Search: 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 find the amplitude, frequency, and steady-state solution of the given system, we will follow these steps:
Step 1: Write the equation of motion for the system.
Step 2: Use the given external force to modify the equation of motion.
Step 3: Determine the natural frequency and damping factor.
Step 4: Solve for the steady-state solution.
Step 5: Calculate the amplitude and frequency of the steady-state solution.
Let's go through each step in detail:
Step 1: Write the equation of motion for the system.
The equation of motion for a damped harmonic oscillator is given by:
m * d^2x/dt^2 + c * dx/dt + k * x = F(t)
Where:
m is the mass attached to the spring.
c is the damping constant.
k is the spring constant.
x is the displacement of the mass from its equilibrium position.
F(t) is the external force applied to the system.
In this case, the equation becomes:
8 * d^2x/dt^2 + 3 * dx/dt + 40 * x = F(t)
Step 2: Use the given external force to modify the equation of motion.
The given external force is 2 * sin(2t + PI/4)N. We substitute this into the equation of motion:
8 * d^2x/dt^2 + 3 * dx/dt + 40 * x = 2 * sin(2t + PI/4)
Step 3: Determine the natural frequency and damping factor.
The natural frequency (ω) of the system is calculated using the formula:
ω = √(k / m)
Given that the mass of the system is 8 kg and the spring constant is 40 N/m, we can calculate the natural frequency:
ω = √(40 / 8) = √5 rad/s
The damping factor (ζ) can be determined using the damping constant:
ζ = c / (2 * √(m * k))
Given that the damping constant is 3 N/m, we can calculate the damping factor:
ζ = 3 / (2 * √(8 * 40)) = 3 / (2 * √320) ≈ 0.0667
Step 4: Solve for the steady-state solution.
For the steady-state solution, we assume a solution of the form:
x(t) = X * sin(ωt + φ)
Substituting this into the modified equation of motion, we get:
(-ω^2 * X + 40 * X) * sin(ωt + φ) + (-2ζω * X + 3X) * cos(ωt + φ) = 2 * sin(2t + PI/4)
To solve for X and φ, we equate the coefficients of the sin and cos terms on both sides of the equation.
For the sin terms:
-ω^2 * X + 40 * X = 0 ----> X * (40 - ω^2) = 0
For the cos terms:
-2ζω * X + 3X = 2 ----> X * (3 - 2ζω) = 2
Solving these equations simultaneously gives us the value of X.
Step 5: Calculate the amplitude and frequency of the steady-state solution.
The amplitude (A) and frequency (f) of the steady-state solution can be calculated using the following formulas:
Amplitude: A = X / √[(1 - ζ^2)^2 + (2ζ)^2]
Frequency: f = ω / (2π)
Using the values of X and ω calculated above and substituting in the appropriate formulas, we can determine the amplitude and frequency.
That concludes the step-by-step explanation of how to find the amplitude, frequency, and steady-state solution of the given system. I hope this helps!