**5.2. Forward model**

true if the system is made up of a matrix *M* and a matrix *K*, except in this case, the generalized

It becomes clear that one of the largest difficulties in using inverse eigenvalue theory to solve many physical problems is setting up the appropriate matrix form of the problem. This is achieved by consideration of the modelling approach chosen to model the forward problem. In most cases, the forward solution will utilize some form of discretization, such as finite elements, global elements, finite difference, and so on. The method chosen has a large impact on the structure of the matrix, making the solution easier or more difficult depending on the situation. Also, in discretizing, the method chosen to relate material parameters has a large effect on the number of independent variables. Therefore, it is extremely important to ensure

The characteristic equation for an *n*th order system is an *n*th order polynomial. As discussed above, the Cayley–Hamilton theorem will produce *n*-independent equations. For both partially described and fully described systems, a system of *n*th order polynomials will need to be solved for the unknown quantities. This will require a good polynomial solver or polynomial root-finding algorithm. For the problems presented in the rest of this chapter, the symbolic computer algebra software *Maple* by Maplesoft, Waterloo, Canada, was used to solve the problems although any modern numerical or symbolic software package could similarly be employed. For our simulations, the *Maple* built-in function *fsolve* was used, as well as the freely available package *DirectSearch*. The details of the *DirectSearch* package can be found in the study of Moiseev [17]. It was observed that *DirectSearch* produced many more (valid) solutions than did *fsolve* (recall that a unique solution to most of these problems does not exist and thus multiple solutions are expected). The existence of multiple solutions is desirable from an engineering design point of view in that it implies that there are many solutions possible and it is up to the designer to choose the most physically realisable one, within the problem constraints. For the Cayley–Hamilton (and determinant method presented in the rest of the chapter) to be effective, a good polynomial solver is required so that the multiple possible

In this section, we demonstrate by way of example how the Cayley–Hamilton method can be used to solve a practical engineering design problem by writing it as an inverse vibrations

Given a desired fundamental frequency, construct a brace–plate system as described by a mass matrix *M* and a stiffness matrix *K*. All dimensional (geometric) properties of the brace–plate system are assumed to be specified and fixed except for the thickness of the brace *hc* and the

Cayley–Hamilton theorem must be used.

that simplification methods are properly considered.

**4.5. Polynomial solvers**

36 Applied Linear Algebra in Action

solutions can indeed be found.

**5.1. Problem statement**

**5. Example with the Cayley–Hamilton inverse method**

problem and thus as an inverse eigenvalue problem.

The first step in the solution process is to generate a finite-dimensional model of the continuous problem. The model is based on an orthotropic plate structurally reinforced by a brace in the weaker plate direction. The model is shown in **Figure 1**.

**Figure 1.** Orthotropic plate reinforced with a rectangular brace.

The forward model is discretized using the assumed shape method. The assumed shape method is an energy method, which uses global plate elements with the kinetic and strain energy plate equations in order to determine the system's equations of motion, from which the mass and stiffness matrices are extracted [18]. For the details of the development of the large mass and stiffness matrices, the reader is referred to the study of Dumond and Bad‐ dour[19]. The system is assumed simply supported, conservative and the material properties are assumed orthotropic. The forward model is created assuming that the mechanical prop‐ erties are all related to Young's moduli in the *y* direction.

#### **5.3. Inverse model**

The goal is to reconstruct the brace–plate system from a desired fundamental frequency. The generalized Cayley–Hamilton theorem inverse eigenvalue method is used as explained in Section 3.2.

A cross section of the fundamental modeshape is shown in **Figure 2**. It is clear that the brace affects the maximum amplitude of this modeshape, thus also affecting the associated frequen‐ cy. In order to adjust the fundamental frequency of the brace–plate system to a desired value, it is necessary to adjust the thickness of the brace [20].

**Figure 2.** Cross section of the brace–plate system's fundamental modeshape.

#### **5.4. Modelling considerations**

The mechanical properties vary based on *Ey*, the Young's modulus in the *y* direction, and the brace thickness controls the brace–plate system's fundamental frequency. Therefore, the forward model is created using the assumed shape method while leaving these two parameters as variables. Thus, the mass matrix *M* is a function of *hc*, the height of the brace–plate system at the (assumed fixed) location of the brace, and the stiffness matrix *K* is a function of *hc* and also *Ey*. Here, 2×2 trial functions are used in the assumed shape method, so it follows that fourth order square matrices are created. The trial functions used are those of the simply supported rectangular plate given by

$$\text{cov}\left(\mathbf{x}, \mathbf{y}, t\right) = \sum\_{n\_{\rm u}=1n\_{\rm \gamma}=2}^{n\_{\rm \gamma}} \sum\_{}^{n\_{\rm \gamma}} \sin\left(\frac{n\_{\rm x} \pi x}{L\_{\rm x}}\right) \sin\left(\frac{n\_{\rm \gamma} \pi y}{L\_{\rm y}}\right) q\_{n\_{\rm u} n\_{\rm \gamma}}\left(t\right) \tag{14}$$

where *m* are the modal numbers, *q* is the time function and *w* is the displacement variable normal to the plate. The displacement variable *w* is then used directly in creating the kinetic and strain energy equations of the simply supported rectangular plate. These equations are broken into three sections as shown in **Figure 1** in order to take into account presence of the brace. This procedure is described in detail in the study of [19]. It is assumed that the *Ey* is known and used as input information into the stiffness matrix. This leaves *hc* as the only unknown parameter, appearing in both the mass and stiffness matrices.

