The coefficients of differential equations of the DPDM can be obtained using Eq. (
5) by eliminating the terms related to the bottom pendulum, i.e., third pendulum and given by
$$\begin{aligned} c_{11}= & {} I_{c1} +\left( {\frac{1}{4}m_{1}+m_{2}} \right) l_{1}^{2},\quad c_{12} =\frac{1}{2}m_{2}l_{1}l_{2}\cos (\theta _{1}-\theta _{2}),\quad c_{21} =\frac{1}{2}m_{2}l_{1}l_{2}\cos (\theta _{1}-\theta _{2}), \nonumber \\ c_{22}= & {} I_{c2} +\left( {\frac{1}{4}m_{2}+m_{3}} \right) l_{2}^{2},\quad d_{11} =\frac{1}{2}m_{2}l_{1}l_{2}\sin (\theta _{1}-\theta _{2})\theta _{2}^{2}+\left( {\frac{1}{2}m_{1}+m_{2}} \right) gl_{1}\sin \theta _{1}, \nonumber \\ d_{22}= & {} -\frac{1}{2}m_{2}l_{1}l_{2}\sin (\theta _{1}-\theta _{2})\theta _{2}^{2}+\frac{1}{2}m_{2}gl_{2}\sin \theta _{2}. \end{aligned}$$
(15)
Following the similar procedure, the coefficients of mass and stiffness for the SPPM may be obtained by eliminating the terms related to the second pendulum of the simple DP system.
The coefficients of [
K] and [
M] matrices for the linearized quadruple pendulum with distributed mass (QPDM) are obtained on the basis of Eq. (
13) as
$$\begin{aligned} d_{11}= & {} \left( {\frac{1}{2}m_{1}+m_{2}+m_{3}+m_{4}} \right) gl_{1},\quad d_{22} =\left( {\frac{1}{2}m_{2}+m_{3}+m_{4}} \right) gl_{2},\quad d_{33} =\left( {\frac{1}{2}m_{3}+m_{4}} \right) gl_{3}, \nonumber \\ d_{44}= & {} \frac{1}{2}m_{4}gl_{4},\quad c_{11} =Ic{ }_{1}+\left( {\frac{1}{4}m_{1}+m_{2}+m_{3}+m_{4}} \right) l_{1}^{2},\quad c_{12} =\left( {\frac{1}{2}m_{2}+m_{3}+m_{4}} \right) l_{1}l_{2}, \nonumber \\ c_{13}= & {} \left( {\frac{1}{2}m_{3}+m_{4}} \right) l_{1}l_{3},\quad c_{14} =\frac{1}{2}m_{4}l_{1}l_{4},\quad c_{21} =\left( {\frac{1}{2}m_{2}+m_{3}+m_{4}} \right) l_{2}l_{1}, \nonumber \\ c_{22}= & {} Ic_{2}+\left( {\frac{1}{4}m_{2}+m_{3}+m_{4}} \right) l_{2}^{2},\quad c_{23} =\left( {\frac{1}{2}m_{3}+m_{4}} \right) l_{2}l_{3},\quad c_{24} =\frac{1}{2}m_{4}l_{2}l_{4}, \nonumber \\ c_{31}= & {} \left( {\frac{1}{2}m_{3}+m_{4}} \right) l_{3}l_{1},\quad c_{32} =\left( {\frac{1}{2}m_{3}+m_{4}} \right) l_{3}l_{2},\quad c_{33} =I_{c3} +\left( {\frac{1}{4}m_{3}+m_{4}} \right) l_{3}^{2},\nonumber \\ c_{34}= & {} \frac{1}{2}m_{4}l_{3}l_{4},\quad c_{41} =\frac{1}{2}m_{4}l_{4}l_{1},\quad c_{42} =\frac{1}{2}m_{4}l_{4}l_{2},\quad c_{43} =\frac{1}{2}m_{4}l_{4}l_{3}, \nonumber \\ c_{44}= & {} I_{c4} +\frac{1}{4}m_{4}l_{4}^{2},\quad I_{c1} =\left( {\frac{m_{1}l_{1}^{2}}{12}} \right) ,\quad Ic_{2}=\left( {\frac{m_{2}l_{2}^{2}}{12}} \right) ,\quad I_{c3} =\left( {\frac{m_{3}l_{3}^{2}}{12}} \right) ,\quad I_{c4} =\left( {\frac{m_{4}l_{4}^{2}}{12}} \right) . \end{aligned}$$
