## A D Truss

The truss structure in Fig. 1.2 is a more complex example of a structure, which can still easily be simulated by the reader, if necessary. For this structure, 1 15 cm, 2 20 cm are dimensions of truss components. Each truss has a cross-sectional area of 1 cm2, elastic modulus of 2.0x107 N cm2, and mass density of 0.00786 kg cm3. This structure has 32 states (or 16 degrees of freedom). Its stiffness and mass matrices are given in Appendix C.1.

## 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 X (For convenience in further discussion we denote by 2 the placement matrix either of the two- or the infinity-norm.) The vector of the actuator placement indices is defined as aa - & a1 cra2 craS , and its kth entry is the placement index of the kth actuator. In the case of the H2 norm, it is the rms sum of the...

## Collocated Controller Design Examples

Two examples of modal collocated controller design are presented the controller design for the simple flexible system, and for the 2D truss structure. The Matlab code for this example is in Appendix B. The system is shown in Fig. 1.1, with masses m1 m2 m3 1, stiffness k1 10, k2 k4 3, k3 4, and the damping matrix D as a linear combination of the mass and stiffness matrices, D 0.004 + 0.001M. The input force is applied to mass m3 and the output is the rate of the same mass. The poles of the...

## Controller Design Examples

Here we present examples of the design of a modal LQG controller for a simple structure, for the 3D truss structure, and for the Deep Space Network antenna. The Matlab code for this example is in Appendix B. Design the LQG controller for the system shown in Fig. 1.1. The system masses are m1 m2 m3 1, stiffness k1 10, k2 3, k3 4, and k4 3, and a damping matrix D 0.004K + 0.001M, where K, M are the stiffness and mass matrices, respectively. The input force is applied to mass m3 , the output is...

## FfliGrt i n

Wherea> i is the ith natural frequency and . denotes either H2, Hm , or Hankel norms. The above equations show that the norm of the ith mode with an accelerometer sensor is obtained as a product of the norm of the ith mode with a rate sensor and the ith natural frequency. Example 6.11. Consider the truss from the previous example. The longitudinal input force is applied to node 21 and the longitudinal acceleration is measured at node 14. Determine the Hm norms of the modes for the structure...

## Frequency Limited Grammians

In this section we interpret the controllability and observability grammians in frequency domain. In order to do this, note that from the Parseval theorem the time integrals (4.3), for the time span (0, < ), can be substituted with the following integrals in the frequency domain is the Fourier transform of eAt, and H* is a complex-conjugate transpose of H. The above grammians are defined over the entire frequency range. The frequency band (-< , < ) can be narrowed to (- , a> ) where <...

## List of Symbols

Each equation in the book would halve the sales. Stephen Hawking A complex-conjugate transpose of matrix A A-1 inverse of square nonsingular matrix A tr(A) trace of a matrix A, tr(A) . aii A Euclidean (Frobenius) norm of a real-valued matrix A diag(a -) diagonal matrix with elements ai along the diagonal eig(A) eigenvalue of a square matrix A Ai (A) ith eigenvalue of a square matrix A Anax (A) maximal eigenvalue of a square matrix A ai (A) ith singular value of a matrix A max (A) maximal...

## Modal Actuators and Sensors Through Grammian Adjustment

In the method presented above the modal actuator does not depend on the sensors' location. But, the sensors can be located such that the actuated mode can be unobservable. We can notice a similar situation in the modal sensor procedure it is independent on the actuator location. But a sensed mode can be uncontrollable. The method proposed below allows us to avoid that undesirable situation. It was shown in Chapter 4 that in modal coordinates the controllability and observability grammians are...

## Modal Actuators and Sensors Through Modal Transformations

In the above equation R+ is a pseudoinverse of R, R+ VI, 1UT, where U, 2, and V are obtained from the singular value decomposition of R, i.e., from R U I.VT. Note that a structure with a modal actuator excites nm modes only (other modes are uncontrollable) therefore, the implementing modal actuator is equivalent to model reduction, where the structure has been reduced to nm modes, or to 2nm states. The input matrix Bo in (8.2) that defines the modal actuator can be determined alternatively from...

## 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...

## Model with Proof Mass Actuators

Proof-mass actuators are widely used in structural dynamics testing. In many cases, however, the actuator dynamics are not included in the model. The proof-mass actuator consists of mass m and a spring with stiffness k, and they are attached to a structure at node na. This is a reaction-type force actuator, see 144 , 57 . It generates a force by reacting against the mass m, thus force f acts on the structure, and -f acts on the mass m (Fig. 3.5 at position na). Typically, the stiffness of the...

