torque vibration isolatior - aau
TRANSCRIPT
Title: Torque vibration isolator
Theme: Engineering Design of MechanicalSystems Masterβs Project
Project period(first part):
01/09-2015 - 18/12-2015
Project group:
3.223
Department of Mechanical and
Manufacturing Engineering
Fibigerstraede 16DK 9220 Aalborg OestTlf. 9940 9297Fax 9815 1675Web www.ses.aau.dk
Author:
Alexandr Hvatov
Supervisor:
Sergei Sorokin
No. printed Copies: 4
No. of Appendix Pages: 11
Completed: 18/12-2015
Synopsis:
First part of this Master project containspreliminary consideration of the circularmembrane in polar coordinates which willbe used in second part of project, wherethin plate-cylindrical shell model considered.Also curved beam model of vibrationtorque isolator considered. Torque isolationmodelled as series of periodic alternatingBernoulli-Euler. In order to obtain solutionFloquet theory used. Analytical solution wasobtained by using of boundary integralsmethod, Greenβs matrix was derived usingbi-orthogonality conditions.
Table of contents
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1 Ch.1. The concept of a periodic structures . . . . . . . . . . . . . . . . . . . . . . . . 7
2 Ch.2 Circular membrane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.1 Membrane vibration equation. Greenβs function. . . . . . . . . . . . . . . . . . . . 92.2 Periodic structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2.1 Infinite structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2.2 Finite structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3 Ch.3. Bernoulli-Euler curved beam model . . . . . . . . . . . . . . . . . . . . . . . . 213.1 Equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.1.1 In-plane vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.1.2 Out-plane vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2 Boundary integrals method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.2.1 In-plane vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.2.2 Out-plane vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.3 The benchmark periodic structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.3.1 Infinite periodic structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.3.2 Finite periodic structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.3.3 Eigenmodes analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.4 The spatial periodic structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.5 Parametric study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
A App. Π. Circular membrane equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
B App. B. Membrane displacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
C App. Π‘. Axial rod vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47C.1 Greenβs function definition and derivation . . . . . . . . . . . . . . . . . . . . . . . 47C.2 Equation of axial displacement of a rod. Boundary integrals method . . . . . . . . . 48C.3 Boundary integrals method direct application . . . . . . . . . . . . . . . . . . . . . 50
D App. D. Floquet theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3
Introduction
In many industrial applications it is necessary to isolate components from vibration in connectedneighbour components. This is also the case for various rotational transmission systems. It has beenshown for different applications see e.g. [1β8], that periodic structures can generate such a vibrationisolation. An idea for such a torque vibration isolator consisted of periodically repeated substructureswith sufficient static torque stiffness, is sketched here below in Fig.1:
Figure 1: Vibration isolator sketch
Main idea of project is to model vibration isolator sketched above in order to have themain advantages and disadvantages being seen before experiments are conducted. Also shouldbe considered general possibility of construction vibration isolation torque sketched above and howthis kind of structure affects on power flow from a wind turbine generator should be explored.
Models, considered in this work are based on differential equations and in order to find solutionof these equations following instruments are used:
1.Boundary Integral Equations method is used to find solution of a differential equation, usingonly information about function and its derivatives on the boundary of volume considered in givenproblem. In this method Greenβs matrices are commonly used, which are solution of equationwith point force excitation. And in this work, bi-orthogonality conditions (in this work they areused without detail explanation, because this theory lies out of scope of this work, more detaileddescription can be found in [9]) are employed to find Greenβs matrices. In appendix C, applicationof this method to a simple case of axial rod vibrations is shown.
2. Floquet theory is widely used in order to obtain solution for periodically alternating infinitestructures. As shown for example in [7, 8] infinite structure cases are in good correspondence witha finite one and therefore Floquet theory gives good possibility to analyse periodic structures andunderstand mechanisms of wave propagation in periodic structures. In appendix D, application ofthis method to a simple case of axial rod vibrations is shown.
Goal of this work is to model vibrational isolation torque as a periodical structure and provideexperimental data in order to see quality of modelling. All work divided in following subtasks:
5
(I) Show possibility of using Floquet theory in polar coordinates on example of circularmembrane equation, since isolator, shown on a Fig.1 contains part, that is modelled as a part ofthe circular plate, which equation is naturally written in polar coordinate system. Floquet theory iswell studied in cartesian coordinate system [1], but there are no works, where application of Floquettheory to other coordinate systems is shown.
(II) Derivation and evaluation of boundary integral equations for a beam model (first model),based on Bernoulli-Euler flat ring equations
(III) Derivation and evaluation of boundary integral equations for a Kirchoff-Love plate andcylindricall shell model (second model) with use of polar coordinates
(IV) Experimental validation of first and second modelSince it is a long masterβs project in first part only subtasks (I) (Ch.2) and (II) (Ch.3) are
considered. Parts (III) and (IV) will be covered in second part of project.Project is written in comprehensive way and information in appendix can be used as supply for
more clear understanding of methods, that lies in ground of this project. Information in appendicesis not mandatory for understanding ideas of this project.
6
Chapter 1
The concept of a periodic structures
Here common definitions for all periodic structures considered in this work are introduced. First,an infinite periodic structure as shown on Fig.1.1 can be considered:
Figure 1.1: Infinite periodical structure scheme
,where π is the common coordinate (axial coordinate of bar, radial coordinate of a circularmembrane, natural coordinate of spring). Each component of a periodic structure can have differentmaterial parameters (like Young modulus), shape parameters (like curvature) and length repeating inperiodic alternating manner. In order to use Floquet theory at least one parameter excluding lengthshould be different for two segments. In terms of vibrations, different wavenumbers ππ for each partare required.
The existence of frequency stop- and pass-bands in infinite periodic waveguides is well knownand understood since the pioneering work by L.Brillouin [1]. The vibro-acoustics of beams, platesand shells and pipes with periodic attachments or step-wise varying properties has been broadlyexplored by many authors. Classial works [1, 2] and the modern one [3β8] are just a few of those,which illustrate the classical and recent advances in this area of research and that it is of interestto develop methods for various waveguides and make industrial application of theory of periodicstructures.
Infinite structure can be considered as built from finite structures called periodicity cell. Hereonly periodicity cell with one period length are considered. Periodicity cells can be chosen arbitrary,for example as:
Figure 1.2: Different periodicity cells schemes
7
But most interesting properties are of a cell, which can be schematically illustrated as:
Figure 1.3: Symmetrical periodicity cell scheme
Structure, illustrated on Fig.1.3 called symmetrical periodicity cell (half "white"part - "black"parthalf "white"part) and it has some important properties: it has one period length 1 + πΎ and it isbalanced with respect to center of masses, i.e. geometrical and gravity centers are in same point.
Since infinite structure can not be implemented practically, one can consider also a finitestructures, built from symmetrical cells. This kind of finite structures has some interesting properties,which will be described in next chapters. One can consider non-symmetrical cell as a βbuildingblockβ but it does not have such properties and will not be considered in this work.
Properties of a finite and infinite structure are closely related to each other, and therefore bothstructures can be considered simultaneously in order to show full picture of wave propagation andvibrations.
8
Chapter 2
Periodic circular membrane
Torque vibration isolator shown on Fig.1 can be considered as an ensemble of flat ring plates andcylindrical shells, connected in an alternative manner, i.e. as a periodic structure. Equation of motionof flat ring plate is naturally written in the polar coordinate system. Therefore, it is expedient toinvestigate performance of a periodic structure in polar coordinates. Previously in [7] and [8] mostsimple cases of axial and flexural vibrations of periodic beams and vibrations of cylindrical shellwere considered. Main ideas of these articles were shown for cartesian coordinate system. Maingoal of this chapter is to consider different coordinate system and show that all properties of theperiodic structures for cartesian coordinates are preserved in polar coordinate system with simplecircular membrane as an illustrative example.
2.1 Membrane vibration equation. Greenβs function.
General equation of motion for the membrane has the form (see App.A for derivation fromHamiltonβs principle and [10]):(
β + π2)π’ = π’ππ +
1
ππ’π +
1
π2π’ππ + π2π’ = βπ(π, π) (2.1)
,where π(π, π) is the intensity of distributed force.The external force can be presented as:
π (π, π) =+ββπ=0
ππ(π) cos(ππ) (2.2)
Then general solution can also be expanded in Fourier series :
π’ (π, π) =+ββπ=0
ππ(π) cos(ππ) (2.3)
Which substituted into Eq.2.1 gives:
(β + π2
)π’ =
+ββπ=0
(π2ππ(π)
ππ2+
1
π
πππ(π)
ππ+ (π2 β π2
π2)ππ(π)
)cos(ππ) = β
+ββπ=0
ππ(π) cos(ππ)
(2.4)Since cosππ are independent functions, for each circumferential wave number π decoupled
equation can be solved (hereafter, index π is omitted in all equations):
π β²β²(π) +1
ππ β²(π) +
(π2 β π2
π2
)π(π) = βπ(π) (2.5)
9
In order to obtain solution Greenβs function method is used. By definition [11], Greenβs functionis a solution of the equation ( in what follows π
πππΊ(π, π0) = πΊβ²(π, π0)):
πΊβ²β²(π, π0) +1
ππΊβ²(π, π0) +
(π2 β π2
π2
)πΊ(π, π0) = βπΏ(π β π0) (2.6)
,where πΏ(π₯)- Dirac delta function. Here delta function πΏ(πβπ0) has meaning of intensity of pointforce, applied at the point π0 and distributed in the angular coordinate direction as cos(ππ), and πis the observation point. Therefore second derivative πΊβ²β²(π, π0) has dimension of force intensity
Greenβs function of an ordinary second-order differential equation has a following properties [11](for clarity notation of point of excitation and observation preserved).
Symmetry with respect to observation and excitation point property:
πΊ(π, π0) = πΊ(π0, π) (2.7a)
Unit jump in derivative at the excitation point:
π
πππΊ(π0, π0 + π) β π
πππΊ(π0, π0 β π) = 1, π β 0 (2.7b)
Continuity at the excitation point:
πΊ(π0, π0 + π) = πΊ(π0, π0 β π), π β 0 (2.7c)
It should be noted that ,since second derivative πΊβ²β²(π, π0) has dimension of force intensity,ππππΊ(π, π0) in this case has dimension of the force, acting on a radial direction. Therefore, force
should have unit jump at the coordinate π = π0. Also, property of unit jump is a reflection ofmathematical theorem π
ππ₯π(π₯) = πΏ(π₯), where π(π₯) is the Heaviside theta-function. As known, theta-
function has unit jump at the point π₯ = 0.Last two properties Eq.2.7b-Eq.2.7c can be used for obtaining explicit form of Greenβs function
of equation Eq.2.5. The general solution of equation Eq.2.6 with arbitrary excitation point π0 has aform:
π’+(π) = π΄ π»(1)π (ππ) , π > π0
π’β(π) = π΅ π»(2)π (ππ) , π β€ π0
(2.7)
, where π»(1)π (π) and π»
(2)π (π) are Hankelβs functions of order π of first and second kind
respectively and π΄,π΅ are integration constants.In order to consider infinite structures function π’+(π) should satisfy radiation, or Sommerfeld,
condition [12].With properties Eq.2.7a-Eq.2.7c one can obtain system of linear algebraical equations with
respect to constants π΄,π΅:
π’+(π0) = π’β(π0)ππππ’+(π0) β π
πππ’β(π0) = 1
(2.8)
Constants are found as:
π΄ = β π»(2)π (π π0)
π(π»(1)π+1(π π0)π»
(2)π (π π0)βπ»
(2)π+1(π π0)π»
(1)π (π π0))
= β14πππ0π»
(2)π (π π0)
π΅ = β π»(1)π (π π0)
π(π»(1)π+1(π π0)π»
(2)π (π π0)βπ»
(2)π+1(π π0)π»
(1)π (π π0))
= β14πππ0π»
(1)π (π π0)
(2.9)
Thus, Greenβs function for equation 2.5 have form:
10
πΊ(π, π0) =
β1
4πππ0π»
(1)π (π π0)π»
(2)π (π π) π β€ π0
β14πππ0π»
(2)π (π π0)π»
(1)π (π π) π > π0
(2.10)
It should be noted, that this form of Greenβs function in not unique. One can choose an arbitrarycombination of Bessel functions π (π) and π½(π) as the general solution Eq.2.7. Also, any solutionof the homogenous equation Eq.2.5 (with π(π) β‘ 0) can be added to the existing form of a Greenβsfunction.
