Figure 1 shows the schematic view of a rotary inverted pendulum using two rotary encoders as sensors and one DC motor as the actuator. The moving parts of the system may be divided into two groups where the parts at each group are fixed to each other and may be considered a single rigid body. The first group includes beam 1, rotating parts of encoder 1 (output shaft and the code disk), the rotor of the DC motor, the fixed parts of encoder 2 (the housing and electronics), and all coupling and connections used to connect them to each other. The second group is composed of the hanging beam (beam 2), the rotating parts of encoder 2, and the couplings used to connect them. Figure 2 clearly shows these two rigid bodies and the parts of the system from which they are composed. This figure includes a 3D model of the fabricated inverted pendulum used for experimental verification of the results which will be introduced in the next section. The total masses of the rigid bodies are represented by \({m}_{1}\), and \({m}_{2}\), respectively. In addition, \({r}_{1}\) and \({r}_{2}\) indicate the location of the center of gravity of mass \({m}_{2}\), where as shown in Fig. 1 the former denotes the distance of CG from the rotating axis of the motor and the latter is its distance from the rotating axis of encoder 2. To extract the governing dynamical equations of the system, Lagrange’s method is adopted. According to this method, first the total kinetic and potential energies of the system should be determined. The velocity of the center of gravity of \({m}_{2}\) may be written as:
\({\overrightarrow v _{_{2}}}= - \left( {{r_1}\dot {\phi }\cos \theta +{r_2}\dot {\theta }} \right)\hat {i}+\left( {{r_1}\dot {\phi }\sin \theta } \right)\hat {j} - \left( {{r_2}\dot {\phi }\sin \theta } \right)\hat {k}\) (1)
The total angular velocity vector of the second rigid body written with respect to the coordinate system \(xyz\), is:
$${\overrightarrow \omega _{_{2}}}= - \left( {\dot {\phi }\sin \theta } \right)\hat {i} - \left( {\dot {\phi }\cos \theta } \right)\hat {j}+\left( {\dot {\theta }} \right)\hat {k}$$
2
Therefore, the total kinetic energy of the system will be:
$$\begin{gathered} T=\frac{1}{2}{I_\phi }{{\dot {\phi }}^2}+\frac{1}{2}{m_2}\left[ {{{\left( {{r_1}\dot {\phi }\cos \theta +{r_2}\dot {\theta }} \right)}^2}+\left( {{r_1}^{2}+{r_2}^{2}} \right){{\left( {\dot {\phi }\sin \theta } \right)}^2}} \right]+\frac{1}{2}{I_{2x}}{\left( {\dot {\phi }\sin \theta } \right)^2}+\frac{1}{2}{I_{2y}}{\left( {\dot {\phi }\cos \theta } \right)^2} \hfill \\ \,\,\,\,\,\,\,\,\,+\frac{1}{2}{I_{2z}}{{\dot {\theta }}^2}+{I_{2yz}}\dot {\phi }\dot {\theta }\cos \theta \hfill \\ \end{gathered}$$
3
where \({I}_{\varphi }\) designates the the moment of inertia of \({m}_{1}\) about the motor axis. Also, \({I}_{2x}\), \({I}_{2y}\), \({I}_{2z}\), and \({I}_{2yz}\) are the nonzero elements of the inertia tensor of \({m}_{2}\) about its mass center and with respect to a coordinate system connected to beam 2, as shown in Fig. 1.
The total potential energy of the system may be written as:
$$U={m_2}g{r_2}\cos \theta$$
4
By definition, the Lagrangian of the system is:
Choosing \(\varphi\) and \(\theta\) as our generalized coordinates, the Lagrange’s equations of the system will be of the following form:
$$\frac{d}{{dt}}\left( {\frac{{\partial L}}{{\partial \dot {\alpha }}}} \right) - \frac{{\partial L}}{{\partial \alpha }}={Q_\alpha }\,\,\,\,\,\,\,\,\,\alpha =\phi ,{\text{ }}\theta$$
6
The term \({Q}_{\alpha }\) in the above equation is the generalized nonconservative force and may be derived by calculation of the total virtual work done as:
$$\delta {W_{nc}}={Q_\phi }\delta \phi +{Q_\theta }\delta \theta =\tau \delta \phi \,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,{Q_\phi }=\tau ,\,\,\,\,\,{Q_\theta }=0$$
7
where \(\tau\) denotes the generated mechanical torque of the motor.
