Structures
5.3 Norms of a Single Mode
For structures in the modal representation, each mode is independent, thus the norms of a single mode are independent as well (they depend on the mode properties, but not on other modes).
5.3.1 The H2 Norm
Define 'Zi as a half-power frequency at the ith resonance, 'Zi 2] Zi i, see [18], [33]. This variable is a frequency segment at theith resonance for which the value of the power spectrum is one-half of its maximal value. The determination of the half- power frequency is illustrated in Fig. 5.1. The half-power frequency is the width of the shaded area in this figure, obtained as a cross section of the resonance peak at the height of hi/ 2, where is the height of the resonance peak. hi
Consider the ith natural mode and its state-space representation ( , see (2.52). For this representation we obtain the following closed-form expression for the H
, , )
mi mi mi
A B C
2 norm:
Property 5.1. H2 Norm of a Mode. Let be the transfer function of the ith mode. The H
( ) ( ) 1
i mi mi
G Z C j I AZ Bmi
2 norm of the ith mode is
2 2 2 2
2 2
2 2
mi mi mi mi
i i
i i i
B C B C
G J Zi.
] Z Z
# #
' ' (5.21)
Proof. From the definition of the H2 norm and (4.45) we obtain
2 tr( T ) ( 2 2) (2 ).
i mi mi ci mi mi i
G # C C W # B C ] Zi
We determine the norm of the second-order modal representation ( , , , )Z ]i i b ci i by replacing Bmi,Cmi with b ci, i, respectively. Note also that Gi 2is the modal cost of Skelton [124], Skelton and Hughes [126]. The Matlab function norm_H2.m in Appendix A.9 can be used to compute modal H2 norms.
0 0
mode 1 mode 2
2 2
G f G2 f
Z2
' 2
magnitude
1
1 2 3 4
frequency, rad/s
Figure 5.1. The determination of the half-power frequency, H2 norm and Hf norm for the second mode.
Example 5.1. In this example we illustrate the determination of the H2norm for a simple system as in Fig. 1.1. For this system, the masses are m1 11, and
while the stiffness coefficients are
2 5, m
3 10,
m k1 10, k2 50, k3 55, and
The damping matrix is proportional to the stiffness matrix D = 0.01K. The single input u is applied simultaneously to the three masses, such that
4 10.
k
1 ,
f u f2 2 ,u
3 5 ,
f u and the output is a linear combination of the mass displacements, where is the displacement of the ith mass and
1 2
2 2 3
y q q q3, qi fi is the force
applied to that mass.
The transfer function of the system and of each mode is shown in Fig. 5.2. We can see that each mode is dominant in the neighborhood of the mode natural frequency, thus the system transfer function coincides with the mode transfer function near this frequency. The shaded area shown in Fig. 5.3(a) is the H2 norm of the mode. Note that this area is shown in the logarithmic scale for visualization purposes and that most of the actual area is included in the neighborhood of the peak; compare with the same plot in Fig. 5.1 in the linear coordinates. The system H2norm is shown as the shaded area in Fig. 5.3(b), which is approximately a sum of areas of each of the modes.
The H2 norms of the modes determined from the transfer function are
1 2 1.9399,
G G2 2 0.3152, G3 2 0.4405, and the system norm is
2 2.0141.
G It is easy to check that these norms satisfy (5.25) since
2 2 2
2.0141 0.3152 0.4405 2.0141.
10–1 100 101 10 10–2
100 102
frequency, rad/s
10–1 100 101
–400 –300 –200 –100
mode 3
structure structure
mode 1 mode 3
mode 1
mode 2
mode 2 (b)
(a)
magnitude
2
phase, deg
102
frequency, rad/s
Figure 5.2. The transfer function of the structure (solid line) and of each mode: Mode 1 (dashed line), mode 2 (dash–dotted line), and mode 3 (dotted line).
5.3.2 The Hf Norm
The Hf norm of a natural mode can be expressed approximately in the closed-form as follows:
Property 5.2. Hf Norm of a Mode. Consider the ith mode (Ami,Bmi,Cmi) or ( , ,Z ]i i bmi,cmi). Its Hf norm is estimated as
2 2 2 .
