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;