For a DC motor the torque applied to the rotor may be written as a function of its terminal voltage \(V\) and speed \(\dot{\varphi }\) as:
$$\tau =\frac{1}{{{R_m}}}\left( {\frac{V}{{{k_e}}} - \dot {\phi }} \right)$$
8
where \({R}_{m}\) and \({k}_{e}\) are speed regulation and back EMF constants of the motor, respectively. It should be noted that the above equation neglects the inductance of the motor coil which according to the low frequencies involved in most mechanical systems is a reasonable assumption.
To make the pendulum stable at \(\theta =0\), a closed loop control scheme is used in this paper where the controller accepts \(\varphi\), \(\theta\), \(\dot{\varphi }\), and \(\dot{\theta }\) as inputs and applies a voltage equal to \({f}_{c}(\varphi , \theta , \dot{\varphi }, \dot{\theta })\) to the motor. Considering this terminal voltage for the motor then substituting equations (1) to (5), (7), and (8) into (6), the governing equations of the system are derived as:
$$\left\{ \begin{gathered} \left( {{I_1}+{I_2}{{\sin }^2}\theta } \right)\ddot {\phi }+\left( {{I_3}\cos \theta } \right)\ddot {\theta }+\left( {{I_2}\sin 2\theta } \right)\dot {\phi }\dot {\theta } - \left( {{I_3}\sin \theta } \right){{\dot {\theta }}^2}+\frac{1}{{{R_m}}}\dot {\phi }=\frac{{{f_c}\left( {\phi ,{\text{ }}\theta ,{\text{ }}\dot {\phi },{\text{ }}\dot {\theta }} \right)}}{{{R_m}{k_e}}} \hfill \\ \left( {{I_3}\cos \theta } \right)\ddot {\phi }+{I_4}\ddot {\theta } - \frac{1}{2}\left( {{I_2}\sin 2\theta } \right){{\dot {\phi }}^2} - {m_2}g{r_2}\sin \theta =0 \hfill \\ \end{gathered} \right.{\text{ }}$$
9
where:
$${I_1}={I_\phi }+{I_{2y}}+{m_2}{r_1}^{2},{\text{ }}{I_2}={I_{2x}} - {I_{2y}}+{m_2}{r_2}^{2},{\text{ }}{I_3}={I_{2yz}}+{m_2}{r_1}{r_2},{\text{ }}{I_4}{\text{=}}{I_{2z}}+{m_2}{r_2}^{2}$$
10
The equilibrium points of the system may be found by replacing all terms containing time derivatives of the variable by zero which results in:
$$\left\{ \begin{gathered} {f_c}\left( {\phi ,{\text{ }}\theta ,{\text{ }}0,{\text{ }}0} \right)=0\,\,\,\,\,\,\,\,\,(a) \hfill \\ \sin \theta =0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\text{ }}(b) \hfill \\ \end{gathered} \right.$$
11
Therefore, \(\theta =0\) and \(\varphi ={\varphi }_{d}\) is an equilibrium point of the system if and only if \({f}_{c}\left({\varphi }_{d}, 0, 0, 0\right)=0\). As a consequence, assuming the function \({f}_{c}\) to be linear, it may be written as:
$${f_c}\left( {\phi ,{\text{ }}\theta ,{\text{ }}\dot {\phi },{\text{ }}\dot {\theta }} \right)={k_1}\left( {\phi - {\phi _d}} \right)+{k_2}\theta +{k_3}\dot {\phi }+{k_4}\dot {\theta }$$
12
It should be noted that in practice there may be an error or an offset in \(\theta\) measurement and, hence, the point at which the measured \(\theta\) vanishes may not exactly coincide with the angle at which CG is accurately located above the axis of the second encoder. Therefore, Eq. (11-a) will not hold anymore, and consequently \(\theta =0\) will not be an equilibrium point for the closed-loop system. To overcome this problem even if the position angle of the first beam i.e., \(\varphi\) is not intended to be controlled at any desired position \({\varphi }_{d}\) the term \({k}_{1}\varphi\) must exist in \({f}_{c}\). By creating a small error between the angle \(\varphi\) and its desired value \({\varphi }_{d}\) (or with \(0\) when \({\varphi }_{d}\) is not important), this term makes Eq. (11-a) hold. Additionally, by employing this term the system will be able to suspend the pendulum stably at any arbitrary given \({\varphi }_{d}\).
