First Natural Frequency of a Spring-Mass System

Let us consider the spring-supported mass.

First Natural Frequency of a Spring-Mass System, the finite element model with restraints

The finite element model with restraints

The length of the cube edge is L. Let L equal 0.1 m.
The material properties are: the Young's modulus E = 2.1E+011 , Poisson's ratio ν=0.28, the density ρ = 7800 kg / m3.
The mass of the cube M is calculated by the following formula:
M= ρ L3
Thus, M = 7.8 kg.
The spring stiffness k is 1000 N/m .
Analytical solution of this problem is given by the following formula:

Thus, f = 1.802 Hz

After carrying out calculation with the help of AutoFEM, the following result is obtained:

Table 1. Parameters of finite element mesh

Finite Element Type

Number of nodes

Number of finite elements

quadratic tetrahedron

271

783

Table 2. Result "Frequency"*

Numerical solution
Frequency f*, Hz

Analytical solution
Frequency f, Hz

Error δ = 100%*| f* - f| / | f |

1.802

1.802

0.00

First Natural Frequency of a Spring-Mass System

 

*The results of numerical tests depend on the finite element mesh and may differ slightly from those given in the table.

 

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