bending natural frequencies are given in table3.3. These values for the slenderness ratio are plotted in figure 3.10 as a function of the number of bays in the lattice structure. The slenderness ratio is plotted for the first four long wavelength longitudinal modes of the structure. The dashed line represents the Euler-Bernoulli beam theory limit for the first beam mode.
Figure 3.11 shows the slenderness ratio for the first LWMs for bending, torsion and longitudinal motion of the lattice structure as a function of the number of bays in the structure. The nominal slenderness ratio for the structural members is also identified. Clearly for the 10 bay structure considered in this thesis, the right hand side of equation
3.24 is less than lB/∆ for all i, so that βi is less than one. This result is also in
accordance to the results of figure 3.6 where the SWM occur first then the LWM.
3.6
Factors controlling the SW and LWMs
In the expressions for the ratio βi as in equation 3.20, it can be seen that the ratio
βi depends upon two terms; the mode shape number of the LWM and the geometric
properties of the structure. These geometric properties are the length of the batten (or longitudinal members), the variable defining the cross sectional area ∆, and the number of bays, m. It is clear that modifying the number of bays of the structure will have a large influence on the LW natural frequencies as they change with the inverse of the number of bays in the structure. Changing the number of bays in the structure influences the SW natural frequencies slightly, because the boundary conditions of the members are slightly modified. This has been discussed in the introduction to this chapter, where the ends of the structure may have an effect on the boundary conditions of the structural members. Changing the parameter ∆, which is related to the cross section of the members, influences only the SW natural frequencies. This can be seen in equations3.11-3.13, where these expressions have no dependency on the cross section of the members. The other parameter influencing the slenderness ratio is the length of the members. By changing the length of batten members, for instance, both SWM and LWM can be affected (note, the relationship lL=lB =lD/
√
2 is assumed). Because in most cases, the length of the structure is a fixed parameter, modifying the
Chapter Three 3.6. Factors controlling the SW and LWMs number of bays in the structure is only possible by changing the length of the structural members. The parameter ∆, however, can be used for tailoring the spectrum of SWM. It is usually possible to change the stiffness to weight ratio by changing a solid circular cross section to a hollow circular cross section. The parameter ∆ changes from d to
p
d2
2+d21. The practical result of this modification is the possibility to increase the
bending stiffness of the members without addition of weight, if the area of the solid and hollow circular cross sections remain equal (a change in the volume of the members is, however, inevitable).
To demonstrate the influence of these parameters on the dynamics of the structure, nu- merical simulation are presented. In figure3.12, the natural frequencies of the structure are represented by the dots in the frequency range 0 - 1 kHz. These natural frequen- cies were calculated by the finite element method using the three-dimensional beam
element available in the commercial software ANSYS dividing the structural members into 20 element each. These natural frequencies were calculated for different values of the non-dimensional ratio (I/l2BS) and are shown as normalized values by the nominal ratio (I/l2
BS) = 1.2445× 10
−5. It can be seen that as the ratio (I/l2
BS) increases,
some of the dots representing the natural frequencies are shifted in a fashion propor- tional to the square root of this ratio (as β ∝ (I/l2
BS)1/2), in this case, mainly the
SW natural frequencies are changed. The LW natural frequencies, however, remain nearly unchanged. Some differences in the LW natural frequencies are expected when SWM occur at frequencies close to the LWMs. The shifting effect of the SWM can also influence the structural responses to disturbance forces. This is demonstrated for the cost function considered in this work, for a disturbance force applied at joint 4, in the y direction. The sum of squared linear velocities at joints 31, 32 and 33 was calculated for different values of the ratio (I/l2
BS) using the dynamic stiffness method.
The sum of these cost functions over the frequency range 20 Hz - 1 kHz (with 1 Hz frequency resolution) are plotted in figure3.13 as a function of the ratio (I/l2BS) where the values are given in dB. As can be seen in the results of figure 3.13, the sum of the cost function varies by up to 3.5 dB, for the values of (I/l2
BS)/(I/lB2S)nominal varying
between 1 and 25. The conclusion is that although the structural response can be quite different when SWM are shifted in frequency, the overall level of the structure response
Chapter Three 3.7. The analysis of power in the lattice structure