Mechanical Vibrations
Multi Degrees of Freedom System
 Philadelphia University
 Engineering Faculty
 Mechanical Engineering Department
 Professor Adnan Dawood Mohammed
Multi DOF system
Multi-DOF systems are so similar to two-DOF.
Equations of motion:
M x  C x  K x  F
They are obtained using:
1)
2)
3)
[M] is the Mass matrix
[K] is the Stiffness matrix
[C] is the Damping matrix
Vector mechanics (Newton or D’ Alembert)
Hamilton's principles
Lagrange's equations
Un-damped Free Vibration: the eigenvalue problem
Equation of motion:
M q  K q  0
in terms of the generalized D.O.F. qi
Write the matrix equation as:
  Kq  0,
Mq
(1)
where M and K are the Mass and Stiffness matrices respectively.
 and q are the acceleration and displaceme nt vectors respectively.
q
premultipl y equation (1) by M
-1
M KA
-1
-1
. Note that M M  I (unit matrix)
the system matrix. Equation 1 becomes :
  Aq  0
Iq
(2)
Assuming harmonic motion:
q  q, where    2 , Equation (2) becomes
A - I{q}  0
(3)
The characters tic equation of the system is the determinan t
equated to ZERO, or
A - I  0,
(4) , the roots i of the
characters tic equation are called the eigenvalues and the natural
frequencie s of the system are determined from them by the relation
i  i2
(5)
By substituti ng i into the matrix equation (3), we obtain the correspond ing
mode shape X i which is called the eigenvector.
It is also possible to find the eigenvecto rs from the adjoint matrix
of the system. Let B  A - I, and start with the definition of the
inverse
B-1 
adjB
. Premultipl y by B B to obtain,
B
B I  B adj B, or
A - I I  A - IadjA - I
(6)
If now we let   i , an eigenvalue , then the determinan t on the
left side of the equation is zero,
0  A - i IadjA - i I
The above equation is valied for all values i and represents " n"
equations for the n - degrees of freedom system. Comparing
this equation w ith equation (4) for the i th mode
A - i I{q}i  0
, we recognize that the adjoint matrix adjA - i I
must consists of columns, each of which is the eigenvecto r q i
(multiplie d by an arbitraray constant)
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
0 
1 / 2m

1 / m
 0
(3k / 2m)
M 1 K   A  
  ( k / m)
M 1
 ( k / 2 m) 
(k / m) 
By letting    2 , equation (a) becomes
(3k / 2m)  
  ( k / m)

 (k / 2m)   x1  0

(k / m)     x2  0
(b)
The characteristic equation from the determinant of the above matrix is
2
5 k 
k 
2
 
     0,
2 m
m
1 k
k
1 
2  2
2m
m
(c), from which
(d)
The eigenvectors can be found from Eqn.(b) by substituting the above values of
. The adjoint matrix from Eqn. (b) is
(k / m)  i
AdjA  I   
 (k / m)

(3k / 2m)  i 
(k / 2m)
Substituting 1 into Eqn. (e) we obtain:
0.5

1.0
0.5 k

1.0  m
Here each column is already normalized to unity and the first eigenvector is
0.5
X1   
1.0 
Similarly when   2  0.5k/m) the adjoint matrix gives;
 1.0

 1.0
0.5  k

 0.5 m
Normalizing to Unity;
 1.0

 1.0
 1.0 k

1.0  m
The second eigenvector from either column is;
  1 .0 
X 
2  1.0 