A real pendulum is subjected to dissipative forces, which decrease their maximum amplitude over time, see Figure 1 . The total dissipative force is proportional to the bob velocity $L\stackrel{\dot{}}{\theta }$, and the damping term $\lambda L\stackrel{\dot{}}{\theta }$ contains the contributions from all dissipative forces acting on the system. The equation of motion is:

$\ddot{\theta }+\frac{\lambda }{M}\stackrel{\dot{}}{\theta }+{\omega }_0^2\theta =0$ (1)

where, $\theta$ is the oscillation amplitude; $M$ is the mass of the bob; and using the Steiner’s theorem for a cylindrical body moving around a fixed axis perpendicular to the oscillation plane ${\omega }_0^2=\frac{MgL}{I_c+M{L}^2}$ [5] , with $L$ being the length of the pendulum measured from the fixed point to the mass center of the bob, $g$ being the acceleration due to gravity, and $I_c=\frac{1}{4}M{r}^2+\frac{1}{12}M{H}^2$ (where $r$ and $H$ are radius and the height of the bob, respectively).

The solution to Equation (1) is:

$\theta ={\theta }_0{\text{e}}^{-\gamma t}\cos \left(\omega t\right)$, (2)

where ${\theta }_0$ is the initial amplitude, the damping coefficient is $\gamma =\frac{\lambda }{2M}$, and the angular frequency is $\omega =\sqrt{{\omega }_0^2-{\gamma }^2}$. The coefficient $\gamma$ can be determined experimentally, and its value depends on the magnitude of all dissipative forces.

Viscoelastic materials are characterized by complex elastic moduli. The Young’s modulus $Y^{*}$ has the following general form:

$Y^{*}=Y^{\prime }+i{Y}^{″}=Y^{\prime }\left(1+i\eta \right)$, (3)

where $\eta =\frac{Y^{″}}{Y^{\prime }}$, the loss factor.

When a periodic stress $\sigma$ with frequency ${\omega }_0$ is applied to a viscoelastic material, a strain $\epsilon$ is produced. The behavior of the viscoelastic material is described by the equation:

${\sigma }^{*}\left(t\right)=\left(Y^{\prime }+i{Y}^{″}\right)\epsilon \left(t\right)$, (4)

where the real stress is associated with the conservative term ${\sigma }^{\prime }\left(t\right)=Y^{\prime }\epsilon \left(t\right)$, and the imaginary term is linked to the dissipative parameters of the system ${\sigma }^{″}\left(t\right)=Y^{″}\epsilon \left(t\right)$. Consequently, it is possible to determine the complex Young’s modulus of a material through a damping experiment [6] . However, there is an intrinsic difficulty in separating all possible energy loss agents [7] . Therefore, the experimental design must minimize losses other than dissipation in the viscoelastic medium.

The equation of motion of a pendulum with a bob of mass $M$ suspended by a string with variable length is given by:

$L_{0}M\left(1+\epsilon \right)\ddot{\theta }+L_0\left(1+\epsilon \right)\lambda \stackrel{\dot{}}{\theta }+Mg\sin \left(\theta \right)=0$, (5)

where $L_0$ is the initial length of the pendulum and $\epsilon$ the strain of the string subjected to periodic stress caused by the longitudinal component of the bob’s weight. In this equation, $1\gg \epsilon$. Therefore, its solution is known as the solution of a damped oscillator [5] .

The forced oscillation of the viscoelastic string is given by the following equation:

$m{L}_0\ddot{\epsilon }+{\lambda }_2{L}_0\stackrel{\dot{}}{\epsilon }+k_2{L}_0\epsilon =Mg\cos \left(\theta \right)$, (6)

where $m$ is the mass of the string; $k_2$ is the elasticity constant of the string; and ${\lambda }_2{L}_0\stackrel{\dot{}}{\epsilon }$ is the dissipative term, with ${\lambda }_2$ being the damping constant of the coupled system and $Mg\cos \left(\theta \right)$ is the weight component along the string [5] .

Equation (6) can be rewritten as:

$\ddot{\epsilon }+2{\gamma }_2\stackrel{\dot{}}{\epsilon }+{\omega }_2^2\epsilon =\frac{Mg}{m{L}_0}\cos \theta$, (7)

where ${\gamma }_2=\frac{{\lambda }_2}{2m}$ and ${\omega }_2^2=\frac{k_2}{m}$. Substituting the solution of Equation (5) $\left[\theta ={\theta }_0{\text{e}}^{-\gamma t}\cos \left(\omega t\right)\right]$ in Equation (7) yields the following:

$\ddot{\epsilon }+2{\gamma }_2\stackrel{\dot{}}{\epsilon }+{\omega }_2^2\epsilon =\frac{Mg}{m{L}_0}\cos \left[{\theta }_0{\text{e}}^{-\gamma t}\cos \left(\omega t\right)\right]$. (8)

Equation (5) and Equation (6) represent two oscillating systems coupled by the damping and gravitational forces. The driven frequency in Equation (8) is $\omega$. Therefore, both systems oscillate with the coupled frequency $\omega$.

