*7.1.1. The method of assumed modes*

The method of assumed modes was used to find a finite-dimensional model (discrete model) for the beam with attached elements. As is well known for the assumed modes approach, the greater the number of modes used, the more accurate the results obtained. Usually, the vibrational modes of a related but simpler problem are used to find approximate solutions to a more complicated problem, in this example, the modeshapes of the beam with no attached elements are used. A good introduction to the assumed modes can be found in the study of Meirovitch [22].

The assumed method, when applied to a continuous, conservative vibrational system, will result in mass and stiffness matrices that represent the system. The dimensions of these matrices are determined by the degree of discretization selected for the problem. Here lies the main advantage of the assumed mode method over the popular finite element analysis (FEA). It has been shown in the study of Cha [23] that the same level of accuracy can be reached by the assumed modes method using smaller degrees of discretization than with FEA. This implies mass and stiffness matrices that are smaller. This reduced dimensionality of the system is far more relevant to solution of the inverse problem than to the forward problem.

The assumed modes method was chosen to derive the equations of motion for the case of an Euler–Bernoulli beam to which a number of discrete elements are attached. The transverse vibrations of the beam can be written as

$$\text{cov}(\mathbf{x}, t) = \sum\_{j=1}^{N} \phi\_{\slash}(\mathbf{x}) \,\, \eta\_{\slash}(t) \tag{20}$$

Here, *w*(*x*, *t*) is the transverse displacement of the beam, *ϕ<sup>j</sup>* (*x*) is a space-dependent eigenfunc‐ tion, *η<sup>j</sup>* (*t*) is the generalized coordinate and *N* is the number of assumed modes chosen for the problem. It is important to note that *ϕ<sup>j</sup>* (*x*) varies with the choice of the beam and any *ϕ<sup>j</sup>* (*x*) should be chosen to satisfy the required geometric boundary conditions of the selected beam.

#### *7.1.2. Derivation of equations of motion*

In order to derive the equations of motion for the one-dimensional Euler–Bernoulli beam with multiple lumped point-mass attachments as shown in **Figure 3**, expressions for kinetic and potential energies must first be found. The kinetic energy of the beam is given by

**Figure 3.** Beam with multiple lumped mass attachments.

$$T = \frac{1}{2} \sum\_{j=1}^{N} M\_j \dot{\eta}\_j^{\ \ 2} \left( t \right) + \frac{1}{2} m\_l \dot{\mathbf{w}}^2 \left( \mathbf{x}\_l, t \right) + \dots + \frac{1}{2} m\_s \dot{\mathbf{w}}^2 \left( \mathbf{x}\_s, t \right) \tag{21}$$

where *Mj* is generalized masses of the bare beam (no attachments), an over dot indicates derivatives with respect to time and *m*1…*ms* are *s* lumped point masses positioned at *x*1…*xs*, respectively.

Using the same procedure, the equation for the potential energy can be written as

$$V = \frac{1}{2} \sum\_{j=1}^{N} K\_j \left| \eta\_j \right|^2(t) \tag{22}$$

where *Kj* are the generalized stiffnesses of the bare beam. Substituting equation (20) into equation (21), the following equation for kinetic energy is obtained

( ) ( ) () 1

*j j*

should be chosen to satisfy the required geometric boundary conditions of the selected beam.

In order to derive the equations of motion for the one-dimensional Euler–Bernoulli beam with multiple lumped point-mass attachments as shown in **Figure 3**, expressions for kinetic and

( ) ( ) ( ) 2 2 <sup>2</sup>

is generalized masses of the bare beam (no attachments), an over dot indicates

( ) <sup>2</sup>

*j j*

= + ++ å & & & (21)

<sup>=</sup> å (22)

*j j s s*

derivatives with respect to time and *m*1…*ms* are *s* lumped point masses positioned at *x*1…*xs*,

1 1

*T M t mw x t mw x t*

Using the same procedure, the equation for the potential energy can be written as

1 2 *N*

are the generalized stiffnesses of the bare beam.

