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Thank you Problem 1 (Dynamics 75 Pts plus 25 Pts extra credit) Consider the triple mass pendulum system from Homework 3 with nominal configuration...

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Problem 1 (Dynamics 75 Pts plus 25 Pts extra credit)

Consider the triple mass pendulum system from Homework 3 with nominal configuration has masses m1=m2=0.5 kg, m3=2 kg, and lengths L01=1m, L02=0.5m, L03=0.75m. In contrast to the pendulum in HW3, assume:

1. That the pendulum system is free to move in 3 dimensions (rather than 2)

2. That the pendulum links are flexible, e.g., they are springs with constant coefficients k1=k2=k3=5 N/m.

3. That the 3 masses experience air damping forces inversely proportional to their absolute velocity (Fdamp=-Bv, where B=1.0Ns/m is a damping coefficient).

Your tasks are as follows:

a) (5pts) Ignore the second and third links. Using the Newton-Euler approach, write the dynamic equations of motion for a single flexible link pendulum, assuming that the first link length is undeformed at the nominal length L01.

b) (10pts) Write MATLAB program simulating the single flexible pendulum system and animate it for T=50 seconds assuming that the initial joint angle condition is /2.

c) (15Pts) For the 3-link, 3D pendulum system, and assuming that the pendulum links are undeformed at the nominal lengths indicated above, form the Lagrangean dynamic equations of motion. How many degrees of freedoms do you need for this system and what generalized coordinates have you chosen? (Hint1: elastic links store potential energy just like capacitors, Hint2: generalized coordinates are those parameters that completely define the position of the system in space. They may not need to be the link length and angle of the system).

d) (15 Pts) Write MATLAB program simulating this particle system in 3D and animate it for T=50 seconds assuming an initial configuration of your choice which does not assume that the links are coplanar. Clearly specify what the initial configuration is and explain what numerical integration method you used. What is the maximum integration step h that allows for a relatively smooth and stable simulation?

e) (10 pts) Plot the total energy of the system as a function of time for B=1.0Ns/m and B=0.0Ns/m and comment on the results.

f) (10 Pts) Now assume that the link stiffness increases to k1=k2=k3=500000 N/m. If the stiffness of the links increases, you are transforming the particle system into the planar 3D robot with masses concentrated at the joints that you had dealt with in Homework 3. What integration step do you need to obtain a smooth and stable simulation? Explain why you may need a different integration step.

g) (10 Pts) Generalize the equations of motion for a system of N flexible mass-pendulums. Clearly show your generalized coordinates for this system.

h) Extra credit (25 Pts) Animate a 10-mass pendulum for appropriate choices of parameters. Notice that the dynamics of the system starts approaching a continuous flexible chain.

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