optimal spring-damper location for a beam-spring system€¦ · optimal spring-damper location for...

Optimal Spring-Damper Location for a

Beam-Spring System

Hao Han

Advisor: Professor Zhuangyi Liu

University of Minnesota Duluth

Duluth,MN 55812

July 24, 2011

Abstract

This paper studies the solution behavior of a system of two coupled elastic beams connected vertically

by springs. With the viscous damping collocated with the ends of the springs, the system energy decays

exponentially. Our goal is to find the optimal spring-damper location which yields the best energy decay

rate of the beam-spring system.

Contents

1 Introduction 1

1.1 Definitions and Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Abstract Evolution Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Goals of the Project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Finite Dimensional Approximation 5

2.1 Finite Dimensional Space HN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 The Operator AN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3 Numerical Computation 12

3.1 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.2 Optimal Spring-Damper Location - Case of One Spring . . . . . . . . . . . . . . . . . . . . . 193.3 Optimal Spring-damper Locations - Case of Two Springs . . . . . . . . . . . . . . . . . . . . 21

4 Conclusions 32

1

1 Introduction

Our main interest comes from a system with two elastic beams connected by vertical springs, which couldbe represented by the following partial differential equations:

utt = −a1uxxxx − k(x)(u − y) − δk(x)ut (1.1)

ytt = −a2yxxxx + k(x)(u − y) − δk(x)yt (1.2)

u(0, t) = u(π, t) = uxx(0, t) = uxx(π, t) = y(0, t) = y(π, t) = yxx(0, t) = yxx(π, t) = 0 (1.3)

u(x, 0) = u0(x), ut(x, 0) = u1(x), y(x, 0) = y0(x), yt(x, 0) = y1(x) (1.4)

where u(x, t), y(x, t) are the displacements of upper and lower beams; k(x) is the spring location,

k(x) =

{

0 x /∈ (p − s, p + s)1 x ∈ (p − s, p + s)

p ∈ (0, π) is the center of the damper location, 2s is the width of the damper; δ and k(x) is the damp-ing coefficient.

In the system (1.1)-(1.4), equation (1.1) describes the vibration of the upper elastic beam and equation(1.2) describes the vibration of the lower elastic beam. Equation (1.3) is the boundary conditions of the twobeams. (1.4) is the initial conditions. Here, we assume that the beams are simply supported at x = 0 andx = π. Our goal is to find the optimal locations of the springs and dampers which yields the best energydecay rate of the beam spring system.

1.1 Definitions and Theorems

In this section, we will give a list of definitions and theorem which are going to be used in this paper.

Definition 1.1 A Norm is a function defined on a vector space V, normally using ‖ ‖ to represent it. Thenorm satisfies the following properties for ∀x, y ∈ V .1.‖x‖ → R2.‖ax‖ = |a|‖x‖3.‖x + y‖ ≤ ‖x‖ + ‖y‖4.If ‖x‖ = 0 then x is the zero vector.

Definition 1.2 A Hilbert space H is a real or complex inner product space that is also a complete metricspace with respect to the distance function induced by the inner product. For a complex inner product space,H is equipped an inner product 〈x, y〉 associating a complex number to each pair of elements x, y of H. Theinner product satisfies the following properties.1.< x, y >= < y, x >2.< a1x1 + a2x2, y >= a1 < x1, y > +a2 < x2, y >3.< x, x >≥ 04.< x, x >= ‖x‖

2

Definition 1.3 The eigenvalues of matrix A in Hilbert space H are convergent iflim

N→∞

λN = λ

where λ is an eigenvalue of matrix A in Hilbert space and λN is the eigenvalue of matrix AN which is the Ndimensional approximation of A.

2

Definition 1.4 A family S(t)(0 ≥ t < ∞) of bounded linear operators in a Banach space H is called astrongly continuous semigroup (in short, a C0 semigroup) if1.S(t1 + t2) = S(t1)S(t2),∀t1, t2 > 02.S(0) = I3.For each x ∈ H, S(t)x is continuous in t on [0,∞).For such a semigroup S(t), we define an operator A with domain D(A) consisting of points x such that thelimit

Ax = limh→0

S(h)x − x

h, x ∈ D(A)

exists. Then A is called the infinitesimal generator of the semigroup S(t). Given an operator A, if A coin-cides with the infinitesimal generator of S(t), then we say that it generates a strongly continuous semigroupS(t), t ≥ 0. Sometimes we also denote S(t) by eAt.

Definition 1.5 A C0 semigroup eAt is stable on H iflimt→∞ ‖eAtx‖ = 0, for ∀x ∈ H

Definition 1.6 A C0 semigroup eAt is said to be exponentially stable ifthere exist positive constants M ≥ 1, ω < 0 such that ‖eAt‖ ≤ Meωt, for ∀t ≥ 0Here, ω is constant.

