Determine the displacement, Mathematics

Assignment Help:

Example: A 16 lb object stretches a spring 8/9 ft by itself. Here is no damping as well as no external forces acting on the system. The spring is firstly displaced 6 inches upwards from its equilibrium position and specified an initial velocity of 1 ft/sec downward. Determine the displacement at any time t, u(t).

Solution

We first require setting up the IVP for the problem. It needs us to get our hands on m and k.

It is the British system so we'll require calculating the mass.

m = W/g = 16/32 = ½

Here, let's get k. We can use the fact that mg = kL to determine k. keep in mind that we'll need all of our length units similar. We'll utilize feet for the unit of measurement for this problem.

k = (mg)/L = 16/(8/9) = 18

For the initial conditions recall that upward motion/displacement is negative whereas downward motion/displacement is positive. Also, as we decided to do everything in feet we had to change the initial displacement to feet.

Here, to solve it we can either go by the characteristic equation or we can just jump straight to the formula which we derived above. We'll do it that manner. First, we require the natural frequency,

w0 = √(18/(1/2)) =√ (36) = 6

The general solution, with its derivative, is after that,

u (t ) = c1 cos (6t)+ c2 sin (6t)

u′ (t ) = -6c1 sin (6t )+6c2 cos (6t )

Applying the initial conditions provides,

-1/2 = u(0) = c1,           c2 = -1/2

1 = u'(0) = 6c2cos(6t),             c2= 1/6

The displacement at any time t is after that,

u(t) = -1/2cos(6t) + 1/6 sin(6t)

Here, let's convert this to a particular cosine. Firstly let's find the amplitude, R.

R = √((-(1/2))2 + (1/6)2) = √(10)/6 = 0.52705

You can utilize either the accurate value here or a decimal approximation. Frequently the decimal approximation will be simple.

Here let's get the phase shift.

d= tan-1((1/6)/(1/2))  = -0.32175

We require being careful with this part. The phase angle determined above is in Quadrant IV, although there is also an angle in Quadrant II which would work also. We find this second angle by adding d onto the first angle. Therefore, we actually have two angles. Hence they are,

d1= -0.32175

d2= d+ p = 2.81984

We require deciding that of these phase shifts is correct, as only one will be accurate. To do it recall that

c 1 = R cos(d)

c2 = R sin(d)

Currently, as we are assuming that R is positive it means that the sign of cosd will be similar as the sign of c1 and the sign of sind will be similar as the sign of c2. Therefore, for this specific case we must have cosd < 0 and sind > 0. It means that the phase shift should be in Quadrant II and therefore the second angle is the one which we necessitate

Thus, after all of this the displacement at any time t is,

u (t ) =0.52705 cos (6t - 2.81984)

There is a figure out of the displacement for the first 5 seconds.

405_Determine the displacement.png

Here, let's take a look at a slightly more realistic situation. There no vibration will go on forever.  Therefore let's add in a damper and notice what happens now.


Related Discussions:- Determine the displacement

Determine the widest piece of mail, A mailbox opening is 4.5 inches high an...

A mailbox opening is 4.5 inches high and 5 inches wide. Determine the widest piece of mail able to ?t in the mailbox without bending? a. 9.5 inches b. 2.2 inches c. 6.7 in

Evaluate the length of the diagonal of the print, A framed print measures 3...

A framed print measures 36 by 22 in. If the print is enclosed by a 2-inch matting, Evaluate the length of the diagonal of the print? Round to the nearest tenth. See Example.

One-to-one function, One-to-one function: A function is called one-to-one ...

One-to-one function: A function is called one-to-one if not any two values of x produce the same y.  Mathematically specking, this is the same as saying,  f ( x 1 ) ≠ f ( x 2

Comperative statics, Discuss comparative statics,Market model and Nationa i...

Discuss comparative statics,Market model and Nationa income model

Complex eigenvalues, It is the last case that we need to take a look at. Th...

It is the last case that we need to take a look at. Throughout this section we will look at solutions to the system, x?' = A x? Here the eigenvalues of the matrix A are compl

Stats, the automatic hopper loader is set to put 36 tons of coal in each ca...

the automatic hopper loader is set to put 36 tons of coal in each car. the actual weights of coal loaded into each car arw normally distributed with a mean of 36 tons and a standar

What is converse- inverse and contrapositive, What is Converse, Inverse, an...

What is Converse, Inverse, and Contrapositive In geometry, many declarations are written in conditional form "If ...., then....." For Example: "If two angles are right angles,

Write Your Message!

Captcha
Free Assignment Quote

Assured A++ Grade

Get guaranteed satisfaction & time on delivery in every assignment order you paid with us! We ensure premium quality solution document along with free turntin report!

All rights reserved! Copyrights ©2019-2020 ExpertsMind IT Educational Pvt Ltd