Presented by Andrey Kuzmin Mathematical aspects of mechanical systems eigentones Department of Applied Mathematics

Presented by Andrey Kuzmin

Mathematical aspects of mechanical systems eigentones

Department of Applied Mathematics

Joint Advanced Student School St.Petersburg 2006

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Agenda

PART I.Introduction to the theory of mechanical vibrations

PART II.Eigentones (free vibrations) of rod systems

– Forces Method– Example

PART III.Eigentones of plates and shells

– Properties of eigentones– The rectangular plate: linear and nonlinear statement– The bicurved shell

PART I

Introduction to the theory of mechanical vibrations


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1.1 History

• History of development of the linearvibration theory:

– XVIII century“Analytical mechanics” by Lagrange – systems

with several degrees of freedom– XIX century

Rayleigh and others – systems with the infinite number degrees of freedom

– XX centuryThe linear theory has been completed

Intro


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• Today’s problems of the linear vibration theory:

• Vibration problems of mechanical systems

1.2 Problems

– How correctly to choose degrees of freedom?

– How correctly to define external influences?

Choice of the calculated scheme

Linear statement Nonlinear statement

Intro


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• Role of the nonlinear theory:

The phenomena description escaping from a field of vision at any attempt to linearize the considered problem.

• Approximate solution methods of nonlinear problems:

– Poincare and Lyapunov’s Methods– Krylov-Bogolyubov's Method– Bubnov-Galerkin’s Method– and others

1.3 Solution

allow making successive approximations

allow making any approximations

Intro

PART II

Eigentones (free vibrations) of rod systems


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2.1 Forces Method

• Consider rod systems in which the distributed mass is concentrated in separate sections (systems with a finite number of degrees of freedom)

• Define displacements from a unit forces applied in directions of masses vibrations

• Construct the stiffness matrix of system:*0B b fb

the gain matrix depend on the unit forces applied in a direction of masses vibrations in the given system

the stiffness matrix of separate elements

transposition of the matrix equal to the matrix b, constructed for statically definable system

Rod systems


9

• Construct a diagonal masses matrix M, calculate matrix

product D = BM and consider system of homogeneous equations

where

• In the end compute the determinant ,

eigenvalues and corresponding eigenvectors of matrix D

2.1 Forces Method

0 or BM E X DX X (1)

an amplitudes vector of displacements

the unit matrix

frequency of free vibrations of the given system

2

1

0BM E

Rod systems


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2.2 Example: the problem setup

• Define frequencies and forms of the free vibrations of a statically indeterminate frame with two concentrated masses

т1 = 2т, т2 = т and identical stiffnesses of rods at a

bending down (EI = const, where E – Young's modulus; I – Inertia moment of section)

Fig. 1, a. Rod system with two degree of freedoms

Rod systems


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2.3 Example: the problem solution

Fig 1, b.The bending moments

stress diagrams depend on the unit forces applied

in a direction of masses vibrations

Fig 1, c.The stress diagrams

depend on the same unit forces in statically

determinate system

Rod systems


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• Calculation of displacements: evaluation of integrals on the Vereschagin's Method

• Then we construct the stiffness matrix

01 1

11

01 2

12 21

02 1

22

1,708

0,482–

0,905

l

l

l

M Mdx

EI EI

M Mdx

EI EI

M Mdx

EI EI

1,708 0,4821

0,482 0,905

BEI

Rod systems


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• The masses matrix has the form (at т1 = 2т, т2 = т):

• To find eigenvalues and eigenvectors of the matrix D = BM we compute the determinant:


2 0

0 1

M m

3,416 0,482 0

0,964 0,905

j

j

m m

EI EIBM Em m

EI EI

Rod systems


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• Then we obtain a quadratic equation

• Thus we can find frequencies of free vibrations of the frame

• For definition of corresponding forms of vibrations we use (1).

