2*(omega^2)*s23*cos(s23*l1),... -EIequ2*(s24^3)*sin(s24*l1)+Jequ2*(omega^2)*s24*sin(s24*l1),... 0,0,0,0];

N3=[0,0,0,0,EIequ2*(s21^2)*sinh(s21*l2),EIequ2*(s22^2)*cosh(s22*l2),... -EIequ2*(s23^2)*sin(s23*l2),-EIequ2*(s24^2)*cos(s24*l2),-EIequ3*... (s31^2)*sinh(s31*l2),-EIequ3*(s32^2)*cosh(s32*l2),EIequ3*(s33^2)*... sin(s33*l2),EIequ3*(s34^2)*cos(s34*l2)];

N4=[0,0,0,0,... EIequ2*(s21^3)*cosh(s21*l2)+Jequ2*(omega^2)*s21*cosh(s21*l2),... EIequ2*(s22^3)*sinh(s22*l2)+Jequ2*(omega^2)*s22*sinh(s22*l2),... -EIequ2*(s23^3)*cos(s23*l2)+Jequ2*(omega^2)*s23*cos(s23*l2),... EIequ2*(s24^3)*sin(s24*l2)-Jequ2*(omega^2)*s24*sin(s24*l2),... -EIequ3*(s31^3)*cosh(s31*l2)-Jequ3*(omega^2)*s31*cosh(s31*l2),... -EIequ3*(s32^3)*sinh(s32*l2)-Jequ3*(omega^2)*s32*sinh(s32*l2),... EIequ3*(s33^3)*cos(s33*l2)-Jequ3*(omega^2)*s33*cos(s33*l2),... -EIequ3*(s34^3)*sin(s34*l2)+Jequ3*(omega^2)*s34*sin(s34*l2)];

155

N5=[0,0,0,0,0,0,0,0,... EIequ3*(s31^2)*sinh(s31*l3),EIequ3*(s32^2)*cosh(s32*l3),... -EIequ3*(s33^2)*sin(s33*l3),-EIequ3*(s34^2)*cos(s34*l3)];

N6=[0,0,0,0,0,0,0,0,... EIequ3*(s31^3)*cosh(s31*l3)+Jequ3*(omega^2)*s31*cosh(s31*l3),... EIequ3*(s32^3)*sinh(s32*l3)+Jequ3*(omega^2)*s32*sinh(s32*l3),... -EIequ3*(s33^3)*cos(s33*l3)+Jequ3*(omega^2)*s33*cos(s33*l3),... EIequ3*(s34^3)*sin(s34*l3)-Jequ3*(omega^2)*s34*sin(s34*l3)];

JR=[G1; G2; G3; G4; G5; G6; N1; N2; N3; N4; N5; N6]; Jrre_r=rref(JR);

156

function JT1=omega_t1(omega)



l1=20E-3; l2=80E-3; l3=250E-3; L=l3;

Eb=210E9; Ep=67E9;

Gb=82E9; Gp=26E9;

rhob=7850; rhop=7910;

wb=25E-3; wp=15E-3;


d31=-183*10^-12;

Sb=5/6; Sp=Sb;

% Mass properties




% Nutral axis

157






Ipnet=wp*tp*(tb^2)/4-wp*tp*tb*zn+wp*(tp^3)/3+wp*(tp^2)*tb/2-wp*(tp^2)*... zn+wp*tp*(zn^2);

% Equivalent shear correction factor times shear modulus times area (sagequ) sagequ1=Sb*Ab*Gb; sagequ2=Sb*(Ab*Gb+Ap*Gp); sagequ3=Sb*Ab*Gb;

% Equivalent polar moment of inertia of cross section Jequ1=(mb*Ib1)/Ab; Jequ2=(mb*Ib2)/Ab+(mp*Ipnet)/Ap; Jequ3=(mb*Ib3)/Ab;


alpha1t1=((omega.^2)/EIequ1)*(Jequ1+(mequ1*EIequ1)/sagequ1); alpha2t1=((omega.^2)/EIequ2)*(Jequ2+(mequ2*EIequ2)/sagequ2); alpha3t1=((omega.^2)/EIequ3)*(Jequ3+(mequ3*EIequ3)/sagequ3);

zeta1t1=(mequ1*(omega.^2)/(EIequ1))*(Jequ1*(omega.^2)/sagequ1-1); zeta2t1=(mequ2*(omega.^2)/(EIequ2))*(Jequ2*(omega.^2)/sagequ2-1); zeta3t1=(mequ3*(omega.^2)/(EIequ3))*(Jequ3*(omega.^2)/sagequ3-1);

s11t1=((-alpha1t1/2)+((alpha1t1.^2)/4-zeta1t1).^(1/2)).^(1/2); s21t1=((-alpha2t1/2)+((alpha2t1.^2)/4-zeta2t1).^(1/2)).^(1/2); s31t1=((-alpha3t1/2)+((alpha3t1.^2)/4-zeta3t1).^(1/2)).^(1/2);

s12t1=s11t1; s22t1=s21t1; s32t1=s31t1;

s13t1=((alpha1t1/2)+((alpha1t1.^2)/4-zeta1t1).^(1/2)).^(1/2); s23t1=((alpha2t1/2)+((alpha2t1.^2)/4-zeta2t1).^(1/2)).^(1/2); s33t1=((alpha3t1/2)+((alpha3t1.^2)/4-zeta3t1).^(1/2)).^(1/2);

s14t1=s13t1; s24t1=s23t1; s34t1=s33t1;

