Question 10.5.2: A lumped mass structure of n degrees of freedom is undergoin...
A lumped mass structure of n degrees of freedom is undergoing undamped free vibration. Determine the actual period of vibration of the ith mass. Are the actual periods of vibration of the n masses equal to one another?
The solution to the uncoupled equations of motion is (Eqns 10.5-30):
q_1=A_1\sin \omega _1t+B_1\cos \omega _1t, or C_1\sin (\omega _1t+\alpha _1), 0r D_1\cos (\omega _1t+\beta _1)
q_2=A_2\sin \omega _2t+B_2\cos \omega _2t, or C_2\sin (\omega _2t+\alpha _2), 0r D_2\cos (\omega _2t+\beta _2)
\vdots
q_n=A_n\sin \omega _nt+B_n\cos \omega _nt, or C_n\sin (\omega _nt+\alpha _n), 0r D_n\cos (\omega _nt+\beta _n) (10.5-30)
\left[\begin{matrix} q_1 \\ q_2 \\ \vdots \\ q_i \\ \vdots \\ q_n \end{matrix} \right]= \left[\begin{matrix} C_1\sin \left(\omega_1t+\alpha _1\right) \\ C_2\sin \left(\omega_2t+\alpha _2\right) \\ \vdots \\ C_i\sin \left(\omega_it+\alpha _i\right) \\ \vdots \\ C_n\sin \left(\omega_nt+\alpha _n\right) \end{matrix} \right] (10.5-41)
Using the coordinate transformation in Eqn 10.5-32,
x=Zq (10.5-32)
the actual displacements are:
\left[\begin{matrix} x_1 \\ x_2 \\ \vdots \\ x_i \\ \vdots \\ x_n \end{matrix} \right]=\left[\begin{matrix} _1z_1 & _2z_1 & \ldots & _nz_1 \\ _1z_2 & _2z_2 \\ \vdots \\ _1z_i & _2z_i \\ \vdots \\ _1z_n & & & _nz_n \end{matrix} \right]\left[\begin{matrix} q_1 \\ q_2 \\ \vdots \\ q_i \\ \\ q_n \end{matrix} \right]
That is,
x_i= \ _1z_iC_1\sin \left(\omega _1t+\alpha _1\right)+ \ _2z_iC_2\sin \left(\omega _2t+\alpha _2\right)+\cdot \cdot \cdot + \ _iz_iC_i\sin \left(\omega _it+\alpha _i\right)+\cdot \cdot \cdot + \ _nz_iC_n\sin \left(\omega _nt+\alpha _n\right) (10.5-42)
where \omega_1 is the natural circular frequency corresponding to the first normal mode shape, \omega_2 that corresponding to the second normal mode shape, and so on.
In Eqn 10.5-42 the period of the first term on the right-hand side is 2\pi /\omega_1, that of the second term is 2\pi /\omega_2 , and so on. Hence the actual period of the displacement x_i must be the least common multiple of all the periods on the right-hand side. That is,
T_i (actual) = LCM of \frac{2\pi }{\omega_1}, \frac{2\pi }{\omega_2}, \frac{2\pi }{\omega_3}, . . . , \frac{2\pi }{\omega_n} (10.5-43)
provided none of the coefficients _1z_iC_1, _2z_iC_2, . . . , _nz_iC_n are zero. If any of these coefficients is zero, say _rz_iC_r=0, then the term 2\pi /\omega_r will be omitted in determining the LCM. Thus if none of the n \times n quantities,
\begin{matrix} _1z_1C_1 & _2z_1C_2 & _3z_1C_3 & \ldots & _nz_1C_n \\ _2z_1C_1 & _2z_2C_2 & _3z_2C_3 & \ldots & _nz_2C_n \\ \vdots \\ _nz_1C_1 & _nz_2C_2 & _nz_3C_3 & \ldots & _nz_nC_n \end{matrix}
are zero, then the actual periods of vibration of all the masses must be the same, i.e.
T_1 \ (actual) = T_2 \ (actual) = \ ··· = T_n \ (actual)
each being given by Eqn 10.5-43.
Since some of these n \times n quantities may be zero, the actual periods of vibration need not be the same for all the n masses (even though, for each individual normal mode shape, the period of vibration is the same for all the masses).