1 Introduction
2 Introduction of HIVC
2.1 System Compositions
2.2 Mathematical Model of HIVC Force Control System
State vector | Input vector | Output vector | Parameter vector | |
---|---|---|---|---|
\(x_{1} = x_{\text{p}}\)
|
\(u_{1} = F_{r}\)
|
\(Y = F\)
|
\(\alpha_{1} = \omega\)
|
\(\alpha_{8} = A_{\text{p}}\)
|
\(x_{2} = \dot{x}_{\text{p}}\)
|
\(\alpha_{2} = \zeta\)
|
\(\alpha_{9} = \beta_{e}\)
| ||
\(x_{3} = x_{\text{v}}\)
|
\(\alpha_{3} = p_{s}\)
|
\(\alpha_{10} = m_{t}\)
| ||
\(x_{4} = \dot{x}_{\text{v}}\)
|
\(\alpha_{4} = p_{0}\)
|
\(\alpha_{11} = K_{f}\)
| ||
\(x_{5} = p_{\text{1}}\)
|
\(\alpha_{5} = C_{ip}\)
|
\(\alpha_{12} = K_{p}\)
| ||
\(x_{6} = p_{\text{2}}\)
|
\(\alpha_{6} = L\)
|
\(\alpha_{13} = K\)
| ||
\(\alpha_{7} = L_{0}\)
|
\(\alpha_{14} = B_{p}\)
|
Simulation model parameters | Initial value |
---|---|
Servo valve gain \(K_{{\text{axv}}}\) (mm/v) | 0.45 |
Natural frequency of servo valve \(\omega\) (rad/s) | 628 |
Damping ratio of servo valve \(\zeta\) | 0.77 |
Effective piston area of servo cylinder \(A_{\text{P}}\) (mm2) | 336.8 |
Volume of input oil pipe \(V_{{\text{g1}}}\) (mm3) | 620 |
Volume of output oil pipe \(V_{{\text{g2}}}\) (mm3) | 860 |
Total piston stroke of servo cylinder \(L\)(mm) | 50 |
Initial piston position of servo cylinder \(L_{0}\) (mm) | 20 |
System return oil pressure \(p_{\text{0}}\) (MPa) | 0.5 |
Proportional gain \(K_{\text{P}}\) | 8 |
Force sensor gain\(K_{\text{f}} \;(\text{V}/\text{kN})\) | 0.0769 |
External leakage coefficient \(C_{{\text{ep}}}\) (mm3·(s·MPa)− 1) | 0 |
Internal leakage coefficient \(C_{{\text{ip}}}\) (mm3·(s·MPa)− 1) | 238 |
Conversion mass \(m_{\text{t}} \;\text{(kg)}\) | 1.1315 |
Effective bulk modulus \(\beta_{\text{e}}\) MPa | 800 |
Load stiffness \(K\;(\text{kN}/\text{m})\) | 450 |
Viscous damping coefficient \(B_{\text{p}} \;(\text{k}N/(\text{m} \cdot \text{s}^{ - 1} \text{)})\) | 2 |
Conversion coefficient \(K_{\text{d}} \;(\text{m}{\text{m}}^{2} /\text{s)}\) | 124.8 |
3 MSA Mathematical Model
3.1 MSA Mathematical Common Model
3.2 MSA Mathematical Model of HIVC
4 Sensitivity Analysis of HIVC
4.1 Dynamic Sensitivity Analysis
4.1.1 Orthogonal Test Design
No. | System pressure Ps (MPa) | Frequency f (Hz) | Amplitude A (N) |
---|---|---|---|
1 | 6 | 1 | 500 |
2 | 6 | 2 | 1000 |
3 | 6 | 4 | 1500 |
4 | 12 | 1 | 1000 |
5 | 12 | 2 | 1500 |
6 | 12 | 4 | 500 |
7 | 18 | 1 | 1500 |
8 | 18 | 2 | 500 |
9 | 18 | 4 | 1000 |
4.1.2 Nine Working Conditions
Performance index | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | |
