Ultrasonic measurement of tie-bar stress for die-casting machine

Abstract

In die casting, the real-time measurement of the stress of the tie-bar helps ensure product quality and protect the machine itself. However, the traditional magnetic-attached strain gauge is installed in the mold and product operating area, which hinders the loading and unloading of the mold and the collection of die castings. In this paper, a method for real-time measurement of stress using ultrasonic technology is proposed. The stress variation of the tie-bar is analyzed, and a mathematical model between ultrasonic signal and stress based on acoustoelastic theory is established. Verification experiments show that the proposed method agrees with the strain gauge, and the maximum of the difference square is only 1.5678 (MPa)². Furthermore, single-factor experiments are conducted. A higher ultrasonic frequency produces a better measurement accuracy, and the mean of difference squares at 2.5 and 5 MHz are 2.3234 and 0.6733 (MPa)², respectively. Measurement accuracy is insensitive to probe location and tonnage of the die-casting machine. Moreover, the ultrasonic measurement method can be used to monitor clamping health status and inspect the dynamic pulling force of the tie-bar. This approach has the advantages of high precision, high repeatability, easy installation, and noninterference, which helps guide the production in die casting.

Graphical abstract

Keywords

die-casting / tie-bar stress / acoustoelastic theory / ultrasonic measurement / dynamic inspection

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Nomenclature

D	Ultrasonic probe diameter
E	Elastic modulus
f	Ultrasonic probe frequency
K	Acoustoelastic coefficient
K1	Material coefficient
l, m	The third-order elasticity coefficients
n	Number of experiments
Δt	Ultrasonic time difference
r	Standard deviation
rs	Standard deviation of strain gauge
ru	Standard deviation of ultrasonic
s0	Natural length of the tie-bar with no stress
s1	Length of the tie-bar with σ stress
t0	Ultrasonic propagation time with no stress
t1	Ultrasonic propagation time with σ stress
v	Ultrasonic wave speed
v0	Velocity with no stress
vσ	Velocity with σ stress
w	Ultrasonic wavelength
σ	Stress of the tie-bar
σi	Stress measured by the strain gauge
σj	Stress measured by the ultrasonic method
ε	Strain
ρ0	Density of the tie-bar
λ, μ	The second-order elastic coefficients
θ0	Half divergence angle of the ultrasound
\overline{\delta }	Difference square
{\overline{\delta }}_{\mathrm{a}\mathrm{v}\mathrm{e}}	Mean of the difference squares
{\overline{\delta }}_{max}	Maximum of the difference square

Appendix

Table A1 Ultrasonic measurement results of verification experiments

Pulling force/kN	{\sigma }_i/MPa	{\sigma }_j/MPa	\overline{\delta }/(MPa)2	{\overline{\delta }}_{max}/(MPa)2	r_{\text{s}}/MPa	r_{\text{u}}/MPa
100	6.23628	6.8461	1.5678	1.5678	0.0486	0
	6.30124	6.8461
	6.30124	6.8461
	6.23628	6.8461
	6.36620	6.8461
300	18.3840	19.1067	1.3876		0.0118	0
	18.6439	19.1067
	18.7088	19.1067
	18.6439	19.1067
	18.5789	19.1067
500	31.5711	31.9511	0.3181		0.0862	0
	31.8310	31.9511
	31.7011	31.9511
	31.7660	31.9511
	31.7011	31.9511
700	44.1736	44.2117	0.0234		0.0636	0
	44.1736	44.2117
	44.1087	44.2117
	44.1736	44.2117
	44.3035	44.2117
900	57.7505	57.6399	0.0696		0.0411	0
	57.7505	57.6399
	57.8155	57.6399
	57.7505	57.6399
	57.6855	57.6399

Table A2 Measurement results of ultrasonic and strain gauge of #3 tie-bar with different ultrasonic probe frequencies

