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?

Expert Solution

The solution to the differential equation of a forced damped spring system driven by an external force has two parts, a transient part and a steady-state part, which must be used together to fit the physical boundary conditions of the problem.

The general solution is a sum of a transient solution that depends on initial conditions, and a steady state that is independent of initial conditions and depends only on the driving amplitude F0, driving frequency ω.

The transient part must contain a damping part in terms of a negative exponential.

The solution given to us is

?(?) = 2 sin 4? + ? ^-? cos 6?.

Clearly , the second part on the RHS contains the damping part ( e^-t) and hence , it is the transient part. (PART B)

The first part on the RHS is independent of the initial conditions and does not contain any damping part, and hence, it is the steady-periodic part. ( 2 sin 4?) (PART C)

The steady periodic part is due to the driving force. Thus, comparing 2 sin 4? with A sin (wt) gives the angular driving frequency w=4.

Hence, Driving frequency = w/(2*) = 4/(2*) = 2/ (PART A)

genius_generous answered 2 years ago

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...

(a)Use Netwons Second Law of Motion to prove that the equation governing the forced damped harmonic...

(a)Use Netwons Second Law of Motion to prove that the equation governing the forced damped harmonic oscillator (spring-mass system) is: mx"(t) + cx'(t) + kx(t) = F(t): (Explain what the constants m; c; k are and what the function F(t) is. Draw a picture of the system.) (b)Assume m = 1; c = 0; k = 4; that F(t) = cos(2t); and that the object attached to the spring begins from the rest position. Find the position function using the...

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...

(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.

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.

This problem is an example of critically damped harmonic motion.

This problem is an example of critically damped harmonic motion. A mass m=6kg is attached to both a spring with spring constant k=96N/m and a dash-pot with damping constant c=48N⋅s/m . The ball is started in motion with initial position x0=5m and initial velocity v0=−24m/s . Determine the position function x(t) in meters. x(t)= Graph the function x(t) . Now assume the mass is set in motion with the same initial position and velocity, but with the dashpot disconnected (...

Mechanical vibration. Examples of forced damped oscillation with harmonic force.

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...

How is the motion of the mass on the spring similar to the motion of the...

How is the motion of the mass on the spring similar to the motion of the bob? How is it different? Then, why does the period of a pendulum not depend on mass but it does for the spring? Explain carefully. Simple harmonic motion is used to describe physics most complex theories. How could something so simple describe the most complex

1) Sinusoidal Motion Properties in Spring Mass System- A 200g mass hangs from vibrating spring at...

1) Sinusoidal Motion Properties in Spring Mass System- A 200g mass hangs from vibrating spring at lowest point of 3cm above table and at it's highest point at 12cm above table. It's oscillation period is 4seconds. Determine the following: a. The spring constant in terms of T (period) b. The maximum velocity magnitude and maximum acceleration magnitude c. The velocity magnitude at 10cm above table d. The vertical position, velocity magnitude and acceleration magnitude at 5 seconds

Question