## 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...

## 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...

## Examples

In this section we perform the identification of the models of a simple structure (in order to illustrate the method in a straightforward manner) the 2D truss a more complicated structure, and the Deep Space Network antenna where the model is identified from the available field data. The Matlab code for this example is in Appendix B. Analyze a simple system with k1 10, k2 50, 3 50, k4 10, m1 m2 m3 1, and with proportional damping matrix, D 0.005K + 0.1M. The input is applied to the third mass...

## 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...

## 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 in Modal Coordinates

Frequently the order of the nodal representation is unacceptably high. For example, it is not uncommon that the number of degrees of freedom of the finite-element model exceeds 1000. Therefore, the nodal state representation is rarely used in structural dynamics. An alternative approach is to obtain the state-space representation using the modal coordinates and the second-order modal form (2.19), where the number of equations is significantly lower, while the accuracy of the analysis has not...

## 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 Indices and Matrices

Properties 7.1(a),(b) are the basis of the actuator and sensor search procedure of a general plant. The actuator index that evaluates the actuator usefulness in test is defined as follows where Gu 112 puy II + Guz 112, while the sensor index is The indices are the building blocks of the actuator placement matrix 2, The placement index of the kth actuator (sensor) is determined from the kth column of X In the case of the H2 norm it is the rms sum of the kth actuator indexes over all modes, k 4 ,...

## 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...

## 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...

## State Space Representation

For a generalized structure let A be the state matrix of the system, Bu , Bw represent the input matrices of u and w, respectively, and Cy, Cz represent the output matrices of y and z, respectively. For simplicity, we assume the feed-through terms Duy, Dwz, etc., to be equal to zero. Then the state-space representation of the

## The Deep Space Network Antenna

The NASA Deep Space Network antenna structure illustrates a real-world flexible structure. The Deep Space Network antennas, operated by the Jet Propulsion Laboratory, consist of several antenna types and are located at Goldstone (California), Madrid (Spain), and Canberra (Australia). The Deep Space Network serves as a communication tool for space exploration. A new generation of Deep Space Network antenna with a 34-m dish is shown in Fig. 1.5. This antenna is an articulated large flexible...

## The Low Authority LQG Controller

For LQG controllers we modify the definition of the low-authority controller of a structure as known from Chapter 10. Let (A, B, C) be the open-loop modal representation of a flexible structure (in the modal form 1 or 2), and let Ac1 A - BBTSc, Ac2 A - SeCTC be the closed-loop matrices where Sc and Se are the solutions of the CARE and FARE equations, respectively. The LQG controller is of low authority if its closed-loop matrices have the following property eig(Ad ) eig(A - BBTSC ) - eig(A -...

## 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...

## Time Limited Grammians

The steady-state grammians, defined over unlimited time integrals, are determined from the Lyapunov equations 4.5 . The grammians over a finite-time interval T t1, t2 where 0 lt t1 lt t2 lt lt are defined by 4.3 , and can be obtained from the matrix differential equations 4.4 . In many cases these equations cannot be conveniently solved, and the properties of their solutions are not readily visible. However, using the definitions from 4.3 we will derive the closed-form grammians over the finite...

## 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,...