4. B and C modeling - DYNA_VIBRA

In the tests on the diagrams in DYNA_VIBRA, a slight viscous damping of one percent was introduced. In this way, the correct depreciation treatment is validated. It is also an opportunity to validate the quadratic modal calculation.

Modeling B is done with elements DIS_T and modeling C with elements 2D_ DIS_T (the Z axis therefore corresponds to Y for this modeling).

4.1. Analytical solution

William Weaver Jr. Stephen P. Timoshenko and Donovan H. Young provide in Chapter 1.9 of « Vibration Problems in Engineering » the solution to the problem of a mass/spring system with viscous damping subjected to harmonic excitation in chapter 1.9 of « Vibration Problems in Engineering ».

The equation to be solved is an equation of the second order in terms of time on a single degree of freedom in space:

\(\ddot{x}+2\eta \dot{x}+{\omega }_{0}^{2}x=\frac{F}{m}\mathrm{sin}({\omega }_{e}t)\)

where \(x\) is the movement of the mass, \(\dot{x}\) its speed, and \(\ddot{x}\) its acceleration.

\({\omega }_{0}^{2}=\frac{k}{m}\) is the natural pulsation of the system, \(m\) being its mass and \(k\) its stiffness.

\(\eta\) is reduced depreciation.

Finally \(F\) is the amplitude of the excitation force while \({\omega }_{e}\) is its pulsation.

4.2. Tested sizes and results

Calculation	Field type	Instant (or modal method)	Values Reference	tolerance
CALC_MODES		“SORENSEN”
	FREQ		3 Hz	1.E -4%
	AMOR_REDUIT		1E-03	1.E -4%
		“TRI_DIAG”
	FREQ		3 Hz	1.E -4%
	AMOR_REDUIT		1E-03	1.E -4%
EULER	DEPL	0.5 s	0.010785	1.E -2%
With and without REST_GENE_PHYS		0.7 s	-3.745074E-03	1.E -1%
		1.0 s	-0.0125639	1.E -1%
NEWMARK	DEPL	0.5 s	0.010785	1.E -2%
		0.7 s	-3.745074E-03	1.E -1%
		1.0 s	-0.0125639	1.E -1%
DEVOG	DEPL	0.5 s	0.010785	1.E -2%
		0.7 s	-3.745074E-03	1.E -1%
		1.0 s	-0.0125639	1.E -1%