Substituting Eq. (12) into (9) while replacing \(\text{sin}\theta\) and \(\text{cos}\theta\) by their Taylor’s series around \(\theta =0\) and keeping the nonlinear terms up to third order results in:
$$\left\{ \begin{gathered} {I_1}\ddot {\phi }+{I_3}\ddot {\theta }+\frac{1}{{{R_m}}}\dot {\phi } - \frac{{{k_2}}}{{{R_m}{k_e}}}\theta - \frac{{{k_1}}}{{{R_m}{k_e}}}\phi = - \frac{{{k_1}}}{{{R_m}{k_e}}}{\phi _d}+{I_2}\left( {\frac{4}{3}\dot {\theta }\dot {\phi }{\theta ^3}} \right)+{I_3}\left( {\frac{1}{2}\ddot {\theta }{\theta ^2}+{{\dot {\theta }}^2}\theta - \frac{1}{3}{{\dot {\theta }}^2}{\theta ^3}} \right) \hfill \\ - {I_2}\left( {\ddot {\phi }{\theta ^2}+2\dot {\theta }\dot {\phi }\theta } \right)+\frac{1}{{{R_m}}}\left( {\frac{{{k_3}}}{{{k_e}}} - 1} \right)\dot {\phi }+\frac{{{k_4}\dot {\theta }}}{{{R_m}{k_e}}} \hfill \\ {I_3}\ddot {\phi }+{I_4}\ddot {\theta } - {m_2}g{r_2}\theta =\frac{1}{2}{I_2}\left( {2\theta {{\dot {\phi }}^2} - \frac{4}{3}{\theta ^3}{{\dot {\phi }}^2}} \right)+\frac{1}{2}{I_3}\ddot {\varphi }{\theta ^2} - \frac{1}{6}{m_2}g{r_2}{\theta ^3} \hfill \\ \end{gathered} \right.$$
13
The above set of nonlinear equations may be analytically solved employing the perturbation method of multiple scales. To do this, first of all by assuming the amplitude of angular displacements of \(\varphi\) and \(\theta\) to be of order of a small parameter denoted by \(ϵ\), one can write:
$$\phi =\varepsilon {\phi _1}+{\varepsilon ^3}{\phi _3},\,\,\,\,\,\,\,\,\,\,\,\,\theta =\varepsilon {\theta _1}+{\varepsilon ^3}{\theta _3}$$
14
Furthermore, assuming the damping and excitation terms to be of order of \({ϵ}^{3}\), one can let:
$${\phi _d}={\varepsilon ^3}{\Phi _d},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\frac{{{k_3}}}{{{k_e}}} - 1={\varepsilon ^2}{K_3},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\frac{{{k_4}}}{{{k_e}}}={\varepsilon ^2}{K_4}$$
15
New independent variables called time scales may be introduced accordingly as:
$${T_0}=t,\,\,\,\,\,\,\,{T_2}={\varepsilon ^2}t$$
16
Representing the operator denoting partial differentiation with respect to \({T}_{n}\) by \({D}_{n}\) (i.e., \({D}_{n}=\frac{\partial }{\partial {T}_{n}}\)) and using the chain rule of differentiation, one can write operators denoting first and second derivatives with respect to \(t\) as an expansion of \({D}_{n}\)s and their multiplications as:
$$\frac{d}{{dt}}={D_0}+{\varepsilon ^2}{D_2},\,\,\,\,\,\,\,\,\,\,\frac{{{d^2}}}{{d{t^2}}}=D_{0}^{2}+2{\varepsilon ^2}{D_0}{D_2}$$
17
Substituting equations (14), (15), and (17) into (13), then separating terms of different orders of \(ϵ\), one has:
