{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

ASE 365 - HW 5 Solutions

# ASE 365 - HW 5 Solutions - Solutions Homework Set 5 1...

This preview shows pages 1–3. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Solutions: Homework Set 5 1. Problem 4.5. Since F ( t ) = F T [ tu ( t )- ( t- T ) u ( t- T )], the response is x ( t ) = F T [ r ( t )- r ( t- T )] = F kT braceleftbiggbracketleftbigg t- 2 Î¶ Ï‰ n + e âˆ’ Î¶Ï‰ n t parenleftbigg 2 Î¶ Ï‰ n cos Ï‰ d t + 2 Î¶ 2- 1 Ï‰ d sin Ï‰ d t parenrightbiggbracketrightbigg u ( t )- bracketleftbigg t- T- 2 Î¶ Ï‰ n + e âˆ’ Î¶Ï‰ n ( t âˆ’ T ) parenleftbigg 2 Î¶ Ï‰ n cos Ï‰ d ( t- T ) + 2 Î¶ 2- 1 Ï‰ d sin Ï‰ d ( t- T ) parenrightbiggbracketrightbigg Â· u ( t- T ) bracerightbigg For the undamped case, this reduces to x ( t ) = F kT braceleftbiggbracketleftbigg t- 1 Ï‰ n sin Ï‰ n t bracketrightbigg u ( t )- bracketleftbigg t- T- 1 Ï‰ n sin Ï‰ n ( t- T ) bracketrightbigg u ( t- T ) bracerightbigg which agrees with the results from the lecture. 2. Problem 4.7. The response to the trapezoidal pulse is x ( t ) = 2 F T bracketleftbigg r ( t )- r ( t- T 2 )- r ( t- 3 T 2 ) + r ( t- 2 T ) bracketrightbigg where r ( t ) is the ramp response found for problem 4.4, given by r ( t ) = 1 k bracketleftbigg t- 2 Î¶ Ï‰ n + e âˆ’ Î¶Ï‰ n t parenleftbigg 2 Î¶ Ï‰ n cos Ï‰ d t + 2 Î¶ 2- 1 Ï‰ d sin Ï‰ d t parenrightbiggbracketrightbigg u ( t ) 3. Problem 4.9. EOM: Â¨ x + 2 Î¶Ï‰ n Ë™ x + Ï‰ 2 n x = F ( t ) m . Use second form from: x ( t ) = integraldisplay t F ( Ï„ ) g ( t- Ï„ ) dÏ„ = integraldisplay t F ( t- Ï„ ) g ( Ï„ ) dÏ„ where F ( t ) = F e âˆ’ Î±t u ( t ) , g ( t ) = 1 mÏ‰ d e âˆ’ Î¶Ï‰ n t sin Ï‰ d tu ( t ). Response: x ( t ) = F mÏ‰ d integraldisplay t e âˆ’ Î± ( t âˆ’ Ï„ ) u ( t- Ï„ ) e âˆ’ Î¶Ï‰ n Ï„ sin Ï‰ d Ï„u ( Ï„ ) dÏ„ = F e âˆ’ Î±t mÏ‰ d integraldisplay t e ( Î± âˆ’ Î¶Ï‰ n ) Ï„ sin Ï‰ d Ï„ dÏ„ = F m [ Ï‰ 2 d + ( Î±- Î¶Ï‰ n ) 2 ] bracketleftbigg e âˆ’ Î±t + parenleftbigg Î±- Î¶Ï‰ n Ï‰ d sin Ï‰ d t- cos Ï‰ d t parenrightbigg e âˆ’ Î¶Ï‰ n t bracketrightbigg u ( t ) 1 With the provided values, this becomes x ( t ) = (0 . 041771 m) e ( âˆ’ 1 sec- 1 ) t (1- cos(19 . 975 sec âˆ’ 1 ) t ) u ( t ) . (In this result, units are expressed explicitly and carefully with the assumption that the textbook should have given Î± the units sec âˆ’ 1 (or rad/sec).) 4. Problem 4.13. Using these two integrals found in the lecture: integraldisplay sin Ï‰ n ( t- Ï„ ) dÏ„ = 1 Ï‰ n cos Ï‰ n ( t- Ï„ ) integraldisplay Ï„ sin Ï‰ n ( t- Ï„ ) dÏ„ = 1 Ï‰ 2 n sin Ï‰ n ( t- Ï„ ) + Ï„ Ï‰ n cos Ï‰ n ( t- Ï„ ) the results for the various time intervals can be found readily. For 0 < t < T/ 2: x 1 ( t ) = integraldisplay t 2 F T Ï„ 1 mÏ‰ n sin Ï‰ n ( t- Ï„ ) dÏ„ = 2 F mÏ‰ n T bracketleftbigg 1 Ï‰ 2 n sin Ï‰ n ( t- Ï„ ) + Ï„ Ï‰ n cos Ï‰ n ( t- Ï„ ) bracketrightbigg t = 2 F kT bracketleftbigg- 1 Ï‰ n sin Ï‰ n t + t bracketrightbigg For T/ 2 < t < 3 T/ 2: x 2 ( t ) = integraldisplay T/ 2 2 F T Ï„ 1 mÏ‰ n sin Ï‰ n ( t- Ï„ ) dÏ„...
View Full Document

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business â€˜17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. Itâ€™s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania â€˜17, Course Hero Intern

• The ability to access any universityâ€™s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLAâ€™s materials to help me move forward and get everything together on time.

Jill Tulane University â€˜16, Course Hero Intern