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Vibration Problem (Plot Curve for Amplitude of Vibration; Determine Amplitude of Vibration for Given Motor Speed)

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A schematic of a horizontal shaker is shown in figure 1 {see attachment}. The shaker consists of a mass supported on four thin, rectangular springs. The springs are L=200mm long, w=50mm wide and t=4mm thick. The Young's modulus for the springs is 200 GPa. The mass is coupled to the ground via a damper with damping factor C=894Ns/m.

The shaker is driven by a motor with an eccentric mass, mo=50g at an eccentricity of e=150mm. The total mass of the shaker, including motor, is 10kg.

1) Plot the curve for the amplitude of vibration, x to the frequency ration, r for 0<=r<=3.

2) Determine the amplitude of vibration for a motor speed of 1440 rev per min.

(Please see attachment.)

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The solution is provided in an attachment, done step-by-step equationally with short explanations and diagrams.

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Step 1: Finding the elastic constant of the system

If a force (F) pushes horizontally, every spring will be submitted to a quarter of this force, that means (F/4).
The deviation due to this force is like that produced on a beam fixed at an end and submitted to bending by a force acting at the other end, like in figure of below:

From Strength of Materials, we know that the deflection () is given by formula:
( 1)
where I = the moment of inertia of the cross section of beam around the axis of bending
E = Young's modulus
For a rectangular section of a beam defined as in above figure, the moment of inertia is given by formula
( 2)
The elastic constant of the system is defined by the relation
( 3)
Step 2: Solving the differential equation of vibration

The ...

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