In order to solve these matrices from the desired fundamental frequency, we must first create the characteristic polynomial using the desired frequency,

$$p\left(\mathcal{A}\right) = \left(\mathcal{A} - a\right) \cdot \left(\mathcal{A} - b\_{\text{i}}\right) \cdot \left(\mathcal{A} - b\_{\text{i}}\right) \cdot \left(\mathcal{A} - b\_{\text{i}}\right) \tag{15}$$

where *a* is the specified fundamental frequency, and *b1*, *b2* and *b3* are unknown values, which will also need to be found as part of solving the problem. Since we have assumed 2×2 trial functions, so that the mass and stiffness matrices are both 4×4, the characteristic polynomial must be fourth order, as shown in equation (15). Subsequently, *p*(*λ*) is expanded so that the polynomials coefficients can be found. Once the polynomial is created, the Cayley–Hamilton equation can be written by substituting (*M*-*<sup>1</sup> K*) for *λ* into equation (15):

$$p\left(K,M\right) = c\_4\left(M^{-1}K\right)^4 + c\_3\left(M^{-1}K\right)^3 + c\_3\left(M^{-1}K\right)^2 + c\_1\left(M^{-1}K\right) + c\_0I = 0\tag{16}$$

where *cn* are the coefficients of *λ* in *p*(*λ*) determined through equation (15). Equation (16) produces 16 equations, of which only 4 are independent. According to Bézout's theorem, selecting and solving the equations on the main diagonal for the four unknowns (*hb*, *b1*, *b2* and *b3*) produce 44 =256 possible solutions. From the set of all possible solutions, complex-valued solutions can be immediately eliminated as not being physically meaningful. Clearly, further constraints must be added to the solution in order to get a solution which fits within the desired physical limits. These physical limits are based on the maximum and minimum brace dimen‐ sions, which are required to compensate for the range of plate stiffnesses used during the analysis, as well as the range of natural frequencies, which can be obtained using these system dimensions. Thus, the following constraints are implemented into the solution:

$$\begin{aligned} &0.013 \le h\_b \le 0.016 \text{ m} \\ &1 \times 10^7 \le b\_1 \le 9 \times 10^8 \text{ rad/s} \\ &1 \times 10^7 \le b\_2 \le 9 \times 10^8 \text{ rad/s} \\ &1 \times 10^7 \le b\_3 \le 9 \times 10^8 \text{ rad/s} \end{aligned} \tag{17}$$

Solving the four equations obtained from equation (16) within the constraints provided by equation (17) yields, a physically realistic solution satisfies the desired fundamental frequency, as well as the system's parameters.

#### **5.5. Results**

A cross section of the fundamental modeshape is shown in **Figure 2**. It is clear that the brace affects the maximum amplitude of this modeshape, thus also affecting the associated frequen‐ cy. In order to adjust the fundamental frequency of the brace–plate system to a desired value,

The mechanical properties vary based on *Ey*, the Young's modulus in the *y* direction, and the brace thickness controls the brace–plate system's fundamental frequency. Therefore, the forward model is created using the assumed shape method while leaving these two parameters as variables. Thus, the mass matrix *M* is a function of *hc*, the height of the brace–plate system at the (assumed fixed) location of the brace, and the stiffness matrix *K* is a function of *hc* and also *Ey*. Here, 2×2 trial functions are used in the assumed shape method, so it follows that fourth order square matrices are created. The trial functions used are those of the simply supported

( ) ( )

æ ö æ ö <sup>=</sup> ç ÷ ç ÷

*n n x y n x n y w xyt q t L L* p

*x y*

where *m* are the modal numbers, *q* is the time function and *w* is the displacement variable normal to the plate. The displacement variable *w* is then used directly in creating the kinetic and strain energy equations of the simply supported rectangular plate. These equations are broken into three sections as shown in **Figure 1** in order to take into account presence of the brace. This procedure is described in detail in the study of [19]. It is assumed that the *Ey* is known and used as input information into the stiffness matrix. This leaves *hc* as the only

In order to solve these matrices from the desired fundamental frequency, we must first create

 l

where *a* is the specified fundamental frequency, and *b1*, *b2* and *b3* are unknown values, which will also need to be found as part of solving the problem. Since we have assumed 2×2 trial functions, so that the mass and stiffness matrices are both 4×4, the characteristic polynomial

 l= -×- ×- ×- (15)

( ) ( )( )( )( ) <sup>123</sup> *p ab b b*

 l *x y*

ç ÷ è ø è ø å å (14)

*n n*

p

1 2 , , sin sin *<sup>x</sup> <sup>y</sup>*

unknown parameter, appearing in both the mass and stiffness matrices.

the characteristic polynomial using the desired frequency,

ll

= =

*x y*

*m m*

it is necessary to adjust the thickness of the brace [20].