It is convenient to define Greenβs function as a solution of the equation (for more detailedexplanation see App.B):
πΊβ²β²(π, π0) +1
ππΊβ²(π, π0) +
(π2 β π2
π2
)πΊ(π, π0) = βπΏ(π β π0)
π0(2.6β)
From physical point of view, with this definition force resultant πΉ =2πβ«0
πβ«π
β πΏ(πβπ0)π0
πππππ =
β2ππ0
πβ«π
πΏ(π β π0)πππ = β2ππ0π0
= β2π remains constant when point of excitation π0 is changing.
With definition 2.6β Greenβs function has form:
πΊ(π, π0) =
β1
2πππ»
(1)π (π π0)π»
(2)π (π π) π β€ π0
β12πππ»
(2)π (π π0)π»
(1)π (π π) π > π0
(2.11)
One can plot real part of function 2.11 in order to see that all properties of Greenβs function arepresent and Greenβs function was found correctly. Let π β‘ 1 and π0 = 1 therefore jump at the pointπ = 1 of the function πΊβ²(π, π0) should be 1
π0= 1:
Figure 2.1: Functions πΊ (blue) and πΊβ²(orange) for π0 = 1, dashed - imaginary part
As seen, function πΊ(π, 1) is continuous at point π = 1, but function πΊβ²(π, 1) experience unitjump, as were predicted. Imaginary part is continuous in both cases, it does not affect unit jump.
Diracβs delta function πΏ(π) has remarkable property (which in mathematics is the definition ofDiracβs delta function): β« π
π
πΏ(π β π0)π(π)ππ = π(π0) (2.12)
Eq.2.12 is valid for any interval π0 β (π, π), including (ββ,+β)And from equations Eq.2.5 - Eq.2.6β and property Eq.2.12 one can obtain displacement π’(π0) at
any excitation point π0 for arbitrary force intensity π(π) (it is shown in App.B). Displacement of amembrane has the form:
11
π’(π0) = [π’β²(π)πΊ(π, π0) β π’(π)πΊβ²(π, π0)]π
π=π
π=π
+
πβ«π
π(π)πΊ(π, π0)ππ (2.13)
Identity Eq.2.13 is a particular case of Kirchhoff integral in acoustics [12, 13].It should be emphasized, that πΊβ²(π, π0) is not uniquely defined at π = π0 and since we consider
values πΊβ²(π, π0) at the boundary value taken as limit from inside of a membrane. Let π = π, π = πbe the membrane boundaries, then limits have form:
πΊβ²(π, π0)
π0=π
= πΊβ²(π, π + π) , π β 0
πΊβ²(π, π0)
π0=π
= πΊβ²(π, πβ π) , π β 0(2.14)
Let us consider Eq.2.11. Using following series expansion at the point π§ = 0:
π»(1)π (π§) = 1 + 2π(πΎβlog(2))
π+ 2π(log(π§))
π+ π (π§2)
π»(2)π (π§) = 1 β 2π(πΎβlog(2))
πβ 2π(log(π§))
π+ π (π§2)
(2.15)
,where πΎ β 0.577216 is the Euler-Mascheroni constaint.With this Eq.2.11 to the leading order can be rewritten as:
πΊ(π, π0) =
12π
(β2π log
(ππ2
)+ π β 2ππΎ
) (2 log
(ππ02
)β ππ + 2πΎ
)π β€ π0
12π
(2 log
(ππ2
)β ππ + 2πΎ
) (β2π log
(ππ02
)+ π β 2ππΎ
)π > π0
(2.16)
With derivative taken:
π
πππΊ(π, π0) =
β π
ππ
(2 log
(ππ02
)β ππ + 2πΎ
)π β€ π0
πππ
(β2 log
(ππ02
)+ π β 2ππΎ
)π > π0
(2.17)
If we find Greenβs function value at point π = π0 + π
π
πππΊ(π0 + π, π0) = β2π log(π(π0 + π)) + π + 2ππΎ β π log(4)
ππ0(2.18)
If π = 0 then ππππΊ(π0, π0) = β 1
π0β 2π log(ππ0)
ππ0β 2ππΎ
ππ0+ π log(4)
ππ0. After same procedure repeated for
π = π0 β π gives ππππΊ(π0, π0) = 1
π0β 2π log(ππ0)
ππ0β 2ππΎ
ππ0+ π log(4)
ππ0. Thus we obtain value for Greenβs
function at the point π = π0 and this value depends on a limit direction.
π
πππΊ(π0 Β± π, π0) = β 1
π0β 2π log(ππ0)
ππ0β 2ππΎ
ππ0+
π log(4)
ππ0(2.19)
If more terms in the series expansion Eq.2.15 are taken, only precision of common part areincreasing, β 1
π0is only part that changes with change of limit direction.
In what follows, it is assumed that all limits are taken correctly for all parts of membrane.Before considering periodic structure single membrane can be considered in order to show the ideaof boundary integrals equations method.
Let π = π and π = π be circular membrane boundaries. Thus, one writes two boundary integralsin form:
π’(π) = [π’β²(π)πΊ(π, π) β π’(π)πΊβ²(π, π)]π
π=π
π=π
+πβ«π
π(π)πΊ(π, π)ππ
π’(π) = [π’β²(π)πΊ(π, π) β π’(π)πΊβ²(π, π)]π
π=π
π=π
+πβ«π
π(π)πΊ(π, π)ππ
(2.20)
12
Unknown here are two displacements at the boundaries π’(π), π’(π) and two forces π’β²(π), π’β²(π).In order to close system of algebraic equations two more equations should be written. For singlemembrane these are boundary conditions in form:
π’(π) = 0π’(π) = 0
(2.21)
or
π’β²(π) = 0π’β²(π) = 0
(2.22)
Mixed conditions can be stated too. Main principle is that energy functional should beminimized (see. App.A). In this work only symmetrical boundary conditions are considered.
Eq.2.20,Eq.2.21 or Eq.2.20, Eq.2.22 are four algebraical equations with respect to four unknownsπ’(π), π’(π), π’β²(π), π’β²(π) and this system has unique solution. When unknowns are found andsubstituted into Eq.2.13, displacement at any point inside the domain π < π0 < π can be found.
2.2 Periodic structure
2.2.1 Infinite structure
We consider periodic membrane shown on Fig.2.2:
Figure 2.2: Infinite membrane scheme
Where βblackβ and βwhiteβ part have different material parameters: Youngβs modulus, wavepropagation speed and length. Following dimensionless parameters are used in this work:
πΌ =πΈ2
πΈ1
; π½ =β2
β1
; πΎ =π2π1
; π =π2π1
;π =π1β1
; π1π1 = Ξ© (2.23)
Infinite periodical structure is considering with respect to radial coordinate π (see Fig. 1.2)With this eigenfrequencies problem can be rewritten in terms of boundary integration equations.
For each cell we write boundary equations at both ends in form:
π’π(ππ) = [π’β²(π)πΊπ(π, ππ) β π’(π)πΊβ²π(π, ππ)] π|
π=πππ=ππ
+ππβ«ππ
π(π)πΊπ(π, ππ)ππ
π’π(ππ) = [π’β²(π)πΊπ(π, ππ) β π’(π)πΊβ²π(π, ππ)] π|
π=πππ=ππ
+ππβ«ππ
π(π)πΊπ(π, ππ)ππ
(2.24)
13
, where π’π are displacement, ππ are left and ππ are right boundaries of π-th part of membranerespectively. For each part we have 2 unknown displacements and 2 unknown forces at theboundaries.
In order to show main principle one periodicity cell can be considered. Since Bessel π (π)function contains singularity at point π = 0 the central cell, that contains point π = 0 should beexcluded from consideration. Thus, boundary conditions have form:
π’2(π2) = [π’β²(π)πΊ2(π, π2) β π’(π)πΊβ²2(π, π2)] π|
π=π2π=π2
+π2β«π2
π(π)πΊ2(π, π2)ππ
π’2(π2) = [π’β²(π)πΊ2(π, π2) β π’(π)πΊβ²2(π, π2)] π|
π=π2π=π2
+π2β«π2
π(π)πΊ2(π, π2)ππ
π’3(π3) = [π’β²(π)πΊ3(π, π3) β π’(π)πΊβ²3(π, π3)] π|
π=π3π=π3
+π3β«π3
π(π)πΊ3(π, π3)ππ
π’3(π3) = [π’β²(π)πΊ3(π, π3) β π’(π)πΊβ²3(π, π3)] π|
π=π3π=π3
+π3β«π3
π(π)πΊ3(π, π3)ππ
(2.25)
As seen, there are four unknown displacements π’2(π2), π’2(π2), π’3(π3), π’3(π3) and four unknownforces π’β²
2(π2), π’β²2(π2), π’
β²3(π3), π’
β²3(π3). Therefore four more equations should be added. Two equations
are interfacial conditions between two cells, that allows to consider cells in the system and representscontinuity and forces equilibrium at the interface:
π’2(π2) = π’3(π3)πΌπ’β²
2(π2) = π’β²3(π3)
(2.26)
And two last equations are taken from Floquet periodicity theorem. That theorem allows to closesystem in case of infinite structure. Any two points for Floquet conditions can be taken, but withone period distance (1 + πΎ)π between points, i.e. π3 β π2 = (1 + πΎ)π:
π’2(π2) = Ξπ’3(π3)π’β²2(π2) = Ξπ’β²
3(π3)(2.27)
It should be emphasized that if one can consider first cell it is usually written as π’1(π1) insteadof π’2(π2) and π’β²
1(π1) instead of π’β²2(π2). Nevertheless, that is the same point and therefore it does
not affect the solution.Equations Eq.2.25-Eq.2.27 represent system of eight linear algebraical equations. In load-free
case, i.e. when π(π) = 0 system is homogeneous. In order to find non-trivial solution conditionπ·(Ξ,Ξ©) = 0, where π·(Ξ,Ξ©) is the determinant of the system Eq.2.25-Eq.2.27, must be fulfilled.In case of circular membrane π·(Ξ,Ξ©) is the second order polynomial in Ξ and we can find its rootsfor each Ξ© and plot them as shown on Fig.2.3.
In case of cartesian coordinates Ξ = exp(ππΎπ΅) (see [1]) and zones where abs(Ξ) = 1 are calledpass bands. It happens when πΎπ΅ is purely real number. In this case waves can propagate freely,because magnitude of waves are not change throughout the period, changes only phase. But in caseof membrane it is not obviously so, as seen on Fig.2.3 in the zones, similar to cartesian pass bands, the parameter Ξ has magnitude close to 2 and it should have another form, than exp(ππΎπ΅).
At first we consider wave propagation far away from center . This case should fully correspondto axial rod vibrations case in cartesian coordinates , because following limit can be considered:
(β
(2)πππππ + π2
)π’ = π’ππ +
1
ππ’π +
1
π2π’ππ + π2π’ β π’ππ + π2π’ =
(β
(1)ππππ‘ππ πππ + π2
)π’ , π β β (2.28)
Therefore, shape of pass- and gap-bands should repeat shape for the axial beam vibrationconsidered in [8] and App.D.