---|---|---|---|---|---|---|---|---|---|---|
Amplitude attenuation \(\varPhi\) (N) | ||||||||||
Experiment | 32.1 | 76.5 | 101.5 | 31.3 | 40.1 | 30.4 | 29.2 | 18.8 | 38.2 | |
Simulation | 9.5 | 41.7 | 185.9 | 12.1 | 31.2 | 18.6 | 14.2 | 6.1 | 27.1 | |
Phase lag \(\psi\) (°) | ||||||||||
Experiment | 8.7 | 13.8 | 24.5 | 5.2 | 8.9 | 14.4 | 3.9 | 8.1 | 17.9 | |
Simulation | 5.0 | 10.5 | 20.8 | 3.5 | 7.1 | 13.2 | 2.8 | 5.4 | 10.8 |
Performance indexes | Amplitude attenuation \(\varPhi\) (N) | Phase lag \(\psi\) (10‒2(°)) | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
P
s
|
f
|
A
|
P
s
|
f
|
A
| |||||||
Factors | Exp. | Sim. | Exp. | Sim. | Exp. | Sim. | Exp. | Sim. | Exp. | Sim. | Exp. | Sim. |
k
1
| 70.0 | 79.0 | 30.9 | 11.9 | 27.1 | 11.4 | 15.7 | 12.1 | 5.9 | 3.8 | 10.4 | 7.9 |
k
2
| 33.9 | 20.6 | 45.1 | 26.3 | 48.7 | 27.0 | 9.5 | 7.9 | 10.3 | 7.7 | 12.3 | 8.3 |
k
3
| 28.7 | 15.8 | 56.7 | 77.2 | 56.9 | 77.1 | 10.0 | 6.3 | 18.9 | 14.9 | 12.4 | 10.2 |
R
| 41.3 | 63.2 | 25.8 | 65.3 | 29.8 | 65.7 | 6.2 | 5.8 | 13.0 | 11.1 | 2.0 | 2.3 |
4.1.3 PSM
4.1.4 Time-history Comparative Curves of Force Change
4.2 Quantitative Sensitivity Analysis
4.2.1 Sensitivity Indexes
4.2.2 Sensitivity Indexes Histograms
4.2.3 Range Analysis
Parameters |
ω
|
\(\zeta\)
|
P
s
|
C
ip
| ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
Factors |
P
s
|
f
|
A
|
P
s
|
f
|
A
|
P
s
|
f
|
A
|
P
s
|
f
|
A
|
k
1
| 1.5 | 0.2 | 0.5 | − 1.6 | − 0.2 | − 0.5 | − 10.7 | − 0.9 | − 0.9 | 1.4 | 1.0 | 0.5 |
k
2
| 0.8 | 0.7 | 1.0 | − 0.8 | − 0.8 | − 1.1 | − 1.9 | − 3.0 | − 2.9 | 1.0 | 1.1 | 1.0 |
k
3
| 0.8 | 2.2 | 1.6 | − 0.8 | − 2.2 | − 1.6 | − 1.3 | − 10.1 | − 10.1 | 0.8 | 1.0 | 1.6 |
R
| 0.7 | 2.0 | 1.1 | 0.8 | 2.0 | 1.1 | 9.4 | 9.2 | 9.2 | 0.6 | 0.1 | 1.1 |
Parameters |
A
p
|
K
p
|
K
|
B
p
| ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
Factors |
P
s
|
f
|
A
|
P
s
|
f
|
A
|
P
s
|
f
|
A
|
P
s
|
f
|
A
|
k
1
| 0.7 | − 0.6 | 0.6 | − 8.5 | − 1.3 | − 1.5 | − 8.7 | − 0.5 | − 1.5 | 1.9 | 0.2 | 0.5 |
k
2
| 0.6 | 0.3 | 0.9 | − 2.6 | − 3.3 | − 3.5 | − 2.4 | − 2.8 | − 3.3 | 0.8 | 0.8 | 1.1 |
k
3
| 0.6 | 2.4 | 0.5 | − 2.1 | − 8.6 | − 8.2 | − 2 | − 9.7 | − 8.3 | 0.8 | 2.5 | 1.9 |
R
| 0.1 | 3.0 | 0.4 | 6.4 | 7.3 | 6.7 | 6.7 | 9.2 | 6.8 | 1.1 | 2.3 | 1.4 |
Parameters |
ω
|
\(\zeta\)
|
P
s
|
C
ip
| ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
Factors |
P
s
|
f
|
A
|
P
s
|
f
|
A
|
P
s
|
f
|
A
|
P
s
|
f
|
A
|
k
1
| − 5.5 | − 0.2 | − 1.6 | 6.8 | 0.3 | 4.5 | − 70.2 | − 20.2 | − 38.8 | − 0.4 | 0.2 | 0.1 |
