mechanical vibrations multi degrees of freedom system philadelphia university engineering faculty...

Mechanical VibrationsMulti Degrees of Freedom System

Philadelphia University Engineering Faculty Mechanical Engineering Department

Professor Adnan Dawood Mohammed

Multi DOF system

Equations of motion:

[M] is the Mass matrix

[K] is the Stiffness matrix

[C] is the Damping matrix

F xKxCxM

1) Vector mechanics (Newton or D’ Alembert)

2) Hamilton's principles3) Lagrange's equations

They are obtained using:

Multi-DOF systems are so similar to two-DOF.

Un-damped Free Vibration: the eigenvalue problem

Equation of motion:

(2) 0

(1) ,0

:becomes 1Equation matrix. system A theKM

matrix)(unit IMM that Note .Mby (1)equation y premultipl

ly.respective snt vectordisplaceme andon accelerati theare and

ly.respective matrices Stiffness and Mass theareK and M where

1-

1-1-

AqqI

KqqM

qq

0 qKqM

Write the matrix equation as:

in terms of the generalized D.O.F. qi

theof definition thestart with and I,-ABLet system. theof

thefrom rseigenvecto thefind topossible also isIt

. thecalled is which X shape mode

ingcorrespond obtain the we(3),equation matrix theinto ngsubstitutiBy

(5)

relation by the themfrom determined are system theof sfrequencie

natural theand thecalled areequation ticcharacters

theof roots the, (4) ,0I-A

or ZERO, toequated

tdeterminan theis system theofequation ticcharacters The

(3) 0}{I-A

becomes (2)Equation , where,

i

i

2i

i

2

matrix adjoint

reigenvecto

seigenvalue

q

qq

i

Assuming harmonic motion:

constant) arbitrarayan by d(multiplie

qr eigenvecto theis which ofeach columns, of consistsmust

I-Amatrix adjoint that therecognize we, 0}{I-A

mode i for the (4)equation ith equation w this

Comparing system. freedom of degrees-n for the equations

n"" represents and valuesallfor valiedisequation above The

I-AI-A0

zero, isequation theof sideleft

on thet determinan then the,eigenvaluean ,let wenow If

(6) I-AI-AI-A

or ,B adj BIB

obtain, toBBby y Premultipl .B inverse

i

th

i

i

i

1-

iii

i

adjq

adj

adjI

B

adjB

Example:Consider the multi-story building shown in figure. The Equations of motion can be written as:

0

Pre-multiply by the inverse of mass matrix

(b) 0

0

)/( )/(

)2/( )2/3(

becomes (a)equation , lettingBy

)/()/(

)2/( )2/3(

/10

02/1

2

1

2

1

1

x

x

mkmk

mkmk

mkmk

mkmkAKM

m

mM

The characteristic equation from the determinant of the above matrix is

(d) 2 2

1

whichfrom (c), ,02

5

21

22

m

k

m

k

m

k

m

k

The eigenvectors can be found from Eqn.(b) by substituting the above values of The adjoint matrix from Eqn. (b) is

i

i

mkmk

mkmkIAAdj

)2/3( )/(

)2/( )/(

Substituting into Eqn. (e) we obtain:

mk

0.10.1

5.05.0

Here each column is already normalized to unity and the first eigenvector is

0.1

5.01X

Similarly when k/m) the adjoint matrix gives;

mk

5.00.1

5.00.1

Normalizing to Unity;

mk

0.10.1

0.10.1

0.1

0.12X

The second eigenvector from either column is;