Consider a forced spring-mass equation of the form x′′ + x = cos(ωt) with initial conditions...
Consider a forced spring-mass equation of the form x′′ + x = cos(ωt) with initial conditions
x(0) = 1 and x′(0) = 0.
a) Suppose ω doesnt = 1, find the solution to the IVP.
b)If ω = 1, find the solution to the IVP.
c)In which of the two cases does the phenomenon of pure resonance occur? Ex- plain your answer.
d)Verify that with ω = 9/10, x(t) = 100 (cos( 9t ) − 81 cos t) solves the IVP.
e) Recall the trig identity cos(α) − cos(β) = −2 sin( α+β ) sin( α−β ). Observe that 22
x(t) ≈ 100 (cos( 9t ) − cos t). Use this last form and the trig identity above to approximate 19 10
x(t) by a product of two sin functions. Make a sketch of the graph, carefully accounting for the periods of the two functions. What would you call the long-term behavior of this solution??
Homework Answers
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Consider a forced spring-mass equation of the form x′′ + x =
cos(ωt) with initial conditions...
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...
The general solution to the second-order differential equation d2ydt2−4dydt+7y=0 is in the form y(x)=eαx(c1cosβx+c2sinβx). Find the values of α and β, where β>0. Answer: α= and β=
The general solution to the second-order differential equation d2ydt2−4dydt+7y=0d2ydt2−4dydt+7y=0 is in the form y(x)=eαx(c1cosβx+c2sinβx).y(x)=eαx(c1cosβx+c2sinβx). Find the values of αα and β,β, where β>0.β>0.Answer: α=α= and β=β=
#40 a-f B-A. (B+A ". Beats slation Recall the identity cos A-cos Be2-2A)sin(-2A) a. Show that 0-10,a, . 9 and (ii)o_10,us2toverify the identity. In which case do you see Gaph the functions on...
#40 a-f
B-A. (B+A ". Beats slation Recall the identity cos A-cos Be2-2A)sin(-2A) a. Show that 0-10,a, . 9 and (ii)o_10,us2toverify the identity. In which case do you see Gaph the functions on both sides of the equation in part (a) with (i) beats? b. 40 Analysis of the forced damped oscillation equation Consider the equation my"+ey'+ky Fo cos wof, which oscillator. Assume all the parameters in the equation are positive. a. Explain why the solutions of the homogeneous equation...
Consider a wave that is represented by ψ(x, t) = 4 cos (kx − ωt). where...
Consider a wave that is represented by ψ(x, t) = 4 cos (kx − ωt). where k = 2π/λ and ω = 2πf. The aim of the following exercises is to show that this expression captures many of the intuitive features of waves. a) Consider a snapshot of the wave at t = 0. Use the expression to find the possible values of x at which the crests (maximum points) of the wave are located. By what distance are neighboring...
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?...
Consider the general form of the forced spring-mass equation my" +cy' + ky=f(t) (c) Use the...
Consider the general form of the forced spring-mass equation my" +cy' + ky=f(t) (c) Use the method of undetermined coefficients to show that with f(t) Fcos(ot), the particular solution y, to (3.1) is CO (3.2) sinot)
Problem 1.Consider the harmonically forced undamped oscillator described by the following ODE:mx′′+kx=F0cosωt, k >0, m >0,...
Problem 1.Consider the harmonically forced
undamped oscillator described by the following ODE:mx′′+kx=F0cosωt,
k >0, m >0, ω >0, F0∈R.
Problem 1. Consider the harmonically forced undamped oscillator described by the following ODE: mx" + kx = Fo cos wt, k > 0, m > 0,w > 0, F0 E R. (1) a) Suppose wa #k/m. Find the general solution of the ODE ). b) Consider the initial value problem of the ODE () with initial conditions x(0) = 0 and...
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...
An object of mass m is connected to a light spring with a force constant of...
An object of mass m is connected to a light spring with a force constant of kH N/meter which oscillates on a frictionless horizontal surface with Simple Harmonic Motion. At t = 0 the spring was at rest but is compressed x = A meter maximum during oscillation. Write the equation of motion from Newton's 2nd law FH = m·a and Hook's Law FH = -kH·x. Because of the starting position assume a solution is x = A sin(ωt) a...
<---- equation (6) Using the equation (6) in page 569 of our text, obtain the solution of ut-CUx -lx in integral form satisfying the initial condition u(x, 0) = e- 30 1 | [A(p) cos px + B(p)sin px...
<---- equation (6)
Using the equation (6) in page 569 of our text, obtain the solution of ut-CUx -lx in integral form satisfying the initial condition u(x, 0) = e- 30 1 | [A(p) cos px + B(p)sin px) e-c2p2tdp 11 (x, t; p) dp= u(x, t)= 0
Using the equation (6) in page 569 of our text, obtain the solution of ut-CUx -lx in integral form satisfying the initial condition u(x, 0) = e-
30 1 | [A(p) cos...
#40 a-f
B-A. (B+A ". Beats slation Recall the identity cos A-cos Be2-2A)sin(-2A) a. Show that 0-10,a, . 9 and (ii)o_10,us2toverify the identity. In which case do you see Gaph the functions on both sides of the equation in part (a) with (i) beats? b. 40 Analysis of the forced damped oscillation equation Consider the equation my"+ey'+ky Fo cos wof, which oscillator. Assume all the parameters in the equation are positive. a. Explain why the solutions of the homogeneous equation...
Consider the general form of the forced spring-mass equation my" +cy' + ky=f(t) (c) Use the method of undetermined coefficients to show that with f(t) Fcos(ot), the particular solution y, to (3.1) is CO (3.2) sinot)
Problem 1.Consider the harmonically forced
undamped oscillator described by the following ODE:mx′′+kx=F0cosωt,
k >0, m >0, ω >0, F0∈R.
Problem 1. Consider the harmonically forced undamped oscillator described by the following ODE: mx" + kx = Fo cos wt, k > 0, m > 0,w > 0, F0 E R. (1) a) Suppose wa #k/m. Find the general solution of the ODE ). b) Consider the initial value problem of the ODE () with initial conditions x(0) = 0 and...
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...
<---- equation (6)
Using the equation (6) in page 569 of our text, obtain the solution of ut-CUx -lx in integral form satisfying the initial condition u(x, 0) = e- 30 1 | [A(p) cos px + B(p)sin px) e-c2p2tdp 11 (x, t; p) dp= u(x, t)= 0
Using the equation (6) in page 569 of our text, obtain the solution of ut-CUx -lx in integral form satisfying the initial condition u(x, 0) = e-
30 1 | [A(p) cos...
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