7.3 Complex damped natural modes
7.3.1 Viscous damping
7.3.1.2 State space formulation
damping. The terms of the series (7.78) are not quite the same as the usual single-degree-of-freedom frequency response function owing to the i
Tr term in the numerator. Nevertheless, each term can be evaluated independently of all other terms, so the set of modes used in the analysis are uncoupled. Note that the frequency dependence in equation (7.78) is confined to the 2 and i terms. The
Sr ,
Tr and Z terms do not vary with frequency. rThe analytical solution of the quadratic eigenvalue problem is not straightforward. A technique used to circumvent this is to reformulate the original second order equations of motion for an n-degree-of-freedom system into an equivalent set of 2n first order differential equations, known as ‘Hamilton’s canonical equations’. This method was introduced by W. J. Duncan in the 1930’s [7.9] and more fully developed by K. A. Foss in 1958 [7.7].
7.3.1.2 State space formulation
In the terminology of control theory, the system response is defined by a
‘state vector’ of order 2n. In a typical mechanical system, its top n elements give
the displacements and its bottom n elements give the velocities at the n coordinates of the system (or vice-versa, depending how the equations are written).
The equations for the forced vibrations of a viscously damped system are
m
x
c
x
k
x
f
t
. (7.79) If one adds to equation (7.79) the trivial equation
m
x
m
x 0 ,the resulting equations may be written in block matrix form
This matrix equation can also be written as
A
y
B
y N , (7.80)is called state vector.
The great advantage of this formulation lies in the fact that the matrices
A and
B , both of order 2n, are real and symmetric.The solution of (7.80) by modal analysis follows closely the procedure used for undamped systems. Consider first the homogeneous equation where
N 0 :
A
y
B
y 0 . (7.83) The solution of (7.83) is obtained by letting
y t
Y et, (7.84) where
Y is a vector consisting of 2n constant elements.Equation (7.84), when introduced in (7.83), leads to the eigenvalue problem
B
Y
A
Y , (7.85)
which can be written in the standard form
E
Y
Y
1 , (7.86)
where the companion matrix
In general
B will have an inverse except when the stiffness matrix is singular, i.e. when rigid-body modes are present.Equations (7.86) can be written
1
0where
I is the identity matrix of order 2n. They have non-trivial solutions if
1
0which is the characteristic equation.
Solution of equation (7.89) gives the 2n eigenvalues.
Corresponding to each eigenvalue r there is an eigenvector
Y r having 2n components. There are 2n of these eigenvectors. They satisfy equation (7.85)
B Y r r
A Y r . (7.90)
Consider the square complex matrix
Y , constructed having the 2n eigenvectors
Y r as columns, and the diagonal matrix
Λ whose diagonal elements are the complex eigenvalues
Y
Y 1 Y 2
Y 2n,
Λ diag
r . (7.91)Orthogonality of modes
The proof of the orthogonality of eigenvectors can proceed in the same way as for the undamped system.
Write equations (7.91) as
B Y A Y . (7.92)
Premultiply equation (7.92) by
Y T to obtain
Y T B Y Y T A Y . (7.93)
Transpose both sides, remembering that
A and
B are symmetric and
is diagonal, and obtain
Y T B Y Y T A Y . (7.94)
From equations (7.93) and (7.94) it follows that
Y T A Y Y T A Y . (7.95) Thus, if all the eigenvalues r are different, then
Y T A Y is a diagonal matrix, and from equations (7.93) or (7.94) also
Y T B Y isdiagonal.
We can denote
Y T A Y a ,
Y T B Y b , (7.96) which means
Y Tr
A
Y r ar,
Y Tr
B
Y r br,
Y Ts
A
Y r 0,
Y Ts
B
Y r 0, rs.These orthogonality conditions state that both
A and
B are diagonalized by the same matrix
Y . The diagonal matrices
a and
b can be viewed as normalization matrices related by
a 1 b .(7.97) For a complex eigenvector only the relative magnitudes and the differences in phase angles are determined. The matrices
a and
b are complex. Hence the normalization of a complex eigenvector consists of not onlyscaling all magnitudes proportionally, but rotating all components through the same angle in the complex plane as well.
