Determining the dynamic characteristics of structures with viscoelastic dampers according to temperature changes within their specific range requires multiple solving of the nonlinear eigenproblem (
26) because recalculation of the parameters of dampers for each temperature is required. This is quite troublesome because, as mentioned earlier, this task can only be solved by iterative methods. However, the use of the Lazaro assumptions [
19] enables a direct determination of the course of the natural frequency and the dimensionless damping factor functions within a certain temperature range. Namely, the shift factor
\(\hat{{\alpha }}_T \) in Eq. (
29) can be treated as a homotopy coefficient, as does the coefficient
p in Eq. (
38). After performing the corresponding mathematical operations described in Sect.
6, the following equation is obtained:
$$\begin{aligned} \frac{\hbox {d}s}{\hbox {d}\hat{{\alpha }}_T }=-\frac{s^{\alpha }}{2a}D\left( {{\bar{\mathbf{q}}},s,\hat{{\alpha }}_T } \right) , \end{aligned}$$
(61)
where
$$\begin{aligned} D\left( {{\bar{\mathbf{q}}},s,\hat{{\alpha }}_T } \right) ={\bar{\mathbf{q}}}^\mathrm{T}\left( {\hat{{\alpha }}_T } \right) \sum _{r=1}^m {\left( {c_{0r} +\frac{k_{1r} \nu _{1r} }{\left( {\nu _{1r} +\hat{{\alpha }}_T s^{\alpha }} \right) ^{2}}} \right) \mathbf{L}_r } {\bar{\mathbf{q}}}\left( {\hat{{\alpha }}_T } \right) . \end{aligned}$$
(62)
The next step is to expand the function
\(D\left( {{\bar{\mathbf{q}}},s,\hat{{\alpha }}_T } \right) \) in the Taylor series for
\(\hat{{\alpha }}_T =1\):
$$\begin{aligned} D\left( {{\bar{\mathbf{q}}},s,\hat{{\alpha }}_T } \right) =D\left( {{\bar{\mathbf{q}}}_0 ,s_0 ,1} \right) +\left. {\left( {\hat{{\alpha }}_T -1} \right) \frac{\hbox {d}D}{\hbox {d}\hat{{\alpha }}_T }} \right| _{\hat{{\alpha }}_T =1} , \end{aligned}$$
(63)
where
\({\bar{\mathbf{q}}}_0 \) and
\(s_0 \) are the eigenvector and the eigenvalue of the system, respectively, for the reference temperature
\(T_0 \quad \left( {\hat{{\alpha }}_T =1} \right) \). Finally, Eq. (
61) takes the following form:
$$\begin{aligned} \frac{\hbox {d}s}{\hbox {d}\hat{{\alpha }}_T }=-\frac{s^{\alpha }}{2a}\left[ {d_1 +\left( {\hat{{\alpha }}_T -1} \right) d_2 } \right] , \end{aligned}$$
(64)
where
$$\begin{aligned} d_1= & {} {\bar{\mathbf{q}}}_0^\mathrm{T} \sum _{r=1}^m {\left( {c_{0r} +\frac{k_{1r} \nu _{1r} }{\left( {\nu _{1r} +\hat{{\alpha }}_T s_0^\alpha } \right) ^{2}}} \right) } \mathbf{L}_r {\bar{\mathbf{q}}}_0 , \end{aligned}$$
(65)
