Consider an underdamped mass spring system with mass m 1 kg

Consider an underdamped mass spring system with mass m = 1 kg spring constant k = 8 N/m, and damping coefficient, a = 4 N.s/m, stretched past equilibrium by 1 m and released with an initial velocity of 0 m/s. Write the solution of the initial value problem. x(t) = Identify the pseudo-frequency. Find the circular frequency if damping is removed. Determine the first time in seconds in which the mass returns to equilibrium position after being set in motion. () t = s The general equation of a damped mass spring system is m d^2 x/dt^2 + a dx/dt + kx =0, where m is the mass of the object, a is the damping coefficient and k is the spring constant as determined by Hooke\'s Law. What is the initial position of the object in this problem? To compute pseudo-frequency, convert the solution to the phase-amplitude form, x = Ae^-at cos(mu t - phi).

Solution

Motion of a Damped Oscillator

The differential equation of a damped harmonic oscillator with coefficient of damping \'a\' is given by

m d²x/dt² = -a (dx/dt) - kx

where k is spring factor. This equation may be written as d²x/dt² + 2r (dx/dt) +w₀² x =0..............(1)

where 2r = a/m and w₀² =k/m

This equation is equivalent to two differential equations of first order given by

y= dx/dt

and dy/dt = - (2ry + w₀² x) ...........................(2)

Dividing these equations, we get the equation of phase trajectory as dy/dx = -( 2ry +w₀² x)/y

IN CASE OF UNDERDAMPED

If r² <w₀² , the motion of partical is damped oscillatory motion. Substituting w²= w₀² -r²

The general solution of equation (1) is x= Ae^-rt cos (wt + *)

y= x\' = - Ae^-rt[ r cos (wt + * ) + w sin ( wt +*) where A and *( read \'phi\') are arbitrary constants.

Let us search for a solution to eqution (1) of the form x(t) = e^-rt cos (wt - *)

considering x\' = -r e^-rt cos(wt - *) - w e^-rt sin (wt - *),

x\'\' = (r² - w² ) e^-rt cos(wt - *) + 2r w e^-rt sin(wt - *) .

Hence , Equaion (1) becomes 0= [ (r² - w² ) - f r + w₀²] e^-rt cos(wt - *) + [ 2rw- f w] e^-rt sin(wt - *) .

The only way that the preceeding equation can be satisfied at all times is if the (constant) coefficients of exp (-rt) cos(wt- *) and exp(-rt) sin(wt - *) separately equal to zero, so that

  (r² - w² ) - f r + w₀² =0,

2rw- f w =0

These equations can be solved to give r= f/2 [f- read as \'neu\']

and w = (w₀² - f²/4)^1/2

Thus, the solution to the damped harmonic oscillator equation is written x(t) = e^-ft/2 cos ( wt - *) ,

assuming that f<2w₀ ( because w² =w₀² - f²/4 cannot be negative) . we conclude that the effect of a relatively small amount of damping , parameterized by the damping constant f , on a system that exhibit simple harmonic oscillation about a stable equilibrium state is to reduce the angular frequencyof the oscillation from its undamped value w₀ to w = (w₀² - f²/4 )^1/2. and to causethe amplitude of the oscillation in time at the rate f/2