Analysis based on the dynamic stiffness matrix

7.4 Forced monophase damped modes

Apart from the complex forced vibration modes discussed so far, there is another category of damped vibration modes defined as real forced vibration modes. These are described by real monophase vectors whose components are not constant, but frequency dependent, and represent the system response to certain monophase excitation forces. They are independent of the type of damping, viscous, structural, frequency dependent or a combination of these. At each undamped natural frequency, one of the monophase response vectors coincides with the corresponding real normal mode. The real forced vibration modes are particularly useful for the analysis of structures with frequency dependent stiffness and damping matrices. The existence of modes of this general type appears to have been pointed out first by Fraeijs de Veubeke [7.11], [7.12].

7.4.1 Analysis based on the dynamic stiffness matrix

For harmonic excitation, the equations of motion (7.40) and (7.56) of a system with combined viscous and structural damping can be written

 

^m

 

^x

 

^c _¹

 

^{d }

 

^{x }

 

^k

 

^{x }

 

^{fˆ eit}

Assuming a steady-state response (7.112)

 



x t

  

 ~x e^it,

where

 

x~ is a vector of complex displacement amplitudes, equation (7.130) becomes

         



^{k 2 m i  c  d}

 ^{ }

^{~x }

 

^fˆ

or



^Z

 

^i

  

^{~x }

 

^{fˆ , (7.131)}

where



Z

 

i



is referred to as the dynamic stiffness matrix.

This can be written



Z

 

i







Z_R

 





i



Z_I

 





, (7.132) where the real part and the imaginary part are given by



Z_R

 



  

 k ²

 

m ,



Z_I

 







   

c  d .

The same formulation applies in the case of frequency dependent stiffness and damping matrices



^Z_R

 

^



^



^k

 

^



^2

 

^{m ,}



Z_I

 









c

 



  

 d , and, in fact, is independent of the type of damping.

Following the development in [7.13], it will be enquired whether there are (real) forcing vectors

 

fˆ such that the complex displacements in

 

~ are all x in phase, though not necessarily in phase with the force. For such a set of displacements, the vector

 

x~ will be of the form

   

~x  xˆ e^i, (7.133) where

 

xˆ is an unknown vector of real amplitudes and  is an unknown phase lag. Substitution of this trial solution into equation (7.131) yields



^ZRi^ZI

  

^xˆ cosisin





 

^fˆ . Separating the real and imaginary parts, we obtain

   

    

   



^ZZR ^ZZI

 

^ˆxˆx

 

^{fˆ ,,}

R I

















sin cos

0 sin

cos (7.134)

or

     

   

^ZZI ^xˆxˆ

 

^{fˆfˆ .,}

R



 sin

cos



 (7.135)

Denoting frequency-dependent generalized symmetric eigenvalue problem. The eigenvalues

 are solutions of the algebraic equation

   

Mode labeling

Consider now the way in which the phase angles _r and the vectors

 

 _r are labeled. When equation (7.141) was derived from equation (7.139), it was assumed that cos0. If, however, cos0, equation (7.139) becomes

 

Z_R

 

 0,

  

^{k 2}

 

^m

 ^{   }

^{  0 , (7.144)}

and the condition for

 

 to be non-trivial is that

   

 

⁰

det k ² m  . (7.145)

This means that  must be an undamped natural frequency. If then

s

 , the

 

 mode corresponding to this value of  and the solution 0

cos  may be identified with the s-th real normal mode

 

u _s. If the  solution and the corresponding value of

 

 are labeled _s and

 

 _s, respectively, then one may write

2

_s  ,

   

 _s u _s, when _s. (7.146) For this value of  there will also be n1 other

 

 _r modes corresponding to the remaining n1 roots _r of equation (7.141).

Equation (7.146) may be used to give a consistent way of labeling the  and the

 

 for values of  other than the undamped natural frequencies [7.13].

Each of the roots  of equation (7.141) is a continuous function of , so that

 





 , and  

 

 . Equation (7.142) shows that

     

 

^T_r

 

^R_I

 

_r^r

T r r

r Z

Z







 

 tan^1  . (7.147)

When 0, _r is a small positive angle. As  grows and approaches

1, one of the roots 

 

 will tend to zero, and 

 

 will approach the value

 2; let this angle be labeled ₁

 

 .

As  is increased, ₁

 

 grows larger than  2 and when it is increased indefinitely, ₁

 

 will approach the value . The remaining n1 angles _s

 

 may be labeled in a similar way: _s

 

 is that phase shift which has the value  2 when _s. The angles _r are referred to as characteristic phase lags.