$${\varepsilon ^1}:\left\{ \begin{gathered} {I_1}D_{0}^{2}{\phi _1}+{I_3}D_{0}^{2}{\theta _1} - \frac{{{k_1}}}{{{R_m}{k_e}}}{\phi _1} - \frac{{{k_2}}}{{{R_m}{k_e}}}{\theta _1}=0\, \hfill \\ {I_3}D_{0}^{2}{\phi _1}+{I_4}D_{0}^{2}{\theta _1} - {m_2}g{r_2}{\theta _1}=0 \hfill \\ \end{gathered} \right.$$
18
$${\varepsilon ^3}:\left\{ \begin{gathered} {I_1}D_{0}^{2}{\phi _3}+{I_3}D_{0}^{2}{\theta _3} - \frac{{{k_2}}}{{{R_m}{k_e}}}{\theta _3} - \frac{{{k_1}}}{{{R_m}{k_e}}}{\phi _3}={f_1} \hfill \\ {I_3}D_{0}^{2}{\phi _3}+{I_4}D_{0}^{2}{\theta _3} - {m_2}g{r_2}{\theta _3}={f_2} \hfill \\ \end{gathered} \right.$$
19
where:
$$\begin{gathered} {f_1}= - 2{I_1}{D_0}{D_2}{\phi _1} - 2{I_3}{D_0}{D_2}{\theta _1} - \frac{{{k_1}}}{{{R_m}{k_e}}}{\Phi _d}\,+{I_2}\left( {2{D_0}{\theta _1}{D_0}{\phi _1}{\theta _1}+{\theta _1}^{2}D_{0}^{2}{\phi _1}} \right) \hfill \\ \,\,\,\,\,\,\,\,\,{\text{ }}+{I_3}\left[ {{{\left( {{D_0}{\theta _1}} \right)}^2}{\theta _1}+\frac{1}{2}D_{0}^{2}{\theta _1}{\theta _1}^{2}} \right]+\frac{{{K_3}}}{{{R_m}}}{D_0}{\phi _1}+\frac{{{K_4}}}{{{R_m}}}{D_0}{\theta _1} \hfill \\ {f_2}= - 2{I_3}{D_0}{D_2}{\phi _1} - 2{I_4}{D_0}{D_2}{\theta _1}+{I_2}{\left( {{D_0}{\phi _1}} \right)^2}{\theta _1}+\frac{1}{2}{I_3}D_{0}^{2}{\phi _1}{\theta _1}^{2} - \frac{1}{6}{m_2}g{r_2}{\theta _1}^{3} \hfill \\ \end{gathered}$$
20
The general solution of Eq. (18) may be written as:
$$\begin{gathered} {\phi _1}={A_1}\exp \left( {i{\omega _1}{T_0}} \right)+{A_2}\exp \left( {i{\omega _2}{T_0}} \right)+cc \hfill \\ {\theta _1}={\alpha _1}{A_1}\exp \left( {i{\omega _1}{T_0}} \right)+{\alpha _2}{A_2}\exp \left( {i{\omega _2}{T_0}} \right)+cc \hfill \\ \end{gathered}$$
21
where \(cc\) denotes the complex conjugate terms. The above solution assumes that the following equation from which \({\omega }_{1}\) and \({\omega }_{2}\) may be calculated, has four real roots:
$$\left( {{I_1}{I_4} - {I_3}^{2}} \right){\omega ^4}+\left[ {\frac{1}{{{R_m}{k_e}}}\left( {{k_1}{I_4} - {k_2}{I_3}} \right)+{I_1}{m_2}g{r_2}} \right]{\omega ^2}+\frac{{{k_1}}}{{{R_m}{k_e}}}{m_2}g{r_2}=0$$
22
This condition needs \({k}_{1}\) and \({k}_{2}\) to be chosen so that the following inequalities hold:
$$\begin{gathered} {k_1}>0 \hfill \\ {k_2}>\frac{{{k_1}{I_4}+{R_m}{k_e}{I_1}{m_2}g{r_2}}}{{{I_3}}} \hfill \\ {\left[ {\frac{1}{{{R_m}{k_e}}}\left( {{k_1}{I_4} - {k_2}{I_3}} \right)+{I_1}{m_2}g{r_2}} \right]^2}>\frac{{4{k_1}}}{{{R_m}{k_e}}}{m_2}g{r_2}\left( {{I_1}{I_4} - {I_3}^{2}} \right) \hfill \\ \end{gathered}$$
23
The coefficients \({\alpha }_{1}\) and \({\alpha }_{2}\) in Eq. (21) are as follows:
