Consider a damped forced mass-spring system with m = 1, γ = 2, and k =...
Consider a damped forced mass-spring system with m = 1, γ = 2, and k = 26, under the influence of an external force F(t) = 82 cos(4t).
a) (8 points) Find the position u(t) of the mass at any time t, if u(0) = 6 and u 0 (0) = 0.
b) (4 points) Find the transient solution uc(t) and the steady state solution U(t). How would you characterize these two solutions in terms of their behavior in time?
c) (4 points) Find the amplitude R and the phase angle δ for this motion and express U(t) as a single trigonometric term: U(t) = R cos(ωt − δ).
d) (4 points) Justify the following description of what happens: “the transient motion uc(t) dies out with the passage of time, leaving only the steady state periodic motion U(t)
Homework Answers
Any doubt in aamy step then comment below.. i will explain you..
In part b .. we see clearly that both solution have oscillatory motion but in transient solutionn, amplitude of motion decrease exponentially with time...
In part d) ... Transient solution have exponential term with negative exponents... so thts why it decreases with time and tends to 0 as t tends to infinity...so we have only steady solution...
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Consider a damped forced mass-spring system with m = 1, γ = 2, and k =...
damped forced mass-spring system with m 2, and k 26, under the 2 Consider a influence...
damped forced mass-spring system with m 2, and k 26, under the 2 Consider a influence of an external force F(t)= 82 cos (4t) 1, 7 = a) (8 points) Find the position u(t) of the mass at any time t, if u(0) 6 and u'(0) = 0. b) (4 points) Find the transient solution u(t) and the steady state solution U(t). How would you characterize these two solutions in terms of their behavior in time? We were unable to...
Consider a damped forced mass-spring system with m = 1, γ = 2, and k =...
Consider a damped forced mass-spring system with m = 1, γ = 2, and k = 26 under the influence of an external force F(t) = 82 cos(ωt). We can prove that the amplitude of this motion is given by R(ω) = p F0 m2 (ω0 2 − ω2 ) 2 + γ 2ω2 = 82 √ ω4 − 48ω2 + 76 For what value of ω will the maximum amplitude occur? When resonance will occur and how would you...
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In solving for the damped forced circuit 1 oscillations (particular solution) has the form 6. +R ...
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damped forced mass-spring system with m 2, and k 26, under the 2 Consider a influence of an external force F(t)= 82 cos (4t) 1, 7 = a) (8 points) Find the position u(t) of the mass at any time t, if u(0) 6 and u'(0) = 0. b) (4 points) Find the transient solution u(t) and the steady state solution U(t). How would you characterize these two solutions in terms of their behavior in time? We were unable to...
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and the motion takes place in a viscous fluid that offers a
resistance numerically equal to the magnitude of the instantaneous
velocity. If the system is driven by an external force of (27 cos
3t − 18 sin 3t) N, determine the steady state response. Express
your answer in the form R cos(ωt − δ). (Let u(t) be the
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The displacement of an object in a spring-mass system in free damped oscillation is 4y'' +40y' + 164y = 0 - 15e cos(4t 0.57) and has solution 1 if the motion is under-damped. If we apply an impulse of the form f(t) = ad(t - T) then the differential equation becomes 4y'' +40y' + 164y = ad(t - T) and has solution y = - 15e cos(4t - 0.57) au(t - T)w(t - r) where w(t) L 482 40s 164...
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In solving for the damped forced circuit 1 oscillations (particular solution) has the form 6. +R q-Vocos(r) + dt C we found the pers istent cos (st where γ -, LC 21 2 Define rso that the amplitude of the response is proportional toT(r)- where we (1-r2)2 +4γ assume γ is a set constant (i) Show that T has a maximum Tmax-T(rmax) if γく1. In such a case find "max and T,nax (ii) Plot in T vs. r for the...
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