**Figure 2.** Cross section of the brace–plate system's fundamental modeshape.

**5.4. Modelling considerations**

38 Applied Linear Algebra in Action

rectangular plate given by

The material properties of the brace and plate used during the analysis are given in **Table 1**.


**Table 1.** Material properties.


The dimensions used for the model throughout the analysis of the brace–plate system are shown in **Table 2**.

**Table 2.** Dimensions of brace-plate model.

These dimensions refer to those shown in **Figure 1**, where '*p*' refers to the plate's dimensions, '*b*' refers to the brace's dimensions and '*c*' refers to the dimensions of the combined system.

As a basis for comparison, a plate having a Young's moduli of *Ey*=850 MPa to which a brace is attached with a combined brace–plate thickness of *hc*=0.015 m is found to have a fundamental natural frequency of 687 Hz, as calculated using the forward model. The analysis is then performed using the inverse Cayley–Hamilton method. As *Ey* of the plate is varied, the thickness of the brace–plate section is calculated such that the fundamental frequency of the brace–plate system is kept consistent at 687 Hz. The results of the computations are presented in **Table 3**.


**Table 3.** Results of the inverse model analysis.

Clearly, adjusting the thickness of the brace also has an effect on the other natural frequencies. These are presented in **Table 4**.


**Table 4.** Calculated frequencies of the inverse model analysis.

Interestingly, the constraints indicated in equation (17), although physically strict, allow for more than one solution in certain cases. An example is shown in **Table 5**.


**Table 5.** Alternate brace thickness solution satisfying the physical constraints.

#### **5.6. Discussion**

The dimensions used for the model throughout the analysis of the brace–plate system are

These dimensions refer to those shown in **Figure 1**, where '*p*' refers to the plate's dimensions, '*b*' refers to the brace's dimensions and '*c*' refers to the dimensions of the combined system.

As a basis for comparison, a plate having a Young's moduli of *Ey*=850 MPa to which a brace is attached with a combined brace–plate thickness of *hc*=0.015 m is found to have a fundamental natural frequency of 687 Hz, as calculated using the forward model. The analysis is then performed using the inverse Cayley–Hamilton method. As *Ey* of the plate is varied, the thickness of the brace–plate section is calculated such that the fundamental frequency of the brace–plate system is kept consistent at 687 Hz. The results of the computations are presented

Clearly, adjusting the thickness of the brace also has an effect on the other natural frequencies.

**Fundamental frequency**

*a* **(Hz)**

**Brace thickness**

*hc* **(m)**

 0.01576 687 0.01536 687 0.01527 687 0.01500 687 0.01466 687 0.01435 687

**Table 3.** Results of the inverse model analysis.

These are presented in **Table 4**.

**Dimensions Values** Length, *Lx* (m) 0.24 Length, *Ly* (m) 0.18 Length, *Lb* (m) 0.012 Reference, *x1* (m) *Lx*/2–*Lb*/2 Reference, *x2* (m) *x1*+*Lb* Thickness, *hp* (m) 0.003 Thickness, *hb* (m) 0.012 Thickness, *hc* (m) *hp*+*hb*

shown in **Table 2**.

40 Applied Linear Algebra in Action

**Table 2.** Dimensions of brace-plate model.

in **Table 3**.

**Young's modulus** *ER* **(MPa)**

From these results, it is evident that it is possible to design a brace–plate system starting with a desired fundamental frequency in conjunction with using the proposed Cayley–Hamilton method. Table 3 clearly shows that by adjusting the thickness of the brace by small increments (10-5 m, machine limit), it is possible to compensate for the variation in the cross-fibre stiffness (*Ey*) of the plate so that the fundamental frequency of the combined system is equal to that of the benchmark value of 687 Hz. The results obtained using the Cayley–Hamilton theorem algorithm match those values obtained using the forward model exactly. However, since no account has been taken of the other frequencies during the analysis, Table 4 shows that frequencies *b1*–*b3* vary considerably from those values obtained for *Ey*=850 MPa. Therefore, it is important to ensure that there is a good understanding of what a given model can control. Moreover, it is interesting to note that within the strict physical constraints of equation (17), there is more than one brace–plate system (solution) that satisfies the Cayley–Hamilton theorem of equation (16). From Table 5, it can be seen that an alternate solution to the system exists, different from the one presented in Table 3, for a plate having an *Ey* of 750 MPa. In this case, by reducing the thickness of the brace, it is still possible to achieve a system having the desired frequency of 687Hz. However, the desired frequency is no longer the fundamental frequency but rather becomes the second frequency, and the fundamental has been replaced with a fundamental frequency of 570 Hz. It is important to keep this phenomenon in mind while designing a system. This is especially true if the order in the spectrum of a certain frequency associated with a certain modeshape is absolutely critical
