Modal Models

Structures

2.2 Second-Order Structural Models

2.2.2 Modal Models

The second-order models are defined in modal coordinates. These coordinates are often used in the dynamics analysis of complex structures modeled by the finite elements to reduce the order of a system. It is also used in the system identification procedures, where modal representation is a natural outcome of the test.

Modal models of structures are the models expressed in modal coordinates. Since these coordinates are independent, it leads to a series of useful properties that simplify the analysis (as will be shown later in this book). The modal coordinate representation can be obtained by the transformation of the nodal models. This transformation is derived using a modal matrix, which is determined as follows.

Consider free vibrations of a structure without damping, i.e., a structure without external excitation (u { 0) and with the damping matrix D = 0. The equation of motion (2.7) in this case turns into the following equation:

(2.8) Mq + Kq = 0.

The solution of the above equation is q Ie^{j tZ} . Hence, the second derivative of the solution is q Z I² e^{j tZ} . Introducing the latter q and into (2.8) givesq

(KZ2M e)I ^{j tZ} 0. (2.9)

This is a set of homogeneous equations, for which a nontrivial solution exists if the determinant of KZ²M is zero,

det(KZ2M) 0. (2.10)

The above determinant equation is satisfied for a set of n values of frequency Z. These frequencies are denoted Z Z₁, ₂,...,Z_n, and their numbern does not exceed the number of degrees of freedom, i.e., n nd _d. The frequency Z_i is called the ith natural frequency.

Substituting Z_i into (2.9) yields the corresponding set of vectors

^

I I1, ,...,2 I_n

`

that satisfy this equation. The ith vector I_i corresponding to the ith natural frequency is called the ith natural mode, or ith mode shape. The natural modes are not unique, since they can be arbitrarily scaled. Indeed, if I_i satisfies (2.9), so does DIi, where D is an arbitrary scalar.

For a notational convenience define the matrix of natural frequencies

1 2

0 0

0 0

0 0 _n

Z Z

Z

ª º

« »

« »

: « »

« »

« »

¬ ¼

"

"

" " " "

"

(2.11)

and the matrix of mode shapes, or modal matrix ), of dimensions which consists of nnatural modes of a structure

d , n un

> @

11 21 1

12 22 2

1 2

1 2

...

d d d

n n n

n n nn

I I I

I I I

I I I

I I I

ª º

« »

« »

) « »

« »

« »

¬ ¼

!

!

" " " "

!

, (2.12)

whereI_ij is the jth displacement of the ith mode, that is,

1 2 i i i

in

I I I

I

­ ½

° °

° ° ® ¾

° °

° °

¯ ¿

# . (2.13)

The modal matrix ) has an interesting property: it diagonalizes mass and stiffness matricesMandK,

T ,

Mm ) M) (2.14)

T .

Km ) K) (2.15)

The obtained diagonal matrices are called modal mass matrix and modal stiffness matrix ( The same transformation, applied to the damping matrix

(M_m)

m).

K

T ,

Dm ) D) (2.16)

gives the modal damping matrix , which is not always obtained as a diagonal matrix. However, in some cases, it is possible to obtain diagonal. In these cases the damping matrix is called a matrix of proportional damping. The proportionality of damping is commonly assumed for analytical convenience. This approach is justified by the fact that the nature of damping is not known exactly, that its values are rather roughly approximated, and that the off-diagonal terms in most cases—as will be shown later—have negligible impact on the structural dynamics. The damping proportionality is often achieved by assuming the damping matrix as a linear combination of the stiffness and mass matrices; see [18], [70],

Dm

Dm

1 2

D D KD M, (2.17)

whereD₁ and D₂ are nonnegative scalars.

Modal models of structures are the models expressed in modal coordinates. In order to do so we use a modal matrix to introduce a new variable, , called modal displacement. This is a variable that satisfies the following equation:

qm m.

q )q (2.18)

In order to obtain the equations of motion for this new variable, we introduce (2.18) to (2.7) and additionally left-multiply (2.7) by )^T, obtaining

, .

