2. Analytical solution of the structural frequency

Figure 1 presents the bar model of a structure in free vibration. Consider the following trigonometric function, taken as valid throughout its domain:

$$\phi(\mathbf{x}) = \mathbf{1} - \cos\left(\frac{\pi\mathbf{x}}{2L}\right),\tag{1}$$

where x is the location of the calculation, originating at the base of the cantilever, and L is the length of the column.

That model represents a column under an axial compressive load, Ns(x), with either constant or variable properties along its length. These properties include the geometry, elasticity/viscoelasticity, and density. Applied springs of variable stiffness kso(x) act as the lateral soil resistance until the foundation elevation Gr.

Using Dynamic Analysis to Adjust the Rheological Model of Three Parameters… DOI: http://dx.doi.org/10.5772/intechopen.88665

Figure 1. Frame element model in free vibration.

The system is under the action of gravitational normal forces, originating from the distributed mass along the length of the column and of a lumped mass at the tip m0.

In the case of vibration of a cantilevered column that is clamped at its base and free at its tip, the shape function given in Eq. (1) satisfies the boundary conditions of the problem. The use of Eq. (1) as a shape function for an actual structure with varying geometry has been validated by [10]. This validation involved a comparison with a computational solution derived using computational modeling by finite element method (FEM) and other mathematical expressions.

By applying the principle of virtual work and its derivations, the dynamic properties of the subject system are obtained. The elastic/viscoelastic conventional stiffness is given by

$$k\_{0\varepsilon}(t) = \int\_{L\_{\varepsilon}}^{L\_{\varepsilon}} E\_{\varepsilon}(t) I\_{\varepsilon}(\infty) \left(\frac{d^2 \phi(\infty)}{d\mathbf{x}^2}\right)^2 d\mathbf{x}, \text{ with } K\_0(t) = \sum\_{\varepsilon=1}^n k\_{0\varepsilon}(t),\tag{2}$$

where for a segment s of the structure, Es(t) is the viscoelastic modulus of the material with respect to time; Is(x) is the variable moment of inertia of the section along the segment in relation to the considered movement, obtained by interpolation of the previous and following sections and if it is constant, it is simply Is; k0s(t) is the temporal term for the stiffness; K0(t) is the final conventional stiffness

#### Dynamical Systems Theory

varying over time; and n is the total number of segment intervals given by the structural geometry. In Eq. (2), obviously, t vanishes when the analysis considers a material with purely elastic, time-independent behavior. The geometric stiffness appears as a function of the axial load, including the self-weight contribution and is expressed as

$$k\_{\rm gc}(m\_o) = \int\_{L\_{\rm e-1}}^{L\_{\rm e}} \left[ N\_0(m\_o) + \sum\_{j=s+1}^n N\_j + \overline{m}\_i(\infty)(L\_s - \infty) \mathbf{g} \right] \left( \frac{d\phi(\infty)}{d\mathbf{x}} \right)^2 d\mathbf{x} \text{ and } \tag{3}$$

$$K\_{\mathfrak{F}}(m\_0) = \sum\_{s=1}^{n} k\_{\mathfrak{F}^s}(m\_0),\tag{4}$$

where kgs(m0) is the geometric stiffness in segment s, Kg(m0) is the total geometric stiffness of the structure with n as defined previously, and N0(m0) is the concentrated force at the top, all of which are dependent on the mass m<sup>0</sup> at the tip, given by

$$N\_0(m\_0) = m\_0 \text{g.}\tag{5}$$

Further, Nj is the normal force from the upper segments, obtained by

$$N\_{\circ} = \int\_{L\_{r-1}}^{L\_{r}} \overline{m}\_{\circ}(\infty) \mathbf{g} d\infty. \tag{6}$$

Then, the total generalized mass is given by

$$M(m\_0) = m\_0 + m,\tag{7}$$

considering that

$$m = \sum\_{\iota=1}^{n} m\_{\iota} \text{ with } m\_{\iota} = \int\_{L\_{\iota-1}}^{L\_{\iota}} \overline{m}\_{\iota}(\mathbf{x}) (\phi(\mathbf{x}))^2 d\mathbf{x}, \text{ and } \overline{m}\_{\iota}(\mathbf{x}) = A\_{\iota}(\mathbf{x}) \rho\_{\iota}, \tag{8}$$

where msð Þ x is the mass distributed to each segment s, which is obtained by multiplying the cross-sectional area, As(x), by the density, ρs, of the material in the respective interval. Therefore, msð Þ x is the mass per unit length, and m is the generalized mass of the system owing to the density of the material, with n as previously defined. If the cross section has a constant area over the interval, As(x) will be just As; consequently, the distributed mass will also be constant. Similarly, if the mass m<sup>0</sup> does not vary, all the other parameters that depend on it will also be constant.

One approach for considering the participation of the soil in the vibration of the system is to consider it as a series of vertically distributed springs that act as a restorative force on the system. With kSos(x) denoting the spring parameter, the effective soil stiffness (as a function of the location x along the length) is generally defined as

$$K\_{\rm So} = \sum\_{\varepsilon=1}^{n} k\_{\rm is} \text{ with } k\_{\varepsilon} = \int\_{L\_{\rm -1}}^{L\_{\rm }} k\_{\rm So}(\mathbf{x}) \phi(\mathbf{x})^2 d\mathbf{x}, \text{ where } k\_{\rm So}(\mathbf{x}) = \mathbf{S}\_{\rm as} D\_{\rm s}(\mathbf{x}), \tag{9}$$

Using Dynamic Analysis to Adjust the Rheological Model of Three Parameters… DOI: http://dx.doi.org/10.5772/intechopen.88665

where the parameter KSo is an elastic characteristic consisting of the sum of kSos(x) along the foundation depth, which depends on the geometry of the foundation Ds(x) and the soil parameter Sos. Considering the normal force as positive, the total structural stiffness is obtained as

$$K(t, m\_0) = K\_0(t) - K\_{\rm g}(m\_0) + K\_{\rm So}.\tag{10}$$

Finally, the natural frequency (in Hertz), as a function of the time and the mass at the tip, is calculated according to Eq. (11). The great advantage of using that equation in terms of two independent variables is that it can be employed to calculate the critical load of buckling as well, because all the generalized parameters are expressed as a function of the mass at the top. Details of this analytical procedure can be seen in [11]:

$$f(t, m\_0) = \frac{1}{2\pi} \sqrt{\frac{K(t, m\_0)}{M(m\_0)}}.\tag{11}$$
