A 8 kg mass is attached to a spring hanging from the ceiling and allowed to come to rest. assume that the spring constant is 40 n/ m and a damping constant is 3 N/m. At t=0 and external force of 2 sin(2t+PI/4)N is applied to the system. determine amp, frequency and steady state solution.
To determine the amplitude, frequency, and steady-state solution of the given system, we need to solve the differential equation that describes the motion of the mass attached to the spring.
The differential equation for this damped harmonic oscillator can be written as:
m * x'' + c * x' + k * x = F(t)
Where:
- m is the mass (8 kg)
- x is the displacement of the mass from its equilibrium position
- c is the damping constant (3 N/m)
- k is the spring constant (40 N/m)
- F(t) is the external force applied to the system (2 sin(2t + π/4) N)
To find the amplitude, frequency, and steady-state solution, we can solve this differential equation step by step:
Step 1: Find the natural frequency (ω) of the system.
The natural frequency is given by:
ω = √(k / m)
Substituting the given values:
ω = √(40 N/m / 8 kg)
= √5 rad/s
Step 2: Find the damping ratio (ζ).
The damping ratio is given by:
ζ = c / (2 * √(m * k))
Substituting the given values:
ζ = 3 N/m / (2 * √(8 kg * 40 N/m))
= 3 / (2 * √(320))
≈ 0.132
Step 3: Find the natural frequency of the damped system (ωd).
The natural frequency of the damped system is given by:
ωd = ω * √(1 - ζ^2)
Substituting the values:
ωd = √5 rad/s * √(1 - 0.132^2)
≈ √(0.9986) * √5 rad/s
≈ √(4.9928) rad/s
≈ 2.235 rad/s
Step 4: Find the amplitude of the steady-state solution.
The amplitude (A) of the steady-state solution is given by:
A = F(t) / (m * ωd^2)
Substituting the given values:
A = (2 sin(2t + π/4) N) / (8 kg * (2.235 rad/s)^2)
≈ (2 sin(2t + π/4) N) / (8 * 4.9928 kg * rad^2/s^2)
≈ sin(2t + π/4) / 11.15678 m
Therefore, the amplitude of the steady-state solution is sin(2t + π/4) / 11.15678 m.
Step 5: Find the frequency of the steady-state solution.
The frequency (f) of the steady-state solution is the same as the frequency of the external force:
f = 2 rad/s
So, the frequency of the steady-state solution is 2 rad/s.
In summary:
- The amplitude of the steady-state solution is sin(2t + π/4) / 11.15678 m.
- The frequency of the steady-state solution is 2 rad/s.
- The damping constants and mass are used to calculate the natural frequency, damping ratio, and natural frequency of the damped system.