14
Figure 2.3: Floquet zones
In order to reach limit several consequent cells considered. Finite structure consisting of π cellsis taken and then Floquet conditions used in order to transfer to infinity. Hereafter, number π iscalled a number of precalculated cells. Therefore, we prescribe 2π pair of the interfacial conditions:
π’1(π) = π’2(π)π’β²
1(π) = πΌπ½π’β²2(π)
π’2((1 + πΎ)π) = π’3((1 + πΎ)π)πΌπ½π’β²
2((1 + πΎ)π) = π’β²3((1 + πΎ)π)
...π’2πβ1(πΏ2πβ1) = π’2π(πΏ2πβ1)π’β²
2πβ1(πΏ2πβ1) = πΌπ½π’β²2π(πΏ2πβ1)
(2.29)
,where πΏπ = ((1+πΎ)n+1)π - distance from 0 to end of n-th periodicity cell.And, since an infinite membrane is considered, we use Floquet conditions to transfer from n-th
cell to infinity:
π’2πβ1(πΏπβ1) = Ξπ’2π+1(πΏ2πβ1 + (1 + πΎ)π)π’β²
2πβ1(πΏπβ1) = Ξπ’β²2π+1(πΏ2πβ1 + (1 + πΎ)π)
(2.30)
Equations Eq.2.29-Eq.2.30 together with boundary integrals in form Eq.2.24 for each part ofmembrane compose the system of homogenous linear algebraical equations with respect to unknowndisplacements and forces at the borders and condition of non-triviality of solution can be used forfinding Floquet zones. For three precalculated cells it is shown on Fig.2.4
Gap band at the proximity of Ξ© = 0 appears due to singularity of a function π»(1)π (π) at the point
π = 0. Greenβs function including function π»(1)π (ππ) and wave number is function of frequency
π = π(Ξ©). Moreover, π(Ξ©) β 0 when Ξ© β 0. Thus at the point Ξ© = 0 argument in π»(1)π (ππ) is
zero. And therefore gap bang at proximity of the point Ξ© = 0 can not be considered as real gap-bandand waves in this frequency region have the properties of pass band waves.
15
Figure 2.4: Floquet zones for n=3
The effect is vanishing when wave propagation far away from center is considered and whensufficient number of precalculated cells is taken, Floquet zones picture for circular membrane inpolar coordinates fully repeats one for axial beam rod vibrations in cartesian coordinates:
Figure 2.5: Floquet zones (axial vibrations β blue, membrane (n=100) β green)
That fact, that πππ (Ξ) = 1 even in pass bands means that Ξ = exp(ππΎπ΅) as in cartesiancoordinates case. In first model it differs and difference depends on number of pre-calculated cells.
Since(
β(2)πππππ + π2
)π’ β
(β
(1)ππππ‘ππ πππ + π2
)π’ at π β +β with order of 1
πthen one can suppose
that
16
πΎππππππ΅ = πΎππππ‘
π΅ + 1ππβππ
Ξπππππ = Ξππππ‘ exp(
1ππβππ
)= exp(ππΎπ΅) exp
(1
ππβππ
) (2.31)
,where ππβππis a characteristic length of the model. For first model was found that ππβππ = 1ππΎ
and if corrected Floquet zones are plotted (π = 20,π = 0) it shows good correspondence:
Figure 2.6: Corrected Floquet zones (π = 20,π = 0)
Besides number π of precalculated cells there is second parameter π β number of circumferentialwaves. As shown on the Fig.2.7, parameter π does not change value of πππ (Ξ) for given π, it affectsonly on width of zero βgapβ band. With parameter π increasing increases width of zero gap bandand increases difference from cartesian case in low frequency range:
Figure 2.7: Floquet zones for n=5 and m=0 (green), m=5 (blue),m=10 (orange)
17
As seen, main patterns, found for cartesian case are preserved for infinite waveguide case inpolar coordinates. But one should consider each case individually, because form of fundamentalsolution is different for each coordinate system (for example, exponents in cartesian coordinates,Bessel and Hankel functions in polar).
2.2.2 Finite structures
Every finite structure can be considered as series of a βunitβ symmetrical periodicity cell (seeFig 1.2). In order to find eigenfrequencies interfacial conditions should be stated:
π’1(π) = π’2(π)π’β²
1(π) = πΌπ½π’β²2(π)
π’2((1 + πΎ)π) = π’3((1 + πΎ)π)πΌπ½π’β²
2((1 + πΎ)π) = π’β²3((1 + πΎ)π)
(2.32)
And two types of boundary conditions (see App.A for Hamiltonβs principle explanation):
π’1(π/2) = 0π’3((3/2 + πΎ)π) = 0
(2.33)
βπ’β²1(π/2) = 0
π’β²3((3/2 + πΎ)π) = 0
(2.34)
Equations Eq.2.32-Eq.2.33 defines eigenfrequency problem with βfixedβ ends or A-type borderconditions, equations Eq.2.32 and Eq.2.34 defines problem with βfreeβ ends or B-type borderconditions (existence of two such types of conditions were predicted by J. Mead in [2]). Roots,obviously, can be found only numerically, since there are no inverse of the Bessel function exists.Eigenfrequencies in both problems converges to gap band borders with increasing of the frequency:
Figure 2.8: Eigenfrequency of single symmetric periodicity cells (red β eigenfrequencies of βfixedβproblem, blue β βfreeβ)
As in axial case, all eigenfrequencies of more than one periodicity cell appears in pass bandsand all eigenfrequencies from a lesser structure exists in a bigger:
18
Figure 2.9: Eigenfrequency of structure with 10 symmetrical periodicity cells
In this chapter main patterns which are typical for periodical structure are shown. Behaviorof eigenfrequencies of finite structure has very common pattern. Eigenfrequencies of axial andflexural rod vibrations, cylindrical shell single symmetrical cell also covers gap bands borders[7, 8]. Eigenfrequencies of several periodicity cells appears in pass bands in all cases. Also, showntechniques (Floquet theorem, boundary integrals method) that will be used in the next chapters. Asseen, main patterns valid for cartesian coordinate system preserves in polar coordinates, but someadjustments required in order to show exact correspondence between these coordinate systems.Since membrane equation is closely related to a thin plate equation it gives reason to say that Floquettheory is working for circular thin plate equation, which will be used for modelling in second partof this work. Moreover, it gives reason to say that Floquet theory is coordinate independent and canbe, with some adjustment,used for any coordinate system.
19
Chapter 3
Bernoulli-Euler curved beam model
In this chapter a simplified model of a torque vibration isolator is considered. Methods fromprevious chapter (Floquet theorem, boundary integrals method) have broad range of applicability.In this chapter all these methods are used for more complicated case of system of differentialequations. This chapter has theoretical and practical goals. Practical goal is to build first model oftorque vibration isolator in the form of a system of a differential equations. Theoretical goal is toshow the difference between the picture of Floquet zones of a single differential equation and asystem of differential equations. In order to achieve that, methods, considered in Ch.2 are expandedand full range of tools for system of differential equations is obtained.
Torque vibration isolator has following periodicity cell with two different parts illustrated on theFig. 3.1
Figure 3.1: Vibration torque part
Part one is the segment of a circular plate, which is similar to a membrane considered in Ch.2with a local polar coordinates (π,Ξ) and part two is the segment of a thin cylindrical shell with localcartesian coordinate π₯ and local polar angle π. Each of these elements are more stiffer in transversedirection π and π₯ than in longitudinal Ξ and π. Thus,as first approximation, the curved Bernoulli-Euler beam model can be used, which excludes deformation of the plate segment in π-coordinateand of the shell segment in π₯-coordinate.
As βproof-of-conceptβ a curved beam with circular cross-section will be considered, because itsconsideration is obviously simpler. A curved beam has only one local coordinate π which is shownon Fig.3.2. Coordinate π is the natural parameter of a curve that connects geometrical center of eachcross-section:
21
Figure 3.2: First model scheme
For each segment of the structure, shown on Fig.3.2 in-plane and out-plane vibrations can beconsidered separately.
3.1 Equations of motion
3.1.1 In-plane vibrations
With coordinate system shown on Fig.3.10 equations of motion of flat ring in-plane vibrationshave the form [14]:
ππ΄π2π’ππ‘2
= πππ₯
ππ + 1
π ππ§ + ππ’ β πππ½
ππ
ππ΄π2π€ππ‘2
= πππ§
ππ β 1
π ππ₯ + ππ€ + 1
π ππ½
(3.1)
,where π’ - π₯-axis displacement, π€ - π§-axis displacement, π½ - rotation with respect to π¦ axis,ππ, ππ - components of force vector, ππ, ππ-components of moment vector, π is the radius ofcurvature. Generalized forces and rotation with Bernoulli-Euler assumptions have form:
ππ¦
πΈπΌπ¦= ππ½
ππ , ππ₯ = βπππ¦
ππ ππ§
πΈπ΄= ππ€
ππ β 1
π π’ , π½ = ππ’
ππ + 1
π π€
(3.2)
For brevity no external loading considered, i.e. ππ’ = ππ€ = ππ½ β‘ 0Time and space dependence for both in-plane and out-plane vibration has the form:
π’(π , π‘), π£(π , π‘), π€(π , π‘), πΎ(π , π‘)π = , π , ,Ξπ ππ₯π(πdimπ β πππ‘) (3.3)
Following dimensionless parameters used:
Ξ© =ππ
π, π = πdimπ, π =
π
π, π =
π, π =
π
π,π =
π, π =
π
π (3.4)
22
With time and space dependence Eq.3.3 and dimensionless parameters Eq.3.4substituted into thesystem Eq.3.1-Eq.3.2 following system is obtained:
π(16kπβ k3π
)+ π
(βk4 β 16π2 + 16Ξ©2
)= 0
π(β16kπ + k3π
)+ π
(16k2 + k2π2 + 16Ξ©2
)= 0
(3.5)
Determinant of system of algebraical linear equations Eq.3.5 gives 6th order polynomial in π,which is called dispersion relation:
π6 + π4Ξ©2 + 2π4π2 β 16π2Ξ©2 + π2π4 β π2Ξ©2π2 β 16Ξ©4 + 16Ξ©2π2 (3.6)
When it is solved with respect to π it gives 6 wavenumbers π(ππ)π :
Figure 3.3: Wave numbers π(ππ), real parts
Figure 3.4: Wave numbers π(ππ), imaginary parts
For applications it is of interest to consider low-frequency range, therefore it shows pictures,that is characteristic only for Bernoulli-Euler curved beam:
23
Figure 3.5: Wave numbers π(ππ), imaginary parts
Different forms of wave number ππ represents different form of waves. At the given frequencyΞ©, value π(Ξ©) can be pure real, pure imaginary and complex number. Since space dependenceexp(π π ) considered, pure imaginary π(Ξ©) represents propagating wave (wave that does not changeits absolute value when π is changing). Main theorem of algebra state that if ππππ is the root ofdispersion relation, then βππππ also the root. Root ππππ defines wave propagating from left to right(with time dependence exp(βπππ‘)) and root βππππ defines wave propagating from right to left. Purereal value πππ represents evanescent waves (waves that exponentially decreasing from left to rightor from right to left):
Figure 3.6: Different form of waves: pure imaginary ππππ (blue), pure real πππ > 0 (orange), purereal πππ < 0 (green)
Complex value represents wave πππππ = π(ππ)ππππ + ππ
(ππ)ππππ, propagating from left to right or from
right to left (depends on a sign of imaginary part π(ππ)ππππ) modulated by exp(π
(ππ)πππππ ),i.e. oscillating
between two curves exp(π(ππ)πππππ ) and β exp(π
(ππ)πππππ ) (shown dashed on the following figure) :
24
Figure 3.7: Different form of waves πππππ = π(ππ)ππππ + ππ
(ππ)ππππ: π
(ππ)ππππ > 0 (blue), π(ππ)
ππππ < 0 (green)
It is also convenient to introduce modal coefficient ππ€ = ππ
, which is found from equationsEq.3.5 for each wave number individually. In that case ππ’ = π
π= 1. Also, modal coefficients for
forces, moments and rotation introduced as:
ππ½ = (πππ’ + πππ€)πππ¦ = (πππ½)πππ§ = 16 (πππ€ β πππ’)πππ₯ = βππππ¦
(3.7)
3.1.2 Out-plane vibrations
Equations of motion of flat ring in-plane vibrations have the form:
ππΌππ2πΎππ‘2
= πππ§
ππ β 1
π ππ₯ + ππΎ
ππ΄π2π£ππ‘2
= πππ¦
ππ + ππ£ + πππΌ
ππ
(3.8)
With generalized forces and rotation with Bernoulli-Euler assumptions in form:
ππ₯
πΈπΌπ₯= ππΌ
ππ + 1
π πΎ , ππ§
πΊπΌπ= ππΎ
ππ β 1
π πΌ
ππ¦ = πππ₯
ππ + 1
π ππ§ , πΌ = βππ£
ππ
(3.9)
Where, π£ - π¦-axis displacement, πΌ - rotation with respect to π₯-axis, πΎ - rotation with respectto π§-axis. With time and space dependence Eq.3.3 and dimensionless parameters Eq.3.4substitutedinto the system Eq.3.8-Eq.3.9 following system is obtained:
Ξ(
π2
π+1+ 2Ξ©2 β π2
)+ π
(π2ππ+1
+ π2π)
= 0
Ξ(
π2ππ+1
+ π2π)
+ π(βπ4 + π2π2
π+1+ 16Ξ©2
)= 0
(3.10)
Determinant of the system Eq.3.10:
βπ6β2π4πΞ©2β2π4Ξ©2β2π4π2+16π2Ξ©2βπ2π4+2π2Ξ©2π2+32πΞ©4+32Ξ©4β16πΞ©2π2β16Ξ©2π2 (3.11)
25
Also gives 6 wavenumbers π(ππ’π‘)π :
Figure 3.8: Wave numbers π(ππ’π‘), real parts
Figure 3.9: Wave numbers π(ππ’π‘), imaginary parts
Modal coefficients in this case have form:
ππΎ = 1 ππ£ = πΞ
ππΌ = βπππ£ πππ¦ = (πππΌ + πππΎ)πππ§ = 1
1+π(πππΎ β πππΌ) πππ¦ = ππππ₯ + ππππ§
(3.12)
3.2 Boundary integrals method
Definition of Greenβs matrix for a system of a differential equation is the same as for Greenβsfunction for a single equation, but in case of π equations with π variable functions π load casesshould be considered. In each case one load is represented by delta-function whereas other are zero.Therefore, Greenβs matrix has dimensions πΓπ.