1

*j V Kt* h =

11 1 , .... , 22 2

potential energies must first be found. The kinetic energy of the beam is given by

 h

(*t*) is the generalized coordinate and *N* is the number of assumed modes chosen for the

<sup>=</sup> å (20)

(*x*) varies with the choice of the beam and any *ϕ<sup>j</sup>*

(*x*) is a space-dependent eigenfunc‐

(*x*)

*N*

*j w xt x t* f

=

,

Here, *w*(*x*, *t*) is the transverse displacement of the beam, *ϕ<sup>j</sup>*

problem. It is important to note that *ϕ<sup>j</sup>*

*7.1.2. Derivation of equations of motion*

**Figure 3.** Beam with multiple lumped mass attachments.

1

h =

*j*

*N*

tion, *η<sup>j</sup>*

44 Applied Linear Algebra in Action

where *Mj*

where *Kj*

respectively.

$$T = \frac{1}{2} \sum\_{j=1}^{N} M\_j \dot{\eta}\_j^{\;\;2} \left( t \right) + \frac{1}{2} m\_l \left[ \sum\_{j=1}^{N} \phi\_j \left( \mathbf{x}\_l \right) \dot{\eta}\_j \left( t \right) \right]^2 + \dots + \frac{1}{2} m\_s \left[ \sum\_{j=1}^{N} \phi\_j \left( \mathbf{x}\_s \right) \dot{\eta}\_j \left( t \right) \right]^2 \tag{23}$$

Equation (22) for the potential energy remains unchanged from a bare beam since no elastic element was added to the beam.

Having found the expressions for kinetic and potential energies in terms of *ϕ* and *η* , these are then substituted into the Lagrange's equations to yield the equations of motion. Lagrange's equations are given by

$$\frac{d}{dt}\left(\frac{\partial T}{\partial \dot{\eta}\_i}\right) - \frac{\partial T}{\partial \eta\_i} + \frac{\partial V}{\partial \eta\_i} = 0 \qquad i = 1, 2, \dots, N \tag{24}$$

where *N* corresponds to the number of generalized coordinates and hence the number of differential equations.

Substituting equations (23) and (22) into equation (24) and converting the system of equations into a matrix representation, the matrix equation of motion is given by

$$\mathbf{M}\ddot{\underline{\eta}} + \mathbf{K}\underline{\eta} = \underline{0} \tag{25}$$

where **M** and **K** are the system mass and stiffness matrices, respectively. The mass matrix is given by

$$\mathbf{M} = \mathbf{M}^d + m\_\mathbf{i} \cdot \underline{\boldsymbol{\phi}}\_\mathbf{i} \cdot \underline{\boldsymbol{\phi}}\_\mathbf{i}^\top + \dots + m\_s \cdot \underline{\boldsymbol{\phi}}\_s \cdot \underline{\boldsymbol{\phi}}\_s^\top \tag{26}$$

In equation (26), *ϕ* ¯ <sup>1</sup>…*ϕ* ¯ *<sup>s</sup>* are *N*-dimensional column vectors of the *N* eigenfunctions evaluated at the attachment points *x*1 …*xs*, so that for example

$$
\underline{\phi}\_{\mathbb{h}} = \begin{bmatrix}
\phi\_{\mathbb{h}}\left(\mathbf{x}\_{1}\right) \\
& \cdot \\
& \cdot \\
& \cdot \\
\phi\_{\mathbb{N}}\left(\mathbf{x}\_{1}\right)
\end{bmatrix} \tag{27}
$$

**M***<sup>d</sup>* is a diagonal matrix whose diagonal components are the generalized masses *Mi* , and *m*1… *ms* are the masses of the lumped attachments.

