Effect of the End Coil Shape of the Helical Compression Spring on Its Stiffness and Distribution of Transverse Reactions During Axial Loading

Robert BARAN; Krzysztof MICHALCZYK; Mariusz WARZECHA

doi:10.2478/ama-2025-0055

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Effect of the End Coil Shape of the Helical Compression Spring on Its Stiffness and Distribution of Transverse Reactions During Axial Loading

Acta Mechanica et Automatica

Volume 19 (2025): Issue 3 (September 2025)

By: Robert BARAN , Krzysztof MICHALCZYK and Mariusz WARZECHA

Open Access

|Sep 2025

Figures & Tables

The testing machine used to spring axial stiffness researches

Samples of springs (a) used to axial stiffness researches; (b) graphical representation of contact length for two example lengths s = 0 and s = 0.25

Axial stiffness k for different numbers of active coils *n_a* depending on the contact length per one side of spring s for a) C = 5, b) C = 7

Tab. 1.

Results of Young’s Modulus E measurement

Wire number	Initial wire length L₀ [mm]	Elongation L₁ [mm]	Elongation L₂ [mm]	Stress σ₁ [MPa]	Stress σ₂ [MPa]	Young's modulus [MPa]
1	60.23	0.031	0.207	120	720	205330
2	60.82	0.031	0.211	120	720	202733
Average Young's modulus E						204031

Kirchhoff's transverse elasticity modulus G measurement station

Measurement of the friction coefficient for steel-steel friction pair

Spring with parameters C = 5, d = 5 mm, *n_a* = 2.5, s = 0.25 with supports a) modeled in the Design Modeler module of Ansys Workbench software, b) mesh used to stiffness calculation modeled in Static Structural module

Tab. 2.

Comparison of experimental data with numerical results.¥

Spring index C	Active coils n_a	Contact length per ending s	Experimental stiffness k [N/mm]	FEM stiffness k_FEM [N/mm]	Error [%]	Data range [%]	MAPE [%]
5	2.5	0	140.8	143.6	-2.0	8.7	3.6
		0.25	155.9	148.5	6.7
		0.5	142.2	139.0	2.2
	3	0	128.0	121.4	5.1	2.2	3.7
		0.25	130.0	126.0	3.1
		0.5	121.2	117.7	2.9
7	2.5	0	51.4	51.8	-0.8	6.1	2.3
		0.25	51.6	49.9	3.3
		0.5	49.5	50.9	-2.8
	3	0	45.2	44.3	1.9	1.6	1.2
		0.25	43.9	43.8	0.4
		0.5	45.0	44.3	1.4

Force-displacement curve for a spring with C = 4, na = 1, γ = 10° for coil contact length s = 0.5

Correlation plot of axial stiffness k versus spring index C with coefficient r = -0.81

Correlation plot of axial stiffness k versus number of active coils *n_a* with coefficient r = -0.29

Correlation plot of axial stiffness k versus helix angle γ with coefficient r = -0.02

Correlation plot of axial stiffness k versus contact length per one ending s with coefficient r = -0.0002

Tab. 3.

Coefficient of variation of springs axial stiffness depending on spring angle, number of active coils and spring index

Spring index C	Active coils n_a	Helix angle γ
Spring index C	Active coils n_a	5°	10°	15°
4	1	1.12%	0.25%	0.40%
	1.25	1.38%	0.62%	0.30%
	1.75	1.17%	0.30%	0.59%
	2.5	0.49%	0.51%	0.21%
	3.5	2.82%	0.20%	0.12%
8	1	0.23%	0.33%	0.13%
	1.25	0.81%	0.04%	0.22%
	1.75	0.64%	0.36%	0.31%
	2.5	0.34%	0.28%	0.10%
	3.5	0.14%	0.13%	0.17%
12	1	0.48%	0.42%	0.30%
	1.25	0.29%	0.33%	0.38%
	1.75	0.35%	0.17%	0.92%
	2.5	0.35%	0.20%	0.13%
	3.5	0.13%	0.22%	0.22%

Axial stiffness for index C = 8, point contact (s = 0) for spring angles of 5, 10 and 15 degrees

Axial stiffness distribution depending on the spring index C and the number of active coils *n_a* for the point contact of end coils

Axial stiffness k for the index C = 8, angle γ = 5°, point contact (s = 0) in comparison with selected analytical methods

Tab. 4.