The forced modes are labeled accordingly. At any frequency _s, the shape of the s-th forced mode is given by the solution





 





_s of

   



ZR ^tan1s ZI

 ^{   }

 _s ^{ 0} . (7.148) Thus,

 

 _s is the forced mode which coincides with

 

u _s when _s. Orthogonality

It may be shown that the modal vectors satisfy the orthogonality conditions

    

    

Z . , Z

I r T

s R r T

s

0 0













rs (7.149)

These conditions imply that

       

 ^T_s  _r   ^T_r  _s0, rs (7.150) hence an excitation modal vector

 

 _r introduces energy into the system only in the corresponding response modal vector

 

 _r.

Damped modal coordinates

If a square matrix

 

 is introduced, which has the monophase response modal vectors as columns

 

^{ }_^

   

^ ₁ ^ ₂ 

 

^ _n^, (7.151) then the motion of the system can be expressed in terms of the component motions in each of the forced modes

 

 _r. Thus the vector of complex displacements

 

x~ may be written

       





 ⁿ

r

r~pr

~p

~x

1



 , (7.152)

where the multipliers ~ are the damped modal coordinates. The linear p_r transformation (7.152) is used to uncouple the equations of motion (7.130).

Steady state response

Substituting (7.152) into (7.131) and premultiplying by

 

^{ T} we obtain

 

^{ T}



^Z

 

^i

  

^

 

^{p~ }

 

^{ T}

 

^{fˆ , (7.153)}

or

 

^zR

 





i



^zI

 



   



 

^~p 

 

 ^T

 

^fˆ , (7.154) where, due to the orthogonality relations (7.149),



z_R

 



      

  ^T Z_R  ,



z_I

 



      

  ^T Z_I  , (7.155) are both diagonal matrices.

The solution of the r-th uncoupled equation (7.154) is

   

Substituting the damped modal coordinates (7.156) into (7.152) we obtain the solution in terms of the monophase response modal vectors

       

The response modal vectors

 

 _r can be normalized to unit length

   

 ^T_r  _r 1. (7.158)

using the frequency dependent scaling factors [7.14]

 

_r

 

_I^r

 

_r where _rs is the Kronecker delta. The two sets of frequency dependent monophase modal vectors form a bi-orthogonal system. While the response vectors are right

eigenvectors of the matrix pencil

    

Z_R , Z_I



, the excitation vectors are left eigenvectors of that pencil.

Equation (7.163) implies

which, using (7.163), can be written

     

monophase response modal vectors as columns

 

 _^

   

 ₁  ₂ 

 

 _n^, (7.167) equations (7.166) yield

     

 ^T Z_R   cos_r



, (7.168, a)

     

 ^T Z_I   sin_r



, (7.168, b) and

    

^{ T Z  }

 

^{eir . (7.169)}

The dynamic stiffness matrix is given by

   

^{Z   T}

 

^eir

^{ }

^{ 1. (7.170)}

Its inverse, the dynamic flexibility or frequency response function (FRF) matrix, is

 

Z ^1

   

H  

 

e^ir

 

 ^T. (7.171) The FRF matrix has the following modal decomposition

 

ⁿ

   

^T_r

or, in terms of the unscaled vectors used to illustrate some of the concepts presented so far. The system parameters are given in appropriate units



kg,N m,Ns m



.

The system has non-proportional damping.

Fig. 7.3

The undamped natural frequencies _r 2 and the real normal modes

 

u _r (of the associated conservative system) are given in Table 7.1.

Table 7.1. Modal data of the 4-DOF undamped system

Mode 1 2 3 4

r, Hz ^{1.1604 2.0450 3.8236 4.7512}

1 -0.26262 -0.06027 -0.02413

Modal 0.37067 1 -0.17236 -0.06833

vector 0.18825 0.18047 0.78318 1

0.08058 0.07794 1 -0.40552

The real normal modes are illustrated in Fig. 7.4.

Fig. 7.4

The damped natural frequencies, the modal damping ratios and the magnitude and phase angle of the complex mode shapes are given in Table 7.2.

Table 7.2. Modal data of the 4-DOF damped system

Mode 1 2 3 4

r, Hz 1.1598 2.0407 3.8228 4.7423

r 0.0479 0.0606 0.0313 0.0500

1 0 0.26473 -173.3 0.06210 -167.3 0.02378 -178.5

Modal 0.37067 3.66 1 0 0.17322 -174.4 0.06882 -171.9

vector 0.18825 3.52 0.18047 2.06 0.77950 -6.86 1 0

0.08058 7.33 0.07794 3.88 1 0 0.40241 172.29

Fig. 7.5

Fig. 7.6 (from [7.16])

Fig. 7.7 (from [7.16])

Fig. 7.8

The real eigenvalues of the generalized eigenproblem (7.142) are plotted versus frequency in Fig. 7.5. The monophase modal response vectors are illustrated in Fig. 7.6. The monophase modal excitation vectors are presented in Fig. 7.7. The characteristic phase lags are shown as a function of frequency in Fig. 7.8.

7.4.2 Analysis based on the dynamic flexibility matrix

In document Mircea Rades - Mechanical Vibrations 2, Structural Dynamic Modeling (Page 36-47)