Definition 1.7 A linear evolution equation{

Z = AZZ(0) = Z0

is well posed on a Hilbert space, if A generates an associated C0 semigroup eAt. The unique solution ofthe evolution equation isZ(t) = eAtZ0

Definition 1.8 The solution of evolution equation is said to be exponential stable if there exist M ≥ 1, w < 0such that‖z(t)‖ ≤ Meωt‖z0‖ for ∀t > 0 and ∀z0 ∈ H

Definition 1.9 Suppose a semigroup eAt is exponentially stable. Its growth rate is defined asω0 = min{ω|‖eAt‖ ≤ Meωt, ω < 0}

Definition 1.10 let σ(A) be the spectrum of an operator A on a Hilbert space, and denoter0 = max{Reλ|λ ∈ σ(A)}We say system satisfies the spectrum determined growth property if r0 = w0.

1.2 Abstract Evolution Equation

Let’s first convert system (1.1)-(1.4) into a first-order abstract evolution equation on a Hilbert space.

Multiplying equation (1.1) by ut, then integrating both sides on [0, π], we have

∫ π

0

uttutdx = −a

∫ π

0

uxxxxutdx −

∫ π

0

k(x)(u − y)utdx − δ

∫ π

0

k(x)u2t dx (1.5)

This can be further written as

1

2

d

dt

∫ π

0

u2t dx = −

1

2

d

dt

∫ π

0

au2xxdx −

∫ π

0

k(x)(u − y)utdx − δ

∫ π

0

k(x)u2t dx (1.6)

3

Similarly, we apply the procedure to equation (1.2)

∫ π

0

yttytdx = −a

∫ π

0

yxxxxytdx +

∫ π

0

k(x)(u − y)ytdx − δ

∫ π

0

k(x)y2t dx (1.7)

which further leads to

1

2

d

dt

∫ π

0

y2t dx = −

1

2

d

dt

∫ π

0

ay2xxdx +

∫ π

0

k(x)(u − y)ytdx − δ

∫ π

0

k(x)y2t dx. (1.8)

Hence, the sum of (1.6) and (1.8) gives

1

2

d

dt

∫ π

0

(u2t + y2

t + au2xx + ay2

xx + k(x)(u − y)2)dx = −δ

∫ π

0

(k(x)u2t + k(x)y2

t )dx. (1.9)

Denote

E(t) =1

2

∫ π

0

(u2t + y2

t + au2xx + ay2

xx + k(x)(u − y)2)dx (1.10)

where 12 (u2

t + y2t ) represents the kinetic energy of the two beams; 1

2a(u2xx + ay2

xx) represents the potentialenergy of the two beams; 1

2k(x)(u− y)2 represents the potential energy of the springs. Then, equation (1.9)represents the changing rate of the energy of the elastic system. From the right hand side of equation (1.9),we see that the changing rate of energy is negative if δ > 0, the energy of the system would keep decreasingas the time t increases; the system is conservative if δ = 0, i.e, E(t) is a constant.

Define a Hilbert Space H. H = U × V × Y × W , where

U = (H10 (0, π) ∩ H2(0, π)), V = L2(0, π), Y = (H1

0 (0, π) ∩ H2(0, π)), W = L2(0, π)

and

L2(0, π) = {f(x)|

∫ π

0

f2(x)dx < ∞},

Hk(0, π) = {f(x)|f(x).f ′(x)...f (k)(x) ∈ L2(0, π)},

H10 (0, π) = {f(x) ∈ H1(0, π)|f ′(0) = f ′(π) = 0}.

For any which Z = (u, v, y, w)T ∈ H,Z1 = (u1, v1, y1, w1)T ∈ H, the Hilbert Space H has inner prod-

uct

〈Z,Z1〉 = 〈(u, v, y, w)T , (u1, v1, y1, w1)T 〉 = a< uxx, u1,xx >

L2 + a< yxx, y1,xx >L2

+< v, v1 >L2 + < w,w1 >L2 + < k(x)(u − y), u1 − y1 >L2 (1.11)

4

with the usual L2 inner product,

< f, g >L2 =

∫ π

0

f.gdx

.

Note that ‖Z‖2 = 2E(t).

Introducing v = ut and w = yt, equations (1.1) and (1.2) then can be formulated as a first order sys-tem

dudt

= vdvdt

= −auxxxx − k(x)(u − y) − k(x)δvdydt

= wdwdt

= −ayxxxx + k(x)(u − y) − k(x)δw

. (1.12)

From here, we use dot (i.e x) to represent the time derivative.