The values of coefficient a and b of equation (13) depending on the spring index C and helix angle γ together with the given R-square parameter

C	γ [°]	a	b	R²
4	5	4.550	0.6462	0.8685
	10	4.328	0.5942	0.9889
	15	3.946	0.4576	0.9917
8	5	4.409	0.5350	0.9164
	10	4.168	0.4938	0.9726
	15	3.701	0.3970	0.9961
12	5	4.393	0.5183	0.9021
	10	4.055	0.4600	0.9751
	15	3.587	0.3937	0.9910

Tab. 5.

The values of coefficient k1 k2, m1, m2 of equations (14) and (15) depending on the spring index with the given R-square parameter

C	k₁	k₂	_R2	m₁	m₂	R₂
4	-0.060	4.879	0.9771	-0.019	0.755	0.9371
8	-0.071	4.801	0.9672	-0.014	0.613	0.9487
12	-0.081	4.818	0.9914	-0.012	0.582	0.9986
Av.	-0.071	4.833	0.9786	-0.015	0.650	0.9615

Tab. 6.

Comparison of parameters a and b calculated using equations (16) and (17) with the target values

C	γ [°]	a	a (16)	b	b (17)
4	5	4.550	4.478	0.6462	0.575
	10	4.328	4.123	0.5942	0.5
	15	3.946	3.768	0.4576	0.425
8	5	4.409	4.478	0.5350	0.575
	10	4.168	4.123	0.4938	0.5
	15	3.701	3.768	0.3970	0.425
12	5	4.393	4.478	0.5183	0.575
	10	4.055	4.123	0.4600	0.5
	15	3.587	3.768	0.3937	0.425
		R²	0.9786	R²	0.9615

Tab. 7.

Comparison of the accuracy of selected methods for calculating the axial stiffness of compression springs

Axial stiffness formula	Mean absolute percentage error (MAPE)
EN 13906-1:2013(E) norm (1)	23.40%
Vogt (1934) (2)	5.91%
Paredes (2016) (2)	3.62%
Krużelecki and Życzkowski (1990) (3)	22.17%
Yıldırım (2016) (4)	18.95%
Liu and Kim (2009) (5)	13.93%
kχ (18)	1.38%

Tab. 8.

Comparison of axial stiffness values calculated using the most accurate methods and the commonly used Eq. (1) with bench test results

Spring index C	Active coils n_a	Contacting coils per ending s	Experimental mean stiffness [N/mm]	k_N (1)	kχ (18)	k_P (2)
Spring index C	Active coils n_a	Contacting coils per ending s	Experimental mean stiffness [N/mm]	Error to experimental value [%]
5	2.5	0	139.9	-15	1	-1
		0.25	156.3	-3	12	10
		0.5	142.0	-13	3	1
	2.75	0	138.4	-6	8	6
		0.25	140.5	-4	10	7
		0.5	130.8	-12	3	1
	3	0	129.4	-4	10	7
		0.25	130.0	-3	10	8
		0.5	122.7	-9	5	2
MAPE [%]				8	7	5
Gap [%]				12	10	11
7	2.5	0	51.2	-15	1	-1
		0.25	51.3	-14	1	0
		0.5	49.2	-19	-4	-5
	2.75	0	48.1	-11	3	2
		0.25	47.3	-13	1	0
		0.5	47.1	-13	1	-1
	3	0	45.5	-8	5	4
		0.25	44.1	-11	2	1
		0.5	45.0	-9	4	3
MAPE [%]				13	2	2
Gap [%]				12	9	8

Plots of the *R_rel* dependency on the contact length and the helix pitch angle for springs with a given spring index C and number of active coils *n_a*

Plots of the *R_rel* dependency on the number of active coils *n_a* for helix pitch angles γ: 5°, 10°, 15°, for springs with C = 8 oraz s = 0

Preliminary approximation plots of *R_rel* as a function of the number of active coils *n_a* for helix pitch angles of 5° 10° and 15°, for springs with C = s and = 0

Tab. 9.

