After discussing the problem of a plate with fixed corners, we have a look at the problem of a plate with two mass attachment types:

In the present analysis, only the isotropic plate material system was studied. The same procedure could be followed for orthotropic plate system.

For case 1, the concentrated mass (*M*) is mounted at the plate center point $\begin{array}{}{\displaystyle (x\phantom{\rule{1em}{0ex}}=\phantom{\rule{1em}{0ex}}\frac{a}{2},y\phantom{\rule{1em}{0ex}}=\phantom{\rule{1em}{0ex}}\frac{b}{2}),}\end{array}$ as shown in Figure 3(a). For case 2, as depicted in Figure 3(b), four equal masses (the mass of each is *M*) are attached at the centers of the plate edges
$\begin{array}{}{\displaystyle \left\{\left(x=\frac{a}{2},y=0\right),\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\left(x=a,\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}y=\frac{b}{2}\right),\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\left(x=\frac{a}{2},y=b\right),\right.\left.\left(x=0,y=\frac{b}{2}\right)\right\}.}\end{array}$

Figure 3 Corner-clamped plate with mass attachments; (a) case 1, (b) case 2.

This mass addition will have no effect on the structure’s potential energy. Generally, however, for a plate with attached mass (either for case 1 or case 2), the kinetic energy of the system could be expressed as:

$$\begin{array}{}{\displaystyle T=\frac{\rho h}{2}\underset{A}{\overset{}{\iint}}\left[{\left(\dot{w}\left(x,y\right)\right)}^{2}\right]dxdy+\frac{M}{2}\sum _{i=1}^{N}\dot{{w}^{2}}({\zeta}_{i},{\eta}_{i})}\end{array}$$(30)

where *ẇ*(*ζ*_{i},*η*_{i}) are the velocities of the plate at the mass fix-up locations, and *N* is the total number of attached masses.

Following the previously described natural frequency analysis procedure, the system’s fundamental natural frequency is:

$$\begin{array}{}{\displaystyle \omega =\frac{{\lambda}^{2}}{{a}^{2}}\sqrt{\frac{D}{\alpha \frac{M}{{a}^{2}}+\rho h}}}\end{array}$$(31)

where *α* is the reduced coefficient of the mass, and it is function of plate aspect ratio *b*/*a*, and Poisson’s ratio *ν*. Again, assuming that *ν* = 0.3, then the values of *α* for several plate aspect ratios, for both cases, are summarized in .

Table 2 Reduced coeficient of the mass for several plate aspect ratios for plate with center mass *ν*=0.3).

It is important to mention at this point that this mass addition has no effect on the mode shape constants and plate eigenvalues, and their values are still equal to those listed in .

As done previously, a general equation to calculate the reduced coefficient of the mass, for both cases, was formulated. The suggested general form for this equation is:

$$\begin{array}{}{\displaystyle \alpha ={c}_{1}{\left(\frac{1}{\frac{b}{a}}\right)}^{{c}_{2}}}\end{array}$$(32)

where *c*_{1} and *c*_{2} are coefficients determined using a nonlinear regression analysis as *c*_{1} = 4.35, *c*_{2} = 1.32 for case 1, and *c*_{1} = 6.85, *c*_{2} = 0.72 for case 2. A comparison between the fitted formula and the actual (analytical) data of *α* is shown in Figure 4. For both cases, the fitted formulas are in excellent match with the analytically derived data.

Figure 4 The reduced coefficient of the mass vs plate aspect ratio for (a) case 1 and (b) case 2.

As a special configuration, we shall consider the mass (*M*) of the added mass (or masses) to be a function of the plate mass. Thus:

$$\begin{array}{}{\displaystyle M=nbah\rho}\end{array}$$(33)

where *n* is the plate mass fraction, and it can be any positive number: i.e., if *n* = 0.5 or *n* = 2, then the total added mass is half of the plate mass or double of it, respectively. Now, substitute Eq. (33) into Eq. (31), and after some simple mathematical manipulation we achieve:

$$\begin{array}{}{\displaystyle \omega =\frac{{\lambda}^{2}}{{a}^{2}}\sqrt{\frac{1}{n\alpha \frac{b}{a}+1}}\sqrt{\frac{D}{\rho h}}}\end{array}$$(34)

or:

$$\begin{array}{}{\displaystyle \omega =\frac{{\mathit{\Lambda}}^{2}}{{a}^{2}}\sqrt{\frac{D}{\rho h}}}\end{array}$$(35)

where *Λ*^{2} = *λ*^{2}$\begin{array}{}{\displaystyle \sqrt{\frac{1}{n\alpha \frac{b}{a}+1}}}\end{array}$ is the massed plate eigenvalue.

From Eq. (35) above, *Λ*^{2} is now function of *b*/*a*, *ν* and *α*, as well as *n*. Luckily, *α* is function of *b*/*a* (besides to *ν*), for both cases. Therefore, we can implicitly say that *Λ*^{2} is a function of *b*/*a*, *ν* and *n* only. As before, assuming *ν* = 0.3, and summarize the values of *Λ*^{2} for different *b*/*a* and *n* combinations for case 1 and case 2, respectively.

Table 3 Values of *Λ*^{2} at different b/a and n for case 1 (*ν*=0.3).

Table 4 Values of *Λ*^{2} at different b/a and n for case 2 (*ν*=0.3).

To establish a formula to calculate *Λ*^{2} as a function of the plate aspect ratio and of mass fraction factor, for both cases, a general non-linear form, which is a combination of Eq. (27) and Eq. (35), was selected as follows:

$$\begin{array}{}{\displaystyle {\mathit{\Lambda}}^{2}={a}_{1}{\left(\frac{1}{\frac{b}{a}}\right)}^{{a}_{2}}\sqrt{\frac{1}{{d}_{1}n{\left(\frac{1}{\frac{b}{a}}\right)}^{{d}_{2}}+1}}}\end{array}$$(36)

Here, *a*_{1} and *a*_{2}, for case 1 and case 2, are similar to those of Eq. (27). Also, *d*_{1} and *d*_{2} are two unknown coefficients. To obtain them, a multi-parametric fit was performed considering both case 1 and case 2. The resultant constant values are (*d*_{1} = 4.33, *d*_{2} = 0.29) and (*d*_{1} = 6.89, *d*_{2} = –0.26) for case 1 and case 2, respectively. Figure 5 shows a comparison between the analytically derived data and the fitted formulas for both mass attachment cases, for several plate aspect ratios. In conclusion, the *Λ*^{2} values for case 1 and case 2, from both solutions, are in a great agreement.

Figure 5 The massed plate eigenvalues (*Λ*^{2}) vs. the plate mass fraction (*n*) at different *b*/*a* for (a) case 1 and (b) case 2.

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