The matrix
Y can be viewed as a transformation matrix which relates the system coordinates
y to a set of modal coordinates
z
y
Y
z . (7.98)Steady-state harmonic response
Consider now the non-homogeneous equations (7.80) and determine the steady-state response due to sinusoidal excitation
f
fˆ eit. For
N
Nˆ eit,
y ~y eit,
z z~ eit, (7.99) equation (7.80) can be written as
A
~y
B
y~
Nˆ
i . (7.100)
Substituting (7.98) into (7.100), premultiplying by
Y T and taking into account the orthogonality properties (7.96), we obtain
i a b
z~
Y T
Nˆ . (7.101)This is a set of 2n uncoupled equations, from which
z~ can be obtained as
z~
i
a b
1
Y T
Nˆ (7.102)and
~ from equation (7.98) as y
~y
Y
i
a b
1
Y T
Nˆ . (7.103)Since in the underdamped case, in which we are primarily interested, all eigenvectors are complex and occur in conjugate pairs, based on (7.69) and (7.82), the matrix
Y can be partitioned as follows
Y , (7.104)
where
is a diagonal matrix of order n, which contains the complex eigenvalues with positive imaginary part, and
is called the complex modal matrix of order n, which contains the complex vectors of modal displacements, corresponding to the eigenvalues in
. Matrices
and
are the complex conjugates of
and
, respectively.From equations (7.102), (7.103) and (7.104) it follows that the top n
equation (7.105) can also be written
Equation (7.107) represents the steady-state response to sinusoidal forces of amplitudes
fˆ in terms of the complex modes
r and
r
r1,2,..,n
.Comparison of complex and real modes
Complex modes
r can be represented in the complex plane by vector diagrams, in which each component of the modal vector is represented by a line of corresponding length and inclination, emanating from the origin. Figure 7.2 shows the ‘compass plots’ of two almost real modesFig. 7.2
In the case of non-proportional damping, the complex conjugate eigenvectors are of the form
r r i
r,
r ri r. (7.108) The free vibration solution can be written as the sum of two complex eigensolutions associated with the pair of eigenvalues and eigenvectors
components of rotating vectors. The contribution of the r-th mode to the motion of a point j can be expressed as rotate at the same angular velocity r, and all decay in amplitude at the same rater, but each has a different phase angle in general, while the position relative to the other coordinates is preserved.
The motion is synchronous, but each coordinate reaches its maximum excursion at a different time than the others. However, the sequence in which the coordinates reach their maximum remains the same for each cycle. Furthermore, after one complete cycle, the coordinates are in the same position as at the beginning of the cycle. Therefore, the nodes (if they may be termed as such) continuously change their position during one cycle, but during the next cycle the pattern repeats itself. Of course, the maximum excursions decay exponentially from cycle to cycle.
Complex modes exhibit non-stationary zero-displacement points, at locations that change in space periodically, at the rate of vibration frequency.
In the case of proportional damping, the complex mode shape
r in equation (7.108) is replaced by a real mode
u r. The angle r is either 0 or 0 180 depending on the sign of 0 ujr. The components of
x
t
r in the complex plane rotate at the same angular velocity r with amplitudes decaying exponentially with time and uniformly over the system, at a rate r, but lie on the same line (they are in phase or 180 out of phase with each other). All points reach 0 maximum departures from their equilibrium positions or become zero at the same instants.Real modes exhibit stationary nodes.
7.3.2 Structural damping
Consider the equations of the forced harmonic motion of a system with structural damping
m
x
k i d
x
~f eit, (7.111)where
m and
kid
are symmetric matrices of order n,
~f is a complex vector of excitation forces and
x x~ eit, (7.112) where
x~ is the vector of complex displacement amplitudes.Denoting 2, consider the homogeneous equation
kid m
x~ 0 . (7.113) Equations of this sort for structural damping are usually regarded as being without physical meaning, because they are initially set up on the understanding that the motion they represent is forced harmonic. However, there is no objection [7.10] to defining damped principal modes
r of the system as being the eigenvectors of this equation, corresponding to which are complex eigenvalues r satisfying the homogeneous equation
kid r m
r 0 . (7.114) In the following it is considered that the n eigenvalues are all distinct, and the corresponding modal vectors are linearly independent.Mead [7.10] showed that such complex damped modes do have a clear physical significance if they are considered damped forced principal modes.
Indeed, let take
~f to be a column of forces equal to gi times the inertia forces corresponding to the harmonic vibration (7.112)
f~ ig2
m
~x . (7.115)Substituting (7.112) and (7.115) into equation (7.111) we find
kid 2 1ig m
x~ 0 . (7.116)Consider first the special case of proportional damping, in which
d k . We then have
1i k 2 1ig m
x~ 0 (7.117)which is satisfied by g and real vectors
~x u .By equating to zero both real and imaginary parts we have
k 2 m
u 0 (7.118)which yields the undamped principal modes and natural frequencies. Thus, if damping is distributed in proportion to the stiffness of the system, the undamped modes can be excited at their natural frequencies by forces which are equal to gi times the inertia forces. The damped forced normal modes are then identical to the undamped normal modes.
When the damping matrix is not proportional to the stiffness matrix, there is no longer a unique value of . Equation (7.117) must be retained in its general form, and the complex eigenvalues r2
1igr
and corresponding complex modal vectors
r satisfy the equation
k r2 1igr m i d
r 0 . (7.119) It is easy to show that the modal vectors satisfy the orthogonality conditions
k d
, , mr T
s
r T
s
0 i
0
rs (7.120)
which do not contain the frequency of excitation or the natural frequencies of modes.
Equation (7.119) may be premultiplied by
Tr to show that
r
rare the ‘complex modal mass’ and the ‘complex modal stiffness’, respectively.
As the n eigenvectors are linearly independent, any vector
~ in their x space can be expressed as linear combination of the eigenvectors
Substituting (7.123) in (7.111) and using equations (7.120)-(7.122), we get
resonating at the frequency r with the loss factor g . The system vibrates in the r complex mode
r.Hence, the solution of equation (7.111) is
The displacement at coordinate j produced by a harmonic force applied at coordinate is given by
Denoting the complex modal constant
the complex displacement amplitude (7.127) becomes