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