158

a1t1=mequ1*(omega.^2)/sagequ1; a2t1=mequ2*(omega.^2)/sagequ2; a3t1=mequ3*(omega.^2)/sagequ3;

b1t1=(Jequ1*(omega.^2)-sagequ1)/EIequ1; b2t1=(Jequ2*(omega.^2)-sagequ2)/EIequ2; b3t1=(Jequ3*(omega.^2)-sagequ3)/EIequ3;

c1t1=sagequ1/EIequ1; c2t1=sagequ2/EIequ2; c3t1=sagequ3/EIequ3;


G1t1=[0,1,0,1,0,0,0,0,0,0,0,0];

G2t1=[(-1/b1t1)*((s11t1^3)+(a1t1+c1t1)*s11t1),0,(-1/b1t1)*(-(s13t1^3)+... (a1t1+c1t1)*s13t1),0,0,0,0,0,0,0,0,0];

G3t1=[sinh(s11t1*l1),cosh(s12t1*l1),sin(s13t1*l1),cos(s14t1*l1),... -sinh(s21t1*l1),-cosh(s22t1*l1),-sin(s23t1*l1),-cos(s24t1*l1),0,0,0,0];

G4t1=[0,0,0,0,sinh(s21t1*l2),cosh(s22t1*l2),sin(s23t1*l2),cos(s24t1*l2),... -sinh(s31t1*l2),-cosh(s32t1*l2),-sin(s33t1*l2),-cos(s34t1*l2)];

G5t1=[(-1/b1t1)*((s11t1^3)*cosh(s11t1*l1)+(a1t1+c1t1)*s11t1*... cosh(s11t1*l1)),(-1/b1t1)*((s12t1^3)*sinh(s12t1*l1)+(a1t1+c1t1)*... s12t1*sinh(s12t1*l1)),(-1/b1t1)*(-(s13t1^3)*cos(s13t1*l1)+... (a1t1+c1t1)*s13t1*cos(s13t1*l1)),(-1/b1t1)*((s14t1^3)*... sin(s14t1*l1)-(a1t1+c1t1)*s14t1*sin(s14t1*l1)),(1/b2t1)*... ((s21t1^3)*cosh(s21t1*l1)+(a2t1+c2t1)*s21t1*cosh(s21t1*l1)),... (1/b2t1)*((s22t1^3)*sinh(s22t1*l1)+(a2t1+c2t1)*s22t1*... sinh(s22t1*l1)),(1/b2t1)*(-(s23t1^3)*cos(s23t1*l1)+(a2t1+c2t1)*... s23t1*cos(s23t1*l1)),(1/b2t1)*((s24t1^3)*sin(s24t1*l1)-(a2t1+c2t1)*... s24t1*sin(s24t1*l1)),0,0,0,0];

G6t1=[0,0,0,0,... (-1/b2t1)*((s21t1^3)*cosh(s21t1*l2)+(a2t1+c2t1)*s21t1*... cosh(s21t1*l2)),(-1/b2t1)*((s22t1^3)*sinh(s22t1*l2)+(a2t1+c2t1)*... s22t1*sinh(s22t1*l2)),(-1/b2t1)*(-(s23t1^3)*cos(s23t1*l2)+... (a2t1+c2t1)*s23t1*cos(s23t1*l2)),(-1/b2t1)*((s24t1^3)*... sin(s24t1*l2)-(a2t1+c2t1)*s24t1*sin(s24t1*l2)),(1/b3t1)*... ((s31t1^3)*cosh(s31t1*l2)+(a3t1+c3t1)*s31t1*cosh(s31t1*l2)),... (1/b3t1)*((s32t1^3)*sinh(s32t1*l2)+(a3t1+c3t1)*s32t1*... sinh(s32t1*l2)),(1/b3t1)*(-(s33t1^3)*cos(s33t1*l2)+(a3t1+c3t1)*... s33t1*cos(s33t1*l2)),(1/b3t1)*((s34t1^3)*sin(s34t1*l2)-(a3t1+c3t1)*... s34t1*sin(s34t1*l2))];

N1t1=[EIequ1*((s11t1^2)*sinh(s11t1*l1)+a1t1*sinh(s11t1*l1)),... EIequ1*((s12t1^2)*cosh(s12t1*l1)+a1t1*cosh(s12t1*l1)),... EIequ1*(-(s13t1^2)*sin(s13t1*l1)+a1t1*sin(s13t1*l1)),... EIequ1*(-(s14t1^2)*cos(s14t1*l1)+a1t1*cos(s14t1*l1)),... -EIequ2*((s21t1^2)*sinh(s21t1*l1)+a2t1*sinh(s21t1*l1)),...

159

-EIequ2*((s22t1^2)*cosh(s22t1*l1)+a2t1*cosh(s22t1*l1)),... -EIequ2*(-(s23t1^2)*sin(s23t1*l1)+a2t1*sin(s23t1*l1)),... -EIequ2*(-(s24t1^2)*cos(s24t1*l1)+a2t1*cos(s24t1*l1)),0,0,0,0];

N2t1=[0,0,0,0,... EIequ2*((s21t1^2)*sinh(s21t1*l2)+a2t1*sinh(s21t1*l2)),... EIequ2*((s22t1^2)*cosh(s22t1*l2)+a2t1*cosh(s22t1*l2)),... EIequ2*(-(s23t1^2)*sin(s23t1*l2)+a2t1*sin(s23t1*l2)),... EIequ2*(-(s24t1^2)*cos(s24t1*l2)+a2t1*cos(s24t1*l2)),... -EIequ3*((s31t1^2)*sinh(s31t1*l2)+a3t1*sinh(s31t1*l2)),... -EIequ3*((s32t1^2)*cosh(s32t1*l2)+a3t1*cosh(s32t1*l2)),... -EIequ3*(-(s33t1^2)*sin(s33t1*l2)+a3t1*sin(s33t1*l2)),... -EIequ3*(-(s34t1^2)*cos(s34t1*l2)+a3t1*cos(s34t1*l2))];

