14 PSD Random Vibration Strength and Fatigue Analysis …
181
Mises stress is divided into three intervals according to a 3σ distribution, with each
interval corresponding to an occurrence probability of 68.3%, 95.4% and 99.7%.
As the stresses for the case greater than 3σ occur only 0.27% of the time, it is
assumed that they do not cause any damage.
From Miner’s law, the formula for the fatigue life under PSD random vibration
can be obtained as Eq. (14.3) [15]:
D =
n 1σ
N 1σ
+
n 2σ
N 2σ
+
n 3σ
N 3σ
(14.3)
In Eq. (14.3), D is the fatigue damage cycle ratio; n1σ, n2σ, n3σ is the actual
number of cycles under the corresponding 1σ, 2σ, 3σ level; N1σ, N2σ, N3σ is the
number of cycles under the corresponding 1σ, 2σ, 3σ solution, which can be found
in JB 4732. Since n 1σ = 0.6831v
+
0 T ,n 2σ = 0.272v
+
0 T ,n 3σ = 0.4331v
+
0 T [16], it is
easy to transform Eq. (14.3) into Eq. (14.4):
D =
0.6831v
+
0 T
N 1σ
+
0.272v
+
0 T
N 2σ
+
0.04331v
+
0 T
N 3σ
(14.4)
In Eq. (14.3), v
+
0 is the average frequency of vibration, Hz; T is the expected life
of the structure, s.
Based on relevant requirement, assuming that the value of expected life T is
1.08 × 10
4 s, the average frequency v
+
0 is 5 Hz, and the maximum stresses at 1σ,
2σ and 3σ are respectively 93.256 MPa, 186.51 MPa and 279.77 MPa. Based on JB
4732 C-1, the corresponding N1σ, N2σ, and N3σ can be calculated as 4.96 × 10
5 ,
3.06 × 10
4 and 8.23 × 10
3 , respectively.
Substituting these values into Eq. (14.4) yields Eqs. (14.5) and (14.6):
D =
0.6831
4.96 × 10 5 +
0.272
3.06 × 10 4 +
0.04331
8.23 × 10 3
× 1.08 × 10
4
× 5
(14.5)
D = 0.834 < 1
(14.6)
Therefore, the structure can meet the fatigue design requirements.
14.6 Conclusions
In this paper, finite element simulation was performed to analyze the strength of
a tube trailer under PSD stochastic excitation. The main conclusions are drawn as
follows.
(1) The first 9 orders natural frequencies are obtained from modal analysis and only
the first natural frequency is covered by the speed limitation of 80 km/h for the
181
Mises stress is divided into three intervals according to a 3σ distribution, with each
interval corresponding to an occurrence probability of 68.3%, 95.4% and 99.7%.
As the stresses for the case greater than 3σ occur only 0.27% of the time, it is
assumed that they do not cause any damage.
From Miner’s law, the formula for the fatigue life under PSD random vibration
can be obtained as Eq. (14.3) [15]:
D =
n 1σ
N 1σ
+
n 2σ
N 2σ
+
n 3σ
N 3σ
(14.3)
In Eq. (14.3), D is the fatigue damage cycle ratio; n1σ, n2σ, n3σ is the actual
number of cycles under the corresponding 1σ, 2σ, 3σ level; N1σ, N2σ, N3σ is the
number of cycles under the corresponding 1σ, 2σ, 3σ solution, which can be found
in JB 4732. Since n 1σ = 0.6831v
+
0 T ,n 2σ = 0.272v
+
0 T ,n 3σ = 0.4331v
+
0 T [16], it is
easy to transform Eq. (14.3) into Eq. (14.4):
D =
0.6831v
+
0 T
N 1σ
+
0.272v
+
0 T
N 2σ
+
0.04331v
+
0 T
N 3σ
(14.4)
In Eq. (14.3), v
+
0 is the average frequency of vibration, Hz; T is the expected life
of the structure, s.
Based on relevant requirement, assuming that the value of expected life T is
1.08 × 10
4 s, the average frequency v
+
0 is 5 Hz, and the maximum stresses at 1σ,
2σ and 3σ are respectively 93.256 MPa, 186.51 MPa and 279.77 MPa. Based on JB
4732 C-1, the corresponding N1σ, N2σ, and N3σ can be calculated as 4.96 × 10
5 ,
3.06 × 10
4 and 8.23 × 10
3 , respectively.
Substituting these values into Eq. (14.4) yields Eqs. (14.5) and (14.6):
D =
0.6831
4.96 × 10 5 +
0.272
3.06 × 10 4 +
0.04331
8.23 × 10 3
× 1.08 × 10
4
× 5
(14.5)
D = 0.834 < 1
(14.6)
Therefore, the structure can meet the fatigue design requirements.
14.6 Conclusions
In this paper, finite element simulation was performed to analyze the strength of
a tube trailer under PSD stochastic excitation. The main conclusions are drawn as
follows.
(1) The first 9 orders natural frequencies are obtained from modal analysis and only
the first natural frequency is covered by the speed limitation of 80 km/h for the