By developing the cosine of the right side of Equation (8) in its Taylor’s series and considering only up to the quadratic term, we get:

$\ddot{\epsilon }+2{\gamma }_2\stackrel{\dot{}}{\epsilon }+{\omega }_2^2\epsilon =\frac{Mg}{m{L}_0}\cos \left[{\theta }_0{\text{e}}^{-\gamma t}\cos \left(\omega t\right)\right]$. (9)

We propose the following solution for Equation (9):

$\epsilon \left(t\right)=A_0\left[1-A_1{\text{e}}^{-\beta t}{\cos }^2\left(\omega t\right)-A_2{\text{e}}^{-\beta t}{\sin }^2\left(\omega t\right)\right]$, (10)

where ${\epsilon }_0$, $A_1$, $A_2$ and $\beta$ are constants; $A_1\ne A_2$; and ${\omega }_2\gg \omega$. To verify this as the solution, we substituted it into Equation (9) and equated the coefficients multiplying the monomials on each side of the equation as follows:

1) Constant monomials:

$A_0=\frac{Mg}{m{L}_0{\omega }_2^2}$ (11)

2) $\sin \omega t\cos \omega t$ monomials under the condition $A_1\ne A_2$:

$\beta ={\gamma }_2$ (12)

3) ${\sin }^2\left(\omega t\right)$ monomials:

$A_2=-\frac{{\omega }^2{\theta }_0^2}{{\omega }_2^2-4{\omega }^2}=-\frac{{\omega }^2}{{\omega }_2^2}\frac{{\theta }_0^2}{1-4\frac{{\omega }^2}{{\omega }_2^2}}\approx 0$ (13)

4) ${\cos }^2\left(\omega t\right)$ monomials:

$A_1=\left(\frac{{\theta }_0^2}{2}\right)\frac{{\omega }_2^2-2{\omega }^2}{{\omega }_2^2-4{\omega }^2}=\left(\frac{{\theta }_0^2}{2}\right)\frac{1-2\frac{{\omega }^2}{{\omega }_2^2}}{1-4\frac{{\omega }^2}{{\omega }_2^2}}\approx \frac{{\theta }_0^2}{2};{\gamma }_2=2\gamma$ (14)

On the other hand, the constitutive equation for the stress state on the viscoelastic fiber can be expressed as follows:

${\sigma }^{\prime }\left(t\right)=\frac{F_k\left(t\right)}{A}=Y^{\prime }\epsilon \left(t\right)$ (15)

${\sigma }^{″}\left(t\right)=\frac{F_d\left(t\right)}{A}=Y^{″}\epsilon \left(t\right)$ (16)

In the absence of energy dissipation, the elastic force is $F_k\left(t\right)=L_0{k}_2\epsilon \left(t\right)$, where $\epsilon \left(t\right)$ is the solution of the undamped driven string. This means $\epsilon \left(t\right)=\frac{M}{m}\frac{{\omega }^2}{{\omega }_2^2-{\omega }^2}\cos \left(\omega t\right)$, while the damping force is $F_d\left(t\right)={\lambda }_0{L}_o\stackrel{\dot{}}{\epsilon }\left(t\right)=2{\gamma }_{0}m{L}_0\stackrel{\dot{}}{\epsilon }\left(t\right)$, where $1/{\gamma }_0$ is the damping time of the viscoelastic string. This property is independent of the experiment used for this calculation. The temporal change of the strain $\stackrel{\dot{}}{\epsilon }\left(t\right)$ is calculated from Equation (10). Considering that $A_2\ll A_1$, ${\text{e}}^{-{\gamma }_{2}t}\approx 1$, and ${\gamma }_2\ll 2\omega$ and the $\omega$ dependence of $\stackrel{\dot{}}{\epsilon }\left(t\right)$, we get the following:

$\frac{Y^{″}}{Y^{\prime }}=\frac{{\lambda }_0{L}_0\stackrel{\dot{}}{\epsilon }\left(t\right)}{k_2{L}_0\epsilon \left(t\right)}\cong \frac{4{\gamma }_0\omega \sin \left(\omega t\right)\cos \left(\omega t\right)}{\frac{M}{m}\frac{{\omega }^2}{1-\frac{{\omega }^2}{{\omega }_2^2}}\cos \left(\omega t\right)}$. (17)

In Equation (17), ${\omega }^2/{\omega }_2^2\ll 1$. In the limit case, where the damping of the pendulum is solely due to viscoelasticity in the string, we propose ${\gamma }_2={\gamma }_{0}m/M=2{\gamma }_{\text{lim}}$, because ${\gamma }_0$ is a property of the viscoelastic string and the coupling between systems is given by the ratio $m/M$. In this case, the damping coefficient of the pendulum is ${\gamma }_{\text{lim}}$. Thus, we get the following:

$\eta =\frac{Y^{″}}{Y^{\prime }}=\frac{4{\gamma }_{\text{lim}}}{\omega }\sin \left(\omega t\right)$ (18)

In this case, the Young’s complex modulus will be as follows:

$Y^{*}=Y^{\prime }\left(1+i\eta \right)$ (19)