$${\alpha _i}= - \frac{{{I_3}{\omega _i}^{2}}}{{\left( {{m_2}g{r_2}+{I_4}{\omega _i}^{2}} \right)}}\,\,\,\,\,\,\,\,i=1,2$$
24
To analyze resonance occurrence in the neighborhood of the natural frequencies of the system, one may let:
$${\Phi _d}={E_1}\cos \left( {{\Omega _1}t} \right)+{E_2}\cos \left( {{\Omega _2}t} \right)$$
25
where \({\varOmega }_{1}\) and \({\varOmega }_{2}\) differ \({\omega }_{1}\) and \({\omega }_{2}\) by small parameters \({\sigma }_{1}\) and \({\sigma }_{2}\), respectively, as:
$$\begin{gathered} {\Omega _1}={\omega _1}+{\varepsilon ^2}{\sigma _1} \hfill \\ {\Omega _2}={\omega _2}+{\varepsilon ^2}{\sigma _2} \hfill \\ \end{gathered}$$
26
To eliminate the secular terms in Eq. (18), the coefficients of the terms \(\text{e}\text{x}\text{p}\left(i{\omega }_{i}{T}_{0}\right)\) in the following expressions must vanish:
$${\beta _i}{f_1}+{f_2}$$
27
where:
$${\beta _i}= - \frac{{{I_3}\omega _{i}^{2}}}{{\left( {{I_1}\omega _{i}^{2}+\frac{{{k_1}}}{{{R_m}{k_e}}}} \right)}}\,\,\,\,\,\,\,\,i=1,2$$
28
Substituting equations (21), (25), and (26) into (27), one obtains:
$$\left\{ \begin{gathered} i{c_{11}}{D_2}{A_1}+\frac{{{\beta _1}{k_1}}}{{2{R_m}{k_e}}}{E_1}\exp \left( {i{\sigma _1}{T_2}} \right) - \frac{{{\beta _1}}}{{{R_m}}}({K_3}+{K_4}{\alpha _1}){A_1}i{\omega _1}+{c_{21}}{A_1}^{2}{{\bar {A}}_1}+{c_{31}}{A_1}{A_2}{{\bar {A}}_2}=0 \hfill \\ i{c_{12}}{D_2}{A_2}+\frac{{{\beta _2}{k_1}}}{{2{R_m}{k_e}}}{E_2}\exp \left( {i{\sigma _2}{T_2}} \right) - \frac{{{\beta _2}}}{{{R_m}}}({K_3}+{K_4}{\alpha _3}){A_2}i{\omega _2}+{c_{32}}{A_1}{{\bar {A}}_1}{A_2}+{c_{22}}{A_2}^{2}{{\bar {A}}_2}=0 \hfill \\ \end{gathered} \right.$$
29
where:
$$\begin{gathered} {c_{11}}=2{\omega _1}\left[ {{\beta _1}{I_1}+{I_3}\left( {1+{\alpha _1}{\beta _1}} \right)+{I_4}{\alpha _1}} \right] \hfill \\ {c_{21}}={\alpha _1}\left\{ {\left[ {{I_2}\left( {1+{\beta _1}{\alpha _1}} \right)+\frac{1}{2}{I_3}{\alpha _1}\left( {3+{\beta _1}{\alpha _1}} \right)} \right]{\omega _1}^{2}+\frac{1}{2}{m_2}g{r_2}{\alpha _1}^{2}} \right\} \hfill \\ {c_{31}}=\left[ {{\beta _1}\left( {2{I_2}+{I_3}{\alpha _1}} \right)+{I_3}} \right]{\alpha _2}^{2}{\omega _1}^{2}+2\left( {{I_2}+{I_3}{\alpha _2}} \right){\alpha _1}{\omega _2}^{2}+{m_2}g{r_2}{\alpha _1}{\alpha _2}^{2} \hfill \\ {c_{12}}=2{\omega _2}\left[ {{\beta _2}{I_1}+{I_3}\left( {1+{\alpha _2}{\beta _2}} \right)+{I_4}{\alpha _2}} \right] \hfill \\ {c_{22}}={\alpha _2}\left\{ {\left[ {{I_2}\left( {1+{\beta _2}{\alpha _2}} \right)+\frac{1}{2}{I_3}{\alpha _2}\left( {3+{\beta _2}{\alpha _2}} \right)} \right]{\omega _2}^{2}+\frac{1}{2}{m_2}g{r_2}{\alpha _2}^{2}} \right\} \hfill \\ {c_{32}}=\left[ {{\beta _2}\left( {2{I_2}+{I_3}{\alpha _2}} \right)+{I_3}} \right]{\alpha _1}^{2}{\omega _2}^{2}+2\left( {{I_2}+{I_3}{\alpha _1}} \right){\alpha _2}{\omega _1}^{2}+{m_2}g{r_2}{\alpha _2}{\alpha _1}^{2} \hfill \\ \end{gathered}$$