Let Z = (u, v, y, w)T , Z0 = (u0, v0, y0, w0)T . Then system (1.12) could be rewritten as

{

Z = AZZ(0) = Z0

(1.13)

Here, the operator A : D(A) ⊂ H → H is defined as

A =

0 I 0 0−aD4 − k(x) −k(x)δ k(x) 0

0 0 0 Ik(x) 0 −aD4 − k(x) −k(x)δ

(1.14)

and

D(A) = {Z ∈ H|u, y ∈ H4(0, π); v, w ∈ H10 (0, π) ∩ H2(0, π);u′′(0) = u′′(π) = y′′(0) = y′′(π) = 0}. (1.15)

The following two theorems can be found in the reference [2]:

Theorem 1.11 The operator A generates a C0-semigroup of construction, eAt, on the Hilbert space H.Therefore, system is well-posed,i.e, for any Z0 ∈ H, the unique solution to the system is Z(t) = eAtZ0, t > 0.

Theorem 1.12 The semigroup eAt is exponentially stable.

Moreover, the next theorem can be proved based on reference [3].

Theorem 1.13 The system (1.13) satisfies the spectrum determined growth property.

5

1.3 Goals of the Project

Our goal is to analyze how the locations of the collocated springs and the dampers influence the decay rateof the energy of the elastic system. However, the growth rate ω0 is very difficult to compute based on itsdefinition, not mention that in our case it also depends on the damper location p. Fortunately, our systemsatisfies the spectrum determined growth property. We can compute r0(p) instead of ω0(p) for each locationp. The corresponding eigenvalue problem is

λ2u = −au′′′′ + k(x)(u − y) − δk(x)λuλ2y = −ay′′′′ − k(x)(u − y) − δk(x)λyu(0) = u(π) = u′′(0) = u′′(π) = y(0) = y(π) = y′′(0) = y′′(π) = 0

. (1.16)

If an explicit expression of r0(p) could be derived from the above eigenvalue problem, then we would beable to find the minimum of r0(p) over all admissible p. However, this is another difficult task.To get around these difficulties, we will take a practical approach to compute r0(p) and min r0(p). First,we choose a sequence of finite dimensional subspace HN of H such that limN→∞HN = H. Next, we projectsystem (1.12) onto HN to get

{

ZN = ANZN

ZN (0) = ZN (0). (1.17)

The eigenvalues of the matrix AN can be computed numerically, which leads to rN0 (p), the maximum of the

real part of all eigenvalues for a given p.Finally, we compute rN

0 (p) at a set of mesh points of p to obtain the optimal locations of the dampers.

2 Finite Dimensional Approximation

In order to construct a finite dimensional approximation of the infinite dimensional system, we first define afinite dimensional subspace HN of H. Then, project the system onto this space.

2.1 Finite Dimensional Space HN

We define a finite dimensional subspace HN of H by

HN = UN × V N × Y N × WN

where

UN = span

{

√

2

π

1

i2sin(ix)

}N

i=1

,

V N = span

{

√

2

πsin(ix)

}N

i=1

,

Y N = span

{

√

2

π

1

i2sin(ix)

}N

i=1

,

6

WN = span

{

√

2

πsin(ix)

}N

i=1

.

The truncated Fourier sine series is chosen because it satisfies the simply supported boundary conditions.Therefore, a basis of HN could be written as below:

√

2π

112 sin(x)

000

,

√

2π

122 sin(2x)

000

, · · · ,

√

2π

1n2 sin(Nx)

000

,

0√

2π

sin(x)

00

,

0√

2π

sin(2x)

00

, · · · ,

0√

2π

sin(Nx)

00

,

00

√

2π

112 sin(x)

0

,

00

√

2π

122 sin(2x)

0

, · · · ,

00

√

2π

1N2 sin(Nx)

0

,

0

0√

2π

sin(x)

,

000

√

2π

sin(2x)

, · · · ,

000

√

2π

sin(Nx)

. (2.1)

Based on the defined basis, we may approximate u, v, y, w by

uN (x, t) =∑N

i=1

√

2π

1i2

ui(t) sin(ix)

vN (x, t) =∑N

i=1

√

2πvi(t) sin(ix)

yN (x, t) =∑N

i=1

√

2π

1i2

yi(t) sin(ix)

wN (x, t) =∑N

i=1

√

2πwi(t) sin(ix)

(2.2)

Therefore, ZN and ZN could be written out as

7

ZN =

uN (x, t)vN (x, t)yN (x, t)wN (x, t)

=

∑N

i=1

√

2π

1i2

ui(t) sin(ix)∑N

i=1

√

2πvi(t) sin(ix)

∑Ni=1

√

2π

1i2

yi(t) sin(ix)∑N

i=1

√

2πwi(t) sin(ix)

, (2.3)

ZN =d

dtZN =

uN (x, t)vN (x, t)yN (x, t)wN (x, t)

=

∑N

i=1

√

2π

1i2

ui(t) sin(ix)∑N

i=1

√

2πvi(t) sin(ix)

