A spring-mass system has a spring constant of 3 Nm. A mass of 2 kg is attached to the spring, 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 27cos(3t)−18sin(3t) N,determine the steady-state response in the form Rcos(ωt−δ).

Question

27-02-2020
Physics

contestada

A spring-mass system has a spring constant of 3 Nm. A mass of 2 kg is attached to the spring, 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 27cos(3t)−18sin(3t) N,determine the steady-state response in the form Rcos(ωt−δ).

Respuesta :

Otras preguntas

All of the following are advantages of purchasing a single-dose unit of a topical anesthetic except one. which one is the exception?a. Dose manageable b. Cost-e

What is meant by functional finance?

A coin is dropped from the top of a tower, and its height, in meters, after t seconds is modeled by the function h(t). what notion could be used to show that 2

how many moles of asa are present in a 5 ml aliquot (measured sample) of this solution?

A line pae through point (-6,5) and ha a lope of 2. Write an equation in AxBy=C form for thi line

Chevrolet Corvette Sting Ray developed 360 horsepower and hada displacement of 327 cubic inches. Express this displacement inliters (L) by using only the conver

brainteaser there are 5 people sitting arund a table with different birthdays, what is the probability they sit in the right order both clockwise and counter cl

kanye west pulled out of which big music festival in april, two weeks before he was scheduled to appear?

When making a routine request, the body of your message should ________. A) put your most important request at the end B) explain and justify your request C) ad

Kaeuper ay they ued mafia tactic while attacking other citie or town why might thi intimidation be what the knight wanted?

oeerivona oeerivona · Answer 1 · 2020-03-01T16:32:16+01:00

Answer:

The answer to the question

The steady state response is u₂(t) = -[tex]\frac{3\sqrt{2} }{2}[/tex]cos(3t + π/4)

of the form R·cos(ωt−δ) with R = [tex]-\frac{3\sqrt{2} }{2}[/tex], ω = 3 and δ = -π/4

Explanation:

To solve the question we note that the equation of motion is given by

m·u'' + γ·u' + k·u = F(t) where

m = mass = 2.00 kg

γ = Damping coefficient = 1

k = Spring constant = 3 N·m

F(t) = externally applied force = 27·cos(3·t)−18·sin(3·t)

Therefore we have 2·u'' + u' + 3·u = 27·cos(3·t)−18·sin(3·t)

The homogeneous equation 2·u'' + u' + 3·u is first solved as follows

2·u'' + u' + 3·u = 0 where putting the characteristic equation as

2·X² + X + 3 = 0 we have the solution given by [tex]\frac{-1+/-\sqrt{23} }{4} \sqrt{-1}[/tex] =[tex]\frac{-1+/-\sqrt{23} }{4} i[/tex]

This gives the general solution of the homogeneous equation as

u₁(t) = [tex]e^{(-1/4t)} (C_1cos(\frac{\sqrt{23} }{4}t) + C_2sin(\frac{\sqrt{23} }{4}t)[/tex]

For a particular equation of the form 2·u''+u'+3·u = 27·cos(3·t)−18·sin(3·t) which is in the form u₂(t) = A·cos(3·t) + B·sin(3·t)

Then u₂'(t) = -3·A·sin(3·t) + 3·B·cos(3·t) also u₂''(t) = -9·A·cos(3·t) - 9·B·sin(3·t) from which 2·u₂''(t)+u₂'(t)+3·u₂(t) = (3·B-15·A)·cos(3·t) + (-3·A-15·B)·sin(3·t). Comparing with the equation 27·cos(3·t)−18·sin(3·t) we have

3·B-15·A = 27

3·A +15·B = 18

Solving the above linear system of equations we have

A = -1.5, B = 1.5 and u₂(t) = A·cos(3·t) + B·sin(3·t) becomes 1.5·sin(3·t) - 1.5·cos(3·t)

u₂(t) = 1.5·(sin(3·t) - cos(3·t) = [tex]-\frac{3\sqrt{2} }{2}[/tex]·cos(3·t + π/4)

The general solution is then u(t) = u₁(t) + u₂(t)

however since u₁(t) = [tex]e^{(-1/4t)} (C_1cos(\frac{\sqrt{23} }{4}t) + C_2sin(\frac{\sqrt{23} }{4}t)[/tex] ⇒ 0 as t → ∞ the steady state response = u₂(t) = [tex]-\frac{3\sqrt{2} }{2}[/tex]·cos(3·t + π/4) which is of the form R·cos(ωt−δ) where

R = [tex]-\frac{3\sqrt{2} }{2}[/tex]

ω = 3 and

δ = -π/4