N3t1=[(sagequ1/b1t1)*((s11t1^3)*cosh(s11t1*l1)+(a1t1+b1t1+c1t1)*s11t1*... cosh(s11t1*l1)),(sagequ1/b1t1)*((s12t1^3)*sinh(s12t1*l1)+... (a1t1+b1t1+c1t1)*s12t1*sinh(s12t1*l1)),(sagequ1/b1t1)*(-(s13t1^3)*... cos(s13t1*l1)+(a1t1+b1t1+c1t1)*s13t1*cos(s13t1*l1)),(sagequ1/b1t1)*... ((s14t1^3)*sin(s14t1*l1)-(a1t1+b1t1+c1t1)*s14t1*sin(s14t1*l1)),... (-sagequ2/b2t1)*((s21t1^3)*cosh(s21t1*l1)+(a2t1+b2t1+c2t1)*s21t1*... cosh(s21t1*l1)),(-sagequ2/b2t1)*((s22t1^3)*sinh(s22t1*l1)+... (a2t1+b2t1+c2t1)*s22t1*sinh(s22t1*l1)),(-sagequ2/b2t1)*(-(s23t1^3)*... cos(s23t1*l1)+(a2t1+b2t1+c2t1)*s23t1*cos(s23t1*l1)),... (-sagequ2/b2t1)*((s24t1^3)*sin(s24t1*l1)-(a2t1+b2t1+c2t1)*s24t1*... sin(s24t1*l1)),0,0,0,0];

N4t1=[0,0,0,0,... (sagequ2/b2t1)*((s21t1^3)*cosh(s21t1*l2)+(a2t1+b2t1+c2t1)*s21t1*... cosh(s21t1*l2)),(sagequ2/b2t1)*((s22t1^3)*sinh(s22t1*l2)+... (a2t1+b2t1+c2t1)*s22t1*sinh(s22t1*l2)),(sagequ2/b2t1)*(-(s23t1^3)*... cos(s23t1*l2)+(a2t1+b2t1+c2t1)*s23t1*cos(s23t1*l2)),(sagequ2/b2t1)*... ((s24t1^3)*sin(s24t1*l2)-(a2t1+b2t1+c2t1)*s24t1*sin(s24t1*l2)),... (-sagequ3/b3t1)*((s31t1^3)*cosh(s31t1*l2)+(a3t1+b3t1+c3t1)*s31t1*... cosh(s31t1*l2)),(-sagequ3/b3t1)*((s32t1^3)*sinh(s32t1*l2)+... (a3t1+b3t1+c3t1)*s32t1*sinh(s32t1*l2)),(-sagequ3/b3t1)*(-(s33t1^3)*... cos(s33t1*l2)+(a3t1+b3t1+c3t1)*s33t1*cos(s33t1*l2)),... (-sagequ3/b3t1)*((s34t1^3)*sin(s34t1*l2)-(a3t1+b3t1+c3t1)*s34t1*... sin(s34t1*l2))];

N5t1=[0,0,0,0,0,0,0,0,... EIequ3*((s31t1^2)*sinh(s31t1*l3)+a3t1*sinh(s31t1*l3)),... EIequ3*((s32t1^2)*cosh(s32t1*l3)+a3t1*cosh(s32t1*l3)),... EIequ3*(-(s33t1^2)*sin(s33t1*l3)+a3t1*sin(s33t1*l3)),... EIequ3*(-(s34t1^2)*cos(s34t1*l3)+a3t1*cos(s34t1*l3))];

N6t1=[0,0,0,0,0,0,0,0,... (sagequ3/b3t1)*((s31t1^3)*cosh(s31t1*l3)+(a3t1+b3t1+c3t1)*s31t1*... cosh(s31t1*l3)),(sagequ3/b3t1)*((s32t1^3)*sinh(s32t1*l3)+... (a3t1+b3t1+c3t1)*s32t1*sinh(s32t1*l3)),(sagequ3/b3t1)*(-(s33t1^3)*... cos(s33t1*l3)+(a3t1+b3t1+c3t1)*s33t1*cos(s33t1*l3)),(sagequ3/b3t1)*... ((s34t1^3)*sin(s34t1*l3)-(a3t1+b3t1+c3t1)*s34t1*sin(s34t1*l3))];

JT1=[G1t1; G2t1; G3t1; G4t1; G5t1; G6t1; N1t1; N2t1; N3t1; N4t1; N5t1;... N6t1];

160

function Jrre_t1=coeff_t1(omega)



l1=20E-3; l2=80E-3; l3=250E-3; L=l3;

Eb=210E9; Ep=67E9;

Gb=82E9; Gp=26E9;

rhob=7850; rhop=7910;

wb=25E-3; wp=15E-3;


d31=-183*10^-12;

Sb=5/6; Sp=Sb;

% Mass properties




% Nutral axis

161







% Equivalent shear correction factor times shear modulus times area (sagequ) sagequ1=Sb*Ab*Gb; sagequ2=Sb*(Ab*Gb+Ap*Gp); sagequ3=Sb*Ab*Gb;

