(1)
The time for vibration of a viscous-damped spring mass system having a mass of 20kg is 0.30s. When the mass is stationary the upper end is made to move vertically downward at a velocity of 100mm/s. neglecting the mass of the spring and taking the damping factor to be 0.1625. Determine;
i. The displacement and the velocity of the mass as a function of time.
ii. The amplitude of steady state vibration when an external force
F=15cos10t is imparted
iii. The phase shift.
To solve this problem, we need to use the equation of motion for a damped spring mass system:
m * x''(t) + c * x'(t) + k * x(t) = F(t)
Where:
- m is the mass of the system,
- x(t) is the displacement of the mass,
- c is the damping factor,
- k is the spring constant, and
- F(t) is the external force applied.
Let's break down each part of the problem:
i. The displacement and velocity of the mass as a function of time.
To find the displacement and velocity as a function of time, we first need to find the natural frequency (ωn) of the system, which is given by:
ωn = √(k / m)
From the given information, we know that the time period of vibration (T) is 0.30 seconds. Therefore, we can find the natural frequency using the formula:
T = 2π / ωn
Substituting the values:
0.30 = 2π / ωn
From this equation, we can solve for ωn:
ωn = 2π / 0.30
Next, let's find the damping ratio (ζ), which is given by:
ζ = c / (2 √(m * k))
Substituting the given values:
0.1625 = c / (2 √(20 * k))
Solving for c:
c = 0.1625 * 2 √(20 * k)
Now we have the necessary values to find the displacement and velocity using the general solution for a damped spring mass system:
x(t) = A * e^(-ζωn * t) * cos(ωd * t + ϕ)
v(t) = -A * e^(-ζωn * t) * (ζωn * sin(ωd * t + ϕ) + ωd * cos(ωd * t + ϕ))
Where:
- A is the amplitude of the system,
- ωd = ωn√(1 - ζ^2) is the damped angular frequency, and
- ϕ is the phase angle.
We need to find A and ϕ using the initial condition given in the problem.
ii. The amplitude of steady-state vibration when an external force F = 15cos(10t) is imparted.
In steady state, the system will vibrate at the same frequency as the external force. So, ωn = 10.
The amplitude (A) can be calculated using the formula:
A = F / (k * √((1 - ω^2 / ωn^2)^2 + (2ζω / ωn)^2))
Substituting the given values:
A = 15 / (k * √((1 - 10^2 / ωn^2)^2 + (2ζ * 10 / ωn)^2))
iii. The phase shift.
The phase shift can be determined by comparing the displacement of the mass with the phase of the external force. We can calculate it using the formula:
ϕ = arctan((2ζω / ωn) / (1 - ω^2 / ωn^2))
Substituting the given values:
ϕ = arctan((2ζ * 10 / ωn) / (1 - 10^2 / ωn^2))
Solving these equations will give us the values for A, ϕ, x(t), and v(t) as functions of time.