30
Eliminating the excitation and nonlinear terms from the above expressions, the condition for linear stability of the system is derived as:
$$\frac{{{\beta _1}\left( {{K_3}+{K_4}{\alpha _1}} \right){\omega _1}}}{{{R_m}{c_{11}}}}cript>$$
31
Amplitudes \({A}_{1}\) and \({A}_{2}\) may be represented by polar form as:
$${A_1}=\frac{1}{2}{a_1}{e^{i{\psi _1}}},\,\,\,\,\,\,\,\,\,\,\,\,\,{A_2}=\frac{1}{2}{a_2}{e^{i{\psi _2}}}$$
32
Substituting the above equation into Eq. (29) one has:
$$\left\{ \begin{gathered} \frac{{i{c_{11}}}}{2}\left( {{a_1}^{\prime }+i{\psi _1}^{\prime }{a_1}} \right)+\frac{{{\beta _1}{k_1}}}{{2{R_m}{k_e}}}{E_1}\exp \left( {i{\sigma _1}{T_2} - i{\psi _1}} \right) - \frac{{{\beta _1}}}{{2{R_m}}}\left( {{K_3}+{K_4}{\alpha _1}} \right)i{\omega _1}{a_1}+\frac{1}{8}{c_{21}}{a_1}^{3}+\frac{1}{8}{c_{31}}{a_1}{a_2}^{2}=0 \hfill \\ \frac{{i{c_{12}}}}{2}\left( {{a_2}^{\prime }+{a_2}i{\psi _2}^{\prime }} \right)+\frac{{{\beta _2}{k_1}}}{{2{R_m}{k_e}}}{E_2}\exp \left( {i{\sigma _2}{T_2} - i{\psi _2}} \right) - \frac{{{\beta _2}}}{{2{R_m}}}\left( {{K_3}+{K_4}{\alpha _2}} \right)i{\omega _2}{a_2}+\frac{1}{8}{c_{22}}{a_2}^{3}+\frac{1}{8}{c_{32}}{a_1}^{2}{a_2}=0 \hfill \\ \end{gathered} \right.$$
33
By the separation of real and imaginary parts in Eq. (33) and introduction of the following new variables:
$$\begin{gathered} {\gamma _1}={\sigma _1}{T_2} - {\psi _1} \hfill \\ {\gamma _2}={\sigma _2}{T_2} - {\psi _2} \hfill \\ \end{gathered}$$
34
the following set of autonomous equations are found:
$$\left\{ \begin{gathered} {c_{11}}{a_1}^{\prime }=\frac{{{\beta _1}}}{{{R_m}}}\left( {{K_3}+{K_4}{\alpha _1}} \right){\omega _1}{a_1} - \frac{{{\beta _1}{k_1}}}{{2{R_m}{k_e}}}{E_1}\sin {\gamma _1} \hfill \\ {c_{11}}{a_1}{\gamma _1}^{\prime }={c_{11}}{a_1}{\sigma _1} - \frac{1}{4}{c_{21}}{a_1}^{3} - \frac{1}{4}{c_{31}}{a_1}{a_2}^{2} - \frac{{{\beta _1}{k_1}}}{{2{R_m}{k_e}}}{E_1}\cos {\gamma _1} \hfill \\ {c_{12}}{a_2}^{\prime }=\frac{{{\beta _2}}}{{{R_m}}}\left( {{K_3}+{K_4}{\alpha _2}} \right){\omega _2}{a_2} - \frac{{{\beta _2}{k_1}}}{{2{R_m}{k_e}}}{E_2}\sin {\gamma _2} \hfill \\ {c_{12}}{a_2}{\gamma _2}^{\prime }={c_{12}}{a_2}{\sigma _2} - \frac{1}{4}{c_{22}}{a_2}^{3} - \frac{1}{4}{c_{32}}{a_1}^{2}{a_2} - \frac{{{\beta _2}{k_1}}}{{2{R_m}{k_e}}}{E_2}\cos {\gamma _2} \hfill \\ \end{gathered} \right.$$