∑N

i=1

√

2π

1i2

yi(t) sin(ix)∑N

i=1

√

2πwi(t) sin(ix)

. (2.4)

This ZN satisfies equation

ZN = AZN (2.5)

on the finite dimensional space HN with

AZN =

0 I 0 0−aD4 − k(x) −k(x)δ k(x) 0

0 0 0 Ik(x) 0 −aD4 − k(x) −k(x)δ

∑Ni=1

√

2π

1i2

ui(t) sin(ix)∑N

i=1

√

2πvi(t)sin(ix)

∑N

i=1

√

2π

1i2

yi(t) sin(ix)∑N

i=1

√

2πwi(t)sin(ix)

=

∑Ni=1

√

2πvi(t) sin(ix)

p(x, t)∑N

i=1

√

2πwi(t)sin(ix)

q(x, t)

(2.6)

where,

p(x, t) = −a

N∑

i=1

√

2

πi2ui(t) sin(ix) − k(x)

N∑

i=1

√

2

π

1

i2ui(t) sin(ix) − k(x)δ

N∑

i=1

√

2

πvi(t) sin(ix)

+k(x)

N∑

i=1

√

2

π

1

i2yi(t) sin(ix) ,

and

q(x, t) = −a

N∑

i=1

√

2

πi2yi(t) sin(ix) − k(x)

N∑

i=1

√

2

π

1

i2yi(t) sin(ix) − k(x)δ

N∑

i=1

√

2

πwi(t) sin(ix)

+k(x)

N∑

i=1

√

2

π

1

i2ui(t) sin(ix) .

8

2.2 The Operator AN

We take the inner product of each side of ZN = AZN with every element in the basis of the Hilbert Space HN .

First, the inner product with the elements in the first line of (2.1),

√

2π

1j2 sin(jx)

000

j = 1, 2, ..., N ,

denoted by ejU , is

〈ZN , ejU 〉HN = 〈AZN , ej

U 〉HN . (2.7)

The left hand side of (2.7) is

LHS =

⟨

∑Ni=1

√

2π

1i2

ui(t) sin(ix)∑N

i=1

√

2πvi(t) sin(ix)

∑N

i=1

√

2π

1i2

yi(t) sin(ix)∑N

i=1

√

2πwi(t) sin(ix)

,

√

2π

1j2 sin(jx)

000

⟩

HN

, j = 1, 2, ...N (2.8)

= a⟨

−∑N

i=1

√

2π

1i2

ui(t) sin(ix),−√

2π

sin(jx)⟩

L2

+⟨

k(x)∑N

i=1

√

2π

1i2

(ui(t) − yi(t)) sin(ix),√

2π

1j2 sin(jx)

⟩

L2

= auj(t) +∑N

i=11i2

(ui(t) − yi(t))⟨

k(x)√

2π

sin(ix),√

2π

1j2 sin(jx)

⟩

L2

= auj(t) +∑N

i=1(ui(t) − yi(t))dij .

(2.9)

Here, we use dij to denote⟨

k(x) 1i2

√

2π

sin(ix),√

2π

1j2 sin(jx)

⟩

L2

.

The right hand side of (2.7) is

RHS =

⟨

∑N

i=1

√

2πvi(t) sin(ix)

p(x,t)∑N

i=1

√

2πwi(t) sin(ix)

q(x,t)

,

√

2π

1j2 sin(jx)

000

⟩

HN

, j = 1, 2, ...N (2.10)

=⟨

−∑N

i=1

√

2πi2vi(t) sin(ix),−

√

2π

sin(jx)⟩

L2

+⟨

k(x)∑N

i=1

√

2π(vi(t) − wi(t)) sin(ix),

√

2π

1j2 sin(jx)

⟩

L2

= aj2vj(t) +∑N

i=1 i2(vi(t) − wi(t))dij .

(2.11)

Therefore, we obtain from (2.9) and (2.11) that

9

auj(t) +

N∑

i=1

(ui(t) − yi(t))dij = aj2vj(t) +

N∑

i=1

i2(vi(t) − wi(t))dij . (2.12)

The inner product with the elements in the second line of (2.1),

0√

2π

sin(jx)

00

, j = 1, 2, ..., N , denoted

by ejV , is

⟨

ZN , ejV

⟩

HN

=⟨

AZN , ejV

⟩

HN

. (2.13)

Then, we have

LHS =

⟨

∑Ni=1

√

2π

1i2

ui(t) sin(ix)∑N

i=1

√

2πvi(t) sin(ix)

∑N

i=1

√

2π

1i2

yi(t) sin(ix)∑N

i=1

√

2πwi(t) sin(ix)

,

0√

2π

sin(jx)

00

⟩

HN

, j = 1, 2, ...N (2.14)

