**2.4 The dynamic effect**

Before examining the durability behavior of the two structures, it is imperative to understand how they react dynamically to any set of loads. The study in this chapter proposes simply to contrast the distinctions of these two structures in a dynamic domain. For this reason, the modal analysis can, in a roundabout way, show how each structure reacts to dynamic conditions. The lower the mode, the

**Figure 5.** *Torsion axle: First normal mode of 33 Hz.*

*Dynamic Effect in Fatigue on High-Deflection Structures DOI: http://dx.doi.org/10.5772/intechopen.88988*

**Figure 6.** *Sub-frame: First normal mode of 283 Hz.*

more probable the structure may enter in resonance during a given random load input. When this happens, the system starts to store kinetic energy and oscillate in a higher amplitude, affecting the durability performance.

The two structures presented in this chapter, the sub-frame (pure structure) and the torsion axle (mechanism-like structure), were submitted to a simple modal analysis using a standard commercial finite element analysis software. It is possible to see how each of them has different first natural modes (**Figure 5**).

Because of the lower compliance, the sub-frame has a higher first normal frequency than the torsion axle, 283 Hz (**Figure 6**) for the sub-frame and 33 Hz for the torsion axle. Until reaching the first normal frequency of the sub-frame (283 Hz), the torsion axle will have also other seven normal frequencies, representing seven different modes to resonate in a frequency sweep from 0 to 283 Hz, while the subframe will have only one mode.

### **2.5 Fatigue assessment**

### *2.5.1 Process introduction*

The fatigue assessment combines the stress/strain results with the repetitions from the event to calculate the damage/life. The stress/strain results in this chapter came from the finite element mode and the load magnitude, and repetitions came from the multi-body model running the Belgian block track.

#### *2.5.2 Static load history approach*

The durability analysis utilizing the load history approach is the less difficult and most customary method for computing fatigue of a given event. It is performed by applying a unitary static load to each interface point of the structure and after that, consolidating the results of stress/strain utilizing direct superposition (**Figure 7**).

The models are loaded with linear unit magnitude static loads (forces and moments) in each degree of freedom (6 in total), and the result is the elemental stress.

These stresses are then used by the fatigue solver. The elemental stress result is combined to calculate the fatigue damage of the event, in this case, the Belgian blocks.

**Figure 7.** *Load history (load × time) of one hard point of the torsion axle in the X-axis.*

#### *2.5.3 Transient and frequency domain modal superposition*

The equation for dynamic motion of a system with linear single degree of freedom is given by the following:

$$\mathbf{x} = \mathbf{x} \tag{1}$$

$$\mathbf{M}\_{\text{mass}} \frac{\partial^2 \mathbf{x}}{\partial t^2}(t) + \mathbf{C}\_{damping} \frac{\partial \mathbf{x}}{\partial t}(t) + K\_{stiffness} \mathbf{x}(t) = L\_{load}(t) \tag{1}$$

The output of the system is a rotation or displacement, in function of a time t. Eq. (1) is a second-order ordinary differential equation that can describe the motion of the system, by its acceleration (second derivative of displacement in respect of time), velocity (derivative of displacement in respect of time), and displacement.

The Duhamel integral can provide a solution for a given *Lload*(*t*) for any instant greater than zero, i.e.,

$$\mathcal{X}(t) = \int\_0^t L\_{load}(t)h(t-\tau)d\tau \tag{2}$$

Equation (2) can be rewritten, considering that *h*(*t*) is the unit impulse (Dirac's delta) response function.

$$\mathcal{K}(\mathbf{t}) = L\_{load}(\mathbf{t}) \* h(\mathbf{t}) \tag{3}$$

Equation (2) will only have solutions in very simple systems. In general conditions, Eq. (2) will only have solutions when using algorithms to perform numerical integrations. These solutions for Eq. (2) should introduce a few volatilities that force the utilization of little steps of time (t). Moreover, the procedure can have a high cost (computational) regarding handling a large amount of time fractions (since the time step is small, a larger number of time fractions will be needed to represent the entire event). For this, two fundamental arrangements are known in general as modal transient response and direct transient response.

On the other hand, by taking the Fourier transform of Eq. (3) and using the concepts of spectral analysis [3–10], it is possible to simplify the time involution integration by multiplying in time domain, i.e.,

taking the Fourier transform of Eq. (3) permits simplifying the solution of the time convolution integration in a simple multiplication in frequency (ω) domain, i.e.,

$$X\{\alpha\_{\text{frequency}}\} = P\{\alpha\_{\text{frequency}}\} H\{\alpha\_{\text{frequency}}\} \tag{4}$$

Here, the functions *Lload*(*t*), *x*(*t*), *h*(*t*) are replaced by its Fourier transform pair, i.e., if considering *x*(*t*),

$$X\left(\mathfrak{u}\_{\{requence\}}\right) = \frac{1}{2\pi} \int\_{-\infty}^{\infty} \propto (t) \, e^{-i\mathfrak{a}\_{\{requence\}}t} \, dt \tag{5}$$

$$\mathbf{x}(\mathbf{t}) = \int\_{-\circ \circ}^{\circ} X \left( \mathbf{o}\_{\text{frequency}} \right) e^{i \mathbf{o}\_{\text{primary}} \cdot \mathbf{t}} \, d \, \mathbf{o}\_{\text{frequency}} \tag{6}$$

Here, ω*frequency* = 2*f*, where ω*frequency* and *f* are the linear and circular frequency variables, expressed, respectively, in [Hz] and [rad/s].

The last equation, Eq. (6), an inverse Fourier transform, and the normal Fourier transform in Eq. (5) can be numerically evaluated using FFT algorithms (fast Fourier transform) which are well-known and widely used.

