Question

In: Advanced Math

*PLEASE SHOW ALL WORK* Consider a damped, forced mass/spring system. Let t denote time (in seconds)...

*PLEASE SHOW ALL WORK*

Consider a damped, forced mass/spring system. Let t denote time (in seconds) and let x(t) denote the position (in meters) of the mass at time t, with x = 0 corresponding to the equilibrium position. Suppose the mass m = 1 kg, the damping constant c = 3 N·s/m, the spring constant k = 2 N/m, the external force is F (t) = 20 cos(2t), the initial position

x(0) = 1 m, and the initial velocity x′(0) = 2 m/s. a. Find the position function x(t).

b. What part of the solution is the transient part and what part is the steady-state part?

Solutions

Expert Solution

Colby Messinger answered 1 year ago
Next > < Previous

Related Solutions

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...
The motion of a forced damped spring system is: ?(?) = 2 sin 4? + ?...
The motion of a forced damped spring system is: ?(?) = 2 sin 4? + ? ^-? cos 6?. A) What is the driving frequency? B) What is the transient part? C) What is the steady-periodic part? Could someone provide explanations as to how to obtain these solutions as I am lost?
(10pts) Consider the damped forced harmonic oscillator with mass 1 kg, damping coefficient 2, spring constant...
(10pts) Consider the damped forced harmonic oscillator with mass 1 kg, damping coefficient 2, spring constant 3, and external force in the form of an instantaneous hammer strike (Section 6.4) at time t = 4 seconds. The mass is initially displaced 2 meters in the positive direction and an initial velocity of 1 m/s is applied. Model this situation with an initial value problem and solve it using the method of Laplace transforms.
Consider an mass-spring system with the following IVP for its disagreement y (t) at time t...
Consider an mass-spring system with the following IVP for its disagreement y (t) at time t greater than or equal to 0. You may assume it is underdamped. y" + y' + 5y = 0 , y (0) = -2 , y '(0) = -1 (a) Convert this to a DE system IVP in displacement y and velocity v. (b) Without using technology or solving the second order DE, make a rough sketch of the system solution on a phase...
A car and the suspension system are a damped spring, mass system, with a natural frequency...
A car and the suspension system are a damped spring, mass system, with a natural frequency 2 Hz and a damping coefficient 0.2. The car drives at speed V over a road with sinusoidal roughness. The roughness wavelength is 20 m, with a amplitude of 50 cm. What speed does the maximum amplitude of vibration occur? What is the corresponding vibration amplitude? Please do not use an older solution.
The displacement of an object in a spring-mass system in free damped oscillation is: 2y''+12y'+26y=0 and...
The displacement of an object in a spring-mass system in free damped oscillation is: 2y''+12y'+26y=0 and has solution: y1 =−5e^(−3t)*cos(2t−0.5π) if the motion is under-damped. If we apply an impulse of the form f(t) = αδ(t−τ) then the differential equation becomes : 2y''+12y'+26y=αδ(t−τ) and has solution y =−5e^(−3t)cos(2t−0.5π)+αu(t−τ)w(t−τ) where w(t) = L(^−1)*(1/(2s^2+12s+26)) a. When should the impulse be applied? In other words what is the value of τ so that y1(τ) =0 . Pick the positive time closest to t...
The motion of a​ mass-spring system with damping is governed by y''(t) + by'(t) + 64...
The motion of a​ mass-spring system with damping is governed by y''(t) + by'(t) + 64 y(t) = 0; y(0) = 3, and y'(0) = 0. Find the equation of motion for b = 0,14,16, and 20. .
Consider a mass-spring-dashpot system with mass 5kg, a spring which is stretched 3 meters by a...
Consider a mass-spring-dashpot system with mass 5kg, a spring which is stretched 3 meters by a force of 10N, and a dashpot which provides a 4N resistance for each m/s of velocity. The mass is also acted on by a periodic force 5 cos(ωt) for some number ω. • For what value of ω (if any) does practical resonance occur? • If the mass starts at rest, at equilibrium, find a formula for x(t) (in terms of ω), the distance...
Consider a mass-spring-dashpot system with mass 5kg, a spring which is stretched 6 meters by a...
Consider a mass-spring-dashpot system with mass 5kg, a spring which is stretched 6 meters by a force of 20N, and a dashpot which provides 4N resistance for each m/s of velocity. The mass is also acted on by a periodic force 5 cos(ωt) for some number ω. For what value of ω (if any) does practical resonance occur? If the mass starts at rest, at equilibrium, find a formula for x(t) in terms of ω, the distance between the mass...
Consider a mass spring system. The spring has a constant k=30N/m and mass=3kg. The mass oscillates...
Consider a mass spring system. The spring has a constant k=30N/m and mass=3kg. The mass oscillates w/amplitude of 10cm. a.)what is the frequency of oscillation b.) what is the displacement at time t=0 c.) when is the first time the mass is at maximum displacement? (t=?) d.) what is the maximum acceleration felt by the mass? Where in the motion does this occur? e.)what is the minimum acceleration felt by the mass? Where in the motion does this occur? f.)What...
ADVERTISEMENT
Subjects
ADVERTISEMENT
Latest Questions
ADVERTISEMENT