For the stiffness matrix, since elastic elements are not added to the beam, the stiffness matrix remains a diagonal matrix whose elements are the generalized stiffnesses of the beam. Hence, the stiffness matrix is given by

$$\mathbf{K} = \mathbf{K}^d \tag{28}$$

#### *7.1.3. Choice of eigenfunctions*

As can be seen in equations (26) and (27), eigenfunctions constitute a major component of the mass matrix of the combined system. It is solely a function of position, *x* and can take many different forms depending on the beam being considered. The most basic requirement of any eigenfunction simulating the vibrational behaviour of a beam is its ability to accommodate and satisfy the beam boundary conditions. Here, the eigenfunctions for an Euler–Bernoulli beam with simply supported boundary conditions are given by Cha [23]

$$\phi\_n(\mathbf{x}) = \sqrt{\frac{2}{\rho L}} \sin \left( \frac{n \pi \mathbf{x}}{L} \right) \tag{29}$$

where *ρ* is the mass per unit length of the beam and *L* represents the length of the beam.

#### *7.1.4. Frequencies and modeshapes*

In order to solve equation (25), a system of *N* second-order differential equations, the vector of generalized coordinates *η* ¯ is written as

$$
\underline{\eta} = \vec{\eta}e^{i\alpha\sigma} \tag{30}
$$

Here, *ω* is the frequency of vibration of the system. Also, the inclusion of the complex number "*i*" is justified given the fact that the system is conservative, and it is expected that the vibrations are purely oscillatory.

Substituting equation (30) into equation (25) and taking derivatives yields

$$\left(-\phi^2 \mathbf{M} + \mathbf{K}\right)\ddot{\boldsymbol{\eta}} = \underline{\mathbf{0}}\tag{31}$$

In order for equation (31) to have a non-trivial solution, the following equation must hold

Structured Approaches to General Inverse Eigenvalue Problems http://dx.doi.org/10.5772/62284 47

$$\det\left(-\phi^2 \mathbf{M} + \mathbf{K}\right) = 0\tag{32}$$

Substituting *λ* = *ω*<sup>2</sup> in equation (32), an equivalent expression is obtained:

$$\det\left(\mathbf{K} - \lambda \mathbf{M}\right) = 0\tag{33}$$

Equation (33) can be solved for either the squared frequencies of the system *λ* (forward problem) or the coefficients of **K** and **M** matrices (inverse problem). In this section, the case of inverse problem is investigated for the case of an Euler–Bernoulli beam with a single mass attachment det(**K**).

#### *7.1.5. Inverse problem*

**M***<sup>d</sup>* is a diagonal matrix whose diagonal components are the generalized masses *Mi*

For the stiffness matrix, since elastic elements are not added to the beam, the stiffness matrix remains a diagonal matrix whose elements are the generalized stiffnesses of the beam. Hence,

As can be seen in equations (26) and (27), eigenfunctions constitute a major component of the mass matrix of the combined system. It is solely a function of position, *x* and can take many different forms depending on the beam being considered. The most basic requirement of any eigenfunction simulating the vibrational behaviour of a beam is its ability to accommodate and satisfy the beam boundary conditions. Here, the eigenfunctions for an Euler–Bernoulli

> p

beam with simply supported boundary conditions are given by Cha [23]

f

is written as

¯

( ) <sup>2</sup> sin *<sup>n</sup> n x <sup>x</sup> L L*

r

æ ö <sup>=</sup> ç ÷

where *ρ* is the mass per unit length of the beam and *L* represents the length of the beam.

In order to solve equation (25), a system of *N* second-order differential equations, the vector

*i t e* w

Here, *ω* is the frequency of vibration of the system. Also, the inclusion of the complex number "*i*" is justified given the fact that the system is conservative, and it is expected that the vibrations

> h**M K** <sup>0</sup> <sup>r</sup>

In order for equation (31) to have a non-trivial solution, the following equation must hold

h h

Substituting equation (30) into equation (25) and taking derivatives yields

( ) <sup>2</sup> -+= w

*<sup>d</sup>* **K K**= (28)

è ø (29)

<sup>=</sup> <sup>r</sup> (30)

*ms* are the masses of the lumped attachments.

the stiffness matrix is given by

46 Applied Linear Algebra in Action

*7.1.3. Choice of eigenfunctions*

*7.1.4. Frequencies and modeshapes*

of generalized coordinates *η*

are purely oscillatory.