% Equivalent polar moment of inertia of cross section Jequ1=(mb*Ib1)/Ab; Jequ2=(mb*Ib2)/Ab+(mp*Ipnet)/Ap; Jequ3=(mb*Ib3)/Ab;





s12t1=s11t1; s22t1=s21t1; s32t1=s31t1;


s14t1=s13t1; s24t1=s23t1; s34t1=s33t1;

162





G1t1=[0,1,0,1,0,0,0,0,0,0,0,0];

G2t1=[(-1/b1t1)*((s11t1^3)+(a1t1+c1t1)*s11t1),0,(-1/b1t1)*(-(s13t1^3)+... (a1t1+c1t1)*s13t1),0,0,0,0,0,0,0,0,0];



G5t1=[(-1/b1t1)*((s11t1^3)*cosh(s11t1*l1)+(a1t1+c1t1)*s11t1*... cosh(s11t1*l1)),(-1/b1t1)*((s12t1^3)*sinh(s12t1*l1)+(a1t1+c1t1)*... s12t1*sinh(s12t1*l1)),(-1/b1t1)*(-(s13t1^3)*cos(s13t1*l1)+... (a1t1+c1t1)*s13t1*cos(s13t1*l1)),(-1/b1t1)*((s14t1^3)*... sin(s14t1*l1)-(a1t1+c1t1)*s14t1*sin(s14t1*l1)),(1/b2t1)*... ((s21t1^3)*cosh(s21t1*l1)+(a2t1+c2t1)*s21t1*cosh(s21t1*l1)),... (1/b2t1)*((s22t1^3)*sinh(s22t1*l1)+(a2t1+c2t1)*s22t1*... sinh(s22t1*l1)),(1/b2t1)*(-(s23t1^3)*cos(s23t1*l1)+(a2t1+c2t1)*... s23t1*cos(s23t1*l1)),(1/b2t1)*((s24t1^3)*sin(s24t1*l1)-(a2t1+c2t1)*... s24t1*sin(s24t1*l1)),0,0,0,0];

G6t1=[0,0,0,0,... (-1/b2t1)*((s21t1^3)*cosh(s21t1*l2)+(a2t1+c2t1)*s21t1*... cosh(s21t1*l2)),(-1/b2t1)*((s22t1^3)*sinh(s22t1*l2)+(a2t1+c2t1)*... s22t1*sinh(s22t1*l2)),(-1/b2t1)*(-(s23t1^3)*cos(s23t1*l2)+... (a2t1+c2t1)*s23t1*cos(s23t1*l2)),(-1/b2t1)*((s24t1^3)*... sin(s24t1*l2)-(a2t1+c2t1)*s24t1*sin(s24t1*l2)),(1/b3t1)*... ((s31t1^3)*cosh(s31t1*l2)+(a3t1+c3t1)*s31t1*cosh(s31t1*l2)),... (1/b3t1)*((s32t1^3)*sinh(s32t1*l2)+(a3t1+c3t1)*s32t1*... sinh(s32t1*l2)),(1/b3t1)*(-(s33t1^3)*cos(s33t1*l2)+(a3t1+c3t1)*... s33t1*cos(s33t1*l2)),(1/b3t1)*((s34t1^3)*sin(s34t1*l2)-(a3t1+c3t1)*... s34t1*sin(s34t1*l2))];

N1t1=[EIequ1*((s11t1^2)*sinh(s11t1*l1)+a1t1*sinh(s11t1*l1)),... EIequ1*((s12t1^2)*cosh(s12t1*l1)+a1t1*cosh(s12t1*l1)),... EIequ1*(-(s13t1^2)*sin(s13t1*l1)+a1t1*sin(s13t1*l1)),... EIequ1*(-(s14t1^2)*cos(s14t1*l1)+a1t1*cos(s14t1*l1)),... -EIequ2*((s21t1^2)*sinh(s21t1*l1)+a2t1*sinh(s21t1*l1)),...

163

-EIequ2*((s22t1^2)*cosh(s22t1*l1)+a2t1*cosh(s22t1*l1)),... -EIequ2*(-(s23t1^2)*sin(s23t1*l1)+a2t1*sin(s23t1*l1)),... -EIequ2*(-(s24t1^2)*cos(s24t1*l1)+a2t1*cos(s24t1*l1)),0,0,0,0];

N2t1=[0,0,0,0,... EIequ2*((s21t1^2)*sinh(s21t1*l2)+a2t1*sinh(s21t1*l2)),... EIequ2*((s22t1^2)*cosh(s22t1*l2)+a2t1*cosh(s22t1*l2)),... EIequ2*(-(s23t1^2)*sin(s23t1*l2)+a2t1*sin(s23t1*l2)),... EIequ2*(-(s24t1^2)*cos(s24t1*l2)+a2t1*cos(s24t1*l2)),... -EIequ3*((s31t1^2)*sinh(s31t1*l2)+a3t1*sinh(s31t1*l2)),... -EIequ3*((s32t1^2)*cosh(s32t1*l2)+a3t1*cosh(s32t1*l2)),... -EIequ3*(-(s33t1^2)*sin(s33t1*l2)+a3t1*sin(s33t1*l2)),... -EIequ3*(-(s34t1^2)*cos(s34t1*l2)+a3t1*cos(s34t1*l2))];