k
2
| − 1.8 | − 1.2 | − 1.8 | 2.5 | 3.8 | 2.5 | − 40.2 | − 43.0 | − 44.9 | 0.1 | 0.1 | 0.1 |
k
3
| − 1.2 | − 7.0 | − 5.0 | 4.0 | 9.2 | 6.2 | − 31.2 | − 78.3 | − 57.9 | 0.1 | − 0.6 | − 0.4 |
R
| 4.3 | 6.8 | 3.4 | 4.3 | 8.9 | 3.7 | 39.0 | 58.1 | 19.1 | 0.5 | 0.8 | 0.5 |
Parameters |
A
p
|
K
p
|
K
|
B
p
| ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
Factors |
P
s
|
f
|
A
|
P
s
|
f
|
A
|
P
s
|
f
|
A
|
P
s
|
f
|
A
|
k
1
| 83.3 | 33.4 | 72.1 | − 94.5 | − 33.8 | − 69.5 | − 90.1 | − 32.6 | − 66.5 | − 2.9 | − 0.1 | − 1.7 |
k
2
| 70.2 | 63.6 | 69.8 | − 69.5 | − 66.8 | − 71.9 | − 66.5 | − 64.5 | − 69.0 | − 1.6 | − 0.5 | − 1.3 |
k
3
| 58.5 | 115.0 | 70.2 | − 56.6 | − 120.1 | − 79.3 | − 54.3 | − 113.8 | − 75.4 | − 1.2 | − 5 | − 2.6 |
R
| 24.8 | 81.6 | 2.3 | 37.9 | 86.3 | 9.8 | 35.8 | 81.2 | 8.9 | 1.7 | 4.9 | 1.3 |
5 Experiments
Sensitivity indexes | S1 (N) | S2 (10−2 (°)) | ||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Factors |
P
s
|
f
|
A
|
P
s
|
f
|
A
| ||||||||||||
Exp. | Sim. | Err. | Exp. | Sim. | Err. | Exp. | Sim. | Err. | Exp. | Sim. | Err. | Exp. | Sim. | Err. | Exp. | Sim. | Err. | |
k
1
| − 9.5 | − 10.7 | 1.2 | − 1.1 | − 0.9 | 0.2 | − 1.0 | − 0.9 | 0.1 | − 66.3 | − 70.2 | 3.9 | − 21.2 | − 20.2 | 1.0 | − 41.0 | − 38.8 | 3.2 |
k
2
| − 2.2 | − 1.9 | 0.3 | − 3.5 | − 3.0 | 0.5 | − 3.2 | − 2.9 | 0.3 | − 40.8 | − 40.2 | 0.6 | − 44.1 | − 43.0 | 1.1 | − 50.3 | − 44.9 | 5.4 |
k
3
| − 1.3 | − 1.3 | 0 | − 8.4 | − 10.1 | 1.7 | − 8.7 | − 10.1 | 1.4 | − 34.5 | − 31.2 | 3.3 | − 76.5 | − 78.3 | 1.8 | − 48.7 | − 57.9 | 9.2 |
R
| 8.2 | 9.4 | 1.2 | 7.3 | 9.2 | 1.9 | 7.7 | 9.2 | 1.5 | 31.8 | 39.0 | 7.2 | 55.3 | 58.1 | 2.8 | 7.7 | 19.1 | 12.1 |
Sensitivity indexes | S1 (N) | S2 (10−2 (°)) | ||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
P
s
|
f
|
A
|
P
s
|
f
|
A
| |||||||||||||
Factors | Exp. | Sim. | Err. | Exp. | Sim. | Err. | Exp. | Sim. | Err. | Exp. | Sim. | Err. | Exp. | Sim. | Err. | Exp. | Sim. | Err. |
k
1
| − 7.5 | −8.5 | 1.0 | − 1.5 | − 1.3 | 0.2 | − 1.9 | − 1.5 | 0.4 | − 92.1 | − 94.5 | 2.4 | − 41.7 | − 33.8 | 7.9 | − 71.8 | − 69.5 | 2.3 |
k
2
| − 2.7 | − 2.6 | 0.1 | − 3.7 | − 3.3 | 0.4 | − 4.0 | − 3.5 | 0.5 | − 70.5 | − 69.5 | 1.0 | − 75.4 | − 66.8 | 8.6 | − 74.8 | − 71.9 | 2.9 |
k
3
| − 2.6 | − 2.1 | 0.5 | − 7.8 | − 8.6 | 0.8 | − 7.0 | − 8.2 | 1.2 | − 57.9 | − 56.6 | 1.3 | − 105.8 | − 120.1 | 14.3 | − 75.4 | − 79.3 | 3.9 |
R
| 4.9 | 6.4 | 1.5 | 6.3 | 7.3 | 1.0 | 5.1 | 6.7 | 1.6 | 34.2 | 37.9 | 3.7 | 64.1 | 86.3 | 22.2 | 3.6 | 9.8 | 6.2 |