=⟨

∑Ni=1

√

2πvi(t) sin(ix),

√

2π

sin(jx)⟩

L2

= vi(t) ,(2.15)

and

RHS =

⟨

∑N

i=1

√

2πvi(t) sin(ix)

p(x,t)∑N

i=1

√

2πwi(t) sin(ix)

q(x,t)

,

0√

2π

sin(jx)

00

⟩

HN

, j = 1, 2, ...N (2.16)

=⟨

p(x, t),√

2π

sin(jx)⟩

L2

= −aj2uj(t) −∑N

i=1 i2j2(ui(t) + δvi(t) −1i2

yi(t))dij .(2.17)

Therefore, we obtain from (2.15) and (2.17) that

10

vi(t) = −ai2j2uj(t) −N

∑

i=1

j2(ui(t) + δvi(t) −1

i2yi(t))dij . (2.18)

In the same way, the inner products with the elements ejY =

00

√

2π

1j2 sin(jx)

0

, ejW =

000

√

2π

sin(jx)

, j =

1, 2, ..., N are

⟨

ZN , ejY

⟩

HN

=⟨

AZN , ejY

⟩

HN

,⟨

ZN , ejW

⟩

HN

=⟨

AZN , ejW

⟩

HN

, (2.19)

which lead to

ayj(t) +

N∑

i=1

(yi(t) − ui(t))dij = aj2wj(t) +

N∑

i=1

i2(wi(t) − vi(t))dij . (2.20)

and

wi(t) = −aj2yj(t) −N

∑

i=1

i2j2(yi(t) + δwi(t) −1

i2ui(t))dij . (2.21)

From all above, we have

auj(t) +∑N

i=1(ui(t) − yi(t))dij = aj2vj(t) +∑N

i=1 i2(vi(t) − wi(t))dij

vi(t) = −aj2uj(t) −∑N

i=1 i2j2(ui(t) + δvi(t) −1i2

yi(t))dij

ayj(t) +∑N

i=1(yi(t) − ui(t))dij = aj2wj(t) +∑N

i=1 i2(wi(t) − vi(t))dij

wi(t) = −aj2yj(t) −∑N

i=1 i2j2(yi(t) + δwi(t) −1i2

ui(t))dij

. (2.22)

The equations could be written as MN˙ZN = ANZN ,

where

11

MN =

M11 0 M13 00 I 0 0

M13 0 M11 00 0 0 I

˙ZN =

u1

u2

...˙uN

v1

v2

...˙vN

y1

y2

...˙yN

w1

w2

...wN

AN =

0 A12 0 A14

A21 A22 A23 0

0 A14 0 A12

A23 0 A21 A22

ZN =

u1

u2

...uN

v1

v2

...vN

y1

y2

...yN

w1

w2

...wN

In the matrix MN ,

M11 =

a + d11 d21 · · · dN1

d12 a + d22 · · · dN2

......

. . ....

d1N d2N · · · a + dNN

M13 =

−d11 −d21 · · · −dN1

−d12 −d22 · · · −dN2

......

. . ....

−d1N −d2N · · · −dNN

In matrix A,

A12 =

a + d11 4d21 · · · N2dN1

d12 4a + 4d22 · · · N2dN2

......

. . ....

d1N 4d2N · · · N2a + N2dNN

A14 =

−d11 −d21 · · · −dN1

−d12 −d22 · · · −dN2

......

. . ....

−d1N −d2N · · · −dNN

A21 =

−a − d11 −4d21 · · · −N2dN1

−4d12 −4a − 16d22 · · · −4N2dN2

......

. . ....

−N2d1N −4N2d2N · · · −N2a − N4dNN

A22 =

−δd11 −4δd21 · · · −N2δdN1

−4δd12 −16δd22 · · · −4N2δdN2

......

. . ....

−N2δd1N −4N2δd2N · · · −N4δdNN

A23 =

d11 d21 · · · dN1

4d12 4d22 · · · 4dN2

......

. . ....

N2d1N N2d2N · · · N2dNN .

Then this could be transformed to ˙ZN = M−1N ANZN . Therefore, AN is actually M−1

N AN .

12

3 Numerical Computation

We now have the operator AN = M−1N AN . Therefore, if N, k(x), a and δ are given, the eigenvalues of AN

can be calculated by Matlab software.

3.1 Convergence

Our first experiment is for the case of one spring. Assume N = 32, p = 0.1, s = 0.05, a = 1, δ = 1. Wecompute the eigenvalues and list the first 20 eigenvalues of lower frequencies. (i.e their imaginary numberpart have the smallest values.)

