VM209

VM209
Static Analysis of Double Bellows Air Spring

Overview

Reference:

Dale T.Berry, Henry T.Y.Yang, Formulation and Experimental Verification of a Pneumatic Finite Element, International Journal For Numerical Methods in Engineering, Vol.39, pp 1097-1114 (1996)

Analysis Type(s):

Static Analysis (ANTYPE = 0)

Element Type(s):

2-Node Axisymmetric Shell Element (SHELL208)

2D Hydrostatic Fluid element (HSFLD241)

2D Smeared Reinforcing Element (REINF263)

Input Listing:

vm209.dat

Test Case

A double bellows-type air spring made of two rubber membranes shaped like toroids (radius Rt, angle of rotation θ degrees) is attached to a rigid plate (height h, radius R) at each end and connected in the middle by a rigid ring (height 2h, radius R). The rubber membrane is reinforced by polyester cords (cross sectional area A) spaced length w apart. The total air spring load with respect to height is measured at pressure loads 20, 40 and 60 PSI.

Figure 327: Multiple Species Flow Problem Sketch

Material Properties	Geometric Properties	Loading
E_rubber = 1000 psi Nu_rubber = 0.49 Density_gas = 4.4256e-5 lb-sec²/in⁴ E_polyester = 40000 psi Nu_polyster = 0.37 E_plate = 3.0467e7 psi Nu_plate = 0.3	Rt = 2.2 inches R = 4 inches θ = 90 h = 0.2 inches A = 1.96e-3 in2 w = 0.05 inches	T = 20 degree Celsius P = 20, 40,60 psi X = 1.5 inches

Material Properties

Geometric Properties

E_rubber = 1000 psi

Nu_rubber = 0.49

Density_gas = 4.4256e-5 lb-sec²/in⁴

E_polyester = 40000 psi

Nu_polyster = 0.37

E_plate = 3.0467e7 psi

Nu_plate = 0.3

Rt = 2.2 inches

R = 4 inches

θ = 90

h = 0.2 inches

A = 1.96e-3 in2

w = 0.05 inches

T = 20 degree Celsius

P = 20, 40,60 psi

X = 1.5 inches

Table 3: PVDATA points for Fluid material model

PVDATA Points (absolute pressure, volume)
20 psi	40 psi	60 psi
(34.7, 238.931)	(34.7, 377.9675418)	(34.7, 517.9960634)
(44.7, 185.4788747)	(44.7, 293.4110447)	(44.7, 402.7732752)
(54.7, 151.5704881)	(54.7, 239.771)	(54.7, 328.6007934)
(74.7, 110.9893668)	(74.7, 175.5752838)	(74.7, 240.622)
(94.7, 87.54916262)	(94.7, 138.4949704)	(84.7, 212.2132633)
(414.7, 19.99253846)	(114.7, 114.345891)	(94.7, 189.8042598)
(1014.7, 8.170795013)	(314.7, 41.67611598)	(104.7, 171.6758682)
	(414.7, 31.62641355)	(164.7, 109.1345683)
	(514.7, 25.48178298)	(214.7, 83.71897252)
	(714.7, 18.35101959)	(264.7, 67.9050374)
	(914.7, 14.3385522)	(314.7, 57.11618494)
	(1014.7, 12.9254693)	(364.7, 49.28561393)
		(414.7, 43.3432925)
		(464.7, 38.67971465)
		(514.7, 34.92221372)
		(564.7, 31.8301105)
		(614.7, 29.24103367)
		(664.7, 27.04146743)
		(714.7, 25.14966196)
		(764.7, 23.50524833)
		(814.7, 22.06267755)
		(864.7, 20.78693852)
		(914.7, 19.65066514)
		(964.7, 18.63217933)
		(1014.7, 17.71406662)

Figure 328: Finite Element Model of Problem

Analysis Assumptions and Modeling Notes

Analysis is performed using both GAS and PVDATA options. Smeared reinforcing element REINF263 is used with a base element to provide evenly spaced reinforcing fibers. A one-quarter axisymmetric (HSFLD241 keyopt (3) =1) model is brought to a reference temperature of T degrees Celsius and subjected to a degree of freedom inflation pressure of P applied at the pressure node. Then, the top rigid plate is lowered a distance X. The CNVTOL command is used to set convergence values for the nonlinear analysis.

In the attached input file (vm209.dat), we have only solved for pressure 20 PSI. Modify the D command and use the correct material number (PV points) to solve for pressure 40 and 60 PSI (D, 1, HDSP, 40/Material number 5 and D, 1, HDSDP, 60/Material number 6)

Results Comparison

Table 4: 20 PSI Applied-Gas

UY (in)	Target Compressive Load (lbs)	Mechanical APDL	Ratio
.25	1231	1226.55	0.996
.50	1692	1609.80	0.951
.75	2230	2050.35	0.919
1.00	2769	2581.33	0.932
1.25	3384	3246.09	0.959
1.5	4230	4109.82	0.972

Table 5: 40 PSI Applied-Gas

UY (in)	Target Compressive Load (lbs)	Mechanical APDL	Ratio
.25	2640	2405.88	0.911
.50	3350	3089.12	0.922
.75	4050	3849.41	0.950
1.00	5000	4738.90	0.948
1.25	6000	5825.32	0.971
1.5	7333	7209.64	0.983

Table 6: 60 PSI Applied-Gas

UY (in)	Target Compressive Load (lbs)	Mechanical APDL	Ratio
.25	3875	3596.30	0.928
.50	4650	4578.78	0.985
.75	6000	5658.43	0.943
1.00	7200	6906.07	0.959
1.25	8750	8413.20	0.962

Table 7: 20 PSI Applied-PVDATA

UY (in)	Target Compressive Load (lbs)	Mechanical APDL	Ratio
.25	1231	1249.79	1.015
.50	1692	1650.75	0.976
.75	2230	2077.76	0.932
1.00	2769	2613.12	0.944
1.25	3384	3263.09	0.964
1.5	4230	4238.18	1.002

Table 8: 40 PSI Applied-PVDATA

UY (in)	Target Compressive Load (lbs)	Mechanical APDL	Ratio
.25	2640	2455.05	0.930
.50	3350	3185.58	0.951
.75	4050	3950.18	0.975
1.00	5000	4742.42	0.948
1.25	6000	5923.76	0.987
1.5	7333	7251.99	0.989

Table 9: 60 PSI Applied-PVDATA

UY (in)	Target Compressive Load (lbs)	Mechanical APDL	Ratio
.25	3875	3613.49	0.933
.50	4650	4580.66	0.985
.75	6000	5674.34	0.946
1.00	7200	6922.04	0.961
1.25	8750	8780.85	1.004

Note: Results Comparison chart displacements have been scaled by a factor of 4 due to quarter symmetry.

Figure 329: Results Using GAS Option

Figure 330: Results Using PVDATA Option