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A mass-spring-system has the following parameters. Stiffness K = 800 N/m Mass M = 3 kg Damping Coefficient k = 20 Ns/m Calculate the time constant, critical damping coefficient and the damping ratio. Derive the equation for the force required when the piston is accelerating. Use the equation to evaluate the static deflection when F = 12 N. Use the equation 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. ​

User KimKha
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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
User Insilenzio
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