(16)
And the coefficients of mass and stiffness matrices for the simple quintuple pendulum with distributed mass (QtPDM) are obtained using Eq. (
16) as
$$\begin{aligned} d_{11}= & {} \left( {\frac{1}{2}m_{1}+m_{2}+m_{3}+m_{4}+m_{5}} \right) gl_{1},\quad d_{22} =\left( {\frac{1}{2}m_{2}+m_{3}+m_{4}+m_{5}} \right) gl_{2},\nonumber \\ d_{33}= & {} \left( {\frac{1}{2}m_{3}+m_{4}+m_{5}} \right) gl_{3},\quad d_{44} =\left( {\frac{1}{2}m_{4}+m_{5}} \right) gl_{4},\quad d_{55} =\frac{1}{2}m_{5}gl_{5},\nonumber \\ c_{11}= & {} Ic{ }_{1}+\left( {\frac{1}{4}m_{1}+m_{2}+m_{3}+m_{4}+m_{5}} \right) l_{1}^{2},\quad c_{12} =\left( {\frac{1}{2}m_{2}+m_{3}+m_{4}+m_{5}} \right) l_{1}l_{2},\nonumber \\ c_{13}= & {} \left( {\frac{1}{2}m_{3}+m_{4}+m_{5}} \right) l_{1}l_{3},\quad c_{14} =\left( {\frac{1}{2}m_{4}+m_{5}} \right) l_{1}l_{4},\quad c_{15} =\frac{1}{2}m_{5}l_{1}l_{5},\nonumber \\ c_{21}= & {} \left( {\frac{1}{2}m_{2}+m_{3}+m_{4}+m_{5}} \right) l_{2}l_{1},\quad c_{22} =Ic_{2}+\left( {\frac{1}{4}m_{2}+m_{3}+m_{4}+m_{5}} \right) l_{2}^{2},\quad c_{23} =\left( {\frac{1}{2}m_{3}+m_{4}+m_{5}} \right) l_{2}l_{3}\nonumber \\ c_{24}= & {} \left( {\frac{1}{2}m_{4}+m_{5}} \right) l_{2}l_{4},\quad c_{25} =\frac{1}{2}m_{5}l_{2}l_{5},\quad c_{31} =\left( {\frac{1}{2}m_{3}+m_{4}+m_{5}} \right) l_{3}l_{1},\quad c_{32} =\left( {\frac{1}{2}m_{3}+m_{4}+m_{5}} \right) l_{3}l_{2},\nonumber \\ c_{33}= & {} I_{c3} +\left( {\frac{1}{4}m_{3}+m_{4}+m_{5}} \right) l_{3}^{2},\quad c_{34} =\left( {\frac{1}{2}m_{4}+m_{5}} \right) l_{3}l_{4},\quad c_{35} =\frac{1}{2}m_{5}l_{3}l_{5},\quad c_{41} =\left( {\frac{1}{2}m_{4}+m_{5}} \right) l_{4}l_{1},\nonumber \\ c_{42}= & {} \left( {\frac{1}{2}m_{4}+m_{5}} \right) l_{4}l_{2},\quad c_{43} =\left( {\frac{1}{2}m_{4}+m_{5}} \right) l_{4}l_{3},\quad c_{44} =I_{c4} +\left( {\frac{1}{4}m_{4}+m_{5}} \right) l_{4}^{2},\quad c_{45} =\frac{1}{2}m_{5}l_{4}l_{5},\nonumber \\ c_{51}= & {} \frac{1}{2}m_{5}l_{5}l_{1},\quad c_{52} =\frac{1}{2}m_{5}l_{5}l_{2},\quad c_{53} =\frac{1}{2}m_{5}l_{5}l_{3},\quad c_{54} =\frac{1}{2}m_{5}l_{5}l_{4},\quad c_{55} =I_{c5} +\frac{1}{4}m_{5}l_{5}^{2}, \nonumber \\ I_{c1}= & {} \left( {\frac{m_{1}l_{1}^{2}}{12}} \right) ,\quad I_{c2} =\left( {\frac{m_{2}l_{2}^{2}}{12}} \right) ,\quad I_{c3} =\left( {\frac{m_{3}l_{3}^{2}}{12}} \right) ,\quad I_{c4} =\left( {\frac{m_{4}l_{4}^{2}}{12}} \right) ,\quad I_{c5} =\left( {\frac{m_{5}l_{5}^{2}}{12}} \right) . \end{aligned}$$