N3t1=[(sagequ1/b1t1)*((s11t1^3)*cosh(s11t1*l1)+(a1t1+b1t1+c1t1)*s11t1*... cosh(s11t1*l1)),(sagequ1/b1t1)*((s12t1^3)*sinh(s12t1*l1)+... (a1t1+b1t1+c1t1)*s12t1*sinh(s12t1*l1)),(sagequ1/b1t1)*(-(s13t1^3)*... cos(s13t1*l1)+(a1t1+b1t1+c1t1)*s13t1*cos(s13t1*l1)),(sagequ1/b1t1)*... ((s14t1^3)*sin(s14t1*l1)-(a1t1+b1t1+c1t1)*s14t1*sin(s14t1*l1)),... (-sagequ2/b2t1)*((s21t1^3)*cosh(s21t1*l1)+(a2t1+b2t1+c2t1)*... s21t1*cosh(s21t1*l1)),(-sagequ2/b2t1)*((s22t1^3)*sinh(s22t1*l1)+... (a2t1+b2t1+c2t1)*s22t1*sinh(s22t1*l1)),(-sagequ2/b2t1)*(-(s23t1^3)*... cos(s23t1*l1)+(a2t1+b2t1+c2t1)*s23t1*cos(s23t1*l1)),... (-sagequ2/b2t1)*((s24t1^3)*sin(s24t1*l1)-(a2t1+b2t1+c2t1)*s24t1*... sin(s24t1*l1)),0,0,0,0];

N4t1=[0,0,0,0,... (sagequ2/b2t1)*((s21t1^3)*cosh(s21t1*l2)+(a2t1+b2t1+c2t1)*s21t1*... cosh(s21t1*l2)),(sagequ2/b2t1)*((s22t1^3)*sinh(s22t1*l2)+... (a2t1+b2t1+c2t1)*s22t1*sinh(s22t1*l2)),(sagequ2/b2t1)*(-(s23t1^3)*... cos(s23t1*l2)+(a2t1+b2t1+c2t1)*s23t1*cos(s23t1*l2)),(sagequ2/b2t1)*... ((s24t1^3)*sin(s24t1*l2)-(a2t1+b2t1+c2t1)*s24t1*sin(s24t1*l2)),... (-sagequ3/b3t1)*((s31t1^3)*cosh(s31t1*l2)+(a3t1+b3t1+c3t1)*s31t1*... cosh(s31t1*l2)),(-sagequ3/b3t1)*((s32t1^3)*sinh(s32t1*l2)+... (a3t1+b3t1+c3t1)*s32t1*sinh(s32t1*l2)),(-sagequ3/b3t1)*(-(s33t1^3)*... cos(s33t1*l2)+(a3t1+b3t1+c3t1)*s33t1*cos(s33t1*l2)),... (-sagequ3/b3t1)*((s34t1^3)*sin(s34t1*l2)-(a3t1+b3t1+c3t1)*... s34t1*sin(s34t1*l2))];

N5t1=[0,0,0,0,0,0,0,0,... EIequ3*((s31t1^2)*sinh(s31t1*l3)+a3t1*sinh(s31t1*l3)),... EIequ3*((s32t1^2)*cosh(s32t1*l3)+a3t1*cosh(s32t1*l3)),... EIequ3*(-(s33t1^2)*sin(s33t1*l3)+a3t1*sin(s33t1*l3)),... EIequ3*(-(s34t1^2)*cos(s34t1*l3)+a3t1*cos(s34t1*l3))];

N6t1=[0,0,0,0,0,0,0,0,... (sagequ3/b3t1)*((s31t1^3)*cosh(s31t1*l3)+(a3t1+b3t1+c3t1)*s31t1*... cosh(s31t1*l3)),(sagequ3/b3t1)*((s32t1^3)*sinh(s32t1*l3)+... (a3t1+b3t1+c3t1)*s32t1*sinh(s32t1*l3)),(sagequ3/b3t1)*(-(s33t1^3)*... cos(s33t1*l3)+(a3t1+b3t1+c3t1)*s33t1*cos(s33t1*l3)),(sagequ3/b3t1)*... ((s34t1^3)*sin(s34t1*l3)-(a3t1+b3t1+c3t1)*s34t1*sin(s34t1*l3))];

JT1=[G1t1; G2t1; G3t1; G4t1; G5t1; G6t1; N1t1; N2t1; N3t1; N4t1; N5t1;... N6t1]; Jrre_t1=rref(JT1);

164

function JT2=omega_t2(omega)



l1=20E-3; l2=80E-3; l3=250E-3; L=l3;

Eb=210E9; Ep=67E9;

Gb=82E9; Gp=26E9;

rhob=7850; rhop=7910;

wb=25E-3; wp=15E-3;


d31=-183*10^-12;

Sb=5/6; Sp=Sb;

% Mass properties




% Nutral axis

165















s12t2=s11t2; s22t2=s21t2; s32t2=s31t2;


166

s14t2=s13t2; s24t2=s23t2; s34t2=s33t2;





G1t2=[(1/(a1t2*c1t2))*((s11t2^3)+(b1t2+c1t2)*s11t2),0,... (1/(a1t2*c1t2))*(-(s13t2^3)+(b1t2+c1t2)*s13t2),0,0,0,0,0,0,0,0,0];

G2t2=[0,1,0,1,0,0,0,0,0,0,0,0];