No. Eigenvalue1 -.2054399138e-3-1.000000861i2 -.2054399138e-3+1.000000861*i3 -.2019552097e-3-1.000408144*i4 -.2019552097e-3+1.000408144*i5 -.3197559943e-2-4.002476485*i6 -.3197559943e-2+4.002476485*i7 -.3147368278e-2-4.004060607*i8 -.3147368278e-2+4.004060607*i9 -.1548308465e-1-9.014673631*i10 -.1548308465e-1+9.014673631*i11 -.1526013867e-1-9.018076289*i12 -.1526013867e-1+9.018076289*i13 -.4610481934e-1-16.04791398*i14 -.4610481934e-1+16.04791398*i15 -.4550144921e-1-16.05359105*i16 -.4550144921e-1+16.05359105*i17 -.1047360576-25.11621176*i18 -.1047360576+25.11621176*i19 -.1035012988-25.12441592*i20 -.1035012988+25.12441592*i

Below is a graph with the eigenvalues in the given range,

We also list the first 10 eigenvalues of operator AN of lower frequencies with positive imaginary part, whenN=32, 64, 128, 256 and 512.

13

−20 −18 −16 −14 −12 −10 −8 −6 −4 −2 0−1000

−800

−600

−400

−200

0

200

400

600

800

1000

Real Number

Imag

inar

y N

umbe

r

Figure 3.1: The eigenvalues in the given range, N = 32, p = 0.1, s = 0.05, a = 1, δ = 1

No. N=32 N=64 N=128 N=256 N=5121 -.205440e-3 -.186201e-3 -.179336e-3 -.175816e-3 -.174041e-3

+1.00000*i +1.00000*i +1.00000*i +1.00000*i +1.00000*i2 -.201955e-3 -.183244e-3 -.176563e-3 -.173136e-3 -.171408e-3

+1.00041*i +1.00037*i +1.00036*i +1.00035*i +1.00035*i3 -.319756e-2 -.290175e-2 -.279627e-2 -.274217e-2 -.271490e-2

+4.00248*i +4.00224*i +4.00216*i +4.00212*i +4.00210*i4 -.314737e-2 -.285900e-2 -.275612e-2 -.270335e-2 -.267674e-2

+4.00406*i +4.00368*i +4.00355*i +4.00348*i +4.00344*i5 -.154831e-1 -.140781e-1 -.135778e-1 -.133211e-1 -.131917e-1

+9.01467*i +9.01329*i +9.01280*i +9.01255*i +9.01242*i6 -.152601e-1 -.138873e-1 -.133982e-1 -.131473e-1 -.130208e-1

+9.01808*i +9.01639*i +9.01579*i +9.01548*i +9.01533*i7 -.461048e-1 -.420237e-1 -.405740e-1 -.398301e-1 -.394550e-1

+16.0479*i +16.0434*i +16.0418*i +16.0410*i +16.0406*i8 -.455014e-1 -.415043e-1 -.400842e-1 -.393556e-1 -.389881e-1

+16.0536*i +16.0486*i +16.0468*i +16.0459*i +16.0455*i9 -.104736 -.957224e-1 -.925332e-1 -.908971e-1 -.900718e-1

+25.1162*i +25.1054*i +25.1016*i +25.0996*i +25.0986*i10 -.103501 -.946527e-1 -.915221e-1 -.899161e-1 -.891060e-1

+25.1244*i +25.1129*i +25.1089*i +25.1068*i +25.1057*i

In this table, eigenvalues are ranked by the value of imaginary part. Look at row 1 and row 2, whendimension of the space increases from 32 to 64, 64 to 128, 128 to 256, 256 to 512, the changes of the real partand imaginary part of the eigenvalue becomes smaller and closer to zero. For example, when N increases from

14

64 to 128, the real part increases by 0.006865 and the imaginary part’s change close to 0; when N increasesfrom 128 to 256, the real part increases by 0.00352 and the imaginary part’s change is close to 0. There-fore, it could be claimed that the 1st eigenvalue is going to converge to a certain point when N goes to infinity.

It is observed that every row has the same properties as row 1 and row 2. Therefore, it could be claimedthat the eigenvalues of the operator AN would converge to certain points when N approaches to infinity.

15

Below is the graphs of the convergence of the 1st and 2nd eigenvalues in the table.

−2.1 −2.05 −2 −1.95 −1.9 −1.85 −1.8 −1.75 −1.7

x 10−4

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Real Number

Imag

inar

y N

umbe

r

Figure 3.2: Showing the convergence of the 1st eigenvalue, p = 0.1, s = 0.05, a = 1, δ = 1

−2.05 −2 −1.95 −1.9 −1.85 −1.8 −1.75 −1.7

x 10−4

1.0003

1.0004

1.0004

1.0004

1.0004

1.0004

1.0004

1.0004

1.0004

1.0004

Real Number

Imag

inar

y N

umbe

r

Figure 3.3: Showing the convergence of the 2nd eigenvalue, p = 0.1, s = 0.05, a = 1, δ = 1

16

Next, we consider a case of two springs. Take N = 64, p1 = 0.1, p2 = 2.1, s = 0.05, a = 1, δ = 1. We listthe first 20 eigenvalues of lower frequencies.