26
Since in boundary integrals method Greenβs matrix for infinite structure considered, Greenβsmatrix should satisfy radiation and decay conditions at the infinity [12]. Therefore, not allwavenumbers can be considered. Detailed explanation of principle of choice and all necessaryreferences contained in [14]. Among six roots of both dispersion relations we choose one complexwith negative real part (π π(π) < 0 ) and two purely imaginary roots with πππππ’π = πππππ
ππ> 0 for
each in-plane and out-plane equations.For Greenβs matrix derivation bi-orthogonality condition will be used. It can be derived from
reciprocity theorem, which is well known for most commonly used linear differential equations andfor elastic helical spring is described in [13]. Derivation of all bi-orthogonality conditions used herecan be found in [9].
3.2.1 In-plane vibrations
Bi-orthogonality condition for this case has the form:
ππ΄π¦ (π )π½π΅(π ) + ππ΄
π§ (π )π€π΅(π ) β π’π΄(π )ππ΅π₯ (π ) = 0 (3.13)
Or with modal coefficients introduced in Eq. 3.10:
π(π)ππ¦ π
(π)π½ + π
(π)ππ§ π
(π)π€ βπ(π)
π’ π(π)ππ₯ = 0 (π = π) (3.14)
Equations Eq.3.13-Eq.3.24 are called bi-orthogonality conditions for flat ring in-plane vibrations.And with reciprocity theorem they are widely used in Greenβs functions theory and for boundaryintegral equations derivation.
In in-plane ring vibrations case each string of Greenβs matrix represents solution for each of aloading case ππ = πΏ(π ), ππ = 0 (π = π ; π, π β π’,π€, π½)
In order to find Greenβs matrix, property of force unit jump of Greenβs function also used.Three load cases considered: load case 1 ππ§(0) = β1
2π πππ(π ) (it represents case ππ€ = πΏ(π ) ), load
case 2 ππ¦(0) = β12π πππ(π ) (ππ½ = πΏ(π )) and load case 3 ππ₯(0) = β1
2π πππ(π ) (ππ’ = πΏ(π )). With
that loading cases introduced it is convenient to split Forces/Moments and displacements into twogroups:
π€(π ), π½(π ), ππ₯(π )π’(π ), ππ§(π ),ππ¦(π )
(3.15)
If functions of first group are even, then functions of second group are odd and vice versa.Since Heaviside theta-function is odd function, only continuity of functions should be considered.Even functions have property of continuity at zero, and therefore, functions of second group arecontinuous.
Let us consider loading case 1. Significant part of Greenβs matrix has a following form:
π’(π ), ππ§(π ),ππ¦(π ) =3β
π=1
π(π)π’ ,π
(π)ππ§,π
(π)ππ¦Ξ
(1)π exp(ππ πππ (π ))π πππ(π ) (3.16)
Properties of Greenβs matrix have following form:
π’(0) = 0ππ¦(0) = 0ππ§(0) = β1
2π πππ(π )
(3.17)
Substituting 3.16 to 3.17 and multiplying each string to π(π)ππ₯ , π
(π)π½ , π
(π)π€ (π = 1, 2, 3)
respectively:
27
3βπ=1
Ξ(1)π π
(π)π’ π
(π)ππ₯ = 0
3βπ=1
Ξ(1)π π
(π)ππ¦π
(π)π½ = 0
3βπ=1
Ξ(1)π πππ§
(π)π(π)π€ = β1
2
(3.18)
Summing all equations in 3.18:
3βπ=1
Ξ(1)π (π(π)
π’ π(π)ππ₯ + π
(π)ππ¦ π
(π)π½ + π
(π)ππ§ π
(π)π€ ) = β1
2(3.19)
Expression in brackets is exactly bi-orthogonal condition 3.24 and it is not zero only when π = π.Thus one can obtain explicit form of each coefficient Ξ
(1)π .
In same way all Greenβs matrix coefficient entries can be found as:
Ξ(ππ)(1)π = β1
2π
(π)π€
π(π)ππ¦ π
(π)π½ +πππ§π π
(π)π€ βπ
(π)π’ π
(π)ππ₯
Ξ(ππ)(2)π = β1
2
π(π)π½
π(π)ππ¦ π
(π)π½ +πππ§π π
(π)π€ βπ
(π)π’ π
(π)ππ₯
Ξ(ππ)(3)π = 1
2π
(π)π’
π(π)ππ¦ ππ½π+πππ§π π
(π)π€ βπ
(π)π’ π
(π)ππ₯
(3.20)
After coefficients found green matrix formed as (on example of loading case 1):
πΊππ1 (π , π 0) = π’(π)(π ), π€(π)(π ), π½(π)(π ), π
(π)π₯ (π ), π
(π)π§ (π ),π
(π)π¦ (π ) =
3βπ=1
π πππ(π )π(π)π’ ,π
(π)π€ ,π
(π)π½ ,π
(π)ππ₯, π πππ(π )πππ§π, π πππ(π )π
(π)ππ¦Ξ(ππ)
(1)π exp(πππ
π πππ (π β π 0))
(3.21)After Greenβs matrix found, one can obtain displacement, expressed in terms of Greenβs matrix
(analogous to 2.13) in same way as it done in App.B. It can be written as ( [13]):
πΏ1ππ€(π 0) + πΏ2ππ½(π 0) + πΏ3ππ’(π 0) =[πΊππ
π (π , π 0). ππ₯(π ), ππ§(π ),ππ¦(π ),βπ’(π ),βπ€(π ),βπ½(π )]π =π
π =π(3.22)
,where πΏππ -Kroneckerβs delta and . - dot product of two vectors. Equations Eq.?? calledboundary integrals for ring in-plane vibrations.
3.2.2 Out-plane vibrations
In same way, with three loading cases: load case 1 ππ¦(0) = β12π πππ(π ) (ππ’ = πΏ(π ) ), load case
2 ππ§(0) = β12π πππ(π ) (ππΎ = πΏ(π )) and load case 3 ππ₯(0) = β1
2π πππ(π ) (ππΌ = πΏ(π )).
Two groups of functions:
π£(π ), πΎ(π ),ππ₯(π )πΌ(π ), ππ§(π ), ππ¦(π )
(3.23)
And bi-orthogonality condition:
π(π)ππ§ π
(π)πΎ + π
(π)ππ¦ π
(π)π£ βπ(π)
πΌ π(π)ππ₯ = 0 (π = π) (3.24)
One can obtain Greenβs matrix coefficients in form:
28
Ξ(ππ’π‘)(1)π = β1
2π
(π)π£
π(π)ππ§ π
(π)πΎ +π
(π)ππ¦ π
(π)π£ βπ
(π)πΌ π
(π)ππ₯
Ξ(ππ’π‘)(2)π = β1
2
π(π)πΎ
π(π)ππ§ π
(π)πΎ +π
(π)ππ¦ π
(π)π£ βπ
(π)πΌ π
(π)ππ₯
Ξ(ππ’π‘)(3)π = 1
2π
(π)πΌ
π(π)ππ§ π
(π)πΎ +π
(π)ππ¦ π
(π)π£ βπ
(π)πΌ π
(π)ππ₯
(3.25)
Greenβs matrix in form:
πΊππ’π‘1 (π , π 0) = π£(π)(π ), πΌ(π)(π ), πΎ(π)(π ), π
(π)π¦ (π ),π
(π)π§ (π ), π
(π)π§ (π ) =
3βπ=1
π(π)π£ , π πππ(π )π
(π)πΌ ,ππΎπ, π πππ(π )π
(π)ππ¦,π
(π)ππ₯, π πππ(π )π
(π)ππ§Ξ(ππ’π‘)
(1)π exp(πππ’π‘
π πππ (π β π 0))
(3.26)And boundary equations in form:
πΏ1ππ£(π 0) + πΏ2ππΎ(π 0) + πΏ3ππΌ(π 0) =[πΊππ’π‘
π (π , π 0). ππ¦(π ),ππ₯(π ), π π§(π ),βπ’(π ),βπΌ(π ),βπΎ(π )]π =π
π =π(3.27)
3.3 The benchmark periodic structure
In order to validate the theory and the Wolfram Mathematica codes, an auxiliary problem hasbeen considered first.
3.3.1 Infinite periodic structure
With respect to natural coordinate π ring can be considered as periodic structure shown onthe Fig.1.2. Similar problem was considered in [3] for a helical spring and flat ring, but it wasconsidered within Timoshenko beam theory. In this work, as stated above Bernoulli-Euler theoryconsidered.
