Help With Point-Mass/Spring Damper Problem Mathematica

The assignment is below, I am not familiar with mathematica and am having difficulty solving the problem. Help with either would be appreciated.

1

PROJECT 1: BUNGEE CORD

Course: Numerical Methods

Instructor: Dr. Hooman Tafreshi Due: November 15, 2018 at 12:30 pm

Consider a 1 m long flexible cord (e.g., bungee cord) represented with an array of 14 point-masses connected to one another by springs and dampers as shown in Figure 1.

Figure 1: Mass-Spring-Damper (MSD) representation of a flexible chain.

The forces acting on a point mass 𝑖 are shown in the free body diagram illustrated in Figure 2. In addition to gravity, there are spring and damper forces that act on each point-mass. Expressions for these forces are listed below:

Figure 2: Free body diagram illustrating forces acting on point mass i.

   1

, 1 1

1

s i i

s ri i i i

i i

p p f k p p l

p p



 



    

 (1)

   1

, 1 1

1

s i i

s ri i i i

i i

p p f k p p l

p p



 



    

 (2)

 1, 1 d

i idi i f k u u     (3)

 1, 1 d

i idi i f k u u     (4)

where , 1 s

i i f

 , , 1 d

i i f

 , , 1 s

i i f

 , and , 1 d

i i f

 are the spring and damper forces acting on point-mass 𝑖 by its neighboring point-

masses, and ip , 1ip  , and 1ip  are the position vectors of point-masses 𝑖, 𝑖 − 1, and 𝑖 + 1, respectively. sk and dk are the

spring and damping constants, respectively. The un-stretched length of the springs is shown with 𝑙𝑟. The velocity vectors

for point-masses 𝑖, 𝑖 − 1, and 𝑖 + 1, are shown with iu , 1iu  , and 1iu  , respectively. The instantaneous distance between neighboring point-masses are

21 3 4 5 6 7 8 9 10 11 12 13 14

y

x

( , )

( , ) ( , )

2

   1 2 2

1 1i i ii ii x x yp p y

      (5)

   1 2 2

1 1i i ii ii x x yp p y

      (6)

Therefore, the fractions on the right-hand side of Equations 1 and 2 are the unit vectors for the distance between the corresponding point-masses. The position and velocity of the point-mass can be obtained by solving Newton’s 2nd law written for each point-mass:

   

   

   

1 1

1 1

1 1

1 1

i i i i

i s r s ri i i i

i i i i

i i i id d

p p p p ma k p p l k p p l

p p p p

k u u k u u mg

 

 

 

 

        

 

    

(7)

where g is the gravitational acceleration, ia is the acceleration of the point-mass 𝑖, and m is the mass of each point-

mass (see [1] for more information).

What you should submit:

1. Develop a Mathematica code that solves the above equations for each point-mass. 2. Assume zero initial velocity and zero stretching for the cord. Plot the profile and velocity of the cord (the 𝑥 − 𝑦

coordinates and velocity of each point-mass) when it falls under gravity at 10 different times starting from t=0

until the system stops moving (steady-state position) for /sk m and /dk m values of 100 / .N m kg and 10

. / .N s m kg , respectively.

3. Repeat step 2 but for when the cord has an upward parabolic initial velocity with a peak value of 25 m/s. 4. Repeat step 2 but for when the cord is initially pulled up from the middle to a height of 0.25 m above the resting

position. 5. Write a short, but yet clean and professional report describing your work. Up to 25% of your grade will be based

solely on the style and formatting of your report. Use proper heading for each section of your report. Be consistent

in your font size. Use Times New Roman only. Make sure that figures have proper self-explanatory captions and are

cited in the body of the report. Make sure that your figures have legends as well as x and y labels with proper and

consistent fonts. Don’t forget that any number presented in the report or on the figures has to have a proper unit.

Equations and pages in your report should be numbered. Embed your figures in the text. Make sure they do not

have unnecessary frames around them or are not plotted on a grey background (default setting of some software

programs!).

Note: While you can work together on your projects, what you submit should be YOUR OWN original work.

References:

1-D.G. Venkateshan, M.A. Tahir, H.V. Tafreshi, and B. Pourdeyhimi, Modeling Effects of Fiber Rigidity on Thickness and Porosity of Virtual Electrospun Mats, Materials and Design, 96, 27 (2016)