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Mechanical Vibration and Springs

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The machine shown in the figure is rigidly anchored on a rigid concrete block and the mass of both is 3000 kg. The block has height h = 1 m and square horizontal section with a = 1.2 m and is supported at the corners by 12 springs of stiffness k = 2000 N/m.

i) Derive the equations of motion of the system using both Newton's law of motion and Lagrange's equations methods.

ii) Find the natural frequencies and mode shapes of the system using both hand calculations and Matlab.

iii) A 200-kg mass falls on the middle of one edge of the block and stayed in contact with it. Neglecting the change in the mass, determine the resulting motion.

iv) If the block is isolated by layers of elastic materials in a rigid concrete base. Design the isolation if the machine has parts that rotate at 250, 500, and 1500 rpm.

v) If there is unbalanced mass of 0.1 kg on the machine at a distance of 0.25 m from the axis of rotation and angular velocity of 1750 rpm. Find the resulting vibration.

The problem has 3 degrees of freedom x, y, and theta.
The moment of inertia for the system can be calculated by using standard formula, however, you can use a value of 640 kgm^2 for the block moment of inertia.

See attachment for diagram.

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Solution Summary

This in-depth solution contains step-by-step calculations and two Matlab scripts to derive equations and find the natural frequencies, resulting motion, isolation of the machine, and resulting vibration.