, and *m*1…

(31)

The problem of imposing two frequencies on a dynamical system consisting of a beam with a single attached lumped mass is considered here which leads to the following system of equations:

$$\begin{cases} \det\left(\mathbf{K} - \mathbb{A}\_{\boldsymbol{u}} \mathbf{M}\right) = 0\\ \det\left(\mathbf{K} - \mathbb{A}\_{\boldsymbol{b}} \mathbf{M}\right) = 0 \end{cases} \tag{34}$$

where *λa* and *λb* are the desired natural frequencies squared to be imposed on the system (design variables). Due to the fact that no stiffness element is added to the beam, **K** remains intact and can be determined using equation (28), whereas the mass matrix of the combined system **M** is given by

$$\mathbf{M} = \mathbf{M}^d + m\_\mathbf{i} \underline{\phi\_\mathbf{i}} \underline{\phi\_\mathbf{i}}^T \tag{35}$$

where *m*1 is the magnitude of the lumped mass and *ϕ* ¯ 1 is a function of the mass position *x*1 and is defined by equation (27). *m*1 and *x*1 are the unknown variables of the inverse problem. Substituting equation (35) into (34), a system of two equations and two unknowns is obtained whose solution is presented in the next section for the case of a simply supported beam.

In using the determinant method, each desired natural frequency is substituted into a deter‐ minant equation of the form of equation (34). Therefore, each desired natural frequency produces a single equation. Therefore, *n* desired natural frequencies require *n* "design degrees of freedom" or parameters to be controlled, such as added masses or springs, or their location on the beam. The value of the added mass (or spring) is considered one design degree of freedom and its unknown location is another. The form of the equations produced through imposing a desired frequency on the determinant depends on the choice of eigenfunctions in the assumed modes method and also how the unknown desired parameter (e.g. position of the mass) is included in those eigenfunctions. Here, we have chosen the traditional trigono‐ metric functions as the eigenfunctions because they are the most well-known choices for the Euler–Bernoulli beam. Other choices of eigenfunctions may lead to better computational results with this method.

### **7.2. Results**

The following assumptions are considered in defining the inverse problem:


Maple (Maplesoft) was used to perform the numerical calculations.

The major steps in solving, as well as coding, the inverse eigenvalue problems are outlined here. First, two natural frequencies are chosen as desired input frequencies (*λa*, *λb*). These are the frequencies we seek to impose on the beam with its mass attachment. To insure a solvable problem, we chose known values from previously solved forward problem. The generalized mass and stiffness matrices must be formed. (**M***<sup>d</sup>* , **K***<sup>d</sup>* ) whose diagonal elements are given by

$$M\_{\parallel} = \mathbf{l} \tag{36}$$

$$K\_i = \frac{\left(i\pi\right)^4 EI}{\left(\rho L^4\right)}\tag{37}$$

The eigenfunction vector, *ϕ* ¯ , must also be built using the simply supported beam eigenfunction equation (29). The stiffness matrix is unaffected because no stiffness elements are added. The mass matrix of the combined system is affected by the presence of lumped masses, whereas the stiffness matrix is not. Hence, the matrices are given by

$$\mathbf{K}\_t = \mathbf{K}^d \tag{38}$$

$$\mathbf{M}\_{\phantom{e}} = \mathbf{M}^{d} + \mathbf{c}\rho L \underline{\phi}\_{\mu l} \underline{\phi}\_{\mu l}^{\tag{39}} \tag{39}$$

where *c* is the mass coefficient and vector *ϕ* ¯ *pL* is defined as

#### Structured Approaches to General Inverse Eigenvalue Problems http://dx.doi.org/10.5772/62284 49

$$\underline{\phi}\_{pL} = \begin{bmatrix} \phi\_1(pL) \\ \cdot \\ \cdot \\ \cdot \\ \phi\_N(pL) \end{bmatrix} \tag{40}$$