Let, for example, X1 = 1. From the first equation we find Х2 for

each value of λj:


22 4,321 2,627 0

j j

m m

EI EI

1

2

3,5891

0,7319

m

EIm

EI

1 21 2

1 10,5278 ; 1,1689

EI EI

m m

withroots

12

22

3,416 3,5891 1 0,482 0

3,416 0,7319 1 0,482 0

m m mX

EI EI EI

m m mX

EI EI EI

Rod systems


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11

5,569-0,359


• Solving each equations separately, we find eigenvectors ν1

and ν2:

• Then we obtain forms of the free vibrations

1 2

1 1;

0,359 5,569

v v

Rod systems

Fig. 1, d. The main forms of the free vibrations

PART III

Eigentones of plates and shells


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3.1 Properties of eigentones

• Properties of linear eigentones (free vibrations):

– Plates and shells – systems with infinite number degrees of freedom. That is:

• number of eigenfrequencies is infinite • each frequency corresponds a certain form of vibrations

– Amplitudes do not depend on frequency and are determined by initial conditions:

• deviations of elements of a plate or a shell from equilibrium position

• velocities of these elements in an initial instant

Plates and shells


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3.1 Properties of eigentones

• Properties of nonlinear eigentones:– Deflections are comparable to thickness of a plate:

Rigid plates / shells Flexible plates / shells

– Frequency depends on vibration amplitude

transform

Fig. 2. Possible of dependence between

the characteristic deflection and nonlinear eigentones frequency

Plates and shells

a) Thin system b) Soft system

Skeletal line

1 1

A A


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System with infinite number degrees of freedom

System with one degree of freedom

3.2 Solution of nonlinear problems

Approximation

Plates and shells


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3.3 The rectangular plate, fixed at edges: a linear problem

• Let a, b – the sides of a plate

h – the thickness of a plate

• Linear equation for a plate:

where

24

20

D ww

h g t

(2)

The rectangular plate

3

212 1

EhD

4 4 44

4 4 2 22

x y x y

w – function of the deflection – density of the plate materialg – the free fall accelerationD – cylindrical stiffnessE – Young's modulus – the Poisson's ratio

4 – the differential functional


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3.4 Solution of the linear problem

24

20 0

sin sin 0a b D w m x n y

w dxdyh g a bt

Integration

220,2

0mn

d

dt

( )f t

h where

• Approximation of the deflection on the Kantorovich's Method:

• Substituting the equation (2) instead of function f(t):

( )sin sinm x n y

w f ta b

some temporal function



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3.4 Solution of the linear problem

Fig. 3. Character of wave

formation of the rectangular plate at

vibrations of the different form

a

b

224 4 2 2 2

220, 2 2 2 2

1

12 1mn

nm c h

m

a b

where

• The square of eigentones frequency at small deflections has form:


Egc

the velocity of spreading of longitudinal

elastic waves in a material of the plate

m = n = 1

a) the first form

m = 2, n = 1

b) the second form

m = n = 2

b) the third form


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• Examine vibrations of a plate at amplitudes which are comparable with its thickness

• Assume that the ratio of the plate sides is within the limits of

• We take advantage of the main equations of the shells theory

at kx = ky = 0:

where

3.5 The rectangular plate, fixed at edges: a nonlinear problem

a

b

1 2

24

2

4

( , )

1 1( , )

2

D ww L w

h g t

L w wE

(3)

(4)


Equilibrium equation

2 2 2 2 2 2

2 2 2 2( , ) 2

A B A B A BL A B

x y x yx y y x

differential functional

a stress function

Deformation equation

the main shell curvatures


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• Set expression (approximation) of a deflection

• Substituting (5) in the right member of the equation (4), we shall obtain the equation, which private solution is:

• Define , , where Fx and Fy – section areas

of ribs in a direction of axes x and y

3.6 Solution of the nonlinear problem

1

2 2cos cos

x yA B

a b

2 2

2

2 2

2

32

32

f aA E

b

f bB E

a

where

( )sin sinx y

w f ta b

(5)

yy

hv

Fx

x

hv

F



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• Then the solution of a homogeneous equation will have the form:

• Finally

3.6 Solution of the nonlinear problem

22 2 22 2 2cos cos

32 2 2yx

p xp yf a x b yE

b a a b

22

2 2 2yx

p xp y

2

2 22

2 2

2

2 22

2 2

1

8 1 1

1

8 1 1

y

x

x y

x

y

x y

b v

ap E fb v v

bv

ap E fb v v

where

the stresses applied to the plate through boundary ribs (they are considered as positive at a tensioning)

4 0



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3.7 Solution: the first stage of approximation

• Apply the Bubnov-Galerkin’s Method to the equation (3) for

some fixed instant t• Suppose X has the form

• Generally we approximate functions w(x,y,t) in the form of series

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2( , )

D wX w L w

h g t

1

n

i ii

w f

some given and independent functions which satisfy to boundary conditions of a problem

the parameters depending on t



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• On the Bubnov-Galerkin’s Method we write out n equations of type

• In our solution η1 has the form

0, 1,2,...,i

F

X dxdy i n (6)

