Overview

Test Case

This test case models a system composed of two masses and two springs, illustrated in Figure 188. Both masses are 2 kg, the stiffness of spring 1 is 6 N/m, and the stiffness of spring 2 is 16 N/m. Mass 1 is subjected to a pulse load F(t) with amplitude F₀ of 50 N and duration t_d of 1.8 s. The objective is to validate the displacement profile of each mass. The spring length is arbitrarily defined as 1.0 m.

This problem is also presented in test case VM182 in the Mechanical APDL Verification Manual.

Figure 188: Schematic of the test case, including domain geometry, main dimensions, and boundary conditions

The following table lists the main parameters of the test case, which follow the International System of Units (length in m, time in s, mass in kg, force in N, and pressure in Pa).

Analysis Assumptions and Modeling Notes

Figure 189: Model setup in LS-DYNA application of the 1D dynamics analysis of a two-mass-spring system

The variables , , , and can be used to simplify the general solution of the system:

(38)

Where and are the concentrated masses, and and are the spring constants of springs 1 and 2. The natural frequency and amplitude ratio of the two modes of the system can then be calculated:

(39)

(40)

Where and are the natural frequencies and amplitude ratios of the system for modes 1 and 2, respectively. During the pulse load (), the motion of the two masses is calculated as:

(41)

(42)

Where and are the displacements of masses 1 and 2, and is the amplitude of each vibration mode of mass 1. After the pulse load (), the motion of the two masses becomes:

(43)

(44)

Two parts are defined to represent the two springs, being meshed with 1D discrete elements. The springs use elastic spring material cards (*MAT_SPRING_ELASTIC) with their respective stiffness constants. Two mass elements of 2.0 kg are defined using *ELEMENT_MASS for the middle and bottom nodes. The top node, connected to spring 2, has its translational and rotational constraints defined with *BOUNDARY_SPC_NODE. A pulse load is prescribed to the bottom node, connected to spring 1, using *LOAD_NODE_POINT. A termination time of 2.40 s is defined using *CONTROL_TERMINATION.

Results Comparison

The displacement profile of each mass is shown in Figure 190.

Figure 190: Displacement of each mass versus time

To quantify the error between the theoretical and LS-DYNA results, the mass displacements at 1.3 s and 2.4 s are calculated with their relative errors and shown in the following table. This comparison verifies the agreement between the displacements.