5.4.5.2. A spring system with 2 DOF

Figure 5.21 A spring model

A spring model shown in Figure 5.21 is a system with two DOF, and the system has two masses, joints and spring entities. Their material properties, spring and damping coefficients are shown in Table 5.5.

Table 5.5 Material properties, spring and damping coefficients

Mass1	5Kg
Mass2	3Kg
Length of m1	300 mm
Stiffness coefficient (k1)	10 N/mm
Stiffness coefficient (k2)	20 N/mm
Damping coefficient (C1)	0 N sec/mm
Damping coefficient (C2)	0 N sec/mm

If the rotational angle \(\theta\) is small, the equation of motion of the system can be derived as:

(5.37)\[\begin{split}\left[ \begin{matrix} {I_\theta} & 0 \\ 0 & {m_2} \\ \end{matrix} \right]\left[ \begin{matrix} \ddot{\theta} \\ \ddot{y} \\ \end{matrix}\right]+\left[ \begin{matrix} {l^2k_1+\frac{l^2}{4}k_2} & \frac{l}{2}k_2 \\ \frac{l}{2}k_2 & k_2 \\ \end{matrix}\right]\left[ \begin{matrix} \theta \\ y \\ \end{matrix}\right]=0\end{split}\]

Substituting the values in Table 5.5, (5.37) becomes

(5.38)\[\begin{split}\left[ \begin{matrix} 0.15 & 0 \\ 0 & 3 \\ \end{matrix} \right]\left[ \begin{matrix} \ddot{\theta} \\ \ddot{y} \\ \end{matrix} \right]+\left[ \begin{matrix} 1350 & 3000 \\ 3000 & 20000 \\ \end{matrix} \right]\left[ \begin{matrix} \theta \\ y \\ \end{matrix} \right]=0\end{split}\]

The analytic natural frequencies of (5.38) are

(5.39)\[\omega_1=\sqrt{3211} = 56.66(rad/sec)=> f_1=9.019(Hz)\]

(5.40)\[\omega_2=\sqrt{12455} = 111.6(rad/sec)=> f_2=17.76(Hz)\]

The eigenvalues of this spring system are validated in Table 5.6.

Table 5.6 Eigenvalues of spring model

Mode Number	Undamped Natural Frequency (Hz)
	RecurDyn/Eigenvalue	Analytic solution
1	9.01862E+00	9.019
2	1.77621E+01	17.76