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16. Rotating spring. Reporter: Reza M. Namin. The problem. A helical spring is rotated about one of its ends around a vertical axis. Investigate the expansion of the spring with and without an additional mass attached to it’s free end. Main approach. Theory Background Theory base

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## Rotating spring

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**16**Rotating spring Reporter: Reza M. Namin**The problem**• A helical spring is rotated about one of its ends around a vertical axis. • Investigate the expansion of the spring with and without an additional mass attached to it’s free end.**Main approach**• Theory • Background • Theory base • Developing the equations • Numerical solution • Experiment • Setup • Parameters, results and comparison • Conclusion**Theory - Background**• Act of a spring due to tensile force: • Hook's law: F = k ∆L • F: Force parallel to the spring • k: Spring constant • ∆L: Change of length • A spring divided to n parts: • F = n k ∆L • μ = kL remains constant • Circular motion • a = rω2 • a: Acceleration • r: Distance from the rotating axis • ω: Angular velocity**Theory - Base**• Effective parameters: • ω: Angular velocity • λ: Spring liner density = m / l • M: Additional mass • μ : Spring module = k l • l, l1, l2: Spring geometrical properties l1 l2 l M**Theory - Base**• Looking for the stable condition in the rotating coordinate system • Accelerated system → figurative force • Acting forces: • Gravity • Spring tensile force • Centrifugal force**Theory – Developing the equations**• Approximation in mass attached conditions: • Considering the spring to be weightless: Fs ω Fc Mg y l M x**Theory – Developing the equations**• Exact theoretical description: • Problem: The tension is not even all over the spring… • Solution: Considering the spring to be consisted of severalsmall springs. M**Theory – Numerical solution**• Numerical method • Finite-volume approximation: • Converting the continuous medium into a discrete medium • Transient (dynamic unsteady) method • Programming developed with QB. M Ti-1 fc Ti+1 w**Theory – Numerical solution**Mesh independency check n: Number of mesh points As n increases, the result will approach to the correct answer**Theory – Numerical solution**Tension in different points of the spring with different additional mass amounts:**Experiment**• Finding spring properties • Direct measurement: Mass & lengths • Suspending weights with the spring to measure k and μ • Changing the angular velocity, measuring the expansion • Change of the angular velocity with different voltages • Measuring the angular velocity with Tachometer • Measuring the length of the rotating spring using a high exposure time photo**Experiment setup**The motor, connection to the spring and the sensor sticker**Experiment setup**The rotating spring and tachometer**Experiment setup**Hold and base**Experiment setup**All we had on the table**Experiments**Suspending weights with the spring Finding k and using that to find μ →K = 33.78 N/m →μ = K l = 1.824 N**Experiments**Expansion increases with increasing angular velocity**Experiments**Measurement of length in different angular velocities Comparison with the numerical theory**Experiments**Comparing the shape of the rotating spring in theory and experiment λ=0.103 kg/m μ =0.369 N l = 16.3 cm l1 =1 cm ω = 120 RPM**Experiments**Investigation of the l-ω plot within different initial lengths**Experiments**Comparison between the physical experiments, numerical results and theoretical approximation within different additional masses**Conclusion**• According to the comparison between the theories and experiments we can conclude: • In case of weightless spring approximation:**Conclusion**• In general, the numerical method may be used to achieve precise description and evaluation. • Some of the results of the numerical method are as follows:**Conclusion**Numerical solution results Change of the spring hardness**Conclusion**Numerical solution results Change of spring density μ =0.3 N l = 10 cm l1 =1 cm**Conclusion**Numerical solution result Change of initial length λ=0.2 kg/m μ =0.3 N l1 =1 cm**Conclusion**Numerical solution results Change in additional mass

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