2 2
mi mi mi mi
i
i i i i
B C b c
G f# ] Z ] Z
2 (5.22)
Proof. In order to prove this, note that the largest amplitude of the mode is approximately at the ith natural frequency; thus,
max 2
max
( )
( ( ))
2 2
mi mi
mi mi
i i i
i i i i
B C
G G V C B
V Z
] Z ] Z
f# 2 .
The modal Hfnorms can be calculated using the Matlab function norm_Hinf.m given in Appendix A.10.
10–1 100 101 10
–2 0 2
frequency, rad/s
10–1
–2
G2 2
G2
G2 f
Gf
(b) (a) 10
magnitude, mode 2
10
10
2
102
magnitude, structure
100
10
0 1 2
10 10 10
frequency, rad/s
Figure 5.3. H2 and Hf norms (a) of the second mode; and (b) of the system.
Example 5.2. In this example we illustrate the determination of the Hfnorm of a simple structure, as in Example 5.1, and of its modes.
The Hf norm of the second mode is shown in Fig. 5.3(a) as the height of the second resonance peak. The Hf norm of the system is shown in Fig. 5.3(b) as the height of the highest (first in this case) resonance peak. The Hf norms of the modes, determined from the transfer function, are G1 f#18.9229, G2 f#1.7454,
3 1.2176,
G f# and the system norm is Gf# G1 f#18.9619.
5.3.3 The Hankel Norm
This norm is approximately evaluated from the following closed-form formula:
Property 5.3. Hankel Norm of a Mode. Consider the ith mode in the state- space form (Ami,Bmi,Cmi), or the corresponding second-order form
( , ,Z ]i i bmi,cmi). Its Hankel norm is determined from
2 2 2 .
4 4
mi mi mi mi
i h i
i i i i
B C b c
G J
] Z ] Z
# 2 (5.23)
The modal Hankel norms can be calculated using the Matlab function norm_Hankel.m given in Appendix A.11.
5.3.4 Norm Comparison
Comparing (5.21), (5.22), and (5.23) we obtain the approximate relationships between H2, Hf, and Hankel norms
2 2
i i h i i
G f# G # ] Z Gi . (5.24)
The above relationship is illustrated in Fig. 5.4, using (5.21), (5.22), and (5.23), assuming the same actuator and sensor locations.
10–2 10–1 100 101 102
10–3 10–3
Hankel H2
Hf 103
102
101
modal norm
100
10–1
10–2
i i
] Z, rad/s Figure 5.4. Modal norms versus ] Zi i.
Example 5.3. The Matlab code for this example is in Appendix B. Consider a truss presented in Fig. 1.2. Vertical control forces are applied at nodes 9 and 10, and the output rates are measured in the horizontal direction at nodes 4 and 5. Determine the H2 and Hf norms for each mode.
The norms are given in Fig. 5.5(a). From (5.24) it follows that the ratio of the H2
and Hf norms is
2 0.707 ;
i
i i i
i
G
G ] Z Z
f
# '
hence, the relationship between the Hf and H2 norms depends on the width of the resonance. For a wide resonant peak (large 'Zi) the H2 norm of the ith mode is larger than the corresponding Hf norm. For a narrow resonant peak (small 'Zi) the Hf norm of the ith mode is larger than the corresponding H2 norm. This is visible in Fig. 5.5(a), where neither norm is dominant.
0 500 1000 1500 2000 2500
natural frequency, rad/s
3000 3500 4000
–4 –2 0 (a) 10
Hf and H2 norms 10
10
Figure 5.5. The 2D truss: (a) H2 (9) and Hf (x) approximate norms; and (b) the exact (9) and the approximate (x) Hankel singular values.
Next, we obtained the exact Hankel singular values, and the approximate values from (5.10) and (5.23), respectively, and they are shown in Fig. 5.5(b), where we can see a good coincidence between the exact and approximate values.