G3t2=[(1/(a1t2*c1t2))*((s11t2^3)*cosh(s11t2*l1)+(b1t2+c1t2)*s11t2*... cosh(s11t2*l1)),(1/(a1t2*c1t2))*((s12t2^3)*sinh(s12t2*l1)+... (b1t2+c1t2)*s12t2*sinh(s12t2*l1)),(1/(a1t2*c1t2))*(-(s13t2^3)*... cos(s13t2*l1)+(b1t2+c1t2)*s13t2*cos(s13t2*l1)),(1/(a1t2*c1t2))*... ((s14t2^3)*sin(s14t2*l1)-(b1t2+c1t2)*s14t2*sin(s14t2*l1)),... (-1/(a2t2*c2t2))*((s21t2^3)*cosh(s21t2*l1)+(b2t2+c2t2)*s21t2*... cosh(s21t2*l1)),(-1/(a2t2*c2t2))*((s22t2^3)*sinh(s22t2*l1)+... (b2t2+c2t2)*s22t2*sinh(s22t2*l1)),(-1/(a2t2*c2t2))*(-(s23t2^3)*... cos(s23t2*l1)+(b2t2+c2t2)*s23t2*cos(s23t2*l1)),(-1/(a2t2*c2t2))*... ((s24t2^3)*sin(s24t2*l1)-(b2t2+c2t2)*s24t2*sin(s24t2*l1)),0,0,0,0];

G4t2=[0,0,0,0,... (1/(a2t2*c2t2))*((s21t2^3)*cosh(s21t2*l2)+(b2t2+c2t2)*s21t2*... cosh(s21t2*l2)),(1/(a2t2*c2t2))*((s22t2^3)*sinh(s22t2*l2)+... (b2t2+c2t2)*s22t2*sinh(s22t2*l2)),(1/(a2t2*c2t2))*(-(s23t2^3)*... cos(s23t2*l2)+(b2t2+c2t2)*s23t2*cos(s23t2*l2)),(1/(a2t2*c2t2))*... ((s24t2^3)*sin(s24t2*l2)-(b2t2+c2t2)*s24t2*sin(s24t2*l2)),... (-1/(a3t2*c3t2))*((s31t2^3)*cosh(s31t2*l2)+(b3t2+c3t2)*... s31t2*cosh(s31t2*l2)),(-1/(a3t2*c3t2))*((s32t2^3)*sinh(s32t2*l2)+... (b3t2+c3t2)*s32t2*sinh(s32t2*l2)),(-1/(a3t2*c3t2))*(-(s33t2^3)*... cos(s33t2*l2)+(b3t2+c3t2)*s33t2*cos(s33t2*l2)),(-1/(a3t2*c3t2))*... ((s34t2^3)*sin(s34t2*l2)-(b3t2+c3t2)*s34t2*sin(s34t2*l2))];



N1t2=[EIequ1*s11t2*cosh(s11t2*l1),... EIequ1*s12t2*sinh(s12t2*l1),...

167

EIequ1*s13t2*cos(s13t2*l1),... -EIequ1*s14t2*sin(s14t2*l1),... -EIequ2*s21t2*cosh(s21t2*l1),... -EIequ2*s22t2*sinh(s22t2*l1),... -EIequ2*s23t2*cos(s23t2*l1),... EIequ2*s24t2*sin(s24t2*l1),0,0,0,0];

N2t2=[0,0,0,0,... EIequ2*s21t2*cosh(s21t2*l2),... EIequ2*s22t2*sinh(s22t2*l2),... EIequ2*s23t2*cos(s23t2*l2),... -EIequ2*s24t2*sin(s24t2*l2),... -EIequ3*s31t2*cosh(s31t2*l2),... -EIequ3*s32t2*sinh(s32t2*l2),... -EIequ3*s33t2*cos(s33t2*l2),... EIequ3*s34t2*sin(s34t2*l2)];

N3t2=[(-sagequ1/c1t2)*((s11t2^2)*sinh(s11t2*l1)+(b1t2+c1t2)*... sinh(s11t2*l1)),(-sagequ1/c1t2)*((s12t2^2)*cosh(s12t2*l1)+... (b1t2+c1t2)*cosh(s12t2*l1)),(-sagequ1/c1t2)*(-(s13t2^2)*... sin(s13t2*l1)+(b1t2+c1t2)*sin(s13t2*l1)),(-sagequ1/c1t2)*... (-(s14t2^2)*cos(s14t2*l1)+(b1t2+c1t2)*cos(s14t2*l1)),... (sagequ2/c2t2)*((s21t2^2)*sinh(s21t2*l1)+(b2t2+c2t2)*... sinh(s21t2*l1)),(sagequ2/c2t2)*((s22t2^2)*cosh(s22t2*l1)+... (b2t2+c2t2)*cosh(s22t2*l1)),(sagequ2/c2t2)*(-(s23t2^2)*... sin(s23t2*l1)+(b2t2+c2t2)*sin(s23t2*l1)),(sagequ2/c2t2)*... (-(s24t2^2)*cos(s24t2*l1)+(b2t2+c2t2)*cos(s24t2*l1)),0,0,0,0];

N4t2=[0,0,0,0,... (-sagequ2/c2t2)*((s21t2^2)*sinh(s21t2*l2)+(b2t2+c2t2)*... sinh(s21t2*l2)),(-sagequ2/c2t2)*((s22t2^2)*cosh(s22t2*l2)+... (b2t2+c2t2)*cosh(s22t2*l2)),(-sagequ2/c2t2)*(-(s23t2^2)*... sin(s23t2*l2)+(b2t2+c2t2)*sin(s23t2*l2)),(-sagequ2/c2t2)*... (-(s24t2^2)*cos(s24t2*l2)+(b2t2+c2t2)*cos(s24t2*l2)),... (sagequ3/c3t2)*((s31t2^2)*sinh(s31t2*l2)+(b3t2+c3t2)*... sinh(s31t2*l2)),(sagequ3/c3t2)*((s32t2^2)*cosh(s32t2*l2)+... (b3t2+c3t2)*cosh(s32t2*l2)),(sagequ3/c3t2)*(-(s33t2^2)*... sin(s33t2*l2)+(b3t2+c3t2)*sin(s33t2*l2)),(sagequ3/c3t2)*... (-(s34t2^2)*cos(s34t2*l2)+(b3t2+c3t2)*cos(s34t2*l2))];