Let us consider a auxiliary problem with parts of periodicity structure connected such that planeof in-plane vibrations of both parts are the same. Finally model can be represented as:
Figure 3.10: Auxiliary problem scheme
Each curved beam segment has the same material and same wire diameter, but differentcurvature radius. Additional dimensionless parameters introduced as:
πΌ =πΈ1
πΈ2
; πΎ =π2π1
; π =π2π1
;π =π1π
; Ξ©1 = Ξ©; Ξ©2 =Ξ©
π; π1 =
π
π 1
; π2 =π
π 2
(3.28)
Hereafter following dimensionless parameters set is considered unless it is stated otherwise:
29
πΌ = 1; πΎ = 0.5; π = 1;π = 5; π1 = 0.2; π2 = 0.02 (3.29)
As in the case of axial and flexural beam vibrations Floquet theory can be used. Followingnotation used:
πππ ππ(π ) = π€π(π ), π½π(π ), π’π(π ), π£π(π ), πΎπ(π ), πΌπ(π )ππππππ(π ) = 1
πΈππΌ(π)π₯
π(π)π₯ (π ), π
(π)π§ (π ),π
(π)π¦ (π ), π
(π)π¦ (π ),π
(π)π₯ (π ), π
(π)π§ (π ) (3.30)
Let us consider three subsequent parts of ring. For each part of ring we define two set ofthree boundary integrals in form Eq.3.29 and Eq.3.35 (totally, 6*6=36 equations) and additionallyinterfacial conditions (4*6=24 equations):
π€1(1) = π€2(1) ; π½1(1) = π½2(1) ;π’1(1) = π’2(1)πΎ1(1) = πΎ2(1) ; π£1(1) = π£2(1) ;πΌ1(1) = πΌ2(1)
π(1)π§ (1) = πΌπππ π
(2)π§ (1) ;π
(1)π¦ (1) = πΌπππ π
(2)π¦ (1) ;π
(1)π₯ (1) = πΌπππ π
(2)π₯ (1)
π(1)π§ (1) = πΌπππ π
(2)π§ (1) ;π
(1)π¦ (1) = πΌπππ π
(2)π¦ (1) ;π
(1)π₯ (1) = πΌπππ π
(2)π₯ (1)
π€2(1 + πΎπππ) = π€3(1 + πΎπππ) ; π½2(1 + πΎπππ) = π½3(1 + πΎπππ) ;π’2(1 + πΎπππ) = π’3(1 + πΎπππ)πΎ2(1 + πΎπππ) = πΎ3(1 + πΎπππ) ; π£2(1 + πΎπππ) = π£3(1 + πΎπππ) ;πΌ2(1 + πΎπππ) = πΌ3(1 + πΎπππ)
π(2)π§ (1 + πΎπππ) = πΌπππ π
(3)π§ (1 + πΎπππ) ;π
(2)π¦ (1 + πΎπππ) = πΌπππ π
(3)π¦ (1 + πΎπππ)
π(2)π₯ (1 + πΎπππ) = πΌπππ π
(3)π₯ (1 + πΎπππ)
π(2)π§ (1 + πΎπππ) = πΌπππ π
(3)π§ (1 + πΎπππ) ;π
(2)π¦ (1 + πΎπππ) = πΌπππ π
(3)π¦ (1 + πΎπππ)
π(2)π₯ (1 + πΎπππ) = πΌπππ π
(3)π₯ (1 + πΎπππ)
(3.31a)Or in vector form:
πππ π1(1) = πππ π2(1)πππππ1(1) = πΌ πππππ2(1)πππ π2(1 + πΎ) = πππ π3(1 + πΎ)πΌπππππ2(1 + πΎ) = πππππ3(1 + πΎ)
(3.31b)
And Floquet periodicity conditions (2*6=12 equations):
π’1(0) = Ξπ’3(1 + πΎ); π£1(0) = Ξπ’3(1 + πΎ); π€1(0) = Ξπ’3(1 + πΎ)πΌ1(0) = ΞπΌ3(1 + πΎ); π½1(0) = Ξ π½3(1 + πΎ); πΎ1(0) = Ξ πΎ3(1 + πΎ)
π(1)π₯ (0) = Ξπ
(3)π₯ (1 + πΎ); π
(1)π¦ (0) = Ξπ
(3)π¦ (1 + πΎ); π
(1)π§ (0) = Ξπ
(3)π§ (1 + πΎ)
π(1)π₯ (0) = Ξπ
(3)π₯ (1 + πΎ); π
(1)π¦ (0) = Ξπ
(3)π¦ (1 + πΎ); π
(1)π§ (0) = Ξπ
(3)π§ (1 + πΎ)
(3.32a)
Or in vector form:
πππ π1(0) = Ξ πππ π3(1 + πΎ)πππππ1(0) = Ξ πππππ3(1 + πΎ) (3.32b)
Boundary integrals with conditions Eq.3.31a-Eq.3.32b defines homogenous system ofalgebraical equations with respect to unknown displacements and forces on borders of the ringparts. Let π·(Ξ,Ξ©) be the determinant of this system. π·(Ξ,Ξ©) is the twelfth order polynomialin Ξ , which defines stop- and pass-bands. It should be empathized, that determinant π·(Ξ,Ξ©)
30
factorizes in two sixth order polynomials in Ξ : in- and out- plane vibrations part, i.e. π·(Ξ,Ξ©) =π·ππ(Ξ,Ξ©) *π·ππ’π‘(Ξ,Ξ©). Therefore these two parts can be considered independently.
It should be noted that both of determinants π·ππ(Ξ,Ξ©) and π·ππ’π‘(Ξ,Ξ©) can be written in form:
π·*(Ξ,Ξ©) = Ξ6 + π5Ξ5 + π4Ξ
4 + π3Ξ3 + π2Ξ
2 + π1Ξ + 1 (3.33)
It has two properties: 1.By Vieta theorem, since free term of polynomial π·*(Ξ,Ξ©) is 1, if Ξ isa root of polynomial Eq.3.33, then Ξβ1 is a root of polynomial too. 2. It has only even powers ofΞ© as coefficients ππ because no damping is considered.
For further consideration it is convenient to write π·ππ as π·π€,π’ with meaning of flexural-axialpart of a Floquet determinant. Also π·ππ’π‘ written as π·πΎ,π£ with meaning of flexural-torsional part.
One can plot dependency Ξ of Ξ©, for example from condition π·π€,π’(Ξ,Ξ©) = 0 . Unlike thecases considered earlier in [8] there exists three pairs (with property Ξ1 * Ξ2 = 1) of branchesof solutions Ξπ(Ξ©): one pair of exponentially increasing and decreasing branches (this couple isnot shown below, because this couple has significantly higher (and lower) magnitude than othertwo branches) and two pair of branches that defines stop- and pass-bands, it preserves for bothdeterminants π·π€,π’ and π·πΎ,π£. Below dependence π·π€,π’(Ξ,Ξ©) = 0 is plotted :
Figure 3.11: Floquet zones for π·π€,π’(Ξ,Ξ©) and different kinds of zones marked with numbers
One can distinct βrealβ stop band (marked as 1), where wave propagation fully blocked,itslocation is defined by intersection of stop-bands of two branches (overlapping of an orange stopband and a red stop band on the Fig.3.11).In zone 1 only exponential increasing-decreasing standingwaves are presented. "Partial"gap band (marked as 2) its location defined by overlapping pass bandand stop band of different branches (overlapping of a red stop band and an orange pass band andvice versa). In zone 2 appears one propagating wave. And pass band (marked as 3) its locationdefined as overlapping of two pass bands of different branches (an orange pass band and a red passband) in zone 3 two propagating waves are presented. All three zones contains one pair of standingexponential increasing-decreasing standing waves, which is described above and not shown on thepicture.
Dependence π·ππ’π‘(Ξ,Ξ©) = 0 also have same form:
31
Figure 3.12: Floquet zones for π·πΎ,π£(Ξ,Ξ©)
Since model considered as whole, both pictures for in-plane and out-plane vibrations should beconsidered simultaneously, but for visual reason they are separated. For comparison, both figuresFig.3.11 and Fig.3.12 shown in one plot:
Figure 3.13: Floquet zones for π·(Ξ,Ξ©)
As seen, there are zones, where all 6 branches have abs(Ξ) = 1 and therefore full wavepropagation is fully blocked. Presence of this kind of stop bands is proved theoretically and shownexperimentally in [3] for flat ring and helical spring in frame of Timoshenko theory and both models,shown here and in article [3] are in agreement with each other.
32
3.3.2 Finite periodic structure
As in previous cases one can introduce concept of a symmetrical periodicity cell shown on theFig.1.3
One can find eigenfrequencies of this structure. With properly selected symmetrical boundaryconditions one can obtain boundary conditions on the gap band borders. In case of the ring in-planevibrations it is two groups of functions Eq. 3.23 discussed earlier. A-type boundary conditions forone periodicity cell defined as [14]:
π€(12), π½(1
2), ππ₯(1
2), π£(1
2), πΎ(1
2),ππ₯(1
2) = 0
π€(32
+ πΎπππ), π½(32
+ πΎπππ), ππ₯(32
+ πΎπππ), π£(32
+ πΎπππ), πΎ(32
+ πΎπππ),ππ₯(32
+ πΎπππ) = 0(3.34)
And B-type boundary conditions as:
π’(12), ππ§(
12),ππ¦(
12), πΌ(1
2), ππ¦(
12), ππ¦(
12) = 0
π’(32
+ πΎπππ), ππ§(32
+ πΎπππ),ππ¦(32
+ πΎπππ), πΌ(32
+ πΎπππ), ππ¦(32
+ πΎπππ), ππ§(32
+ πΎπππ) = 0(3.35)
Boundary integrals with interfacial conditions Eq.3.31a and boundary conditions Eq.3.34 orEq.3.35 defines also homogenous system of linear algebraical equations with respect to unknowndisplacements and forces/moments at the ring parts borders. And by equaling determinant of thissystem to zero we can find eigenfrequencies. Eigenfrequencies of system Eq.3.34 and Eq.3.35 fullycovers stop-band borders as in other cases considered in [8].
Determinant of the system Eq.3.31a, Eq.3.34 or Eq.3.31a, Eq.3.35 also can be factorizedinto two parts: in-plane and out-plane. And also both parts will be shown separately. In planeeigenfrequencies can be plotted as:
Figure 3.14: Eigenfrequencies of the single in-plane periodical cell: boundary conditions A (red)and boundary conditions B (green)
As seen property of eigenfrequencies of a single periodicity cell preserved. Eigenfrequency ofa single periodicity cell appears only in gap bands borders and covers all borders of gap bands. Ifone considers out-plane periodicity cells, same property can be seen:
33
Figure 3.15: Eigenfrequency of single periodicity cell (out-plane part)
One can consider system with more periodicity cells and we see that new eigenfrequenciesappears only in βpureβ pass-bands:
Figure 3.16: Eigenfrequencies of three periodicity cell border conditions A (red) and borderconditions B (green), common EF - black
Out-plane vibrations shows the same picture and therefore, for brevity it is not shown. As seen,properties of eigenfrequencies, considered in [7], [8] and Ch.2 are preserves for Bernoulli-Euler ringcase.
34
3.3.3 Eigenmodes analysis
Eingenmode analysis gives clear understanding of wave propagation picture in infinitewaveguide. Form of waves and energy transmission mode (standing or propagating wave) canbe easily shown when eigenmodes analysis is performed. Also it is first step to strain state analysis,which is often of interest in engineering practice.
First same procedure used and determinant π·(Ξ,Ξ©) obtained from system Eq.3.31a-Eq.3.32b.As shown above in case of a flat ring second branch of Floquet zones appears. And differencebetween full and partial gap-band can be considered (see Fig.3.12).
For each root of polynomial π·ππ(Ξ,Ξ©) one can find eigenmodes of in-plane vibrations. SystemEq.3.31a-Eq.3.32b with parameter Ξ substituted has zero determinant. Therefore, one constantassumed as constant, for example π’1(0) = 1 and one arbitrary of equations of the system excludedfrom consideration. For certainty system Eq.3.31a-Eq.3.32b with first equation changed to π’1(0) = 1called eigenmode of infinite waveguide. In the pure gap-band all waves are standing:
Figure 3.17: Standing waves in pure gap-band Ξ© = 0.037
In all cases all waved coupled as increasing-evanescent or traveling from left to right and right-left in three groups such that Ξ1 * Ξ2 = 1. In partial gap-band appears one pair of propagatingtravelling waves, whereas other two pairs are standing and coupled as increasing-evanescent:
Figure 3.18: One pair of travelling waves in partial gap-band Ξ© = 0.06
And in pure pass-band appears second couple of travelling propagating waves and one couplestanding increasing-evanescent. Waves for pass-band will not be represented here for brevity.
