Mechanical Translational Systems

Transcription

1 QUESTION 1 For the system in Figure 1.1, the springs are at their free lengths when the mass displacements are zero. Complete the following: Figure 1.1 QUESTION 2 For the system in Figure 2.1, the mass displacements are zero when the springs are at their free Use x1(t), x2(t), v1(t), and v2(t) as state variables. c. Repeat part b using x1(t), f2(t), f1(t), and v2(t) as state variables.

2 Figure 2.1 QUESTION 3 For the system in Figure 3.1, the displacement of mass 1 is zero when the spring is at its free length. The position of mass 1 is measured relative to the position of mass 2. Complete the following: Figure 3.1 2

3 QUESTION 4 For the system in Figure 4.1, the mass displacements are zero when the springs are at their free c. Determine the condition for which the equations are valid. d. Determine the mass displacements at equilibrium. e. Define mass positions that are zero when the system is in equilibrium and determine a new set of first order differential equations (i.e., state equations) describing the system dynamics. 2 1 Figure 4.1 QUESTION 5 For the system in Figure 5.1, the springs are at their free lengths when the mass displacements are zero. Complete the following: 3

4 Figure 5.1 QUESTION 6 For the system in Figure 6.1, the mass displacements are zero when the springs are at their free Use x1(t), x2(t), v1(t), and v2(t) as state variables. c. Repeat part b using f1(t), x2(t), v1(t), and f2(t) as state variables. x 2 (t) x 1 (t) 2 1 Figure 6.1 4

5 QUESTION 7 For the system in Figure 7.1, the mass displacements are zero when the springs are at their free lengths. The position of mass 2 is measured relative to the position of mass 1. Complete the following: Figure 7.1 QUESTION 8 For the system in Figure 8.1, the mass displacements are zero when the springs are at their free c. Determine the condition for which the equations are valid. d. Determine the mass displacements at equilibrium. e. Define mass positions that are zero when the system is in equilibrium and determine a new set of first order differential equations (i.e., state equations) describing the system dynamics. 5

6 Figure 8.1 QUESTION 9 For the system in Figure 9.1, the mass displacements are zero when the springs are at their free b. Determine a set of first order differential equations describing its motion. c. Determine the condition for which the equations are valid. d. Determine the mass displacements at equilibrium. e. Define mass positions that are zero when the system is in equilibrium and determine a new set of first order differential equations describing its motion. 6

7 1 Figure 9.1 QUESTION 10 Derive mathematical models relating the displacement of the junction point to the displacement y for the systems in Figures 10.1 and Derive mathematical models relating the displacement of the junction point to the force f for the systems in Figures 10.3 and y Figure 10.1 y Figure

8 f Figure 10.3 f Figure 10.4 QUESTION 11 Determine a model for the vertical motion of a hot air balloon. Describe each variable and list the states and input. QUESTION 12 For the system in Figure 12.1, the springs are at their free lengths when the displacement of mass 1 is zero and the displacement of mass 1 is measured relative to the displacement of mass 2. Complete the following: a. Determine a set of state equations describing the b. If mass 1 is locked to mass 2, determine a differential equation relating the displacement of the combined mass to the applied force. c. For part b determine a differential equation relating the force through both springs to the applied force. 8

9 f(t) Figure 12.1 QUESTION 13 For the dynamic system shown below, determine a set of first order differential equations describing the f(t) Figure