Structure with Inertial Actuators

Structures

6.5 Structures with Actuators and Sensors

6.5.4 Structure with Inertial Actuators

where Z₁ is the fundamental (lowest) frequency of the structure. These conditions say that the actuator natural frequency should be significantly lower than the fundamental frequency of the structure, and that the actuator stiffness should be much smaller than the dynamic stiffness of the structure at any frequency of interest.

If the aforementioned conditions are satisfied, we obtain D_ci#1 for thus, the norms of the structure with the proof-mass actuator are equal to the norms of the structure without the proof-mass actuator. Also, the controllability and observability properties of the system are preserved. In particular, the presence of the proof-mass actuator will not affect the model order reduction. Note also that for many cases, whenever the first condition of (6.25) is satisfied, the second condition (6.30) is satisfied too.

1, , ; i ! n

Example 6.12. Consider the 3D truss as in Example 3.4 with and without the proof-mass actuator. Let the force input act at node 21 in the y-direction, and let the rate without output be measured at node 14 in the y-direction. Determine the Hankel singular values of the truss for the ideal actuator (force applied directly at node 21) and of the truss with a proof-mass actuator. The mass of the proof-mass actuator is

0.1 Ns cm,2

m and its stiffness is k 1 N cm. Its natural frequency is 3.1623 rad s,

Zo much lower than the truss fundamental frequency.

For the ideal force applied in the y-direction of node 21 the Hankel singular values are shown as dots in Fig. 6.13. Next, a proof-mass actuator was attached to node 21 to generate the input force. Circles in Fig. 6.13 denote Hankel singular values of the truss with the proof-mass actuator. Observe that the Hankel singular values are the same for the truss with and without the proof-mass actuator, except for the first Hankel singular value, related to the proof-mass actuator itself.

, 2

1

c c s c o

G G NZ ,

D D

E U

(6.32)

which was derived in Subsection 3.3.2.

0 10 20 30 40 50 60 70

–15 –10

100

10^–5

Hankel singular value

10 10

mode number

Figure 6.13. 3D truss with and without the proof-mass actuator: Hankel singular values with (x) and without (9) the proof-mass actuator; they are identical except for the additional Hankel singular value of the proof-mass actuator itself.

Also, the relationship between the actuator force ( )f_o and the force acting on the structure ( )f was derived in Subsection 3.3.2,

2

, 2

o c c 1

f f NZ

D D .

E U

(6.33)

The above result shows that the structural transfer function with the inertial actuator is proportional to the structural transfer function without the actuator.

Property 6.6. Norms of Modes with Inertial Actuators. The norms of the ith structural mode (G_si), and of the ith structural mode with an inertial actuator

are related as in (6.26); however, the factor

(G_ci), D_ci is now

2 2. 1

ci o

i i

D NZ

E U

(6.34)

Proof. Similar to Property 6.5. ‹

With the conditions in (6.30) satisfied, one obtains D_ci NZ_o² for

thus, the norms of the structure with the inertial actuator are proportional to the norms of the structure without the actuator. This scaling does not influence the results of model reduction, since the procedure is based on ratios of norms rather than their absolute values.

1, , ; i ! n

7

Actuator and Sensor Placement

ª how to set up a test procedure and control strategy

Experimentalists think that it is a mathematical theorem while the mathematicians believe it to be an experimental fact.

—Gabriel Lippman

A typical actuator and sensor location problem for structural dynamics testing can be described as a structural test plan. The plan is based on the available information on the structure itself, on disturbances acting on the structure, and on the required structural performance. The preliminary information on structural properties is typically obtained from a structural finite-element model. The disturbance information includes disturbance location and disturbance spectral contents. The structure performance is commonly evaluated through the displacements or accelerations of selected structural locations. The actuator and sensor placement problem was investigated by many researchers, see, for example, [1], [7], [24], [47], [55], [86], [89], [90], [96], [97], [101], [103], [105], [106], [127], and a review article [131].

It is not possible to duplicate the dynamics of a real structure during testing. This happens, not only due to physical restrictions or a limited knowledge of disturbances, but also because the test actuators cannot often be located at the actual location of disturbances, and sensors cannot be placed at locations where the performance is evaluated. Thus, to conduct the test close to the conditions of a structure in a real environment one uses the available (or candidate) locations of actuators and sensors and formulates the selection criteria and selection mechanisms.

The control design problem of a structure can be defined in a similar manner.

Namely, actuators are placed at the allowable locations, and they are not necessarily

collocated with the locations where disturbances are applied; sensors are placed at

the sensor allowable locations, which are generally outside the locations where

performance is evaluated.

For simple test articles, an experienced test engineer can determine the appropriate sensor or actuator locations in an ad hoc manner. However, for the first- time testing of large and complex structures the placement of sensors and actuators is neither an obvious nor a simple task. In practice heuristic means are combined with engineering judgment and simplified analysis to determine actuator and sensor locations. In most cases the locations vary during tests (in a trial and error approach) to obtain acceptable data to identify target modes.

For a small number of sensors or actuators a typical solution to the location

problem is found through a search procedure. For large numbers of locations the

search for the number of possible combinations is overwhelming, time-consuming,

and gives not necessarily the optimal solution. The approach proposed here consists

of the determination of the norm of each sensor (or actuator) for selected modes, and

then grading them according to their participation in the system norm. This is a

computationally fast (i.e., nonsearch) procedure with a clear physical interpretation.