Homework1_ASE330M

Homework Assignment #1, ASE 330M (UT Austin), Fall 2023 HW#1, ASE 330M Fall 2023, UT Austin Instructor: Dr. Bakolas Due on Thursday, 09/07/23, at 9:00pm; Electronic Submission through Can- vas or Gradescope . Note: Although you can submit your homework after the end of the lecture on 09/07 (until 9:00pm), neither I nor the TAs will answer any questions about the homework problems after the beginning of that lecture (at 12:30pm). The late homework submis- sion policy can be found in the syllabus. Instructions: In each problem, you should bring the equations of motion in the stan- dard form as we have done in class. I am not asking for the state space form since this homework assignment is supposed to cover only Chapter 2 from your textbook. Problem 1 Solve Problem 2.13 from your textbook (read carefully the “Instructions” given above). Problem 2 Draw the free body diagram and find the equations of motion for the translational me- chanical system illustrated in Figure 1. The quantity x 1 corresponds to the (absolute) displacement of mass M 1 . In addition, the quantity x 2 corresponds to the (relative) displacement of the mass M 2 measured with respect to the moving mass M 1 . When x 1 = x 2 = 0 and f α = 0, the system is at static equilibrium (all four springs have their natural lengths). The constant of the two springs attached to mass M 1 is equal to K 1 and the constant of the two springs attached to mass M 2 is equal to K 2 . The coefficient of the friction element between mass M 1 and the ground is equal to B 1 and the coefficient of the friction element between mass M 1 and mass M 2 is equal to B 2 . Copyright 2023, The University of Texas at Austin (instructor: Dr. Efstathios Bakolas), All rights reserved. No distribution without written permission.

Homework Assignment #1, ASE 330M (UT Austin), Fall 2023 Figure 1: Translational mechanical system with horizontal motion (Problem 2). Problem 3 Draw the free body diagram and find the equations of motion for the translational me- chanical system with vertical motion illustrated in Figure 2 and write them in standard form . We assume that both springs have the same natural length ‘ . The quantity x 1 corresponds to the (absolute) displacement of mass M 1 (positive direction is taken to be downwards). In addition, the quantity x 2 corresponds to the (relative) displacement of the mass M 2 measured with respect to the lower end of the spring with constant K 1 . Problem 4 Solve Problem 3 again using this time the following approach: Draw the free body dia- gram and find the equations of motion when the displacements of mass M 1 and mass M 2 , denoted by z 1 and z 2 , respectively, are measured with respect to the corresponding static equilibrium positions for these two masses, which are denoted as x ? 1 and x ? 2 respectively (you can redraw Figure 1 using the new reference frame, if you think that this will help you). Write the equations of motion in standard form . Note that when the two masses are at their static equilibrium positions, both z 1 and z 2 are equal to zero. Copyright 2023, The University of Texas at Austin (instructor: Dr. Efstathios Bakolas), All rights reserved. No distribution without written permission.