Probe frequency/MHz	Pulling force/kN	{\sigma }_i/MPa	\mathrm{\Delta }t/ns	{\sigma }_j/MPa	\overline{\delta }/(MPa)2	{\overline{\delta }}_{\mathrm{a}\mathrm{v}\mathrm{e}}/(MPa)2
2.5	100	6.04139	104.948	6.2621	0.3094	2.3234
		5.97643	104.948	6.2621
		6.04139	104.948	6.2621
		6.04139	104.948	6.2621
		5.97643	104.948	6.2621
	300	19.5533	335.832	19.1062	1.1865
		19.6183	335.832	19.1062
		19.5533	335.832	19.1062
		19.6183	335.832	19.1062
		19.6183	335.832	19.1062
2.5	500	31.7011	566.717	31.9503	0.1754	2.3234
		31.7660	566.717	31.9503
		31.7011	566.717	31.9503
		31.8959	566.717	31.9503
		31.8310	566.717	31.9503
	700	45.4728	797.601	44.7945	2.0490
		45.4079	797.601	44.7945
		45.4079	797.601	44.7945
		45.4728	797.601	44.7945
		45.4079	797.601	44.7945
	900	57.5556	1007.496	56.4709	7.8967
		57.8155	1007.496	56.4709
		57.7505	1007.496	56.4709
		57.7505	1007.496	56.4709
		57.7505	1007.496	56.4709
5	100	6.23628	115.445	6.8461	1.5678	0.6733
		6.30124	115.445	6.8461
		6.30124	115.445	6.8461
		6.23628	115.445	6.8461
		6.36620	115.445	6.8461
	300	18.3840	335.840	19.1067	1.3876
		18.6439	335.840	19.1067
		18.7088	335.840	19.1067
		18.6439	335.840	19.1067
		18.5789	335.840	19.1067
	500	31.5711	566.730	31.9511	0.3181
		31.8310	566.730	31.9511
		31.7011	566.730	31.9511
		31.7660	566.730	31.9511
		31.7011	566.730	31.9511
	700	44.1736	787.125	44.2117	0.0234
		44.1736	787.125	44.2117
		44.1087	787.125	44.2117
		44.1736	787.125	44.2117
		44.3035	787.125	44.2117
	900	57.7505	1028.510	57.6399	0.0696
		57.7505	1028.510	57.6399
		57.8155	1028.510	57.6399
		57.7505	1028.510	57.6399
		57.6855	1028.510	57.6399

Table A3 Measurement results of ultrasonic and strain gauge of #3 tie-bar of different ultrasonic probe locations

Probe location	Pulling force/kN	{\sigma }_i/MPa	\mathrm{\Delta }t/ns	{\sigma }_j/MPa	\overline{\delta }/(MPa)2	{\overline{\delta }}_{\mathrm{a}\mathrm{v}\mathrm{e}}/(MPa)2
Near positioning hole	100	6.23628	115.445	6.8461	1.5678	0.6733
		6.30124	115.445	6.8461
		6.30124	115.445	6.8461
		6.23628	115.445	6.8461
		6.36620	115.445	6.8461
	300	18.3840	335.840	19.1067	1.3876
		18.6439	335.840	19.1067
		18.7088	335.840	19.1067
		18.6439	335.840	19.1067
		18.5789	335.840	19.1067
	500	31.5711	566.730	31.9511	0.3181
		31.8310	566.730	31.9511
		31.7011	566.730	31.9511
		31.7660	566.730	31.9511
		31.7011	566.730	31.9511
	700	44.1736	787.125	44.2117	0.0234
		44.1736	787.125	44.2117
		44.1087	787.125	44.2117
		44.1736	787.125	44.2117
		44.3035	787.125	44.2117
	900	57.7505	1028.510	57.6399	0.0696
		57.7505	1028.510	57.6399
		57.8155	1028.510	57.6399
		57.7505	1028.510	57.6399
		57.6855	1028.510	57.6399
Near radius	100	6.3662	115.445	6.3272	0.0220	0.0105
		6.4312	115.445	6.3272
		6.3012	115.445	6.3272
		6.3012	115.445	6.3272
		6.2363	115.445	6.3272
	300	19.5533	346.335	19.5533	0.0085
		19.6183	346.335	19.5533
		19.5533	346.335	19.5533
		19.5533	346.335	19.5533
		19.4883	346.335	19.5533
	500	31.7011	556.235	31.7141	0.0118
		31.7660	556.235	31.7141
		31.7660	556.235	31.7141
		31.7011	556.235	31.7141
		31.6361	556.235	31.7141
	700	45.4728	808.115	45.4469	0.0051
		45.4079	808.115	45.4469
		45.4079	808.115	45.4469
		45.4728	808.115	45.4469
		45.4728	808.115	45.4469
	900	58.4651	1039.005	58.4911	0.0051
		58.5300	1039.005	58.4911
		58.4651	1039.005	58.4911
		58.5300	1039.005	58.4911
		58.4651	1039.005	58.4911