35
3.4 The spatial periodic structure
Parts of model, shown on a Fig.3.10 are connected such that plane of in-plane vibrations offirst part is perpendicular to a plane of in-plane vibrations of the second part. Whereas boundaryintegrals are not affected by such kind of rotation of coordinate system, interfacial conditions arechanging due to local coordinate system in different parts:
Figure 3.19: Local coordinate systems
Coordinate change is rotation on the angle π/2 with respect to z axis. Therefore, interfacialconditions changing as:
π’ = π£, π£ = β, π€ = πΌ = βπ½, π½ = β, πΎ = πΎ
(3.36)
With this coordinate changing new displacement and force vector definitions are required
πππ ππ1(π ) = π€π1(π ), π½π1(π ), π’π1(π ), π£π1(π ), πΎπ1(π ), πΌπ1(π )ππππππ1(π ) = 1
πΈππΌπ1π₯ππ1
π₯ (π ), π π1π§ (π ),π π1
π¦ (π ), ππ1π¦ (π ),π π1
π₯ (π ), π π1π§ (π )
πππ ππ2(π ) = π€π2(π ),βπΌπ2(π ), π£π2(π ),βπ’π2(π ), πΎπ2(π ),βπ½π2(π )ππππππ2(π ) = 1
πΈππΌπ2π₯ππ2
π¦ (π ), π π2π§ (π ),βπ π2
π₯ (π ),βππ2π₯ (π ),βπ π2
π¦ (π ), π π2π§ (π )
(3.37)
,where π1 = 1, 3, 5, ... are parts with odd numbers and π2 = 2, 4, 6, ... are parts with evennumbers. Parts are coupled with respect to geometry.
With this new definition Floquet problem can be defined as above:
π€1(1) = π€2(1) ; π½1(1) = βπΌ2(1) ;π’1(1) = π£2(1)πΎ1(1) = πΎ2(1) ; π£1(1) = βπ’2(1) ;πΌ1(1) = βπ½2(1)
π(1)π§ (1) = πΌπππ π
(2)π§ (1) ;π
(1)π¦ (1) = βπΌπππ π
(2)π₯ (1) ;π
(1)π₯ (1) = πΌπππ π
(2)π¦ (1)
π(1)π§ (1) = πΌπππ π
(2)π§ (1) ;π
(1)π¦ (1) = βπΌπππ π
(2)π₯ (1) ;π
(1)π₯ (1) = βπΌπππ π
(2)π¦ (1)
π€2(1 + πΎπππ) = π€3(1 + πΎπππ) ; π½2(1 + πΎπππ) = βπΌ3(1 + πΎπππ) ;π’2(1 + πΎπππ) = π£3(1 + πΎπππ)πΎ2(1 + πΎπππ) = πΎ3(1 + πΎπππ) ; π£2(1 + πΎπππ) = βπ’3(1 + πΎπππ) ;πΌ2(1 + πΎπππ) = βπ½3(1 + πΎπππ)
π(2)π§ (1 + πΎπππ) = πΌπππ π
(3)π§ (1 + πΎπππ) ;π
(2)π¦ (1 + πΎπππ) = βπΌπππ π
(3)π₯ (1 + πΎπππ)
π(2)π₯ (1 + πΎπππ) = πΌπππ π
(3)π¦ (1 + πΎπππ)
π(2)π§ (1 + πΎπππ) = πΌπππ π
(3)π§ (1 + πΎπππ) ;π
(2)π¦ (1 + πΎπππ) = βπΌπππ π
(3)π₯ (1 + πΎπππ)
π(2)π₯ (1 + πΎπππ) = βπΌπππ π
(3)π¦ (1 + πΎπππ)
(3.38a)
36
Or in vector form:
πππ π1(1) = πππ π2(1)πππππ1(1) = πΌ πππππ2(1)πππ π2(1 + πΎ) = πππ π3(1 + πΎ)πΌπππππ2(1 + πΎ) = πππππ3(1 + πΎ)πππ π1(0) = Ξ πππ π3(1 + πΎ)πππππ1(0) = Ξ πππππ3(1 + πΎ)
(3.38b)
System of equations Eq.3.38b with definitions Eq.3.37 is define system of linear algebraicequations. Determinant of this system (Ξ,Ξ©) is the twelfth order polynomial, which also factorizesinto two six-order parts (Ξ,Ξ©) = π€,π£(Ξ,Ξ©) * πΎ,π’(Ξ,Ξ©). But, since we consider essentiallyspatial structure, one can not divide vibrations into in-plane and out-plane mode because vibrationsplane canβt be defined for spatial structures. Dependence Ξ of Ξ© also can be plotted as:
Figure 3.20: Floquet zones of π·π€,π’(Ξ,Ξ©)(blue) and π€,π£(Ξ,Ξ©)(green)
As seen, both pictures are matching. This is explained by natural symmetry of problemconsidered, both in-plane and out-plane displacement have same stiffness and therefore theirinterchange does not affect end result. Parts π·πΎ,π£(Ξ,Ξ©) and πΎ,π’(Ξ,Ξ©) also matching and thereforenot represented here.
Thus, auxiliary problem are exact correspond to model, shown on a Fig.3.2. And all results,shown above are valid for a first model.
3.5 Parametric study
In engineering application often of interest to know, how a parameterβs change affects onproperties of system. It is used in optimization process in order to choose proper methods foroptimization and reach best possible optimal properties.
Recalling initial parameters:
37
πΌ = 1; πΎ1 = 0.5; π = 1;π = 5; π1 = 0.2; π2 = 0.02 (3.39)
First, let us consider part curvature ratio π = π1π2
change effect. For parameters considered aboveit has value of π1 = 0.2
0.02= 10. For determinacy parameter π2 is fixated and influence of changing π1
on Floquet zones picture is considered.On the plot below with no color shown full gap bands for a given parameter π and given
frequency Ξ©, whereas color represents partial or full pass band:
Figure 3.21: Floquet zones for different π
As seen, with increasing ratio π pure gap bands tends to move to higher frequency.Second parameter considered is πΎ, following plot also represents full gap bands and partial or
full pass bands as above:
Figure 3.22: Floquet zones for different πΎ
38
For πΎ we have reciprocal tendency, with increasing πΎ gap bands tends to move to lowerfrequency.
This parameter study in showing that model considered can be used for engineering applicationsfor optimization processes and obtaining properties, demanded by the industry and considering basicideas of influence of parameters on pure gap band positions.
In this chapter first model of a torque vibration isolator was considered. It has been modelledas a system of differential equations for an infinite periodic structure. Therefore, methods, used inCh.2, were expanded to a system of a differential equation. Greenβs matrix coefficient with usingbi-orthogonality condition were obtained in analytical form. Difference between Floquet zones ofa single differential equation and system of differential equation was shown. Also, was shownindependence with respect to symmetry transformation of Floquet zones for a structure with naturalsymmetry. Also parameters study was conducted in order to show one of industrial applications ofgiven model.
39
Conclusion and future work
First part of this master project has two main goals: theoretical and practical. In practical point ofview torque simple vibration isolator model was considered and vibration isolation properties of thisstructure were considered. Model shows that full waves propagation block can be reached in lowfrequency region, that of interest in industrial applications. Also, possibility of model optimizationwas shown, but deep investigation of optimization of properties are out of scope of this project.
From theoretical point of view, all main principles of Floquet theory have been developed andshown on examples. Possibility of using of polar coordinates were considered. Also first step inhierarchy of torque vibration isolator models were considered. Model of torque vibration isolatorhas been developed and considered in frame of Floquet zones theory. Tools that developed in thispart of masterβs project have broad range of applicability and they will be used in second part ofwork. Results obtained in this part will be used for comparison with second model, which will beconsidered in second part of masterβs project.
One of supplementary goals was to develop and validate the program that can be used formodelling of an isolator. Wolfram Mathematica software was used for programming, because it fitsbest for analytical modelling.
Also, first parts gives all necessary theory, shown in examples and with all necessary references,that allows one to use differential equations in modelling of vibrations of physical structures.
Goal reached in this part, gives following propositions for goals of second part:(I) Consider second model circular plate-cylindrical shell, using developed for polar coordinate
system Floquet theory from theoretical and practical point of view(II) Compare first and second model in order to consider advantages and disadvantages of choice
of more complicated model. Also it is of interest in theoretical point of view to compare Floquettheory for different coordinate system.
(III)Conduct experiments in order to validate models and developed theory and define practicalapplicability of considered in both parts of mastersβs project models
Both parts allows one to build hierarchy of two models of torque vibration isolator, that useswide range of methods with wide range of applicability.
40
Literature
1. L.Brillouin. Wave Propagation in Periodic Structures second ed. Dover Publications New York,1953.
2. D.J.Mead. Wave propagation in continuous periodic structures. Research contributions fromSouthampton 1964-1995 // Journal of Sound and Vibration. 1996. Vol. 190. P. 495β524.
3. A.SΓΈe-Knudsen, R.Darula, Sorokin S. Theoretical and experimental analysis of the stop-bandbehavior of elastic springs with periodically discontinuous of curvature // Journal of the Acous-tical Society of America. 2012. Vol. 132. P. 1378β1383.
4. A.SΓΈe-Knudsen, Sorokin S. Modelling of linear wave propagation in spatial fluid filled pipesystems consisting of elastic curved and straight elements // Journal of Sound and Vibration.2010. Vol. 329. P. 5116β5146.
5. A.SΓΈe-Knudsen. Design of stop-band filter by use of curved pipe segments and shape optimiza-tion // Structural and Multidisciplinary Optimization. 2011. Vol. 44. P. 863β874.
6. A.SΓΈe-Knudsen. Design of stop-band filters by use of compound curved pipe segmentsand shape optimization // 10th International Conference on Recent Advances in StructuralDynamics. Institute of Sound and Vibration Research, 2010.
7. A.Hvatov, S.Sorokin. Analysis of eigenfrequencies of finite periodic structures in view of loca-tion of frequency pas- and stop-bands // Proceedings of 20th International Congress on Soundand Vibration. 2013.
8. A.Hvatov, S.Sorokin. Free vibrations of finite periodic structures in pass- and stop-bands of thecounterpart infinite waveguides // Journal of Sound and Vibration. 2015. Vol. 347. P. 200β217.
9. S.Sorokin. On the bi-orthogonality conditions for multi-modal elastic waveguides. // Journal ofSound and Vibration. 2013. Vol. 332. P. 5606β5617.
10. Dym C. L., I.H.Shames. Solid Mechanics. A Variational Approach, Augmented Edition.Springer, 2013.
11. Duffy D. G. Greenβs Functions with Applications. Chapman and Hall, 2001.
12. Skudrzyk E. The Foundations of Acoustics. Basic mathematics and basic acoustics. Springer-Verlag, 1971.
13. Achenbach J. D. Reciprocity in Elastodynamics. Cambridge University Press, 2003.
14. S.Sorokin. The Greenβs matrix and the boundary integral equations for analysis of time-harmonic dynamics of elastic helical springs. // Journal of the Acoustical Society of America.2011. Vol. 129. P. 1315β1323.
15. R.Nielsen, S.Sorokin. Periodicity effects of axial waves in elastic compound rods // Journal ofSound and Vibration. 2015. Vol. 353. P. 135β149.
41
Appendix A
Derivation of equation of motion for acircular membrane
Kinetic energy is written in form
π =1
2
β«β«π
ππ’2π‘ππ (A.1)
Potential energy is written in form
π =1
2
β«β«π
π0(π’π₯ + π’π¦)ππ +1
2
β«πΏ
π(πΏ)π’2ππΏ (A.2)
where π0 = ππππ π‘ membrane surface tension,π(πΏ) membrane elasticity module.
