Actuator/Sensor Indices and Modal Indices

Structures

7.3 Placement Indices and Matrices

7.3.3 Actuator/Sensor Indices and Modal Indices

The placement matrix gives an insight into the placement properties of each actuator, since the placement index of the kth actuator is determined as the rms sum of the kth column of 6. (For convenience in further discussion we denote by

6the

placement matrix either of the two- or the infinity-norm.) The vector of the actuator placement indices is defined as and its kth entry is the placement index of the kth actuator. In the case of the H

>

1 2

...

^T

,

a a a aS

V V V V @

2

norm, it is the rms sum of the

kth actuator indexes over all modes,

2 1

, 1,

n

ak ik

i

V ¦ V

k ^{!, ,S}

S

@

(7.16)

and in the case of the H

_f

and Hankel norms it is the largest index over all modes

(7.17)

max( ), 1, , , 1, , ,

ak ik

i i n k

V V

! !

Similarly, we define the vector of the sensor placement indices as and its kth entry is the placement index of the

kth

sensor. In the case of the H

>

1 2

...

^T

,

s s s sR

V V V V

2

norm, it is the rms sum of the

kth sensor indexes over

all modes,

2 1

, 1,

n

sk ik

i

V ¦ V

k ^!

^{, ,}

^R

! R

(7.18)

and in the case of the H

_f

and Hankel norms it is the largest index over all modes

(7.19)

max( ), 1, , , 1, , ,

sk ik

i i n k

V V

!

We define the vector of the mode indices as and its

ith entry is the index of the ith mode. This entry is an rms sum of the ith mode

indices over all actuators

>

1 2

... @

^T

,

m m m mn

V V V V

2 1

, 1,

S

mi ik

k

V ¦ V

i ^!

^{, ,}

ⁿ

^(7.20)

or an rms sum of the ith mode indices over all sensors

2 1

, 1,

R

mi ik

k

V ¦ V

i ^{!, ,n}

^(7.21)

The actuator placement index, V

_ak

, is a nonnegative contribution of the kth actuator at all modes to the H

2

or H

_f

norms of the structure. The sensor placement index, V

_sk

, is a nonnegative contribution of the kth sensor at all modes to the H

2

or H

_f

norms of the structure. The mode index, V

_mi

, is a nonnegative contribution of the

ith mode for all actuators (or all sensors) to the H2

or H

_f

norms of the structure.

We illustrate the determination of the H

_f

actuator and modal indices for the pinned beam in Fig. 7.1. Six actuators are located on the beam and four modes are considered. The second mode index is the rms sum of indices of all actuators for this mode, and the third actuator index is the largest index of this actuator over four modes.

From the above properties it follows that the index V

_ak

⁽ V

sk

⁾ characterizes the importance of the kth actuator (sensor), thus it serves as the actuator (sensor) placement index. Namely, the actuators (sensors) with small index V

_ak

⁽ V

sk

^{) can}

be removed as the least significant ones. Note also that the mode index V

_mi

^{can be}

used as a reduction index. Indeed, it characterizes the significance of the ith mode

for the given locations of sensors and actuators. The norms of the least significant

modes (those with the small index V

_mi

) should either be enhanced by the

reconfiguration of the actuators or sensors, or be eliminated.

Figure 7.1. Determination of the Hf actuator and modal indices of a pinned beam (Ø— actuator location; and ª—actuators used for the calculation of the indices): The mode index is the rms sum of indices of all six actuators for this mode, while the actuator index is the largest of the actuator indices over four modes.

Example 7.1.

Consider the 2D truss from Fig. 1.2. It is excited in the y-direction by an actuator located at node 4. Accelerometers serve as sensors. The task is to find four accelerometer locations within all 16 possible locations, that is, within all but 1 and 6 nodes, in the x- and y-directions. Assume the unit weights for all modes, and chose the 2-norm indices for the analysis.

We calculated the placement indices V

_si

, i = 1,...,16, of each accelerometer location and show them in Fig. 7.2 for lower (2–5) nodes of the truss, and in Fig. 7.3 for upper (7–10) nodes of the truss. The left column of these figures represents the H

2

index V

_si

for the x-direction accelerometers, while the right column represents the index for the y-direction accelerometers. The largest value indices are for nodes 5, 10, 4, and 9, all in the y-direction. Note that the chosen locations are the nodes at the tip in the same direction, and that a single accelerometer would probably do the same job as the four put together. This problem is addressed in the following section.

Example 7.2. Placing two sensors on a beam for the best sensing of up to four modes. The Matlab code for this example for n= 15 elements is in Appendix B.

Consider a beam as in Subsection 1.1.4 for n = 100 elements, and shown in Fig. 7.4

with a vertical force at node

n_a

40 . Using the presented above H

_f

placement technique find the best place for two displacement sensors in the

y-direction to sense

the first, second, third, and fourth mode, and to sense simultaneously the first two modes, the first three modes, and the first four modes.

0 5 10 15

0 0.5

0 5 10 15

0 5 10 15

0 0.5

0 5 10 15

0 0.5

node 3

0 5 10 15

0 0.5

0 5 10 15

0 0.5

node 5 node 4

0.5

00 15

10 5

00

0

x-direction y-direction

0.5

node 2 node 2

node 3 node 4

0.5

node 5

5 10 15

mode number mode number

Figure 7.2. The 2D truss sensor indices for nodes 2–5.

Each node of a beam has three degrees of freedom { , , }

x y

T : horizontal displacement

x, vertical displacementy, and rotation in the figure plane

T . Denote a unit vector that has all zeros except 1 at the ith location, then the displacement output matrix for sensors located at the ith node is

[0,0, ,1, ,0]

ei ! !

3 1

.

qij i

C e

The input matrix is

_{3 1 119}

a

T T

o n

B e e

.

We obtain the H

_f

norm

G_ki

f

for the kth mode (k = 1,2,3,4) and ith sensor location from (7.7) using

B_o

and as above. From these norms we obtain the sensor placement indices for each mode from (7.14), using weight such that

Cqi

max (

_i

V

_fki

) 1.