Damage detection in a fixed-fixed beam using natural frequency changes

2.1. Frequency response function (FRF)

Vibration based damage identification techniques (VBDIT) use change in natural frequency and mode shaps for damage detection in structure. Basic dynamic response equation for n DOF system can be expressed as Eq. (1):

For an external force and displacement $f\left(t\right)=\left\{F\left(\omega \right)\right\}{e}^{j\omega t}$ and $a\left(t\right)=\left\{a\left(\omega \right)\right\}{e}^{j\omega t}$, a damped free condition FRF is expressed as:

The analytical and measured FRFs is presented as [$H\left(\omega \right)]$ and [${H\left(\omega \right)}^{*}]$, where:

It is assumed that mass of structure remains constant and stiffness changes:

When multiplied by ${\left[H\right]}^{*}$, Eq. (4) gives:

Based of analytical and measured FRF, beta ($\beta$) will be calculated as:

2.2. Iterative modal strain energy (IMSE) method

Damage severity will be estimated for damage location identified form FRF, since $M^{*}=M$ and Global stiffness matrix will be written as linear combination of local stiffness matrix for each element:

where $N_d$ is the total number of damaged elements while ${\alpha }_n$and $l_n$ show the damage severity coefficient and the damaged element. Damage severity equation will be as follows:

Structural MSE and elemental MSE changes due to damage which is applied here as an input for estimation of damage extent:

Here ${\mathrm{\Phi }}_i$, ${\mathrm{\Phi }}_i^{*}$, ${\lambda }_i$ and ${\lambda }_i^{*}$ represent the mode shapes and natural frequencies for intact and damaged structure. Using the equation Eqs. (9, 10), Eq. (8) can be written as follow:

For $m$ Eq. (11) can be simplified as:

where $C$, $\alpha$ and $b$ present elemental MSE, damage severity coefficient and change in natural frequency. For $m\ge N_d$, Least square solution method will be used to calculate Damage severity coefficient:

Eq. (14) requires mode shapes for damaged structure at full coordinates which is difficult to obtain, here initially zero damage is assumed for Fixed-Fixed beam and IMSE method is applied to quantify the damage using measured natural frequencies:

Eqs. (14, 15) will be used for each Iteration of IMSE Method. IMSE method consists of four steps for damage quantification.

Step 1: Initialize the solution with ${\alpha }^0=$ 0, calculate ${\mathrm{\Phi }}_i^{*\left(0\right)}\left(K^{*},M\right)$ where $K^{*}=K$.

Step 2: Solve for $\alpha$ using ${\mathrm{\Phi }}_i^{\mathrm{*}\left(0\right)}$, first iteration for IMSE completes here.

Step 3: Compute ${\mathrm{\Phi }}_i^{\mathrm{*}(k-1)}$ from ${\alpha }^{\left(k-1\right)}$, and estimate ${\alpha }^{\left(k\right)}$ using ${\mathrm{\Phi }}_i^{\mathrm{*}(k-1)}$, where $k=\mathrm{}$2, 3….

Step 4: if $\left|{\alpha }^{\left(k\right)}-{\alpha }^{\left(k-1\right)}\right|<r$ damage severity is estimated, otherwise move to step 3, where $r=$0.0001.

2.3. Damage detection algorithm

A damage detection algorithm is proposed based on FRF & IMSE method is shown in Fig. 1.

Fig. 1Damage detection algorithm based on natural frequencies

2.4. Noise-effect

In actual measured modal parameters differ by simulated FE parameters due to noise. In order to simulate this effect on dynamic response of structure, here $k_j^{*}$ and $k_j$ are stiffnesses with and without Gussian Noise for damaged elements localized. Here ${\gamma }_j$ present the random number with standard deviation of 1 and of mean 0, $n$ shows the percentage of noise:

4.1. Damage localization using FRF

FRF for each damage case indicates presence of damage in the structure. Damage localization indicator $\beta \left(1,i\right)$ uses Eq. (6) for damage location at element and its corresponding DOF.

Fig. 3Damage indicator (β) for DCI: a) based on element, b) based on DOF

a)

b)

4.2. Damage quantification using IMSE method

IMSE method utilizes the natural frequencies of Intact and damaged structures. For a multiple damage case it requires only first three natural frequencies for damage quantification.

Fig. 4Damage indicator (β) for DCII: a) based on element, b) based on DOF

a)

b)

Fig. 5Damage indicator (β) for DCIII: a) based on element, b) based on DO

a)

b)

For beam structure single and multiple damage cases FRFs indicate the presence of damage in all three damage cases. For DCI, $\beta$ indicator shows a higher value at 6th element and its corresponding DOFs. Similarly, for DC II and DC III, there exists a higher value of $\beta$ at 3rd, 6th, 8th and 9th Element. This damage localization is used as an input for damage quantification in the damaged elements. For DCI, IMSE Method uses only first natural frequency of damaged structure. The damage severity estimator $\alpha$ is estimated within 10 iterations with a tolerance of 0.0001, similarly DCII and DCIII require first two and three natural frequencies to estimate the damage severity within 9 and 11 iterations.

Fig. 6Damage severity estimator (α) for DCI, DCII and DCIII