Action integral have form:
π» =1
2
β« π‘1
π‘0
β«β«π
(π β π )ππππ‘ =1
2
β« π‘1
π‘0
(
β«β«π
(ππ’2π‘ β π0(π’π₯ + π’π¦))ππ β 1
2
β«πΏ
π(πΏ)π’2ππΏ)ππ‘ (A.3)
Variation of the first summand:
β« π‘1
π‘0
β«β«π
ππ’π‘πΏπ’π‘ππππ‘ =
β«β«π
ππ’π‘πΏπ’π‘1π‘0ππ β
β« π‘1
π‘0
β«β«π
ππ’π‘πΏπ’π‘π‘πΏπ’ππππ‘ = ββ« π‘1
π‘0
β«β«π
ππ’π‘π‘πΏπ’ππππ‘
(A.4)Second summand:
ββ« π‘1
π‘0
β«β«π
π0π’π₯πΏπ’π₯ππππ‘ = ββ« π‘1
π‘0
β«πΏ
π0π’π₯πΏπ’ππ¦ππ‘ +
β« π‘1
π‘0
β«β«π
π0π’π₯π₯πΏπ’ππππ‘ (A.5)
Third summand:
ββ« π‘1
π‘0
β«β«π
π0π’π¦πΏπ’π¦ππππ‘ = ββ« π‘1
π‘0
β«πΏ
π0π’π¦πΏπ’ππ₯ππ‘ +
β« π‘1
π‘0
β«β«π
π0π’π¦π¦πΏπ’ππππ‘ (A.6)
With π’π₯ππ¦ + π’π¦ππ₯ = ππ’ππ
first parts of A.5 and A.6 turns to:
ββ« π‘1
π‘0
β«πΏ
π0ππ’
πππΏπ’ππΏππ‘ (A.7)
Total variation πΏπ» from A.4-A.7
43
πΏπ» =
β« π‘1
π‘0
β«πΏ
(π(πΏ)π’β π0ππ’
ππ)πΏπ’ππΏππ‘ +
β« π‘1
π‘0
β«β«π
(π0(π’π₯π₯ + π’π¦π¦) β ππ’π‘π‘)πΏπ’ππππ‘ (A.8)
From which we obtain membrane equation of motion:
π0βπ’β ππ’π‘π‘ = 0 (A.9)
And two possibly border conditions
(π(πΏ)π’β π0ππ’
ππ)
πΏ
= 0 (A.10)
or
π’
πΏ
= 0 (A.11)
With time dependence exp(βπππ‘), i.e. π’(x, π‘) = π(x) exp(βπππ‘) equation A.9 can be rewrittenas:
βπ +π2π
π0
π = 0 (A.12)
And with designation π2 = π2ππ0
it has final form (uppercase omitted):
(β + π2)π’ = 0 (A.13)
It should be empathized, that potential energy has the form:
π =1
2
β«β«π
(π0 + πΈπ΄)(π’π₯ + π’π¦)ππ +1
2
β«πΏ
π(πΏ)π’2ππΏ (A.14)
And when limit πΈπ΄ β 0 considered one has pure membrane vibrations case. Otherwise, πΈπ΄ >>π0 represents plate in-plane vibration case. Here limit πΈπ΄ β 0 considered from beginning.
44
Appendix B
Derivation of the boundary integral equationfor a circular membrane
Starting with following equation
π’β²β²(π) +1
ππ’(π) +
(π2 β π2
π2
)π’(π) = βπ(π) (B.1)
Multiplying by πΊ(π, π0) and integrating over all membrane area gives:
πΌ =
2πβ«0
πβ«π
(π’β²β²(π) +1
ππ’(π) +
(π2 β π2
π2
)π’(π))πΊ(π, π0)πππππ = β
2πβ«0
πβ«π
π(π)πΊ(π, π0)πππππ (B.2)
Since axi-symmetric case considered:
πΌ = 2π
πβ«π
(π’β²β²(π) +1
ππ’(π) +
(π2 β π2
π2
)π’(π))πΊ(π, π0)πππ = β2π
πβ«π
π(π)πΊ(π, π0)πππ (B.3)
And after cancellation:
πΌ =
πβ«π
(π’β²β²(π) +1
ππ’(π) +
(π2 β π2
π2
)π’(π))πΊ(π, π0)πππ = β
πβ«π
π(π)πΊ(π, π0)πππ (B.4)
Considering following integrals and using by-part integration (arguments omitted for brevity andclarity):
πΌ1 =
πβ«π
π’β²β² πΊπππ = π’β² πΊπ
ππ
βπβ«
π
π’β²(πΊβ² π+πΊ)ππ = [π’β² πΊπβπ’(πΊβ² π+πΊ)]
ππ
+
πβ«π
π’(πΊβ²+πΊβ²+πΊβ²β² π)ππ
(B.5)
πΌ2 =
πβ«π
π’β² πΊππ = π’πΊ
ππ
βπβ«
π
π’πΊβ² ππ (B.6)
45
πΌ3 =
πβ«π
(π2 β π2
π2
)π’πΊ πππ (B.7)
Substituting B.5-B.7 into B.4 gives:
πΌ = πΌ1 + πΌ2 + πΌ3 = [π’β² πΊπ β π’πΊβ² π]
ππ
+
πβ«π
(πΊβ²β² +1
ππΊβ² +
(π2 β π2
π2
)πΊ)π’ πππ (B.8)
Recalling modified Greenβs function definition:
πΊβ²β²(π, π0) +1
ππΊβ²(π, π0) +
(π2 β π2
π2
)πΊ(π, π0) = βπΏ(π β π0)
π0(B.9)
With definition B.9 equation B.8 has the form:
πΌ = [π’β²(π)πΊ(π, π0) β π’(π)πΊβ²(π, π0)]π
π=π
π=π
+
πβ«π
βπΏ(π β π0)
π0π’(π) πππ (B.10)
And with property of delta function it can be rewritten as:
πΌ = [π’β²(π)πΊ(π, π0) β π’(π)πΊβ²(π, π0)]π
π=π
π=π
β π’(π0) π0π0
(B.11)
Here modified Greenβs function used, in term π’(π0) π0π0
cancelled π0 and boundary equation hasmore simple form.Also, it can be rewritten with recalled right hand side of B.4 in form:
π’(π0) = [π’β²(π)πΊ(π, π0) β π’(π)πΊβ²(π, π0)]π
π=π
π=π
+
πβ«π
π(π)πΊ(π, π0)ππ (B.12)
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Appendix C
Axial rod vibrations in boundary equationsmethod point of view
C.1 Greenβs function definition and derivation
Axial beam vibration can be described with following equation:
π’π₯π₯ β1
π2π’π‘π‘ = βπ(π₯, π‘) (C.1)
,where π has meaning of sound of speed (speed of elastic wave propagation) and q(x,t) is theforce density.
With harmonic vibrations state assumed π’(π₯, π‘) = exp[(βπππ‘)π(π₯) Eq.C.1 can be rewritten as:
π’β²β²(π₯) + π2π’(π₯) = βπ(π₯) (C.2)
,where π = ππ
called wave number. In what follows harmonic vibrations state assumed anduppercase of letters is omited.
Greenβs function by definition is a solution of the equation with force density π(π₯) = πΏ(π₯β π₯0),where πΏ(π₯) is Diracβs delta function. In the rod axial vibrations case it has meaning of a point forceacting at point π₯ with observation point π₯0 and Greenβs function equation can be written as:
π2
ππ₯2πΊ(π₯, π₯0) + π2πΊ(π₯, π₯0) = βπΏ(π₯β π₯0) (C.3)
It can be proven that Greenβs function should satisfy following properties:
πΊ(π₯, π₯0) = πΊ(π₯0, π₯)πππ₯πΊ(π₯0, π₯0 + π) β π
ππ₯πΊ(π₯0, π₯0 β π) = β1, π β 0
πΊ(π₯0, π₯0 + π) = πΊ(π₯0, π₯0 β π), π β 0(C.4)
Since common solution of homogenous equation C.2 has the form:
π’(π₯) = πΆ1 exp(πππ₯) + πΆ2 exp(βπππ₯) (C.5)
Solution of equation can be written in form (if one should consider infinite beam it is veryimportant that function πΊ(π₯, π₯0) should satisfy radiation conditions at infinity [12]):
πΊ+(π₯, π₯0) = π΄ exp(πππ₯) , π₯ > π₯0
πΊβ(π₯, π₯0) = π΅ exp(βπππ₯) , π₯ β€ π₯0(C.6)
Greenβs function in form Eq.C.6 substituted into two last properties in C.4 gives:
47
πΊ+(π₯0, π₯0) = πΊβ(π₯0, π₯0)
πππ₯
(πΊ+(π₯, π₯0) βπΊβ(π₯, π₯0))
π₯=π₯0
= β1(C.7)
or in explicit form:
π΄ exp(πππ₯0) = π΅ exp(βπππ₯0)πππ΄ exp(πππ₯0) + πππ΅ exp(βπππ₯0) = β1
(C7β)
System Eq.C7β has the unique solution for unknown coefficients π΄ and π΅:
π΄,π΅ = π
2πexp(βπππ₯0),
π
2πexp(πππ₯0) (C.8)
And finally Greenβs function can be written as:
πΊ(π₯, π₯0) =
π2π
exp(βππ(π₯β π₯0)) π₯ β€ π₯0
π2π
exp(ππ(π₯β π₯0)) π₯ > π₯0
(C.9)
Or in short form:
πΊ(π₯, π₯0) =π
2πexp(ππ abs(π₯β π₯0)) (C.10)
If this function plotted it is seen that all properties Eq.C.4 are fulfilled:
Figure C.1: Greenβs function of axial rod vibration equation(blue) and its derivative(orange) andimaginary parts (dashed)
C.2 Equation of axial displacement of a rod. Boundary integralsmethod
Greenβs function can be used in order to obtain solution of a differential equation with arbitraryload. Recalling axial rod vibrations equation:
π’β²β²(π₯) + π2π’(π₯) = βπ(π₯) (C.11)
48
And Greenβs function definition:
π2
ππ₯2πΊ(π₯, π₯0) + π2πΊ(π₯, π₯0) = βπΏ(π₯β π₯0) (C.12)
In what follows πππ₯πΊ(π₯, π₯0) = πΊβ²(π₯, π₯0)
Multiplying equation C.11 on πΊ(π₯, π₯0) and integrating over beam length
πβ«π
(π’β²β²(π₯) + π2π’(π₯))πΊ(π₯, π₯0)ππ₯ = βπβ«
π
π(π₯)πΊ(π₯, π₯0)ππ₯ (C.13)
First left hand side of an equation C.13 considered:
πΌ =
πβ«π
(π’β²β²(π₯) + π2π’(π₯))πΊ(π₯, π₯0)ππ₯ (C.14)
Considering following integral and using by-part integration:
πΌ1 =πβ«π
π’β²β²(π₯)πΊ(π₯, π₯0)ππ₯ = π’β²(π₯)πΊ(π₯, π₯0) βπβ«π
π’β²(π₯)πΊβ²(π₯, π₯0)ππ₯ =
= [π’β²(π₯)πΊ(π₯, π₯0) β π’(π₯)πΊβ²(π₯, π₯0)]
ππ
+πβ«π
π’(π₯)πΊβ²β²(π₯, π₯0)ππ₯
(C.15)
And second part of integral just rewritten as:
πΌ2 =
πβ«π
π2π’(π₯)πΊ(π₯, π₯0)ππ₯ (C.16)
With using equality πΌ = πΌ1 + πΌ2 Eq.C.14 can be rewritten:
πΌ = [π’β²(π₯)πΊ(π₯, π₯0) β π’(π₯)πΊβ²(π₯, π₯0)]
ππ
+
πβ«π
(πΊβ²β²(π₯, π₯0) + π2πΊ(π₯, π₯0))π’(π₯)ππ₯ (C.17)
With Greenβs function definition Eq.C.12 one can rewrite equation C.17 in form:
πΌ = [π’β²(π₯)πΊ(π₯, π₯0) β π’(π₯)πΊβ²(π₯, π₯0)]
ππ
βπβ«
π
πΏ(π₯β π₯0)π’(π₯)ππ₯ (C.18)
Diracβs delta-function has a property:
πβ«π
π(π₯)πΏ(π₯β π₯0)ππ₯ = π(π₯0) (C.19)
,which is valid for any range (π, π) that contains π₯0 including (ββ,+β).With property C.19 and recalling right hand side of C.13 one can rewrite C.18
π’(π₯0) = [π’β²(π₯)πΊ(π₯, π₯0) β π’(π₯)πΊβ²(π₯, π₯0)]
ππ
+
πβ«π
π(π₯)πΊ(π₯, π₯0)ππ₯ (C.20)
Therefore, if values on a beam boundary are known, one can find displacement at any pointwithin the boundary. Eq.C.20 is called boundary integral equation for an axial rod vibrations.
