Mechanical vibrations, Mathematics

Assignment Help:

This time we are going to take a look at an application of second order differential equations. It's now time take a look at mechanical vibrations. In exactly we are going to look at a mass which is hanging from a spring.

Vibrations can arise in pretty much all branches of engineering and thus what we're going to be doing now can be simply adapted to other situations, generally with just a change in notation.

Let's find the situation setup. We are going to begin with a spring of length l, termed as the natural length, and we're going to hook an object along with mass m up to this. While the object is attached to the spring, it will stretch a length of L. We will identify it the equilibrium position the position of the center of gravity for the object like this hangs on the spring along with no movement.

There is sketch given below, of the spring with and without the object attached to this.

1446_Mechanical Vibrations.png

As denoted in the above sketch we are going to suppose that all velocities, forces and displacements in the downward direction will be positive. All velocities, forces and displacements in the upward direction will be negative.

Also, as demonstrated in the sketch above, we will measure all displacement of the mass by its equilibrium position. Thus, the u = 0 position will corresponding to the center of gravity for the mass as this hangs on the spring and is at rest, which is no movement.

Here, we need to develop a differential equation which will provide the displacement of the object at any time t.  Firstly, recall Newton's Second Law of Motion.

ma = F

In this case we will use the second derivative of the displacement, u, for the acceleration and so Newton's Second Law turns into,

mu′′ = F (t, u, u′)

We now require determining all the forces that will act on the object. There are four forces which we will suppose act upon the object. Two, will all the time act upon the object and two which may or may not act on the object.


Related Discussions:- Mechanical vibrations

..Job, Eddie mkes $15.75 per hour. Estimate how much Eddie will make per ye...

Eddie mkes $15.75 per hour. Estimate how much Eddie will make per year if he works 40 hours per week and 50 weeks per year.

Function that computes the product of two matrices, Write a function that c...

Write a function that computes the product of two matrices, one of size m × n, and the other of size n × p. Test your function in a program that passes the following two matrices t

Composite functions, f(x)=4x-3 and g(x)=(x+3)/4 a)Find the function fg(x) ...

f(x)=4x-3 and g(x)=(x+3)/4 a)Find the function fg(x) b)Hence describe the relationship between the functions f and g c)Write down the exact value of fg(sqrt(3))

Line plots, how to you find the difference between different line plots

how to you find the difference between different line plots

Trignometry, how can i easily solve the trignometry question?

how can i easily solve the trignometry question?

How to grow your brand with existing customers., "To grow your brand, you n...

"To grow your brand, you need to encourage your existing customers to buy your product a liitle more often. It is far more important to maximise the number of times your buyers buy

Fractions, what is 1/3 + 2/9 equal

what is 1/3 + 2/9 equal

Mean is 8.32 find the median, In a frequency distribution mode is 7.88, mea...

In a frequency distribution mode is 7.88, mean is 8.32 find the median.  (Ans: 8.17) Ans:  Mode = 3 median - 2 mean 7.88 = 3 median - 2 x 8.32 7.88 +16.64 = 3 median

MAT201, #There is a balance of $1,234 and this person receive a refund chec...

#There is a balance of $1,234 and this person receive a refund check in the amount of $25 with her paycheck that was deposited into her account for $1500 which made her balance $27

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