T T T T

m m m

oq m ov m

M q D q K q Bo

y C q C q

) ) ) ) ) ) )

) )

u

Assuming a proportional damping, and using (2.14), (2.15), and (2.16) we obtain the above equation in the following form:

, .

m m m m m m T o

oq m ov m

M q D q K q B u

y C q C q

)

) )

Next, we multiply (from the left) the latter equation by M_m¹, which gives

1 1 1

, .

m m m m m m m m T o

oq m ov m

q M D q M K q M B u

y C q C q

)

) )

The obtained equations look quite messy, but the introduction of appropriate notations simplifies them,

(2.19)

2 2 ,

.

m m m

mq m mv m

q q q B

y C q C q

=: :

mu

In (2.19) : is a diagonal matrix of natural frequencies, as defined before. Note, however, that this is obtained from the modal mass and stiffness matrices as follows:

2 M K_m1 _m.

: (2.20)

In (2.19) is the modal damping matrix. It is a diagonal matrix of modal damping,=

1 2

0 0

0 0

0 0 _n

] ]

]

ª º

« »

« »

= « »

« »

« »

¬ ¼

"

"

" " " "

"

, (2.21)

where]_i is the damping of theith mode. We obtain this matrix using the following relationship M D_m¹ _m =:2 , thus,

1 1

2 2

1 1

0.5M D_{m m} 0.5M K_{m m m}

= : D . (2.22)

Next, we introduce the modal input matrix B_m in (2.19),

1 ^T .

m m o

B M) B (2.23)

Finally, in (2.19) we use the following notations for the modal displacement and rate matrices:

mq oq ,

C C ) (2.24)

mv ov .

C C ) (2.25)

Note that (2.19) (a modal representation of a structure) is a set of uncoupled equations. Indeed, due to the diagonality of : and =, this set of equations can be written, equivalently, as

(2.26)

2

1

2

, 1, ,

,

mi i i mi i mi mi

i mqi mi mvi mi n

i i

q q q b u

y c q c q i n

y y

] Z Z

¦

! ,

where b_mi is the ith row of B_m and are the ith columns of and , respectively. The coefficient

mqi,

c c_mvi C_mq

Cmv ]_i is called a modal damping of the ith mode. In the above equations is the system output due to the ith mode dynamics, and the quadruple

yi

( , ,Z ]_{i i} b_mi,c_mi) represents the properties of the ith natural mode. Note that the structural response y is a sum of modal responses yi, which is a key property used to derive structural properties in modal coordinates.

This completes the modal model description. In the following we introduce the transfer function obtained from the modal equations. The generic transfer function is obtained from the state-space representation using (2.6). For structures in modal coordinates it has a specific form.

Transfer Function of a Structure. The transfer function of a structure is derived from (2.19),

(2.27)

2 2 1

( ) ( _{mq mv})( _n 2 ) _m.

GZ C j CZ : Z I j ZZ : B

However, this can be presented in a more useful form, since the matrices: and = are diagonal, allowing for representation of each single mode.

Transfer Function of a Mode. The transfer function of the ith mode is obtained from (2.26),

2 2

( )

( ) .

2

mqi mvi mi

mi

i i

c j c b

G j

Z Z

Z Z ] Z Zi

(2.28)

The structural and modal transfer functions are related as follows:

Property 2.1. Transfer Function in Modal Coordinates. The structural transfer function is a sum of modal transfer functions

(a)

1

( ) ( )

n mi i

GZ

¦

G Z ^(2.29)

or, in other words,

2 2

1

( )

( ) ,

2

n mqi mvi mi

i i

i

c j c b

G j

Z Z

Z Z ] Z Zi

¦

^(2.30)

and the structural transfer function at the ith resonant frequency is approximately equal to the ith modal transfer function at this frequency

(b) ( ₂ )

( ) ( )

2

mqi i mvi mi

i mi i

i i

jc c b

G G Z

Z Z

] Z

# , i 1,!, .n (2.31)

Proof. By inspection of (2.27) and (2.28). ‹

Structural Poles. The poles of a structure are the zeros of the characteristic equations (2.26). The equation s²2] Z_{i i}sZ_i² 0 is the characteristic equation of the ith mode. For small damping the poles are complex conjugate, and in the following form:

1 2

2 2

1 ,

1 .

i i i i

i i i i

s j

s j

] Z Z ]

] Z Z ]

(2.32)

The plot of the poles is shown in Fig. 2.1, which shows how the location of a pole relates to the natural frequency and modal damping.