49
C.3 Boundary integrals method direct application
Let us consider unit length beam with forcing problem written as:
π’(0) = 1π’(1) = 0
(C.21)
For brevity no external loading considered, i.e. π(π₯) β‘ 0. For both boundaries boundary integralshould be written as:
π’(0) = (π’β²(1)πΊ(1, 0) β π’(1)πΊβ²(1, 0)) β (π’β²(0)πΊ(0, 0) β π’(0)πΊβ²(0, 0))π’(1) = (π’β²(1)πΊ(1, 1) β π’(1)πΊβ²(1, 1)) β (π’β²(0)πΊ(0, 1) β π’(0)πΊβ²(0, 1))
(C.22)
Or in implicit form :
12π’(0) β 1
2exp(ππ)π’(1) + π
2ππ’1(0) β π exp(ππ)
2ππ’1(1) = 0
β12
exp(ππ)π’(0) + 12π’(1) + π exp(ππ)
2ππ’1(0) β π
2ππ’1(1) = 0
(C.23)
,where π’(0), π’(1), π’β²(0), π’β²(1) are yet unknown. With boundary conditions added followingsystem has the unique solution:
12π’(0) + 1
2exp(βππ)π’(1) + π
2ππ’1(0) β π exp(βππ)
2ππ’1(1) = 0
12
exp(βππ)π’(0) + 12π’(1) + π exp(βππ)
2ππ’1(0) β π
2ππ’1(1) = 0
π’(0) = 1π’(1) = 0
(C.24)
It can be found as:
π’(0), π’(1), π’β²(0), π’β²(1) = 1, 0, (2 β π cot(π))ππ,βπππ
sin(π) (C.25)
After found constants Eq.C.25 are substituted into Eq.C.20 and displacement can be obtainedfor any point π₯0 within the range (0,1) (here π β‘ 1):
Figure C.2: Displacement found with boundary integrals
This solution can be validated trough transfer matrix method. This method has less computationdifficulty, but also less stabile. In axial rod vibrations case it gives significant computationaladvantage, but in other cases stability question should be considered.
50
Recalling axial rod vibration equation (here also π(π₯) β‘ 0):
π’β²β²(π₯) + π2π’(π₯) = 0 (C.26)
It has common solution in form:
π’(π₯) = πΆ1 exp(πππ₯) + πΆ2 exp(βπππ₯) (C.27)
And considering same boundary problem Eq.C.23 one can write following system:
π’(0) = πΆ1 + πΆ2 = 1π’(1) = πΆ1 exp(ππ) + πΆ2 exp(βππ) = 0
(C.28)
System Eq.C.28 can be solved with respect to unknown integration constaints πΆ1, πΆ2 as:
πΆ1, πΆ2 = 1
1 β exp(2π),
1
1 β exp(β2π) (C.29)
When constants Eq.C.29 are substituted into C.27 displacement can be plotted as:
Figure C.3: Displacement found with boundary integrals(blue) and displacement found by transfermatrix method (red)
As seen, both methods are giving the same result. Advantage of the boundary integrals methodare not seen here, but in more complicated cases which are considered in main part instability of atransfer matrix method is significant and it can not be used for obtaining results. Still,it were usedfor checking validity of found Greenβs functions.
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Appendix D
Floquet theorem. Infinite periodic rod.
Floquet theorem is a very powerful tool for finding solution for a differential equations on ainfinite periodic structures. This is very common theorem for a periodic differential operator andlies out of scope of this work. Nevertheless, it will be shown here for a particular case. Here aninfinite periodic rod axial vibrations case will be considered. Results of this chapter is used in Ch.1for comparison
Infinite rod can be represented as:
Figure D.1: Infinite rod
Each part of rod has its own vibration equation written as (harmonic vibrations state assumedand also, for brevity free vibrations state, i.e. π(π₯) β‘ 0 for any part of infinite rod):
π’β²β²1(π₯) + π2
1π’1(π₯) = 0π’β²β²2(π₯) + π2
2π’2(π₯) = 0...π’β²β²π(π₯) + π2
ππ’π(π₯) = 0...
(D.1)
In this appendix following dimensionless parameters are used:
πΌ = πΈ2
πΈ1; π½ = β2
β1; πΎ = π2
π1;π = π2
π1;π = π1
β1; Ξ© = πβ1
π1(D.2)
Since we have only two kind of sections "white"and "black"we are interested only in twoequations (all quantities are already dimensionless):
π’β²β²π€βππ‘π(π₯) + Ξ©2π’π€βππ‘π(π₯) = 0
π’β²β²πππππ(π₯) + (Ξ©π½
π)2π’πππππ(π₯) = 0
(D.3)
,where π€βππ‘π = 1, 3, 5, ... are number of section with first set of parameters and πππππ =2, 4, 6, ... are number of sections with second set of parameters.
For each section one can find Greenβs function in form Eq.C.10:
53
πΊπ€βππ‘π(π₯, π₯0) = π2Ξ©
exp(βπΞ© abs(π₯β π₯0))
πΊπππππ(π₯, π₯0) = ππ2Ξ©π½
exp(βπΞ©π½π
abs(π₯β π₯0))(D.4)
And boundary integrals in form Eq.C.20 respectively:
π’π(π₯0) = [π’β²π(π₯)πΊπ(π₯, π₯0) β π’π(π₯)πΊβ²
π(π₯, π₯0)]
ππππ
(D.5)
Letβs consider three consequent parts of infinite rod (they are marked with numbers on aFig.D.1). For each boundary boundary integrals can be written:
π’1(0) = (π’β²1(π)πΊ1(π, 0) β π’1(π)πΊβ²
1(π, 0)) β (π’β²1(0)πΊ1(0, 0) β π’1(0)πΊβ²
1(0, 0))π’1(π) = (π’β²
1(π)πΊ1(π, π) β π’1(π)πΊβ²1(π, π)) β (π’β²
1(0)πΊ1(0, π) β π’1(0)πΊβ²1(0, π))
π’2(π) = (π’β²2((1 + πΎ)π)πΊ2((1 + πΎ)π, π) β π’2((1 + πΎ)π)πΊβ²
2((1 + πΎ)π, π))ββ(π’β²
2(π)πΊ2(π, π) β π’2(π)πΊβ²2(π, π))
π’2((1 + πΎ)π) = (π’β²2((1 + πΎ)π)πΊ2((1 + πΎ)π, (1 + πΎ)π) β π’2((1 + πΎ)π)πΊβ²
2((1 + πΎ)π, (1 + πΎ)π))β(π’β²
2(π)πΊ2(π, (1 + πΎ)π) β π’2(π)πΊβ²2(π, (1 + πΎ)π))
π’3((1 + πΎ)π) = (π’β²3((2 + πΎ)π)πΊ3((2 + πΎ)π, (1 + πΎ)π) β π’3((2 + πΎ)π)πΊβ²
3((2 + πΎ)π, (1 + πΎ)π))ββ(π’β²
3((1 + πΎ)π)πΊ3((1 + πΎ)π, (1 + πΎ)π) β π’3((1 + πΎ)π)πΊβ²3((1 + πΎ)π, (1 + πΎ)π))
π’3((2 + πΎ)π) = (π’β²3((2 + πΎ)π)πΊ3((2 + πΎ)π, (2 + πΎ)π) β π’3((2 + πΎ)π)πΊβ²
3((2 + πΎ)π, (2 + πΎ)π))ββ(π’β²
3((1 + πΎ)π)πΊ3((1 + πΎ)π, (2 + πΎ)π) β π’3((1 + πΎ)π)πΊβ²3((1 + πΎ)π, (2 + πΎ)π))
(D.6)Boundary equations gives 6 equations with 6 unknown displacements
π’1(0), π’1(π), π’2(π), π’2((1 + πΎ)π), π’3((1 + πΎ)π), π’3((2 + πΎ)π) and 6 unknown slopes. In order tofind solution one need 6 more equations.
In order to consider rod as the system interfacial conditions are written:
π’1(π) = π’2(π)π’β²1(π) = πΌπ’β²
2(π)π’2((1 + πΎ)π) = π’3((1 + πΎ)π)πΌπ’β²
2((1 + πΎ)π) = π’β²3((1 + πΎ)π)
(D.7)
As seen, this adds only four equations. If four, five,etc. consequent cells considered, twoequations still be missed. In order to get closed system for infinite rod Floquet theorem used.Floquet theorem gives possibility to transfer from finite number of cells to infinity. It statement foraxial rod vibrations case can be formulated as: "displacement of two points on a distance of oneperiod differs on a constant complex multiplier and this multiplier is constant for any cross-section".Period of structure shown on a Fig.D.1 is (1 + πΎ)π and therefore Floquet theorem statement can bewritten as:
π’1(0) = Ξπ’3((1 + πΎ)π)π’β²1(0) = Ξπ’β²
3((1 + πΎ)π)(D.8)
System Eq.D.6-Eq.D.8 is homogenous system of 12 equations with 12 unknown displacementsand slopes. Since it homogenous, in order it to have non-trivial solution, its determinant π·(Ξ,Ξ©)should be zero. In this case π·(Ξ,Ξ©) is the second order polynomial in Ξ and it can we writtenimplicitly as (more detailed properties of this equation is considered in [15] ):
Ξ2 +Ξ (πΌ2π½2 + π2) sin(πΞ©) sin
(πΎπΞ©π
)πΌπ½π
β 2Ξ cos(πΞ©) cos
(πΎπΞ©
π
)+ 1 = 0 (D.9)
54
Figure D.2: Zeroes of determinant π·(Ξ,Ξ©) for πΌ = 2.5 , π½ = 1 , πΎ = 2, π = 1 , π = 1.5
Since free term of the Eq.D.9 is 1, by Vietaβs theorem there are two roots Ξπ(Ξ©), π = 1, 2 suchthat Ξ1 *Ξ2 = 1 and if abs(Ξ1) = 1 then abs(Ξ2) = 1. For each Ξ© one can plot two roots of Eq.D.9as:
Fig.D.2 called picture of Floquet zones. It shows two kind of zones: pass band where abs(Ξ) = 1and gap bands abs(Ξ) = 1. In pass band waves can propagate freely because amplitude of wavedoes not change, changes only phase. In gap band appears two standing waves with increasing anddecreasing amplitude and therefore wave propagation is prohibited. Bloch proposed following formΞ = exp(ππΎπ΅) and πΎπ΅ is called Bloch number. When πΎπ΅ is purely real one have pass band andgap band otherwise. Often Floquet theorem for wave equations called also Bloch theorem, becauseBloch has studied electron movement in 3D space and purposed such wave movement form asdescribed above.
55