No. Eigenvalue1 -.1337119832e-1-1.000063277*i2 -.1337119832e-1+1.000063277*i3 -.1277976021e-1-1.025874325*i4 -.1277976021e-1+1.025874325*i5 -.5714003666e-1-4.043338560*i6 -.5714003666e-1+4.043338560*i7 -.5641546881e-1-4.071337905*i8 -.5641546881e-1+4.071337905*i9 -.1538191124e-1-9.014470408*i10 -.1538191124e-1+9.014470408*i11 -.1518614237e-1-9.017854542*i12 -.1518614237e-1+9.017854542*i13 -.2456095234-16.24894815*i14 -.2456095234+16.24894815*i15 -.2427389370-16.27890557*i16 -.2427389370+16.27890557*i17 -.4464751779-25.45874338*i18 -.4464751779+25.45874338*i19 -.4433719211-25.49363828*i20 -.4433719211+25.49363828*i

Below is a graph with the eigenvalues in the given range,

−200 −180 −160 −140 −120 −100 −80 −60 −40 −20 0−4000

−3000

−2000

−1000

0

1000

2000

3000

4000

Real Number

Imag

inar

y N

umbe

r

Figure 3.4: The eigenvalues in the given range, N = 64, p1 = 0.1, p2 = 2.1, s = 0.05, a = 1, δ = 1

17

We also list the first 10 eigenvalues of operator AN of lower frequencies with positive imaginary number,when N=32, 64, 128, 256 and 512.

No. N=32 N=64 N=128 N=256 N=5121 -.198245e-1 -.188330e-1 -.181685e-1 -.178312e-1 -.176665e-1

+.999944*i +.999950*i +.999954*i +.999956*i +.999957*i2 -.192727e-1 -.183322e-1 -.176995e-1 -.173781e-1 -.172212e-1

+1.03831*i +1.03645*i +1.03520*i +1.03457*i +1.03426*i3 -.937471e-1 -.894759e-1 -.863140e-1 -.847664e-1 -.839977e-1

+4.07312*i +4.06965*i +4.06712*i +4.06588*i +4.06526*i4 -.909324e-1 -.869163e-1 -.839211e-1 -.824542e-1 -.817250e-1

+4.11819*i +4.11275*i +4.10875*i +4.10679*i +4.10581*i5 -.185361 -.175179 -.168760 -.165599 -.164045

+9.16730*i +9.15794*i +9.15217*i +9.14931*i +9.14791*i6 -.183836 -.173825 -.167482 -.164361 -.162827

+9.20754*i +9.19602*i +9.18888*i +9.18535*i +9.18361*i7 -.559137 -.528157 -.509106 -.499404 -.494689

+16.5359*i +16.5058*i +16.4877*i +16.4784*i +16.4738*i8 -.555109 -.524532 -.505680 -.496084 -.491420

+16.6032*i +16.5696*i +16.5492*i +16.5387*i +16.5337*i9 -.314991 -.296111 -.285135 -.279767 -.277087

+25.3288*i +25.3085*i +25.2967*i +25.2909*i +25.2880*i10 -.312553 -.293884 -.283030 -.277724 -.275075

+25.3535*i +25.3317*i +25.3191*i +25.3128*i +25.3097*i

In this table, eigenvalues are ranked by the value of imaginary part. When dimension of the space in-creases from 32 to 64, 64 to 128, 128 to 256, 256 to 512, the changes of the real part and imaginary partof the eigenvalue becomes smaller and closer to zero. Therefore, it could be claimed that every eigenvaluewould converge to certain points when N goes to infinity.

18

Below is the graphs of the convergence of the 1st and 2nd eigenvalues in the table.

−0.02 −0.0195 −0.019 −0.0185 −0.018 −0.01750.9999

0.9999

0.9999

0.9999

1

1

1

1

Real Number

Imag

inar

y N

umbe

r

Figure 3.5: Showing the convergence of the 1st eigenvalue, p1 = 0.1, p2 = 2.1, s = 0.05, a = 1, δ = 1

−0.0195 −0.019 −0.0185 −0.018 −0.0175 −0.0171.034

1.0345

1.035

1.0355

1.036

1.0365

1.037

1.0375

1.038

1.0385

Real Number

Imag

inar

y N

umbe

r

Figure 3.6: Showing the convergence of the 2nd eigenvalue, p1 = 0.1, p2 = 2.1, s = 0.05, a = 1, δ = 1

19

3.2 Optimal Spring-Damper Location - Case of One Spring

As mentioned before, our goal is to find the optimal Spring-damper locations. At each location p, rN0 (p),

the maximum real part of the eigenvalue, would determine the decay rate of the energy of the system. Weneed to find the maximum of |rN

0 (p)| over all p.