The values of the function coefficients Eq. (20), along with the coefficients of determination

γ [°]	c₁	c₂	c₃	c₄	R-square
5	0.0685	-0.1508	2.7048	1.2563	0.9862
10	0.1660	-0.3381	2.1928	0.8522	0.9853
15	0.2731	-0.5805	2.1819	0.9409	0.9948

Tab. 10.

The approximation results with fixed values of coefficients

γ [°]	c₁	c₂	c₃	c₄	R-square
5	0.0787	-0.1529	2.3598	1.0165	0.9760
10	0.1541	-0.3385			0.9796
15	0.2502	-0.5745			0.9934

Approximation plots of coefficients c₁ and c₂ along with the data for springs with C = s and = 0

Distribution of the number of particular cases with specific values of difference *R_relF* — *R_relA*

The coordinate system used to determine the transverse reaction angle *R_rel*. Top view of the spring

Tab. 11.

Results of the transverse reaction angle ψ test due to axial compression – basic 225 results

Active coils n_a	Contact length s	Angle of reaction ψ [°]	Active coils n_a	Contact length s	Angle of reaction ψ [°]
1	0	90	2.25	0	315
1	0.25	180	2.5	0	0
1	0.5	270	2.5	0.25	90
1	1	90	2.5	0.5	180
1.25	0	315	2.5	1	0
1.25	0.25	45	2.75	0	45
1.25	0.5	135	3	0	90
1.25	1	315	3.25	0	315
1.5	0	0	3.5	0	0
1.5	0.25	90	3.5	0.25	90
1.5	0.5	180	3.5	0.5	180
1.5	1	0	3.5	1	0
1.75	0	45	3.75	0	45
1.75	0.25	135	4	0	90
1.75	0.5	225	4.25	0	315
1.75	1	45	4.5	0	0
2	0	90	4.75	0	45

Graph of functions describing the dependence of the transverse reaction angle ψ on the contact length s and the partial number of active coils *n_p*

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DOI: https://doi.org/10.2478/ama-2025-0055 | Journal eISSN: 2300-5319 | Journal ISSN: 1898-4088

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Language: English

Page range: 471 - 484

Submitted on: Feb 10, 2025

Accepted on: Aug 13, 2025

Published on: Sep 30, 2025

Published by: Bialystok University of Technology

In partnership with: Paradigm Publishing Services

Keywords:

helical compression spring,

Related subjects:

Mechanical engineering,

Mechanics,

Bioengineering and biomedical engineering,

Biomechanics,

Civil engineering,

Environmental engineering

© 2025 Robert BARAN, Krzysztof MICHALCZYK, Mariusz WARZECHA, published by Bialystok University of Technology
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Volume 19 (2025): Issue 3 (September 2025)

Effect of the End Coil Shape of the Helical Compression Spring on Its Stiffness and Distribution of Transverse Reactions During Axial Loading

Figures & Tables

Fig. 1.

Fig. 2.

Fig. 3.

Fig. 4.

Tab. 1.

Fig. 5.

Fig. 6.

Fig. 7.

Tab. 2.

Fig. 8.

Fig. 9.

Fig. 10.

Fig. 11.

Fig. 12.

Tab. 3.

Fig. 13.

Fig. 14.

Fig. 15.

Tab. 4.

Tab. 5.

Tab. 6.

Tab. 7.

Tab. 8.

Fig. 16.

Fig. 17.

Fig. 18.

Tab. 9.

Tab. 10.

Fig. 19.

Fig. 20.

Fig. 21.

Tab. 11.

Fig. 22.

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