**7. Example of using the determinant method**

In this section, an example of how the determinant method can be used to solve an inverse vibration problem is presented. In this section, the authors demonstrate a method to impose two desired natural frequencies on a dynamical system of a simply supported Euler–Bernoulli beam to which a single lumped mass is attached by determining the magnitude and mounting position of the mass. The full details of this example are given in the study of Mir Hosseini and Baddour [21].

### **7.1. Modelling**

**6. The determinant method for partially described systems**

Baddour [21], is introduced.

turn leads to two scalar equations, specifically

*dimension of the system*, *are given as input information*.

**2.** *Generate a determinant equation from each desired frequency*:

*as unknown modelling parameters*.

the multiple solutions expected.

**6.1. Algorithm**

matrix entries.

determinant equation

42 Applied Linear Algebra in Action

Although effective, the Cayley–Hamilton method for solving inverse eigenvalue problems described in the previous section may not be the most computationally efficient method when solving partially described systems. Partially described systems are those for which only a select few eigenvalues are specified, as opposed to the entire spectrum being given. In this section, the determinant method, which first appeared in the study of Mir Hosseini and

For clarity, the method is explained for the case where two eigenvalues are required to be a certain value. The method proceeds in the same manner where the number of specified eigenvalues is larger. Solving the characteristic equation of the system requires the use of the

where *λ* is the eigenvalue obtained by finding the roots of the polynomial equation 18. Suppose that two of the possible *n* roots (in this case the eigenvalues *a1* and *a2*) are required to be of a certain value. Replacing *λ* in the characteristic polynomial with the desired *a1* and *a2* each in

> det 0 det 0

The two scalar equations in equation (19) can be simultaneously solved for up to two un‐ knowns. This greatly reduces the computational effort involved. Constraints can be used to create a physically realistic system. Similar to the case for the Cayley–Hamilton theorem, the equations will be polynomials and the use of a good polynomial solver is required to obtain

In the case of Problem C, *m* eigenvalues are given for an *n* dimensional system, so that *n* − *m* is the number of unknown or unspecified eigenvalues for the system. The output of the determinant method consists of the remaining *n* − *m* eigenvalues as well as any of the unknown

Determinant Algorithm. *A certain set of desired eigenvalues*, *a*1, …, *am*, *which is smaller than the*

**1.** *Generate the M and K matrices from the chosen forward modelling technique leaving variable m*

= -=

0 (18)

= -= (19)

*pt K M* () ( ) = -= det l

() ( ) () ( ) 1 1 2 2

*p a K aM p a K aM*
