## Modal Actuators and Sensors

^ how to excite and monitor selected modes

All coordinate systems are equal, but some are more equal than others.

In some structural tests it is desirable to isolate (i.e., excite and measure) a single mode. Such a technique considerably simplifies the determination of modal parameters, see [116]. This was first achieved by using the force appropriation method, also called the Asher method, see [107], or phase separation method, see [21]. In this method a spatial distribution and the amplitudes of a harmonic input force are chosen to excite a single structural mode. Modal actuators or sensors in a different formulation were presented in [38], [93], [75], and [114] with application to structural acoustic problems. In this chapter we present two techniques to determine gains and locations of actuators or sensors to excite and sense a target mode or a set of targeted modes.

In the first technique we determine actuator (or sensor) gains based on the relationship between the modal and nodal coordinates of the actuator or sensor locations; see [43]. This approach is distinct from the force appropriation method since it does not require harmonic input force. Rather, we determine the actuator locations and actuator gains, and the input force time history is irrelevant (modal actuator acts as a filter). The locations and gains, for example, can be implemented as a width-shaped piezoelectric film. Finally, in this approach we can excite and/or observe not only a single structural mode but also a set of selected modes.

The second technique—called an assignment technique—consists of the determination of the actuator (sensor) locations and gains to obtain a balanced system with the prescribed Hankel singular values. By setting the Hankel singular values equal to 1 for certain modes and to 0 for the remaining ones, the obtained sensors will "see" only modes associated with nonzero Hankel singular values. Just these sensors form a set of modal sensors. Similarly, by setting the Hankel singular values equal to 1 for certain modes and to 0 for the remaining, we obtain actuators that excite modes associated with nonzero Hankel singular values. Just these actuators form a set of modal actuators.

In this section we discuss the determination of actuator and sensor locations and gains such that they excite and sense selected structural modes. A structural model in this chapter is described by the second-order modal model, as in Subsection 2.2.2. In modal coordinates the equations of motion of each mode are decoupled; see (2.26). Thus, if the modal input gain is zero, the mode is not excited; if the modal output gain is zero, the mode is not observed. This simple physical principle is the base for the more specific description of the problem in the following sections.

### 8.1.1 Modal Actuators

The task in this section is to determine the locations and gains of the actuators such that nm modes of the system are excited with approximately the same amplitude, where 1 < nm < n, and n is the total number of considered modes. We solve this task using the modal equations (2.19) or (2.26). Note that if the ith row, bmi, of the modal input matrix, Bm, is zero, the ith mode is not excited. Thus, assigning entries of bmi to either 1 or 0 we make the ith mode either excited or not. For example, if we want to excite the first mode only, Bm is a one-column matrix of a form

Bm =[1 0 — 0f. On the other hand, if one wants to excite all modes independently and equally, one assigns a unit matrix, Bm = I.

Given the modal matrix Bm we derive the nodal matrix Bo from (2.23). We rewrite the latter equation as follows:

Matrix R is of dimensions n x nd. Recall that the number of chosen modes is nm < n. If the selected modes are controllable, i.e., the rank of R is nm, the least-squares solution of (8.1) is

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