Concept explainers
Solve Prob. 10-30 using the Goodman-Zimmerli fatigue-failure criterion.
The design parameters for the spring.
Answer to Problem 31P
The specifications of the spring are A313 stainless steel wire.
The wire diameter for the spring is
The outer diameter for the spring is
The free length for the spring is
The total number of the coils for the spring is
Explanation of Solution
Write the expression for the amplitude of alternating component of force.
Here, the maximum load on the spring is
Write the expression for the midrange steady component of the force.
Here, the midrange steady component of the force is
Write the expression for ultimate tensile strength.
Here, the intercept constant is
Write the expression for the maximum allowable stresses for helical springs.
Here, the allowable yield stress for helical springs
Write the expression for the shearing ultimate strength.
Here, the shearing ultimate strength is
Write the expression for the slope of the load line using the goodman fatigue failure criterion.
Here, the load line slope is
Write the expression for the goodman ordinate intercept.
Here, the ordinate intercept for shear is
Write the expression for the amplitude component of the strength.
Write the expression for the back angle.
Here, the back angle is
Write the expression for the free end location angle.
Here, the free end location angle
Write the expression for the spring index.
Here, the spring index is
Write the expression for the mean coil diameter.
Here, the mean coil diameter is
Write the expression for the Bergstrasser factor to compensate the curvature effect.
Here, the Bergstrasser factor is
Write the expression for the alternating shear stress component.
Write the expression for the fatigue factor of the safety.
Here, the fatigue factor of the safety is
Write the expression for the number of the active coils.
Here, the number of the active coils is
Write the expression for the total number of the coils.
Here, the total number of the coils is
Write the expression for the maximum deflection of the spring.
Here, the maximum deflection of the spring is
Write the expression for the deflection under the steady load.
Here, the fractional overrun to closure is
Write the expression for the solid length of the spring.
Here, the solid length of the spring is
Write the expression for the free length of the spring.
Here, the free length of the spring is
Write the expression for the critical free length of the spring.
Here, the critical free length of the spring is
Write the expression for the shear force of the spring.
Here, the shear force of the spring is
Write the expression for the factor of the safety.
Here, the factor of the safety is
Write the expression for the frequency of the fundamental wave.
Here, the acceleration due to gravity is
Write the expression for the outer diameter of the spring.
Here, the outer diameter of the spring is
Conclusion:
Substitute
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Refer to table 10-4 “for estimating minimum tensile strength of the spring wires” to obtain the intercept and slope constants
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Refer to Zimmerli’s endurance data to obtain the amplitude component of the strength and mid range component of the strength as
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Refer to table 10-5 “Mechanical properties of some spring wires” to obtain the modulus of rigidity for A313 stainless wire as
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Since, the free length is lesser than the
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Since the steel is A313 stainless wire. Hence specific weight is
Substitute
Repeat all the steps for other values of the wire diameter. All the calculated values for other values of wire diameter are shown in below table.
The following table shows the first iteration.
1 | 0.080 | 0.0915 | 0.1055 | 0.1205 | |
2 | 0.146 | 0.146 | 0.263 | 0.263 | |
3 | 169 | 169 | 128 | 128 | |
4 | 244.363 | 239.618 | 231.257 | 223.311 | |
5 | 163.723 | 160.544 | 154.942 | 149.618 | |
6 | 85.5 | 83.86 | 80.94 | 78.15 | |
7 | 52.70 | 53.23 | 54.26 | 55.34 | |
8 | 43.40 | 43.56 | 43.63 | 43.69 | |
9 | 29 | 29.04 | 29.09 | 29.12 | |
10 | 2.75 | 2.12 | 1.60 | 1.22 | |
11 | 9.046 | 12.30 | 16.85 | 22.43 | |
12 | 0.723 | 1.126 | 1.778 | 2.703 | |
13 | 1.15 | 1.10 | 1.07 | 1.05 | |
14 | 29 | 29.04 | 29.09 | 29.12 | |
15 | 1.5 | 1.5 | 1.5 | 1.5 | |
16 | 14.26 | 6.45 | 2.89 | 1.40 | |
17 | 16.26 | 8.45 | 4.89 | 3.40 | |
18 | 1.3 | 0.774 | 0.51 | 0.41 | |
19 | 2.17 | 2.17 | 2.17 | 2.17 | |
20 | 4.38 | 3.64 | 3.39 | 3.28 | |
21 | 3.802 | 5.924 | 9.35 | 14.21 | |
22 | 85.74 | 85.87 | 86.02 | 86.13 | |
23 | 0.997 | 0.977 | 0.941 | 0.907 | |
24 | 141.05 | 145.55 | 149.93 | 152.96 | |
Since, the factor of safety lesser than one, Hence the design is not suitable.
Repeat all the steps for second iteration.
The following table shows the second iteration.
1 | 0.080 | 0.0915 | 0.1055 | 0.1205 | |
2 | 0.146 | 0.146 | 0.263 | 0.263 | |
3 | 169 | 169 | 128 | 128 | |
4 | 244.363 | 239.618 | 231.257 | 223.311 | |
5 | 163.723 | 160.544 | 154.942 | 149.618 | |
6 | 85.5 | 83.86 | 80.94 | 78.15 | |
7 | 52.70 | 53.23 | 54.26 | 55.34 | |
8 | 43.40 | 43.56 | 43.63 | 43.69 | |
9 | 21.75 | 21.78 | 21.81 | 21.84 | |
10 | 2.75 | 2.12 | 1.60 | 1.22 | |
11 | 6.995 | 8.86 | 12.29 | 16.48 | |
12 | 0.512 | 0.811 | 1.29 | 1.98 | |
13 | 1.22 | 1.15 | 1.10 | 1.07 | |
14 | 21.756 | 21.78 | 21.81 | 21.84 | |
15 | 2 | 2 | 2 | 2 | |
16 | 40.24 | 17.28 | 7.47 | 3.53 | |
17 | 42.24 | 19.28 | 9.47 | 5.53 | |
18 | 3.37 | 1.76 | 1.00 | 0.667 | |
19 | 2.17 | 2.17 | 2.17 | 2.17 | |
20 | 6.25 | 4.64 | 3.87 | 3.54 | |
21 | 2.69 | 4.26 | 6.82 | 10.44 | |
22 | 64.33 | 64.40 | 64.51 | 64.60 | |
23 | 1.32 | 1.30 | 1.25 | 1.21 | |
24 | 98.93 | 104.82 | 109.340 | 112.409 | |
Substitute
Thus, the outer diameter for the spring is
Thus, the specifications of the spring are A313 stainless steel wire.
The wire diameter for the spring is
The free length for the spring is
The total number of the coils for the spring is
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Chapter 10 Solutions
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