Example 2.2. Determine the modal model of a simple structure from Example 2.1.

The natural frequency matrix is

3.1210 0 0 0 2.1598 0 0 0 0.7708

ª º

« »

: « »

« »

¬ ¼

,

and the modal matrix is

(a)

0.5910 0.7370 0.3280 0.7370 0.3280 0.5910 0.3280 0.5910 0.7370

ª º

« »

) « »

« »

¬ ¼

.

The modes are shown in Fig. 2.2.

] Zi i

arcsin( )

i i

D ]

Zi

Im

0 Re

s2

s1

1 2

i i

Z ]

1 2

i i

Z ] Zi

Figure 2.1. Pole location of theith mode of a lightly damped structure: It is a complex pair with the real part proportional to the ith modal damping; the imaginary part approximately equal to the ith natural frequency; and the radius is the exact natural frequency.

The modal mass is M_m I₃, the modal stiffness is K_m :², and the modal damping, from (2.22), is

0.0156 0 0

0 0.0108 0

0 0 0.0039

ª º

« »

= « »

« »

¬ ¼

.

We obtain the modal input and output matrices from (2.23), (2.24), and (2.25):

0.3280 0.5910 , 0.7370 Bm

ª º

« »

« »

« »

¬ ¼

0.5910 0.7370 0.3280

0 0 0

0 0 0

Cmq

ª º

« »

« »

« »

¬ ¼

,

and

0 0 0

0.5910 0.7370 0.3280 0.3280 0.5910 0.7370 Cmv

ª º

« »

« »

« »

¬ ¼

.

0.591

m

1

m

₂

m

₃

0.328 0.591 0.737

m

1

m

₂

m

₃

0.737 0.328 –0.591

m

1

m

₂

m

₃

m

2

m

₃

m

1

–0.737 0.328

equilibrium

I

1—mode 1

I

2^{—mode 2}

I

3^{—mode 3}

Figure 2.2. Modes of a simple system: For each mode the mass displacements are sinusoidal and have the same frequency, and the displacements are shown at their extreme values (see the equation (a)).

Example 2.3. Determine the first four natural modes and frequencies of the beam presented in Fig. 1.5.

Using the finite-element model we find the modes, which are shown in Fig. 2.3.

For the first mode the natural frequency is Z₁ 72.6rad/s, for the second mode the

natural frequency is Z₂ 198.8rad/s, for the third mode the natural frequency is

3 386.0

Z rad/s, and for the fourth mode the natural frequency isZ₄ 629.7 rad/s.

–1 –0.5 0 0.5 1

mode 3 mode 1

mode 2 mode 4

displacement,y-dir.

0 2 4 6 8 10 12 14

node number

Figure 2.3. Beam modes: For each mode the beam displacements are sinusoidal and have the same frequency, and the displacements are shown at their extreme values.

Example 2.4. Determine the first four natural modes and frequencies of the antenna presented in Fig. 1.6.

We used the finite-element model of the antenna to solve this problem. The modes are shown in Fig. 2.4. For the first mode the natural frequency is

1 13.2

Z rad/s, for the second mode the natural frequency is Z₂ 18.1 rad/s, for the third mode the natural frequency is Z₃ 18.8 rad/s, and for the fourth mode the natural frequency isZ₄ 24.3 rad/s.

Example 2.5. The Matlab code for this example is in Appendix B. For the simple system from Fig. 1.1 determine the natural frequencies and modes, the system transfer function, and transfer functions of each mode. Also determine the system impulse response and the impulse responses of each mode. Assume the system masses m₁ m₂ m₃ 1, stiffnesses k₁ k₂ k₃ 3, k₄ 0, and the damping matrix proportional to the stiffness matrix,D= 0.01K or d_i 0.01k_i,i= 1, 2, 3, 4.

There is a single input force at mass 3 and a single output: velocity of mass 1.