N5t2=[0,0,0,0,0,0,0,0,... EIequ3*s31t2*cosh(s31t2*l3),... EIequ3*s32t2*sinh(s32t2*l3),... EIequ3*s33t2*cos(s33t2*l3),... -EIequ3*s34t2*sin(s34t2*l3)];

N6t2=[0,0,0,0,0,0,0,0,... (-sagequ3/c3t2)*((s31t2^2)*sinh(s31t2*l3)+(b3t2+c3t2)*... sinh(s31t2*l3)),(-sagequ3/c3t2)*((s32t2^2)*cosh(s32t2*l3)+... (b3t2+c3t2)*cosh(s32t2*l3)),(-sagequ3/c3t2)*(-(s33t2^2)*... sin(s33t2*l3)+(b3t2+c3t2)*sin(s33t2*l3)),(-sagequ3/c3t2)*... (-(s34t2^2)*cos(s34t2*l3)+(b3t2+c3t2)*cos(s34t2*l3))];

JT2=[G1t2; G2t2; G3t2; G4t2; G5t2; G6t2; N1t2; N2t2; N3t2; N4t2; N5t2;... N6t2];

168

function Jrre_t2=coeff_t2(omega)



l1=20E-3; l2=80E-3; l3=250E-3; L=l3;

Eb=210E9; Ep=67E9;

Gb=82E9; Gp=26E9;

rhob=7850; rhop=7910;

wb=25E-3; wp=15E-3;


d31=-183*10^-12;

Sb=5/6; Sp=Sb;

% Mass properties




% Nutral axis

169















s12t2=s11t2; s22t2=s21t2; s32t2=s31t2;


s14t2=s13t2;

170

s24t2=s23t2; s34t2=s33t2;





G1t2=[(1/(a1t2*c1t2))*((s11t2^3)+(b1t2+c1t2)*s11t2),0,... (1/(a1t2*c1t2))*(-(s13t2^3)+(b1t2+c1t2)*s13t2),0,0,0,0,0,0,0,0,0];

G2t2=[0,1,0,1,0,0,0,0,0,0,0,0];

G3t2=[(1/(a1t2*c1t2))*((s11t2^3)*cosh(s11t2*l1)+(b1t2+c1t2)*s11t2*... cosh(s11t2*l1)),(1/(a1t2*c1t2))*((s12t2^3)*sinh(s12t2*l1)+... (b1t2+c1t2)*s12t2*sinh(s12t2*l1)),(1/(a1t2*c1t2))*(-(s13t2^3)*... cos(s13t2*l1)+(b1t2+c1t2)*s13t2*cos(s13t2*l1)),(1/(a1t2*c1t2))*... ((s14t2^3)*sin(s14t2*l1)-(b1t2+c1t2)*s14t2*sin(s14t2*l1)),... (-1/(a2t2*c2t2))*((s21t2^3)*cosh(s21t2*l1)+(b2t2+c2t2)*s21t2*... cosh(s21t2*l1)),(-1/(a2t2*c2t2))*((s22t2^3)*sinh(s22t2*l1)+... (b2t2+c2t2)*s22t2*sinh(s22t2*l1)),(-1/(a2t2*c2t2))*(-(s23t2^3)*... cos(s23t2*l1)+(b2t2+c2t2)*s23t2*cos(s23t2*l1)),(-1/(a2t2*c2t2))*... ((s24t2^3)*sin(s24t2*l1)-(b2t2+c2t2)*s24t2*sin(s24t2*l1)),0,0,0,0];

G4t2=[0,0,0,0,... (1/(a2t2*c2t2))*((s21t2^3)*cosh(s21t2*l2)+(b2t2+c2t2)*s21t2*... cosh(s21t2*l2)),(1/(a2t2*c2t2))*((s22t2^3)*sinh(s22t2*l2)+... (b2t2+c2t2)*s22t2*sinh(s22t2*l2)),(1/(a2t2*c2t2))*(-(s23t2^3)*... cos(s23t2*l2)+(b2t2+c2t2)*s23t2*cos(s23t2*l2)),(1/(a2t2*c2t2))*... ((s24t2^3)*sin(s24t2*l2)-(b2t2+c2t2)*s24t2*sin(s24t2*l2)),... (-1/(a3t2*c3t2))*((s31t2^3)*cosh(s31t2*l2)+(b3t2+c3t2)*s31t2*... cosh(s31t2*l2)),(-1/(a3t2*c3t2))*((s32t2^3)*sinh(s32t2*l2)+... (b3t2+c3t2)*s32t2*sinh(s32t2*l2)),(-1/(a3t2*c3t2))*(-(s33t2^3)*... cos(s33t2*l2)+(b3t2+c3t2)*s33t2*cos(s33t2*l2)),(-1/(a3t2*c3t2))*... ((s34t2^3)*sin(s34t2*l2)-(b3t2+c3t2)*s34t2*sin(s34t2*l2))];



N1t2=[EIequ1*s11t2*cosh(s11t2*l1),... EIequ1*s12t2*sinh(s12t2*l1),... EIequ1*s13t2*cos(s13t2*l1),...