Table A4 Measurement results of ultrasonic and strain gauge of #3 tie-bar with different tonnages of die-casting machine

Mechanical tonnage/t	Pulling force/kN	{\sigma }_i/MPa	\mathrm{\Delta }t/ns	{\sigma }_j/MPa	\overline{\delta }/(MPa)2	{\overline{\delta }}_{\text{ave}}/(MPa)2
400	100	6.3662	115.445	6.3272	0.0220	0.0105
		6.4312	115.445	6.3272
		6.3012	115.445	6.3272
		6.3012	115.445	6.3272
		6.2363	115.445	6.3272
	300	19.5533	346.335	19.5533	0.0085
		19.6183	346.335	19.5533
		19.5533	346.335	19.5533
		19.5533	346.335	19.5533
		19.4883	346.335	19.5533
	500	31.7011	556.235	31.7141	0.0118
		31.7660	556.235	31.7141
		31.7660	556.235	31.7141
		31.7011	556.235	31.7141
		31.6361	556.235	31.7141
	700	45.4728	808.115	45.4469	0.0051
		45.4079	808.115	45.4469
		45.4079	808.115	45.4469
		45.4728	808.115	45.4469
		45.4728	808.115	45.4469
	900	58.4651	1039.005	58.4911	0.0051
		58.5300	1039.005	58.4911
		58.4651	1039.005	58.4911
		58.5300	1039.005	58.4911
		58.4651	1039.005	58.4911
800	200	6.1017	298.8505	6.0664	0.0034	0.0332
		6.1369	298.8505	6.0664
		6.0311	298.8505	6.0664
		5.9606	298.8505	6.0664
		6.1017	298.8505	6.0664
	600	21.4793	1065.4670	21.7050	0.0326
		21.7262	1065.4670	21.7050
		21.8320	1065.4670	21.7050
		21.7262	1065.4670	21.7050
		21.7615	1065.4670	21.7050
	1000	34.6702	1721.6388	34.9453	0.0512
		35.0934	1721.6388	34.9453
		35.0229	1721.6388	34.9453
		34.9524	1721.6388	34.9453
		34.9876	1721.6388	34.9453
	1400	48.0022	2390.8040	48.1362	0.0434
		47.9317	2390.8040	48.1362
		48.0375	2390.8040	48.1362
		48.3802	2390.8040	48.1362
		48.3196	2390.8040	48.1362
	1800	64.4026	3196.4010	64.4802	0.0343
		64.6143	3196.4010	64.4802
		64.4379	3196.4010	64.4802
		64.5790	3196.4010	64.4802
		64.3674	3196.4010	64.4802

Acknowledgements

The authors acknowledge the financial support of the Key Project of Science and Technology Innovation 2025 of Ningbo City, China (Grant No. 2020Z018), the “Pioneer” and “Leading Goose” R&D Program of Zhejiang Province, China (Grant No. 2022C01069), the National Natural Science Foundation of China (Grant No. 51875519), and the Project of Innovation Enterprises Union of Ningbo City, China (Grant No. 2021H002).

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