The remaining problems in this section deal with free damped motion. In Problems 15 through 21, a mass m is attached to both a spring (with given spring constant k) and a dash- pot (with given damping constant c). The mass is set in mo- tion with initial position xo and initial velocity vo. Find the position function x(1) and determine whether the motion is overdamped, critically damped, or underdamped. If it is un- derdamped, write the position function in the form x(1) = C₁e-pi cos(@ita₁). Also, find the undamped position function u(t)= Cocos (wot-ao) that would result if the mass on the spring were set in motion with the same initial position and velocity, but with the dashpot disconnected (so c = 0). Fi- nally, construct a figure that illustrates the effect of damping by comparing the graphs of x(t) and u(t). 15. m = , c = 3, k = 4; xo = 2, vo = 0 16. m = 3, c = 30, k = 63; xo = 2, v0 = 2 17. m = 1,c = 8, k = 16; xo 5, vo = -10 18. m = 2, c = 12, k = 50; xp = 0, vo = -8 19. m = 4, c = 20, k = 169; xo = 4, v0 = 16 = =
The remaining problems in this section deal with free damped motion. In Problems 15 through 21, a mass m is attached to both a spring (with given spring constant k) and a dash- pot (with given damping constant c). The mass is set in mo- tion with initial position xo and initial velocity vo. Find the position function x(1) and determine whether the motion is overdamped, critically damped, or underdamped. If it is un- derdamped, write the position function in the form x(1) = C₁e-pi cos(@ita₁). Also, find the undamped position function u(t)= Cocos (wot-ao) that would result if the mass on the spring were set in motion with the same initial position and velocity, but with the dashpot disconnected (so c = 0). Fi- nally, construct a figure that illustrates the effect of damping by comparing the graphs of x(t) and u(t). 15. m = , c = 3, k = 4; xo = 2, vo = 0 16. m = 3, c = 30, k = 63; xo = 2, v0 = 2 17. m = 1,c = 8, k = 16; xo 5, vo = -10 18. m = 2, c = 12, k = 50; xp = 0, vo = -8 19. m = 4, c = 20, k = 169; xo = 4, v0 = 16 = =
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:The remaining problems in this section deal with free damped
motion. In Problems 15 through 21, a mass m is attached
to both a spring (with given spring constant k) and a dash-
pot (with given damping constant c). The mass is set in mo-
tion with initial position xo and initial velocity vo. Find the
position function x(1) and determine whether the motion is
overdamped, critically damped, or underdamped. If it is un-
derdamped, write the position function in the form x(1) =
C₁e pl cos(@ita₁). Also, find the undamped position
function u(t)= Co cos(wotao) that would result if the mass
on the spring were set in motion with the same initial position
and velocity, but with the dashpot disconnected (so c = 0). Fi-
nally, construct a figure that illustrates the effect of damping
by comparing the graphs of x(t) and u(t).
15. m = , c = 3, k = 4; xo = 2, vo = 0
16. m = 3, c = 30, k = 63; xp = 2, v0 = 2
17. m = 1, c = 8, k = 16; x = 5, vo = -10
18. m = 2, c = 12, k = 50; xo = 0, vo = -8
19. m = 4, c = 20, k = 169; xo = 4,0 = 16
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