1 sin sinx y

a b



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• Hence, integrating (6) and passing to dimensionless parameters, we obtain the equation

where the dimensionless parameters K and ζ have the form


2

2 202

1 0d

Kdt

(7)

2 2

2 2 2 2 2 42

2 2

1,5 1 0.75 11 1 11 1

1 11 1 1 1

yx

x y

vK v

v v

(8)

( )f t

h



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• Hence, integrating (6) and passing to dimensionless parameters, we obtain the equation

• Parameter – the square of the main frequency of the plate eigentones:


2

2 202

1 0d

Kdt

(7)

20

24 2 22 20 2 2

1

12 1

hc

ab



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24

2( , )

D ww L w

h g t

2

2 202

1 0d

Kdt

Bubnov-Galerkin’s Method

– the nonlinear differential partial equation of the fourth degree

– the nonlinear differential equation in ordinary derivatives of the second degree

2 stage1 stage

= ?

Integration


• Thus


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• Consider the simply supported plate

• Let's present temporal function in the form

3.8 Solution: the second stage of approximation

0

0

0

x

y

x y

Fx

F

y

p p

v

v

from (8) hence

2 4

22

3 1 1

4 1K


cosA t

vibration frequency

dimensionless amplitude

(9)

that is ribs are absent


32

• Let

• Further integrate Z over period of vibrations

• We obtain the equation expressing dependence between

frequency of nonlinear vibrations ω and amplitude A:


2T

2 /

0

( ) cos( ) 0Z t t dt

2 2 20

31

4KA


2

2 202

( ) 1d

Z t Kdt


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• Define

• Then


0

frequency of nonlinear vibrations

frequency of linear vibrations

2 231

4KA

Fig. 4. A skeletal line of the thin type for ideal

rectangular plate at nonlinear vibrations of the general form


A


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• Now we consider shallow and rectangular

in a plane of the shell

• The main shell curvatures kx, ky are assumed by constants:

3.9 The bicurved shell

1 2

1 1x yk k

R R

Fig. 5. The shallow

bicurved shell.

The bicurved shell

where R1,2 – radiuses of

curvature


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• The dynamic equations of the nonlinear theory of shallow shells have the form:

where the differential functional

• For full and initial deflections are define by

3.10 The bicurved shell: the problem setup

24 2

0 2

4 20 0 0

( , )

1 1( , ) ( , )

2

k

k

D ww w L w

h t

L w w L w w w wE

2 22

2 2k x y

A AA k k

y x

The bicurved shell

0 0( )sin sin sin sinx y x y

w f t w fa b a b


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• Using the method considered above, we obtain the following ordinary differential equation of shell vibrations

• Here

The square of the main frequency of ideal shell eigentones at small deflections has the form

3.11 The bicurved shell: the problem solution

2

2 2 302

0d

dt

(10)

010 1 0

( )

ff tf f f

h h

2 2 220 2 2

c h

a b

22 2 2

2*22 2 2 2

1

12 1 1k

The bicurved shell

where


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Here variables , , have the form


2

2 2 302

0d

dt

(10)

4 * *2 44

02 4 2 4 22 2

16 162 8 81 1 1 1

312 1 1

y xk k

4 * *2 44

02 4 2 4 22 2

16 161 8 1 8 91

2 2 412 1 1

y xk k

2

42

0,75 112

The bicurved shell


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• Thus we obtain the following equation for definition of an amplitude-frequency characteristic

2 28 31

3 4A A

where0

The bicurved shell

Fig. 6. The amplitude-frequency dependences for shallow

shells of various curvature

A

2

4

6

8

0 1 2

shell at

cylindrical shell at

plate at

* * 24x yk k

* * 0x yk k

* *0, 24x yk k


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Applications


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References

• Ilyin V.P., Karpov V.V., Maslennikov A.M. Numerical methods of a problems solution of building mechanics. – Moscow: ASV; St. Petersburg.: SPSUACE, 2005.

• Karpov V.V., Ignatyev O.V., Salnikov A.Y. Nonlinear mathematical models of shells deformation of variable thickness and algorithms of their research. – Moscow: ASV; St. Petersburg.: SPSUACE, 2002.

• Panovko J.G., Gubanova I.I. Stability and vibrations of elastic systems. – Moscow: Nauka. 1987.

• Volmir A.S. Nonlinear dynamics of plates and shells. – Moscow: Nauka. 1972.

Documents

Presented by Andrey Kuzmin Mathematical aspects of mechanical systems eigentones Department of Applied Mathematics