## Models with Rigid Body Modes

Many structures are free or unrestrained they are not attached to a base. An example is the Deep Space Network antenna structure shown in Fig. 1.5 if uncontrolled, it can rotate freely with respect to the azimuth (vertical) axis and its dish can freely rotate with respect to the elevation (horizontal) axis. Modal analysis for such structures shows that they have zero natural frequency, and that the corresponding natural mode shows structural displacements without flexible deformations. A mode...

## Models with Small Nonproportional Damping

The damping properties of structures are often assumed in the modal form, i.e., they are introduced as damping coefficients in the modal equations (2.19) or (2.26). This is done not only for the sake of analytical simplicity, but also because it is the most convenient way to measure or estimate it. This is the way, for example, to estimate the material damping in the finite-element analysis of large flexible structures, where the modal analysis is executed, the low-frequency modes retained, and...

## Norms of a Generalized Structure

Consider a structure as in Fig. 3.10, with inputs w and u and outputs z andy. Let Gwz be the transfer matrix from w to z, let Gwy be the transfer matrix from w to y, let Guz be the transfer matrix from u to z, and let Guy be the transfer matrix from u to y. Let Gwzi, Guyi, Gwyi, and Guzi be the transfer functions of the ith mode. The following multiplicative properties of modal norms hold Property 5.18. Modal Norms of a General Plant. The following norm relationships hold Gwzi Guyi Gwy Guz ,...

## Placement of a Large Number of Sensors

For the placement of a large number of sensors the maximization of the performance index alone is not a satisfactory criterion. These locations can be selected using the correlation of each sensor modal norm. Define the kth sensor norm vector, which is composed of the squares of the modal norms where Guy- denotes the transfer function of the ith mode at the kth sensor. The norm denotes the H2, H , or Hankel norm. We select the sensor locations using the correlation coefficient rik, defined as...

## Reduction in the Finite Time and Frequency Intervals

We introduced the time- and frequency-limited grammians in Chapter 4. They are used in model reduction such that the response of the reduced system fits the response of the full system in the prescribed time and or frequency intervals. This approach is useful, for example, in the model reduction of unstable plants (using time-limited grammians) or in filter design (using band-limited grammians). Figure 6.2. (a) Magnitude of the transfer function and (b) impulse responses of the full (solid...

## Simultaneous Placement of Actuators and Sensors

In this section we present a simultaneous selection of sensor and actuator locations this is an extension of the actuator and sensor placement algorithm presented above. The latter algorithm describes either actuator placement for given sensor locations, or sensor placement for given actuator locations. The simultaneous placement is an issue of some importance, since fixing the locations of sensors while placing actuators (or vice versa) limits the improvement of system performance. The...

## Structures with Rigid Body Modes

Structures with rigid-body modes have poles at zero, therefore they behave like systems with integrators. The corresponding H2, Hm, and Hankel norms for systems with poles at zero do not exist as their values tend to infinity. However, the infinite values of the norms of some modes should not be an obstacle in the reduction process. These values indicate that the corresponding states should be retained in the reduced model, regardless of the norms of other modes. The reduction problem can be...

## Time and Frequency Limited Grammians

The time- and frequency-limited grammians characterize a system in a limited-time interval and in a limited-frequency window. They are obtained from the full time grammians using time and frequency transformation or vice versa. The results are identical in both cases, since the time and frequency transformations commute, as will be shown below. Consider the controllability grammian in the finite-time interval, defined in (4.80). From the Parseval theorem, the grammian (4.97) in the...

## Controllability and Observability of the Discrete Time Structural Model

Consider now a structure in modal coordinates. Similar to the continuous-time grammians the discrete-time grammians in modal coordinates are diagonally dominant, where Wci and W0i are 2 x 2 blocks, such that Wci wCiI2 and W0i w0iI2, see 98 , where IK II2 2 1 - cos fflj At 2 1 - cos fflj At wci TT--Y- wci cont-Yl-- 4.8 Q In the above equations Bmi is the ith block of Bm in modal coordinates, and Cmi is the ith block of Cm in modal coordinates, where Cm CmqQ_1 Cmv , see 2.42 for Zs0. In the...

## Sensor Placement Strategy

Actuator locations are already determined. 2. Select the areas where the sensors can be placed, obtaining the R candidate sensor locations. 3. Determine the sensor placement indices lt rk i for all the candidate sensor locations i 1, , R , and for all the modes of interest k 1, , n . 4. For each mode, select r1 for the most important sensor locations. The resulting number of sensors r2 for all the modes considered i.e., r2 lt n x r1 is much smaller than the number of candidate locations,...