The time constant of a mass-spring-damper system is given by the formula:
τ = M / k
where M is the mass of the object, and k is the damping coefficient. Substituting the given values, we get:
τ = 3 / 20 = 0.15 s
The critical damping coefficient is given by the formula:
c_crit = 2 * sqrt(k * M)
Substituting the given values, we get:
c_crit = 2 * sqrt(800 * 3) = 78.13 Ns/m
The damping ratio is given by the formula:
ζ = c / c_crit
where c is the actual damping coefficient. Substituting the given values, we get:
ζ = 20 / 78.13 = 0.256
The equation for the force required when the piston is accelerating is given by:
F = M * (d^2x / dt^2) + k * (dx / dt) + K * x
where x is the displacement of the mass from its equilibrium position. To evaluate the static deflection when F = 12 N, we set F = 12 and solve for x:
12 = 3 * (d^2x / dt^2) + 20 * (dx / dt) + 800 * x
At equilibrium, the mass is not accelerating, so d^2x / dt^2 = 0 and dx / dt = 0. Therefore, we can simplify the equation to:
12 = 800 * x
Solving for x, we get:
x = 0.015 m
To evaluate the force needed to make the mass accelerate at 4 m/s² at the moment when the velocity is 0.5 m/s, we set a = 4 and v = 0.5, and solve for F:
F = M * a + k * v + K * x
Substituting the given values, we get:
F = 3 * 4 + 20 * 0.5 + 800 * x
We can use the static deflection we calculated earlier:
F = 3 * 4 + 20 * 0.5 + 800 * 0.015
Simplifying, we get:
F = 12.3 N