The Fourier integrals in Eqs. (6), (5) will be valid, depending on the properties of the function considered. This occurs when the system in Eq. (1) is submitted to random loading inputs.

In the context of processes theory [2, 11–13], Eq. (4) must be used to handle input/output relationships when the system is submitted to random excitations.

A random event can be portrayed in frequency domain using spectral density functions. These functions, also known as power spectral densities (PSDs), along with the correlation functions are related by Fourier transform pairs.

The spectral density function *Sxx*(ω), for any stationary and ergodic random variable *x*(*t*), is given by

$$\mathcal{S}\_{\text{xx}}\{\alpha\_{frequency}\} = \frac{1}{2\pi} \int\_{-\infty}^{\infty} R\_{\text{xx}}(t) \, e^{-i\alpha\_{f\alpha\_{pump}\tau}} d\tau \tag{7}$$

with

$$R\_{\text{xx}}(\mathsf{T}) = \int\_{-\infty}^{\infty} \mathsf{S}\_{\text{x}}(\mathsf{o}\_{\text{frequency}}) \, e^{i\mathbf{o}\_{\text{primary}} \cdot \mathsf{T}} \, d\mathsf{o} \,\tag{8}$$

where *Rxx*(τ) is the autocorrelation of *x*(*t*), or in other words, it is the expected value *E*[*x*(*t*)*x*(*t* + τ)], i.e.,

$$R\_{\text{xx}}(\mathbf{r}) = \lim\_{T \to \infty} \frac{1}{T} \int\_0^T \mathbf{x}(t)\mathbf{x}(t + \mathbf{r})dt\tag{9}$$

and, Fourier transform is valid to be applied over *Rxx*(τ), in certain conditions. Equation (1) can be rewritten in form of a matrix for multiple degrees of freedom (MDOF), i.e.,

$$\begin{aligned} \left[\mathbf{M}\_{\text{mass}}\right] \left\{ \frac{\partial^2 \mathbf{x}}{\partial t^2} (t) \right\} + \left[\mathbf{C}\_{damping}\right] \left\{ \frac{\partial \mathbf{x}}{\partial t} (t) \right\} + \left[K\_{\text{eff}\text{f}\text{res}}\right] \left\{ \mathbf{x} (t) \right\} &= \left\{ L\_{\text{load}} (t) \right\} \end{aligned} \tag{10}$$

The multiple load input spectral density also can be expressed in a matrix form given by (where m is the number of load inputs)

 [*Spp*(ω*frequency*)]*m*×*<sup>m</sup>* <sup>=</sup> ⎡ ⎢ ⎣ *S*11(ω*frequency*) ⋯ *S*<sup>1</sup>*m*(ω*frequency*) <sup>⋮</sup> <sup>⋱</sup> <sup>⋮</sup> *Sm*1(ω*frequency*) ⋯ *Smm*(ω*frequency*) ⎤ ⎥ ⎦ (11)

where the off-diagonal terms *Sij*(ω*frequency*) are the spectral densities for the crosscorrelation of the load inputs (*Lloadi* (*t*) and *Lloadj* (*t*)) and the diagonal term *Sii*(ω*frequency*) is the auto spectral density of *Lloadi* (*t*).

Therefore, in the frequency domain, the input/output relation for the matrix system in Eq. (10) is (where n is the number of output response variables)

$$\left[\text{S}\right]\left[\text{xx}\left(\text{o}\_{\text{fraquency}}\right)\right]\_{\text{n }\text{xn}} = \left[H\left(\text{o}\_{\text{fraquency}}\right)\right]\_{\text{n }\text{xn}}\left[\text{S}\_{\text{pp}}\left(\text{o}\_{\text{fraquency}}\right)\right]\_{\text{m }\text{xn}}\left[H\left(\text{o}\_{\text{fraquency}}\right)\right]\_{\text{m }\text{n}}^{T}\tag{12}$$

The "\*" is the complex conjugate and T denotes transpose matrix.

The matrix [*H*(ω*frequency*)] is the transfer function matrix between the input loads and output response variables, i.e.,

$$\begin{aligned} \text{(2.2.2.2)}\\ \text{(} H\text{(co}\_{frequency}\text{))}\text{]} &= \frac{1}{\text{(}\text{[M]}\text{ o}\_{frequency}\text{)}\text{ + i [C]}\text{ o}\_{frequency}\text{+ [K]}}\end{aligned} \tag{13}$$

that can be calculated by standard FE solutions, as a unit modal frequency response. This transfer function becomes an input to the fatigue solver in frequency domain analysis process.

### *2.5.4 Conversion of loads from time to frequency domain*

The loads needed for the frequency domain analysis must be created using a Fourier transformation. Customarily, the correct interpretation of the frequency domain approach depends upon three premises (stationarity, Gaussian, random) completely met. In any case, some flexibility is conceivable. Besides, there are methodology that can be applied to break down nonstationary (and nonrandom, non-Gaussian) data into shorter subcases that do conform in an adequate manner.

The loading channels on both strictures are comprised of the number of hard points (where the structure interfaces with the environment, like constrains and load inputs) times the degrees of freedom. The torsion axle for example has 48 loading channels, 8 hard points (as can be seen in **Figure 8**, hard points are represented as yellow circles) times 6 degrees of freedom. For the frequency domain approach to deal with the correlation between channels, the purported cross-PSD's are additionally required. Each channel requires a real and imaginary PSD (together they compose the complex cross-PSD), and these are observed in **Figure 9**.

**Figure 8.** *The hard points of the torsion axle.*

*Dynamic Effect in Fatigue on High-Deflection Structures DOI: http://dx.doi.org/10.5772/intechopen.88988*


**Figure 9.** *Torsion axle. PSD matrix for event 1.*