Equations (38) and (39) along with the two desired values of *λ* are substituted into equation (34). This gives two equations in two unknowns, which shall be solved for the two un‐ knowns, *c* and *p*, using *fsolve* as well as the *DirectSearch* packages in Maple. *Fsolve* is Maple's built-in equation-solving package. The details of the *DirectSearch* package can be found in the study of Moiseev [17]. The results obtained for *c* and *p* include the anticipated results (already known from the forward problem since we chose the desired *λ* from a known for‐ ward problem) plus additional results for *c* and *p*. To check whether the order of the fre‐ quencies in the frequency spectrum will be conserved or if the results returned by the inverse problem achieve the desired system frequencies, the parameters *c* and *p* must be substituted into the forward code in order to obtain the entire spectrum of system frequen‐ cies. From symmetry of the boundary condition about the beam midpoint, only half of the beam is considered and the results can be extended to the other half. These results are pre‐ sented in the left-hand columns of **Table 7**.

the mass) is included in those eigenfunctions. Here, we have chosen the traditional trigono‐ metric functions as the eigenfunctions because they are the most well-known choices for the Euler–Bernoulli beam. Other choices of eigenfunctions may lead to better computational

**•** The acceptable mass range is a fraction of the mass of the beam, that is, *m* = *cρL* for 0 < *c* < 1.

**•** The degree of discretization using assumed modes is *N*=*10* for the simply supported beam.

**•** The acceptable position range is a fraction of the length of the beam *L*, that is, *pL*, where

The major steps in solving, as well as coding, the inverse eigenvalue problems are outlined here. First, two natural frequencies are chosen as desired input frequencies (*λa*, *λb*). These are the frequencies we seek to impose on the beam with its mass attachment. To insure a solvable problem, we chose known values from previously solved forward problem. The generalized

> ( ) ( )

*i* 4 *i EI*

*K*

4

*L* p

equation (29). The stiffness matrix is unaffected because no stiffness elements are added. The mass matrix of the combined system is affected by the presence of lumped masses, whereas

> *d T <sup>t</sup> pL pL* **M M**= + *c L*

rff

¯ *pL* is defined as

r

, **K***<sup>d</sup>*

) whose diagonal elements are given by

1 *Mi* = (36)

<sup>=</sup> (37)

*<sup>d</sup>* **K K** *<sup>t</sup>* = (38)

(39)

, must also be built using the simply supported beam eigenfunction

The following assumptions are considered in defining the inverse problem:

Maple (Maplesoft) was used to perform the numerical calculations.

mass and stiffness matrices must be formed. (**M***<sup>d</sup>*

¯

where *c* is the mass coefficient and vector *ϕ*

the stiffness matrix is not. Hence, the matrices are given by

The eigenfunction vector, *ϕ*

results with this method.

48 Applied Linear Algebra in Action

**7.2. Results**

0 ≤ *p* ≤ 1 .



**Table 6.** Inverse problem for second and third frequencies for a simply supported beam.



**Table 7.** Inverse problem for second and fourth frequencies for a simply supported beam.

**Input given to inverse determinant method**

50 Applied Linear Algebra in Action

**Input given to inverse determinant**

*λ*2 = 1411.9003, *λ*4 = 20420.5236 (Obtained with *m* = 0.2*ρL l* = 0.4*L* in the forward code)

**method**

**Solution via inverse determinant method**

**Table 6.** Inverse problem for second and third frequencies for a simply supported beam.

*m* = 0.1516*ρL l* = 0.0972*L*

*m* = 0.4695*ρL l* = 0.4268*L*

*Search* package only)

(This result was obtained through *Direct*

(This result was obtained through *Direct*

*Search* and *fsolve* packages)

**Full span of frequency squared spectrum obtained through the solution of forward problem** 18122.3022 55694.1292 1.18846×105 1.91164×105 3.88304×105 5.89699×105 8.69163×105

**squared spectrum obtained**

**via the solution of forward problem**

94.7851 **1411.9003** 6626.2231 **20420.5236** 51205.1150 1.11210×105 2.15698×105 3.82314×105 6.29638×105 9.73288×105

51.2467 **1411.9003** 6082.1204 **20420.5235** 57662.4808 1.00203×105 2.33797×105 3.33408×105 6.17507×105 9.04206×105

**Solution via inverse determinant method Full span of frequency**

The squared frequencies in the left-hand columns of these tables are chosen from the previ‐ ously solved forward problem, and their subscripts indicate their order in the frequency spectrum. The middle columns of these tables contain the values of the masses as well as their positions on the beam obtained after substituting the squared frequencies of the first columns into the inverse problem code. Finally, the right-hand- columns are the full span of frequency spectrum obtained after substituting the masses as well as their positions of the middle column into the forward problem. The desired squared frequencies are in bold face in the right-hand column vectors to make it easier for the reader to compare them with the desired frequencies of the left-hand column.

