The vertical motion of a machine tool can be represented schematically by Fig. 9.10. The mass, m, of 200 kg, is carried on elastic supports, so that the natural frequency for vertical motion is 30 Hz, and the viscous damping coefficient,γ, is 0.1 of critical. A mechanism applies a vertical, periodic

Question

The vertical motion of a machine tool can be represented schematically by Fig. 9.10. The mass, m, of 200 kg, is carried on elastic supports, so that the natural frequency for vertical motion is 30 Hz, and the viscous damping coefficient, $γ$ , is 0.1 of critical. A mechanism applies a vertical, periodic force, F(t), that can be approximated by a symmetrical square wave of period T = 0.1 s, and a magnitude of ± 3000 N.

Plot the vertical displacement time history, z(t), of the machine.

Accepted Answer

The notation used is as defined in the preceding section.

It is first noted that since the natural frequency of the system is 30 Hz, it is likely to be excited by the third harmonic of the excitation, which is a square wave of 1/T = 10 Hz, and that the response to the 10 Hz fundamental will be essentially static.

From Example 9.1 it can be seen that the square wave force input, in newtons, treated as an even function, can be represented by the Fourier series:

[latex]F(t) = 3000 \left(\frac{4}{\pi} \cos \omega_{0} t - \frac{4}{3\pi} \cos 3\omega_{0} t + \frac{4}{5\pi} \cos 5\omega_{0} t - \frac{4}{7\pi} \cos 7\omega_{0} t + ... ight) (A)[/latex]

Using the fact that [latex]- \cos heta = \cos ( heta - \pi ) [/latex], Eq. (A) can be written as:

[latex]F(t) = d_{1} \cos \omega_{0}t + d_{3} \cos (3 \omega_{0} t - \pi) + d_{5} \cos 5 \omega _{0}t + d_{7} \cos (7\omega _{0}t - \pi ) + ... (B)[/latex]

or in the form of Eq. (9.57):

[latex]F(t) = a_{0} + \sum\limits_{n = 1}^{\infty }{\left[d_{n} \cos (n\omega _{0} t - \psi _{n}) ight] } (C) \ \mathrm{where} \ \omega _{0} = 2\pi / T \ a_{0} = 0, (D)\ d_{n} = 0[/latex]

for even values of n.

Values of [latex]d_{n} \mathrm{and} \psi_{n}[/latex] for odd values of n are independent of T, and are as follows:

[latex]d_{1} = (3000 imes 4) /\pi = 3819 N \psi _{1} = 0 \ d_{3} = (3000 imes 4) / (3\pi) = 1273 N \psi _{3} = \pi \ d_{5} = (3000 imes 4) / (5\pi ) = 763.9 N \psi _{5} = 0 \ d_{7} = (3000 imes 4) / (7\pi ) = 545.7 N \psi _{7} = \pi \ . \ . \ . [/latex]

The following numerical values are constant throughout:
[latex]\omega_{u} = (2\pi imes 30) = 60 \pi[/latex] rad/s = natural frequency of system = 30Hz;
m = 200 kg = mass of machine;
[latex]k = m \omega_{u}^{2} = 200(2\pi imes 30)^{2} = 7.106 imes 10^{6}[/latex] N/m = stiffness of supports;
[latex]\gamma = 0.1[/latex] = viscous damping coefficient.
The displacement response is given by Eq. (9.62), which can be written as:

[latex]z(t) = \frac{a_{0}}{k} + \sum\limits_{n=1}^{\infty}{\left[\frac{d_{n}}{k} \cdot \frac{1}{\sqrt{(1 - \Omega^{2}_{n})^{2} + (2\gamma \Omega_{n})^{2}} } \cos (n\omega_{0}t - \psi_{n} - \phi_{n}) ight] } , n = 1, 2, 3, ... (9.62) \ z(t) = \frac{a_{0}}{k} + \sum\limits_{n=1}^{\infty}{\left[\frac{d_{n}}{k} A_{n} \cos (n\omega_{0}t - \psi_{n} - \phi_{n}) ight] } (E) \ \mathrm{where} \ A_{n} = \frac{1}{\sqrt{(1 - \Omega^{2}_{n})^{2} + (2\gamma \Omega_{n})^{2}} } (F) \ \Omega_{n} = \frac{n\omega_{0}}{\omega_{u}} = \frac{2\pi n}{2 \pi (30) T} = \frac{n}{30T} (G)[/latex]

and [latex]\phi_{n}[/latex] is given by Eq. (9.60):
[latex]\phi_{n} = an^{-1} \frac{2\gamma \Omega_{n}}{1 - \Omega^{2}_{n}} (H)[/latex]

Numerical values are shown in Table 9.3.
The displacement time history, z(t), is now given by Eq. (E). It can be seen from Table 9.3 that the largest contribution to the response is at 30 Hz, where the third harmonic of the force waveform coincides with the natural frequency. There are also significant components at 10 and 50 Hz.
Figure 9.11(a) shows the input force waveform, and Figure 9.11(b) the displacement response waveform, including components up to 70 Hz.

Table 9.3 Numerical Values for Example 9.3.
$\phi _{n}$ (rad)	$A_{n}$	$\Omega _{n}$	$\psi_{n}$ (rad)	$d_{n}$ (N)	$n \omega_{0}$ (rad / s)	n	Frequency (Hz)
0.2213	1.1218	0.3333	0	3819	20 $\pi$	1	10
1.5707	5.0000	1.0000	$\pi$	1273	60 $\pi$	3	30
3.029	0.5528	1.6666	0	763.9	100 $\pi$	5	50
3.096	0.2238	2.3333	$\pi$	545.7	140 $\pi$	7	70
3.116	0.1246	3.0000	0	424.4	180 $\pi$	9	90
3.125	0.0802	3.6666	$\pi$	347.2	220 $\pi$	11	110
3.130	0.0561	4.3333	0	293.8	260 $\pi$	13	130

Question 9.3: The vertical motion of a machine tool can be represented sch......

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