Problems and Solutions Section 1.10 (1.102 through 1.114) 1.102 A 2-kg mass connected to a spring of stiffness 10 3 N/m has a dry sliding friction force (F c ) o f 3 N. As the mass oscillates, its am plitude decreases 20 cm. How long does this take? Solution: With m = 2kg, and k = 1000 N/m the natural frequency is just ! n = 1000 2 = 22.36 rad/s From equation (1.101): slope = !2µ mg" n # k = !2 F c " n # k = $ x $t Solving the last equality for !t yields: !t = " ! x # k 2 f c $ n = "(0.20)(# )(10 3 ) 2(3)(22.36) = 4.68 s 1.103 Consider the system of Figure 1.41 with m = 5 kg and k = 9 " 10 3 N/m with a friction force of magnitude 6 N. If the initial a mplitude is 4 cm, determine the amplitude one cycle later as well as the damped frequency. Solution: Given m = 5 kg, k = 9 ! 10 3 N/ m, f c = 6 N, x 0 =0.0 4 m , the amplitude after one cycle is x 1 = x 0 ! 4 f c k = 0.04 ! (4)(6) 9 " 10 3 = 0.0373 m Note that the damped natural frequency is the same as the natural frequency in the case of Coulomb damping, hence ! n = k m = 9 " 10 3 5 = 42.43 rad/s
Welcome message from author
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
8/12/2019 Section1_10
http://slidepdf.com/reader/full/section110 1/17
8/12/2019 Section1_10
http://slidepdf.com/reader/full/section110 2/17
8/12/2019 Section1_10
http://slidepdf.com/reader/full/section110 3/17
8/12/2019 Section1_10
http://slidepdf.com/reader/full/section110 4/17
8/12/2019 Section1_10
http://slidepdf.com/reader/full/section110 5/17
8/12/2019 Section1_10
http://slidepdf.com/reader/full/section110 6/17
8/12/2019 Section1_10
http://slidepdf.com/reader/full/section110 7/17
8/12/2019 Section1_10
http://slidepdf.com/reader/full/section110 8/17
8/12/2019 Section1_10
http://slidepdf.com/reader/full/section110 9/17
8/12/2019 Section1_10
http://slidepdf.com/reader/full/section110 10/17
8/12/2019 Section1_10
http://slidepdf.com/reader/full/section110 11/17
8/12/2019 Section1_10
http://slidepdf.com/reader/full/section110 12/17
8/12/2019 Section1_10
http://slidepdf.com/reader/full/section110 13/17
8/12/2019 Section1_10
http://slidepdf.com/reader/full/section110 14/17
8/12/2019 Section1_10
http://slidepdf.com/reader/full/section110 15/17
8/12/2019 Section1_10
http://slidepdf.com/reader/full/section110 16/17
8/12/2019 Section1_10
http://slidepdf.com/reader/full/section110 17/17