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

BARAN, Robert; MICHALCZYK, Krzysztof; WARZECHA, Mariusz

doi:10.2478/ama-2025-0055

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 na depending on the contact length per one side of spring s for a) C = 5, b) C = 7

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, na = 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

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 na 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

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 na 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

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

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

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

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 RrelF — RrelA

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

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

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

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

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

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%

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

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

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%

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

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

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

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

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

Figures & Tables

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

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

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

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

Comparison of experimental data with numerical results_¥

Results of Young’s Modulus E measurement

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

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

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

The approximation results with fixed values of coefficients

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

Paradigm

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