171

-EIequ1*s14t2*sin(s14t2*l1),... -EIequ2*s21t2*cosh(s21t2*l1),... -EIequ2*s22t2*sinh(s22t2*l1),... -EIequ2*s23t2*cos(s23t2*l1),... EIequ2*s24t2*sin(s24t2*l1),0,0,0,0];

N2t2=[0,0,0,0,... EIequ2*s21t2*cosh(s21t2*l2),... EIequ2*s22t2*sinh(s22t2*l2),... EIequ2*s23t2*cos(s23t2*l2),... -EIequ2*s24t2*sin(s24t2*l2),... -EIequ3*s31t2*cosh(s31t2*l2),... -EIequ3*s32t2*sinh(s32t2*l2),... -EIequ3*s33t2*cos(s33t2*l2),... EIequ3*s34t2*sin(s34t2*l2)];

N3t2=[(-sagequ1/c1t2)*((s11t2^2)*sinh(s11t2*l1)+(b1t2+c1t2)*... sinh(s11t2*l1)),(-sagequ1/c1t2)*((s12t2^2)*cosh(s12t2*l1)+... (b1t2+c1t2)*cosh(s12t2*l1)),(-sagequ1/c1t2)*(-(s13t2^2)*... sin(s13t2*l1)+(b1t2+c1t2)*sin(s13t2*l1)),(-sagequ1/c1t2)*... (-(s14t2^2)*cos(s14t2*l1)+(b1t2+c1t2)*cos(s14t2*l1)),... (sagequ2/c2t2)*((s21t2^2)*sinh(s21t2*l1)+(b2t2+c2t2)*... sinh(s21t2*l1)),(sagequ2/c2t2)*((s22t2^2)*cosh(s22t2*l1)+... (b2t2+c2t2)*cosh(s22t2*l1)),(sagequ2/c2t2)*(-(s23t2^2)*... sin(s23t2*l1)+(b2t2+c2t2)*sin(s23t2*l1)),(sagequ2/c2t2)*... (-(s24t2^2)*cos(s24t2*l1)+(b2t2+c2t2)*cos(s24t2*l1)),0,0,0,0];

N4t2=[0,0,0,0,... (-sagequ2/c2t2)*((s21t2^2)*sinh(s21t2*l2)+(b2t2+c2t2)*... sinh(s21t2*l2)),(-sagequ2/c2t2)*((s22t2^2)*cosh(s22t2*l2)+... (b2t2+c2t2)*cosh(s22t2*l2)),(-sagequ2/c2t2)*(-(s23t2^2)*... sin(s23t2*l2)+(b2t2+c2t2)*sin(s23t2*l2)),(-sagequ2/c2t2)*... (-(s24t2^2)*cos(s24t2*l2)+(b2t2+c2t2)*cos(s24t2*l2)),... (sagequ3/c3t2)*((s31t2^2)*sinh(s31t2*l2)+(b3t2+c3t2)*... sinh(s31t2*l2)),(sagequ3/c3t2)*((s32t2^2)*cosh(s32t2*l2)+... (b3t2+c3t2)*cosh(s32t2*l2)),(sagequ3/c3t2)*(-(s33t2^2)*... sin(s33t2*l2)+(b3t2+c3t2)*sin(s33t2*l2)),(sagequ3/c3t2)*... (-(s34t2^2)*cos(s34t2*l2)+(b3t2+c3t2)*cos(s34t2*l2))];

N5t2=[0,0,0,0,0,0,0,0,... EIequ3*s31t2*cosh(s31t2*l3),... EIequ3*s32t2*sinh(s32t2*l3),... EIequ3*s33t2*cos(s33t2*l3),... -EIequ3*s34t2*sin(s34t2*l3)];

N6t2=[0,0,0,0,0,0,0,0,... (-sagequ3/c3t2)*((s31t2^2)*sinh(s31t2*l3)+(b3t2+c3t2)*... sinh(s31t2*l3)),(-sagequ3/c3t2)*((s32t2^2)*cosh(s32t2*l3)+... (b3t2+c3t2)*cosh(s32t2*l3)),(-sagequ3/c3t2)*(-(s33t2^2)*... sin(s33t2*l3)+(b3t2+c3t2)*sin(s33t2*l3)),(-sagequ3/c3t2)*... (-(s34t2^2)*cos(s34t2*l3)+(b3t2+c3t2)*cos(s34t2*l3))];

JT2=[G1t2; G2t2; G3t2; G4t2; G5t2; G6t2; N1t2; N2t2; N3t2; N4t2; N5t2;... N6t2]; Jrre_t2=rref(JT2);

172

function [Adisc,Bdisc]=state_space(Mdisc,Cdisc,Kdisc,fdisc)

% payman zolmajd, 06.10.2015 % Thesis 7990, summer 2015

% This function file develops a state space formulation for a vibrating % system using MCK matrices and a force vector f. Outputs A and B matrices % only. C and D matrices to be formulated by user.

% Determine size of M, C and K matrices n=size(Mdisc,1);

% Width of f matrix nf=size(fdisc,2);

% Formulate Ahat, Bhat and fhat matrices fhat=zeros(2*n,nf); fhat(n+1:2*n,1:nf)=fdisc;

Ahat=[eye(n), zeros(n); zeros(n,n),Mdisc];

Bhat=[zeros(n), -eye(n); Kdisc, Cdisc];

% Calculate A and B matrices Adisc=Ahat\(-Bhat); Bdisc=Ahat\fhat;

173

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a comparative study of vibration analysis of piezoelectrically …rx... · 2019. 2. 13. · a...

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