(17)
The coefficients of [
K] and [
M] for the simple TPPM can be generated using Eq. (
5) by eliminating the terms related to the moment of inertia and replacing the numerical factors of remaining terms with one. The following coefficients of the TPPM are obtained as
$$\begin{aligned} c_{11}= & {} \left( m_{1}+m_{2}+m_{3}\right) l_{1}^{2},\quad c_{12} =m_{2}l_{1}l_{2}\cos \left( \theta _{1}-\theta _{2}\right) +m_{3}l_{1}l_{2}\cos \left( \theta _{1}-\theta _{2}\right) , \nonumber \\ c_{13}= & {} m_{3}l_{1}l_{3}\cos \left( \theta _{3}-\theta _{1}\right) ,\quad c_{21} =\left( m_{2}+m_{3}\right) l_{1}l_{2}\cos \left( \theta _{1}-\theta _{2}\right) ,\quad c_{22} =\left( m_{2}+m_{3}\right) l_{2}^{2}, \nonumber \\ c_{23}= & {} m_{3}l_{2}l_{3}\cos \left( \theta _{2}-\theta _{3}\right) ,\quad c_{31} =m_{3}l_{1}l_{3}\cos \left( \theta _{3}-\theta _{1}\right) ,\quad c_{32} =m_{3}l_{2}l_{3}\cos \left( \theta _{2}-\theta _{3}\right) ,\quad c_{33} =m_{3}l_{3}^{2}, \nonumber \\ d_{11}= & {} \left( m_{2}+m_{3}\right) l_{1}l_{2}\sin \left( \theta _{1}-\theta _{2}\right) \dot{\theta }_{2}^{2}+m_{3}l_{1}l_{3}\sin \left( \theta _{1}-\theta _{3}\right) \dot{\theta }_{3}^{2}+\left( m_{1}+m_{2}+m_{3}\right) gl_{1}\sin \theta _{1}, \nonumber \\ d_{22}= & {} -\left( m_{2}+m_{3}\right) l_{1}l_{2}\sin \left( \theta _{1}-\theta _{2}\right) \dot{\theta }_{1}^{2}+m_{3}l_{2}l_{3}\sin \left( \theta _{2}-\theta _{3}\right) \dot{\theta }_{3}^{2}+\left( m_{2}+m_{3}\right) gl_{2}\sin \theta _{2}, \nonumber \\ d_{33}= & {} -m_{3}l_{1}l_{3}\sin \left( \theta _{1}-\theta _{3}\right) \dot{\theta }_{1}^{2}-m_{3}l_{2}l_{3}\sin \left( \theta _{2}-\theta _{3}\right) \dot{\theta }_{2}^{2}+m_{3}gl_{3}\sin \theta _{3}. \end{aligned}$$
(18)
The coefficients of [
K] and [
M] for the linearized DPPM are obtained using Eq. (
16) in which the coefficients related to the third pendulum are eliminated obtained as
$$\begin{aligned} c_{11}= & {} (m_{1}+m_{2})l_{1}^{2},\quad c_{12} =m_{2}l_{1}l_{2}\cos (\theta _{1}-\theta _{2}),\quad c_{21} =m_{2}l_{1}l_{2}\cos (\theta _{1}-\theta _{2}),\quad c_{22} =m_{2}l_{2}^{2}, \nonumber \\ d_{11}= & {} m_{2}l_{1}l_{2}\dot{\theta }_{2}^{2}\sin (\theta _{1}-\theta _{2})+(m_{1}+m_{2})gl_{1}\sin \theta _{1},\quad d_{22} =-m_{2}l_{1}l_{2}\dot{\theta }_{1}^{2}\sin (\theta _{1}-\theta _{2})+m_{2}gl_{2}\sin \theta _{2}.\qquad \quad \end{aligned}$$