From **Table 7**, some observations can be made. First, the desired value and order of the two input frequencies were preserved by the inverse problem. Specifically, in Table 7, the two input frequencies remain as the second and fourth system frequencies when the complete frequency spectrum was found for all three possible solutions returned by the inverse problem. Second, for each mass obtained, there must be two corresponding positions that are symmetrical with respect to the middle of the beam. This follows from the symmetry of the boundary conditions of the simply supported beam. Finally, a good equation solver is imperative for this method to work properly. For example, the use of an alternative equation solver (*DirectSearch*) yielded additional parameters *c* and *p* that were not returned by the built-in equation solver of Maple.

In order to verify that the method works when the chosen natural frequencies are not close together, a second example with the simply supported beam is presented. In this second example, the chosen frequencies are the second and eighth natural frequencies. As shown in Table 8, the method worked well in this case. Interestingly, a larger number of possible solutions were obtained in this case, in comparison with the cases shown in **Tables 6** and **7**. It is observed that the larger the separation in chosen order between λ<sup>i</sup> and λ<sup>j</sup> , the greater the number of possible solutions to the inverse problem.



**Table 8.** Inverse problem for second and eighth frequencies for a simply supported beam.

#### **7.3. Conclusion**

is observed that the larger the separation in chosen order between λ<sup>i</sup>

**Solution via inverse determinant method**

*m* = 0.2*ρL l* = 0.6*L*

*m* = 0.0752*ρL l* = 0.8447*L*

*m* = 0.0585*ρL l* = 0.7133*L*

number of possible solutions to the inverse problem.

**Input given to the inverse problem**

52 Applied Linear Algebra in Action

(Obtained with *m* = 0.2*ρL l* = 0.4*L* in the forward code)

*λ*2 = 1411.900275, *λ*8 = 3.773881670 × 105

and λ<sup>j</sup>

**Full span of frequency squared spectrum obtained**

**via the solution of forward problem**

71.3527 **1411.9003** 7164.65111 20420.52363 60880.6819 1.05946×105 2.21752×105 **3.77388×105** 5.73630×105 9.74091×105

7887.5523 22430.0485 56495.1732 1.26101×105 2.13015×105 **3.77388×105** 6.37810×105 9.05601×105

94.2653 **1411.9003** 6953.8085 22707.0792 58428.8453 1.25669×105 2.31928×105 **3.77388×105** 5.86872×105 9.10470×105

90.83562 **1411.9003**

*m* = 0.0807*ρL l* = 0.6615*L* 86.6441 **1411.9003**

, the greater the

The determinant method was used to demonstrate how to impose two natural frequencies on a dynamical system consisting of a beam to which a single lumped mass is attached. In this method, the known (design) variables are the two natural frequencies and the unknown variables are the magnitude of the attached mass as well as its position along the beam. The proposed method is easy to code and can accommodate any kind of eigenfunction. It was shown that the expected values of the added mass and its position are recovered from the inverse problem, in addition to the unexpected possible solutions. Even for this simple problem, a unique solution does not exist. The non-uniqueness can be considered as a benefit from the point of view of design possibilities. An investigation of the inverse problem also reveals that the order of the frequencies in the hierarchy of the whole frequency spectrum is conserved. Ideally, the degree of discretization should be the same for both forward and inverse problem. Although not discussed here, a promising observation made by Mir Hosseini and Baddour [21] through numerical simulations was that using a lower degree of discretiza‐ tion for the inverse problem could still give acceptable results from the point of view of engineering design, especially when lower orders of frequencies are involved.