35
The equilibrium points of the above dynamical system may be found by substituting \({a}_{1}^{\text{'}}={a}_{2}^{\text{'}}={\gamma }_{1}^{\text{'}}={\gamma }_{2}^{\text{'}}=0\), leading to the following set of equations for \({a}_{1}\) and \({a}_{2}\):
$$\left\{ \begin{gathered} {\left[ {\frac{{{\beta _1}}}{{2{R_m}}}({K_3}+{K_4}{\alpha _1}){\omega _1}{a_1}} \right]^2}+{\left( {\frac{1}{2}{c_{11}}{a_1}{\sigma _1} - \frac{1}{8}{c_{21}}{a_1}^{3} - \frac{1}{8}{c_{31}}{a_1}{a_2}^{2}} \right)^2}={\left( {\frac{{{\beta _1}{k_1}}}{{2{R_m}{k_e}}}{E_1}} \right)^2} \hfill \\ {\left[ {\frac{{{\beta _2}}}{{2{R_m}}}({K_3}+{K_4}{\alpha _2}){\omega _2}{a_2}} \right]^2}+{\left( {\frac{1}{2}{c_{12}}{a_2}{\sigma _2} - \frac{1}{8}{c_{22}}{a_2}^{3} - \frac{1}{8}{c_{32}}{a_1}^{2}{a_2}} \right)^2}={\left( {\frac{{{\beta _2}{k_1}}}{{2{R_m}{k_e}}}{E_2}} \right)^2} \hfill \\ \end{gathered} \right.$$
36
The above equation may be solved to calculate vibration amplitudes \({a}_{1}\) and \({a}_{2}\) as functions of the excitations amplitude \({E}_{1}\), \({E}_{2}\) and deviations \({\sigma }_{1}\) and \({\sigma }_{2}\) from the natural frequencies of the system. That is while to depict the response curves, it suffices to simply calculate \({\sigma }_{1}\), \({\sigma }_{2}\), \({E}_{1}\), and \({E}_{2}\) by a rearrangement of Eq. (36) as:
$$\begin{gathered} {\sigma _1}=\frac{{\frac{1}{4}{c_{21}}{a_1}^{3}+\frac{1}{4}{c_{31}}{a_1}{a_2}^{2} \pm \sqrt {{{\left( {\frac{{{\beta _1}{k_1}}}{{2{R_m}{k_e}}}{E_1}} \right)}^2} - {{\left[ {\frac{{{\beta _1}}}{{2{R_m}}}({K_3}+{K_4}{\alpha _1}){\omega _1}{a_1}} \right]}^2}} }}{{{c_{11}}{a_1}}} \hfill \\ {\sigma _2}=\frac{{\frac{1}{4}{c_{22}}{a_2}^{3}+\frac{1}{4}{c_{32}}{a_1}^{2}{a_2} \pm \sqrt {{{\left( {\frac{{{\beta _2}{k_1}}}{{2{R_m}{k_e}}}{E_2}} \right)}^2} - {{\left[ {\frac{{{\beta _2}}}{{2{R_m}}}({K_3}+{K_4}{\alpha _2}){\omega _2}{a_2}} \right]}^2}} }}{{{c_{12}}{a_2}}} \hfill \\ {E_1}=\frac{{{R_m}{k_e}}}{{{\beta _1}{k_1}}}\sqrt {{{\left[ {\frac{{{\beta _1}}}{{{R_m}}}({K_3}+{K_4}{\alpha _1}){\omega _1}{a_1}} \right]}^2}+{{\left( {{c_{11}}{a_1}{\sigma _1} - \frac{1}{4}{c_{21}}{a_1}^{3} - \frac{1}{4}{c_{31}}{a_1}{a_2}^{2}} \right)}^2}} \hfill \\ {E_2}=\frac{{{R_m}{k_e}}}{{{\beta _2}{k_1}}}\sqrt {{{\left[ {\frac{{{\beta _2}}}{{{R_m}}}({K_3}+{K_4}{\alpha _2}){\omega _2}{a_2}} \right]}^2}+{{\left( {{c_{12}}{a_2}{\sigma _2} - \frac{1}{4}{c_{22}}{a_2}^{3} - \frac{1}{4}{c_{32}}{a_1}^{2}{a_2}} \right)}^2}} \hfill \\ \end{gathered}$$
37
