Say you have performed an experiment to measure the vertical displacement of the

Question

Say you have performed an experiment to measure the vertical displacement of the top surface of a square plate (180mm times 180 mm) under vibration (see figure 1) with certain mode You have the following set of data from the experiment. If you plot the displacement pattern, it would look kfce the Green curve n the figure. Now for some other engineenng purpose (which you will be exposed in EMCH 260) it is necessary to understand the mathematical patter of the vibration profile of the plate So you have been asked by your advisor or the boss m your company to explain the vibration pattern of the plate using a representative mathematical function Consequently he also need to find the inflection zones in the plato along the positive and negative x axis. Inflection zones are the points where the displacement pattern switches its signs (i.e. either positive to negative or negative to positive). Can you provide him both this answers following EXPLICIT mathematical steps with reasoning? Writedown, what are the steps that you need to totow to solve the problem, if you need to do any mathematical derivation and/or mathematical steps by hand, please do so by describing each and every step very clearly wth the respective name ot the procedures, also state why you need to do it. Explan why a particular choice ot your function is better than others. How could you justify?

Explanation / Answer

Answer 1)

First we will find mathematical relation between x and U. By looking at the graph, it looks symmetrical along the x-axis.

To get a linear relation we need to divide out total length of x axis in 04 sections.

The process of finding the equation of the curve of best fit, which may be most suitable for predicting the unknown values, is known as curve fitting. Below are the methods used to get mathematical relation between two variables

Here we will use, least square method because in graphical method straight line drawn may not be unique. Other two methods will not give accurate value of constant parameters.

Let us proceed with least square method

For first interval [ -180, -100]

X              U              x * U         x^2

-180          -0.040       7.200        32400

-165          -0.020       3.300        27225

-150          0.010        -1.500       22500

-125          0.040        -5.000       15625

-100          0.075        -7.500       10000

x=-720    U=0.0650         xU=-3.5           x^2=107750

Let the straight line is U = a + b * x

Normal equations are, U= n * a + bx

                                xU= ax + bx^2

Putting these values in normal equations

                                0.065 = a * 5 + b * -720

                                -3.5   = a * -720 + b * 107750

On solving these two equations, we get

a = 0.0014, b = 0.2203

So mathematical relation

For interval [-180,-100] U = 0.0014 * x + 0.2203

For interval [-100,0] U = -6.0 * 10^-4 * x + 0.015

For interval [0,100] U = 6.0 * 10^-4 * x + 0.015 (sign of b changes)

For interval [100,180] U = -0.0014 * x + 0.2203 (sign of b changes)

Answer b) From the given graph and mathematical relations inflection zones where Vertical displacement changes its sign from +ve to –ve or -ve to +ve

First it changes between -165 to -150 and later it changes from 150 to 165 on x axis.

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