Taking 100 or 300 mesh points evenly distributed on [0, π] as the center of the Spring-damper locationp, and then we find out the value of |rN

0 (p)|. Therefore, we would be able to observe the change of the decayrate with the change of p.

The next two graphs are the value of |rN0 (p)| at the mesh points. In the first graph, we take 300 mesh

points and dimension N=128; In the second one, we take 100 mesh points and dimension N=256.

0 0.5 1 1.5 2 2.5 3 3.50

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

p

Figure 3.7: The value of |rN0 (p)| at the mesh points jπ

200 , j = 1, 2, ..., 200, N = 128, s = 0.05, a = 1, δ = 1

20

0 0.5 1 1.5 2 2.5 3 3.50

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

p

Figure 3.8: The value of |rN0 (p)| at the mesh points jπ

100 , j = 1, 2, ..., 100, N = 256, s = 0.05, a = 1, δ = 1

21

3.3 Optimal Spring-damper Locations - Case of Two Springs

Let p1, p2 be the centers of the Spring-dampers location. Taking 300 mesh points evenly distributed on [0, π].We compute |rN

0 (p1+p2

2 )| for all admissible mesh points for d = p2 − p1 = π30 , 2π

30 , ..., π.

The following graphs are the values of |rN0 (p)| at the mesh points and at each given d value. In all these

graphs, we take 300 mash points and dimension N=64. We keep changing the distance of the centers of thetwo springs to get the graphs below.

0 0.5 1 1.5 2 2.5 3 3.50

0.005

0.01

0.015

0.02

0.025

0.03

0.035

(p1+p2)/2

Figure 3.9: |rN0 (p1+p2

2 )| at the mesh points, d = p2 − p1 = π30 , N = 64, s = 0.05, a = 1, δ = 1

22

0 0.5 1 1.5 2 2.5 3 3.50

0.005

0.01

0.015

0.02

0.025

0.03

0.035

(p1+p2)/2


2 )| at the mesh points,d = p2 − p1 = 2π30 , N = 64, s = 0.05, a = 1, δ = 1

23

0 0.5 1 1.5 2 2.5 30

0.005

0.01

0.015

0.02

0.025

0.03

0.035

(p1+p2)/2



24

0 0.5 1 1.5 2 2.5 30

0.005

0.01

0.015

0.02

0.025

0.03

0.035

(p1+p2)/2



25

0 0.5 1 1.5 2 2.5 30

0.005

0.01

0.015

0.02

0.025

0.03

0.035

(p1+p2)/2



26

0 0.5 1 1.5 2 2.5 30

0.005

0.01

0.015

0.02

0.025

0.03

0.035

(p1+p2)/2



27

0 0.5 1 1.5 2 2.5 30

0.005

0.01

0.015

0.02

0.025

0.03

0.035

(p1+p2)/2



28

0 0.5 1 1.5 2 2.5 30

0.005

0.01

0.015

0.02

0.025

0.03

(p1+p2)/2



29

0 0.5 1 1.5 2 2.5 30

0.005

0.01

0.015

0.02

0.025

0.03

(p1+p2)/2



30

0 0.5 1 1.5 2 2.5 30

0.005

0.01

0.015

0.02

0.025

0.03

(p1+p2)/2



31

Since the maximum value of |rN0 (p)| decreases as d = p2 − p1 increases from 10π

30 to π, we dont’t includethem graphs here.

We enlarge the figure 3.13 to identify the maximum |rN0 (p1+p2

2 )|

1.4 1.45 1.5 1.55 1.6 1.65 1.70.03

0.0305

0.031

0.0315

0.032

0.0325

0.033

0.0335

0.034

0.0345

0.035

(P1+P2)/2


2 )| at the mesh points,d = p2 − p1 = 5π30 , N = 64, s = 0.05, a = 1, δ = 1.

32

4 Conclusions

For the case of one spring, from the graphs above, we see that the |rN0 (p)| has the largest value 0.01601 when

p is close to 126π300 and 174π

300 . Therefore, the optimal location should be p=126π300 or p=174π

300 .

For the case of two springs, from the graphs above, we see that |rN0 (p)| has the largest value 0.03324

when p2 − p1 = 5π30 and p1+p2

2 = π2 . Therefore, the optimal locations should be close to 5π

12 and 7π12 .

33

References

[1] Z. Liu and S. Zheng, Semigroups Associated with Dissipative Systems, 398 Research Notes in Mathe-matics, Chapman & Hall/CRC, 1999.

[2] Z. Liu and C. Zhang, Stability of a Beam-Spring System, preprint, 2011

[3] M Renardy , The Type of Certain c0-semigroup, Comm. in PDE vol 108 N0 7-8 pg 1299-1307, 1993

34

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