To explore how the inherent geometric nonlinearity of the pendulum affects the behavior of the system, a linear solution to the governing equations is performed as well. The results of the linear solution are used along with the results of the nonlinear solution when comparing the analytical, numerical, and experimental findings in the next section. The first step toward this linear solution is eliminating the nonlinear terms in Eq. (13) which results in:
$$\left\{ \begin{gathered} {I_1}\ddot {\phi }+{I_3}\ddot {\theta } - \frac{{{k_2}}}{{{R_m}{k_e}}}\theta - \frac{{{k_1}}}{{{R_m}{k_e}}}\phi = - \frac{{{k_1}}}{{{R_m}{k_e}}}{\phi _d}+\frac{1}{{{R_m}}}\left( {\frac{{{k_3}}}{{{k_e}}} - 1} \right)\dot {\phi }+\frac{{{k_4}\dot {\theta }}}{{{R_m}{k_e}}} \hfill \\ {I_3}\ddot {\phi }+{I_4}\ddot {\theta } - {m_2}g{r_2}\theta =0 \hfill \\ \end{gathered} \right.$$
38
where \({\varphi }_{d}\) is a single harmonic excitation as:
$${\phi _d}=E\exp \left( {i\omega t} \right)$$
39
Substituting the above expression into Eq. (37), the steady state solutions for \(\varphi\) and \(\theta\) are found to be of the following form:
$$\begin{gathered} \phi =\Phi \exp \left( {i\omega t} \right) \hfill \\ \theta =\Theta \exp \left( {i\omega t} \right) \hfill \\ \end{gathered}$$
40
where \({\Phi }\) and \({\Theta }\) are complex amplitudes, derived from the following equation:
$$\left\{ \begin{gathered} \left[ {{I_1}{\omega ^2}+\frac{{{k_1}}}{{{R_m}{k_e}}}+\frac{1}{{{R_m}}}\left( {\frac{{{k_3}}}{{{k_e}}} - 1} \right)j\omega } \right]\Phi +\left( {{I_3}{\omega ^2}+\frac{{{k_2}}}{{{R_m}{k_e}}}+\frac{{{k_4}}}{{{R_m}{k_e}}}j\omega } \right)\Theta =\frac{{{k_1}}}{{{R_m}{k_e}}}{E_1} \hfill \\ {I_3}{\omega ^2}\Phi +\left( {{I_4}{\omega ^2}+{m_2}g{r_2}} \right)\Theta =0 \hfill \\ \end{gathered} \right.$$
41
Solving the above equation, one obtains:
$$\left\{ \begin{gathered} \Phi =\frac{{\left( {{I_4}{\omega ^2}+{m_2}g{r_2}} \right){k_1}{E_1}}}{{{R_m}{k_e}\left\{ {\left( {{I_4}{\omega ^2}+{m_2}g{r_2}} \right)\left[ {{I_1}{\omega ^2}+\frac{{{k_1}}}{{{R_m}{k_e}}}+\frac{1}{{{R_m}}}\left( {\frac{{{k_3}}}{{{k_e}}} - 1} \right)j\omega } \right] - \left( {{I_3}{\omega ^2}+\frac{{{k_2}}}{{{R_m}{k_e}}}+\frac{{{k_4}}}{{{R_m}{k_e}}}j\omega } \right){I_3}{\omega ^2}} \right\}}} \hfill \\ \Theta = - \frac{{{I_3}{\omega ^2}{k_1}{E_1}}}{{{R_m}{k_e}\left\{ {\left( {{I_4}{\omega ^2}+{m_2}g{r_2}} \right)\left[ {{I_1}{\omega ^2}+\frac{{{k_1}}}{{{R_m}{k_e}}}+\frac{1}{{{R_m}}}\left( {\frac{{{k_3}}}{{{k_e}}} - 1} \right)j\omega } \right] - \left( {{I_3}{\omega ^2}+\frac{{{k_2}}}{{{R_m}{k_e}}}+\frac{{{k_4}}}{{{R_m}{k_e}}}j\omega } \right){I_3}{\omega ^2}} \right\}}} \hfill \\ \end{gathered} \right.$$
42