$$\begin{aligned} d_2= & {} 2{\bar{\mathbf{q}}}_0^\mathrm{T} \sum _{r=1}^m {\left( {c_{0r} +\frac{k_{1r} \nu _{1r} }{\left( {\nu _{1r} +\hat{{\alpha }}_T s_0^\alpha } \right) ^{2}}} \right) } \mathbf{L}_r \left. {\frac{\hbox {d}{\bar{\mathbf{q}}}_0 }{\hbox {d}\hat{{\alpha }}_T }} \right| _{\hat{{\alpha }}_T =1} + \nonumber \\&-2{\bar{\mathbf{q}}}_0^\mathrm{T} \sum _{r=1}^m {\frac{k_{1r} \nu _{1r} \hat{{\alpha }}_T \alpha s_0^{\alpha -1} }{\left( {\nu _{1r} +\hat{{\alpha }}_T s_0^\alpha } \right) ^{3}}} \mathbf{L}_r {\bar{\mathbf{q}}}_0 \left. {\frac{\hbox {d}s_0 }{\hbox {d}\hat{{\alpha }}_T }} \right| _{\hat{{\alpha }}_T =1} -2{\bar{\mathbf{q}}}_0^\mathrm{T} \sum _{r=1}^m {\frac{k_{1r} \nu _{1r} \alpha s_0^\alpha }{\left( {\nu _{1r} +\hat{{\alpha }}_T s_0^\alpha } \right) ^{3}}} \mathbf{L}_r {\bar{\mathbf{q}}}_0 , \end{aligned}$$
(66)
The sensitivities of the eigenvector and eigenvalue due to changes in the parameter
\(\hat{{\alpha }}_T \) are calculated from the following matrix sensitivity equation:
$$\begin{aligned} \left[ {{\begin{array}{ll} {\mathbf{A}_{11} }&{} {\mathbf{A}_{12} } \\ {\mathbf{A}_{21} }&{} {\mathbf{A}_{22} } \\ \end{array} }} \right] \left\{ {{\begin{array}{l} {\left. {\frac{\hbox {d}{\bar{\mathbf{q}}}}{\hbox {d}\hat{{\alpha }}_T }} \right| _{\hat{{\alpha }}_T =1} } \\ {\left. {\frac{\hbox {d}s}{\hbox {d}\hat{{\alpha }}_T }} \right| _{\hat{{\alpha }}_T =1} } \\ \end{array} }} \right\} =\left\{ {{\begin{array}{l} {\mathbf{b}_1 } \\ {\mathbf{b}_2 } \\ \end{array} }} \right\} , \end{aligned}$$
(67)
where
$$\begin{aligned} \mathbf{A}_{11}= & {} s_0^2 \mathbf{M}+s_0 \mathbf{C}+{\tilde{\mathbf{K}}}+\sum _{r=1}^m {\frac{k_{1r} s_0^\alpha }{\nu _{1r} +s_0^\alpha }} \mathbf{L}_r , \end{aligned}$$
(68)
$$\begin{aligned} \mathbf{A}_{12}= & {} \mathbf{A}_{21} =2s_0 \mathbf{M}+\mathbf{C}+\sum _{r=1}^m {\frac{k_{1r} \nu _{1r} \alpha s_0^{\alpha -1} }{\left( {\nu _{1r} +s_0^\alpha } \right) ^{2}}} \mathbf{L}_r {\bar{\mathbf{q}}}_0 , \end{aligned}$$
(69)
$$\begin{aligned} \mathbf{A}_{22}= & {} \frac{1}{2}{\bar{\mathbf{q}}}_0^\mathrm{T} \left[ {2\mathbf{M}+} \right. \left. {\sum _{r=1}^m {k_{1r} \nu _{1r} \alpha s_0^{\alpha -2} \frac{\left( {\alpha -1} \right) \nu _{1r} +s_0^\alpha \left( {\alpha +1} \right) }{\left( {\nu _{1r} +s_0^\alpha } \right) ^{3}}{} \mathbf{L}_r } } \right] {\bar{\mathbf{q}}}_0 , \end{aligned}$$
(70)
$$\begin{aligned} \mathbf{b}_1= & {} -\sum _{r=1}^m {\alpha s_0^{\alpha -2} \frac{k_{1r} \nu _{1r} }{\left( {\nu _{1r} +s_0^\alpha } \right) ^{2}}{} \mathbf{L}_r {\bar{\mathbf{q}}}_0 } , \end{aligned}$$
(71)
$$\begin{aligned} \mathbf{b}_2= & {} -\frac{1}{2}{\bar{\mathbf{q}}}_0^\mathrm{T} \sum _{r=1}^m {\frac{k_{1r} \nu _{1r} \alpha s_0^{\alpha -1} \left( {\nu _{1r} -s_0^\alpha } \right) }{\left( {\nu _{1r} +s_0^\alpha } \right) ^{3}}{} \mathbf{L}_r {\bar{\mathbf{q}}}_0 } , \end{aligned}$$
(72)