(19)
The coefficients of the simple QPPM related to mass and stiffness matrices are obtained in light of Eq. (
18) as
$$\begin{aligned} d_{11}= & {} (m_{1}+m_{2}+m_{3}+m_{4})gl_{1},\quad d_{22} =(m_{2}+m_{3}+m_{4})gl_{2},\quad d_{33} =(m_{3}+m_{4})gl_{3},\quad d_{44} =m_{4}gl_{4}, \nonumber \\ c_{11}= & {} (m_{1}+m_{2}+m_{3}+m_{4})l_{1}^{2},\quad c_{12} =(m_{2}+m_{3}+m_{4})l_{1}l_{2},\quad c_{13} =(m_{3}+m_{4})l_{1}l_{3},\quad c_{14} =m_{4}l_{1}l_{4}, \nonumber \\ c_{21}= & {} (m_{2}+m_{3}+m_{4})l_{2}l_{1},\quad c_{22} =(m_{2}+m_{3}+m_{4})l_{2}^{2},\quad c_{23} =(m_{3}+m_{4})l_{2}l_{3},\quad c_{24} =m_{4}l_{2}l_{4}, \nonumber \\ c_{31}= & {} (m_{3}+m_{4})l_{3}l_{1},\quad c_{32} =(m_{3}+m_{4})l_{3}l_{2},\quad c_{33} =(m_{3}+m_{4})l_{3}^{2},\quad c_{34} =m_{4}l_{3}l_{4}, \nonumber \\ c_{41}= & {} m_{4}l_{4}l_{1},\quad c_{42} =m_{4}l_{4}l_{2},\quad c_{43} =m_{4}l_{4}l_{3},\quad c_{44} =m_{4}l_{4}^{2}. \end{aligned}$$
(20)
Finally, the coefficients of the linearized simple quintuple pendulum with point mass(QtPPM) on the basis of Eq. (
18) are written as following
$$\begin{aligned} c_{11}= & {} (m_{1}+m_{2}+m_{3}+m_{4}+m_{5})l_{1}^{2},\quad c_{12} =(m_{2}+m_{3}+m_{4}+m_{5})l_{1}l_{2},\quad c_{13} =(m_{3}+m_{4}+m_{5})l_{1}l_{3}, \nonumber \\ c_{14}= & {} (m_{4}+m_{5})l_{1}l_{4},\quad c_{15} =m_{5}l_{1}l_{5},\nonumber \\ c_{21}= & {} (m_{2}+m_{3}+m_{4}+m_{5})l_{2}l_{1},\quad c_{22} =(m_{2}+m_{3}+m_{4}+m_{5})l_{2}^{2},\quad c_{23} =(m_{3}+m_{4}+m_{5})l_{2}l_{3}, \nonumber \\ c_{24}= & {} (m_{4}+m_{5})l_{2}l_{4},\quad c_{25} =m_{5}l_{2}l_{5},\nonumber \\ c_{31}= & {} (m_{3}+m_{4}+m_{5})l_{3}l_{1},\quad c_{32} =(m_{3}+m_{4}+m_{5})l_{3}l_{2},c_{33} =(m_{3}+m_{4}+m_{5})l_{3}^{2},\quad c_{34} =(m_{4}+m_{5})l_{3}l_{4},\nonumber \\ c_{35}= & {} m_{5}l_{3}l_{5}, \nonumber \\ c_{41}= & {} (m_{4}+m_{5})l_{4}l_{1},\quad c_{42} =(m_{4}+m_{5})l_{4}l_{2},\quad c_{43} =(m_{4}+m_{5})l_{4}l_{3},\quad c_{44} =(m_{4}+m_{5})l_{4}^{2},\quad c_{45} =m_{5}l_{4}l_{5},\nonumber \\ c_{51}= & {} m_{5}l_{5}l_{1},\quad c_{52} =m_{5}l_{5}l_{2},\quad c_{53} =m_{5}l_{5}l_{3},\quad c_{54} =m_{5}l_{5}l_{4},\quad c_{55} =m_{5}l_{5}^{2}, \nonumber \\ d_{11}= & {} (m_{1}+m_{2}+m_{3}+m_{4}+m_{5})gl_{1},\quad d_{22} =(m_{2}+m_{3}+m_{4}+m_{5})gl_{2},\quad d_{33} =(m_{3}+m_{4}+m_{5})gl_{3}, \nonumber \\ d_{44}= & {} (m_{4}+m_{5})gl_{4},\quad d_{55} =m_{5}gl_{5}. \end{aligned}$$
(21)