shows a single degree of freedom spring-mass system having a forced vibration and damper, starting from the second Newton’s law of motion, show that the extension of the system is given by
X=Epg wnt sin(wd t + y). Also explain the conditions of the system when the damping factor is greater than, equal to, or less than unity.
To derive the equation for the extension of the spring-mass system with forced vibration and damping, we start with Newton's second law of motion.
The equation is given by:
m(d^2x/dt^2) + c(dx/dt) + kx = F0 sin(ωt + φ)
where:
m is the mass of the system,
x is the displacement of the mass from the equilibrium position,
t is time,
c is the damping coefficient (damping factor multiplied by 2√mk),
k is the spring constant of the system,
F0 is the amplitude of the driving force,
ω is the angular frequency (2πf),
φ is the phase angle.
Assuming we have a harmonic solution of the form x(t) = X sin(ωt + φ), where X is the amplitude of the steady-state motion, and substituting it into the equation, we get:
-mω^2X sin(ωt + φ) + cωX cos(ωt + φ) + kX sin(ωt + φ) = F0 sin(ωt + φ)
Simplifying the equation further, we get:
X(-mω^2 + k)sin(ωt + φ) + cωX cos(ωt + φ) = F0 sin(ωt + φ)
Now, divide both sides of the equation by (-mω^2 + k) to obtain:
X sin(ωt + φ) = (F0 / -mω^2 + k) sin(ωt + φ) + (cωX / -mω^2 + k) cos(ωt + φ)
This equation can be rewritten in terms of a single trigonometric function using the identity sin(A + B) = sin(A)cos(B) + cos(A)sin(B):
X sin(ωt + φ) = (F0 / -mω^2 + k) sin(ωt + φ) + (cωX / -mω^2 + k) √(1 - sin^2(ωt + φ))
Now, let's define the natural angular frequency of the spring-mass system as ωn = √(k/m), and the damping factor as ζ = c / (2√(mk)).
Replacing these values and simplifying the equation further, we get:
X sin(ωt + φ) = (F0 / m(ω^2n - ω^2)) sin(ωt + φ) + (2ζωnX / (ω^2n - ω^2)) √(1 - sin^2(ωt + φ))
Simplifying the equation with trigonometric identities, we obtain:
X sin(ωt + φ) = (F0 / m(ω^2n - ω^2)) sin(ωt + φ) + (2ζωnX / (ω^2n - ω^2)) cos(ωt + φ)
Now, we can see that the extension of the system (X) is given by:
X = (F0 / m(ω^2n - ω^2)) + (2ζωnX / (ω^2n - ω^2)) cos(ωt + φ)
Finally, by using the identity cos(A) = cos(B)cos(C) - sin(B)sin(C), we can rewrite the equation as:
X = (F0 / m(ω^2n - ω^2)) + (2ζωnX / (ω^2n - ω^2)) [cos(φ)cos(ωt) - sin(φ)sin(ωt)]
So, X = Epg wnt sin(wd t + y), where Epg = (F0 / m(ω^2n - ω^2)), wn = √(k/m), wd = √(k/m) √(1 - ζ^2), and y = tan^(-1)(ζ/√(1 - ζ^2)).
Now, moving on to the conditions of the system based on the damping factor (ζ):
1. When the damping factor (ζ) is greater than unity (ζ > 1):
- The system is called an overdamped system.
- The mass moves towards the equilibrium position slowly, without oscillation.
- The amplitude of the motion gradually decreases with time.
- The response of the system is slower compared to other damping conditions.
2. When the damping factor (ζ) is equal to unity (ζ = 1):
- The system is called a critically damped system.
- The mass returns to its equilibrium position in the shortest possible time without oscillation.
- The response of the system is the fastest compared to other damping conditions.
- The amplitude decreases faster than the underdamped case.
3. When the damping factor (ζ) is less than unity (ζ < 1):
- The system is called an underdamped system.
- The mass undergoes oscillations around the equilibrium position before gradually coming to rest.
- The amplitude of the motion gradually decreases with time.
- The response of the system is faster than the overdamped case but slower than the critically damped case.