VM9

VM9
Large Lateral Deflection of Unequal Stiffness Springs

Overview

Reference:

G. N. Vanderplaats, Numerical Optimization Techniques for Engineering Design with Applications, McGraw-Hill Book Co., Inc., New York, NY, 1984, pp. 72-73, ex. 3-1.

Analysis Type(s):

Nonlinear Transient Dynamic Analysis (ANTYPE = 4)

Element Type(s):

Spring-Damper Elements (COMBIN14)

Combination Elements (COMBIN40)

Input Listing:

vm9.dat

Test Case

A two-spring system is subjected to a force F as shown below. Determine the strain energy of the system and the displacements δ_x and δ_y.

Figure 12: Unequal Stiffness Springs Problem Sketch

Geometric Properties

= 10 cm

k₁ = 8 N/cm

k₂ = 1 N/cm

m = 1

Analysis Assumptions and Modeling Notes

The solution to this problem is best obtained by adding mass and using the "slow dynamics" technique with approximately critical damping. Combination elements (COMBIN40) are used to provide damping in the X and Y directions. Approximate damping coefficients c_x and c_y, in the x and y directions respectively, are determined from

where m is arbitrarily assumed to be unity. k_x and k_y cannot be known before solving so are approximated by k_y = k₂ = 1 N/cm and k_x= k_y/2 = 0.5 N/cm, hence c_x = 1.41 and c_y = 2.0. Large deflection analysis is performed due to the fact that the resistance to the load is a function of the deformed position. POST1 is used to extract results from the solution phase.

Results Comparison

	Target	Mechanical APDL	Ratio
Strain Energy, N-cm	24.01	24.011	1.000
Deflection_x , cm	8.631	8.632	1.000
Deflection_y , cm	4.533	4.533	1.000