numerical and continuum analysis1_numerical

8/3/2019 Numerical and Continuum Analysis1_Numerical

http://slidepdf.com/reader/full/numerical-and-continuum-analysis1numerical 1/34

1

Numerical Methods

Topics:

• Introduction to Numerical Methods in Vibration

• Approximate Methods• Holzer Method for Close Coupled Systems

• Holzer Method for Branched Systems

• Myklestad-Prohl Method for Far Coupled Systems

• Finite Element Method



2

•Towards the end of the 19 century , development in rotating machinery led to large

complicated and heavy systems

•Thus the requirement for the methods which could handle large matrices to determine

the Eigen values and Eigen vectors was felt by the industries

•After the II world war , the higher modes of the systems under operation also became

very important in fields such as the aviation and shipping fields

•These all lead to the development of the systems solution in the numerical methods

The important methods are

1.Dunkerley’s method

2.Rayleigh’s method Up to I natural frequency

3.Holzer method4.Myklestad – Prohl method For the higher modes and complicated systems

Introduction to Numerical Methods in Vibrations



3

Numerical Methods

Topics:








4

Determining the first natural frequency i.e, a method for checking the resonance which

were used earlier as an approximate estimate

Dunkerley’s Lower bound Approximation

Consider an EV problem . The frequency equation is given by

For N DOF

Where

For the influence coefficient we define

Where p(i,i) represents NF of system with only i th mass considered

Approximate Methods



5

The equation becomes

Since , we can write

The above Dunkerlays formula gives always the less approximate than the

exact values

The simplicity of the method is in using the reducing the multiple masses to

several SDOF masses individually

Approximate Methods



6

Rayleigh’s Upper Bound Approximation

Consider a MDOF system with matrices M and K representing the mass and

stiffness matrices

Let X be the modal vectors with i th column representing i th model vector with

i th column representing the i th mode shape corresponding to P(i)

The KE and PE of the system are given as

Hence we get

Which give the First NF of the system and is known as Rayliegh’s quotient

This value is always greater the first NF actual value

Approximate Methods



7

Numerical Methods

Topics:








8

Consider a closed coupled system shown in the figure

The convention is to measure displacement and velocity to the right side

internal forces are measured along outward normal

The X and F , displacement and force state vector is given as

Holzer Method for Close Coupled Systems



9

The equation of motion for r’th mass is where

R- right L- Left side of the massFor harmonic force , the above eqn becomes

The displacement of mass mi is

Combining the above equation we get the matrix form of the equations

ie.,

P is the point matrix defines TF to right station in terms of the left station

P is function of mass and frequency ω




10

Considering the force field of the spring we get

Hence the matrix form is given as

Here F is the Field matrix defining the TF across the field , function of stiffness

only

Substituting

We get Transfer matrix




11

Importance is to be given to maintain the order of the matrix multiplication and

station numbering using transfer matrix

The order to be followed is given below

Finally we get

and , where U is the overall transfer matrix

The application of boundary conditions is also important as pet the appropriate system

situation




12

Sequence of Procedure to be Adapted


• Assume Value of ω2 – Desired Natural Frequency. By Making a CrudeModel( Using Dunkerley’s or Rayleigh’s )

• Transfer Matrices [ T ]i for all Stations. At End Points Determine Point or

Field Matrices.

• Determine Overall Transfer Matrix [ U ]• Change ω2 by suitable increment and repeat Steps 1 to 3

• Plot u12 vs ω2 and find value of ω2 for which u12 is Zero. This is a

Natural Frequency

• With {S}0 in the Equation repeat above Steps with the Natural Frequency

obtained in Step 5, to determine the State Vectors at Stations 1,2,…..n.Plot the Amplitudes X1,X2,….Xn to give the Mode Shape for the Frequency

Obtained in Step 5

• Repeat Steps 1 to 6 for obtaining other Natural Frequencies and Mode

Shapes



13

Numerical Methods

Topics:








14

The same method can be extended to the branched systems as well and it is

briefly described below

A branched system

The overall transfer matrices for the three branches are as follows

and at a branch point the following conditions are to satisfied

Holzer Method for Branched Systems



15

The above given system can berepresented in the branched

torsional form as

Using end conditions we have

And hence

Using the branch point conditions

we get

Therefore and




16

Which can be finally written as

and the frequency equation is

Which gives the natural frequency of the system

Example of a geared torsional system




17

Numerical Methods

Topics:




• Myklestad-Prohl Method for Far Coupled Systems• Finite Element Method



18

Consider an n mass system representing the discrete mass of an element as

shown (the masses are considered to be lumped)

The notation system for n mass system is given below

Myklestad-Prohl Method for Far Coupled Systems



19

For i th element of length li and mass mi and S represents the state vector

In the given Xz plane

Slope θ

Bending Moment My

Shear Force Vz

The state vector can be defined left to the station i

Convention

w , θ positive in Y and Z direction

Vz My represent the internal forces and are positive in Y and Z direction in positive

face

A positive face is the one which has outward normal positive x direction




20

The relations for a particular i’th field

are given aside

They are given as

Using the cantilever relations for shear and bending

As well as for slope and deflection




21

Simplifying we get

In short form where S is the state matrix and F is the transfer matrix

Following the relations given for the i’th mass

We can write

that is

Combining these equations we get




22

The transfer matrix is given finally by

and

Which is a short form of

For SS beam

At the ends

For cantilever beam




23

Numerical Methods

Topics:




• Myklestad-Prohl Method for Far Coupled Systems• Finite Element Method



24

Finite Elements Method

Close Coupled Systems

1. Choose Element

2. Define Shape Function3. Derive Element Stiffness

Matrix

4. Assemble Element Stiffness

Matrix

5. Set up Mass Matrix and Eigen

Value Problem

6. Solve Eigen Value Problem

• Matrix Condensation

Scheme• Component mode

Synthesis MethodShape Function

Boundary

Conditions



25


Close Coupled Systems

Element 1

Element 2

Eigen Value Problem



26


Far Coupled Systems

1. Choose Element

2. Define Shape Function3. Derive Element Stiffness Matrix

4. Assemble Element Stiffness

Matrix

5. Set up Mass Matrix and Eigen

Value Problem

6. Solve Eigen Value Problem

• Matrix Condensation

Scheme

• Component mode SynthesisMethod



27


Far Coupled Systems

Satisfying

Shape Functions



28


Far Coupled Systems

Stiffness Matrix



29


Far Coupled Systems



30

Assignment

1



31

Assignment

2



32

Assignment

3



33

Assignment

4



34

Assignment

Obtain the first five natural frequencies

and corresponding mode shapes of the

'Penstock pipe' using Myklestad's

method. Details about the system are

given in the table. Assume ends to be

fixed and two intermediate simplesupports in between.

Description

L1= 15.445m Internal Dia. of the pipe = 5230mm

L2

= 15.000m Pipe material : Boiler Quality plates (IS 2002

Grade 2A)

L3= 14.508 or

4.268m

Pipe thickness = 14 or 16mm

5

numerical and continuum analysis1_numerical

Documents