An engineer is forced by geometric considerations to apply the torque on the spring of Prob. 4–3 at the location x = 0.4l. For a uniform-diameter spring, this would cause one leg of the span to be underutilized when both legs have the same diameter. For optimal design the diameter of each leg should be designed such that the maximum shear stress in each leg is the same. This problem is to redesign the spring of part (b) of Prob. 4–3. Using x = 0.4l, l = 10 in, T = 1500 lbf · in, and G = 11.5 Mpsi, design the spring such that the maximum shear stresses in each leg are equal and the spring has the same spring rate (angle of twist) as part (b) of Prob. 4–3. Specify d1, d2, the spring rate k, and the torque and the maximum shear stress in each leg.
The diameter
The diameter
The spring rate
The individual torque in each leg.
The maximum stress in each leg.
Answer to Problem 4P
The diameter
The diameter
The spring rate is
The torque in the left leg is
The torque in the right leg is
The maximum stress in the left leg is
The maximum stress in the right leg is
Explanation of Solution
Write the expression for the angular displacement of the torsional bar for the distance
Here, the torque in the left leg is
Write the expression for the polar moment of inertia of the torsional bar for the distance
Here, diameter of the spring in the portion
Substitute
Write the expression for the angular displacement of the torsional bar for the distance
Here, the torque in the right leg is
Write the expression for the polar moment of inertia of the torsional bar for the distance
Here, diameter of the spring in the portion
Substitute
Write the expression for maximum shear stress at section of length
Here, the shear stress is
Write the expression for maximum shear stress at section of length
Here, the shear stress is
Since the spring is under uniform stress, so the shear stress in both the legs are considered as equal, that is
Substitute
Write the expression of the net torque on the system.
Here, the net torque is
Write the expression for net angular deflection in the bar.
Here, the length of the shaft is
Since the spring has uniform deflection throughout, so the deflection in the left leg is equal to the overall deflection.
Substitute
Since the spring has uniform deflection throughout, so the deflection in the right leg is equal to the overall deflection.
Substitute
Since the spring has uniform deflection throughout, so the deflection in the right leg is equal to the deflection in the left leg. That is,
Substitute
Write the stiffness of the combine system.
Here,
Conclusion:
Substitute
Substitute
Substitute
Divide Equation (VI) and Equation (XVIII).
Substitute
Substitute
Substitute
Thus, the diameter
Substitute
Thus, the diameter
Substitute
Thus, the torque in the left leg is
Substitute
Thus, the torque in the right leg is
Substitute the value
Substitute the value
Thus, the spring rate is
Substitute the value
Thus, the maximum stress in the left leg is
Substitute the value
Thus, the maximum stress in the right leg is
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