We determine the transfer function from (2.27), using data from Example 2.2. The magnitude and phase of the transfer function are plotted in Fig. 2.5. The magnitude plot shows resonance peaks at natural frequencies Z₁ 0.7708 rad/s, Z₂ 2.1598 rad/s, and Z₃ 3.1210 rad/s. The phase plot shows a 180-degree phase change at each resonant frequency.

(b)

(c) (d)

(a)

Figure 2.4.Antenna modes: (a) First mode (of natural frequency 2.10 Hz); (b) second mode (of natural frequency 2.87 Hz); (c) third mode (of natural frequency 2.99 Hz); and (d) fourth mode (of natural frequency 3.87 Hz). For each mode the nodal displacements are sinusoidal, have the same frequency, and the displacements are shown at their extreme values. Gray color denotes undeformed state.

We determine the transfer functions of modes 1, 2, and 3 from (2.28), and their magnitudes and phases are shown in Fig. 2.6. According to Property 2.1, the transfer function of the entire structure is a sum of the modal transfer functions, and this is shown in Fig. 2.6, where the transfer function of the structure was constructed as a sum of transfer functions of individual modes.

The impulse response of the structure is shown in Fig. 2.7; it was obtained from (2.19). It consists of three harmonics (or responses of three modes) of natural frequencies Z₁ 0.7708 rad/s, Z₂ 2.1598 rad/s, and Z₃ 3.1210 rad/s. The

harmonics are shown on the impulse response plot, but are more explicit at the impulse response spectrum plot, Fig. 2.7, as the spectrum peaks at these frequencies.

Impulse response is the time-domain associate of the transfer function (through the Parseval theorem); therefore, Property 2.1 can be written in time domain as

1

( ) ( )

n i i

h t

¦

h t

whereh(t) is the impulse response of a structure and is the impulse response of the ith mode. Thus, the structural impulse response is a sum of modal responses.

This is illustrated in Fig. 2.8, where impulse responses of modes 1, 2, and 3 are plotted. Clearly the total response as in Fig. 2.7 is a sum of the individual responses.

Note that each response is a sinusoid of frequency equal to the natural frequency, and of exponentially decayed amplitude, proportional to the modal damping

i( ) h t

]i. Note also that the higher-frequency responses decay faster.

10^–1 10⁰ 10¹

10^–2 10^–1 10⁰ 10¹ 10²

Z2

frequency, rad/s

10^–1 10⁰

–400 –200 0 200

Z2

Z1

Z1 ^(b)

3 (a) Z

Z3

10¹ magnitudephase, deg

frequency, rad/s

Figure 2.5. Transfer function of a simple system: (a) Magnitude shows three resonance peaks; and (b) phase shows three shifts of 180 degrees; Z Z₁, ₂, and Z₃ denote the natural frequencies.

10^–1 10⁰ 10¹ 10^–4

10^–2 10⁰ 10²

(a)

frequency, rad/s

10^–1 10

frequency, rad/s

0 10

–500 –400 –300 –200 –100 0 100

(b)

mode 1 mode 1 mode 3

mode 2

structure

structure mode 2

mode 3

1

magnitudephase, deg

Figure 2.6.The transfer functions of single modes and of the structure: (a) Magnitudes; and (b) phases. The plots illustrate that the structure transfer function is a sum of modal transfer functions.

0 1 2

frequency, rad/s

3 4 5

0 0.5 1

Z3

(b)

Z2

Z1

0.5

300

0 50 100 150

time, s

200 250

–0.5 0

(a)

v1,velocity of mass 1

1.5

spectrum of v1

Figure 2.7.Impulse response of the simple system: (a) Time history; and (b) its spectrum.

Both show that the response is composed of three harmonics.

0 20 40 60 80 100 120 140 160 180

time, s

0 20 40 60 80 100 120 140 160 180

time, s

0 20 40 60 80 100 120 140 160 180 200

–0.4 –0.2 0 0.2 0.4

(c)

–0.4 200 –0.2 0 0.2 0.4

(b)

–0.4 200 –0.2 0 0.2 0.4

(a)

mode 1mode 2mode 3

time, s

Figure 2.8.Impulse responses of (a) first mode; (b) second mode; and (c) third mode. All show single frequency time histories.