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작성자 씨앤지소프텍
댓글 0건 조회 6,897회 작성일 21-01-27 15:46

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Single degree of freedom system with harmonic base displacement

A single degree of freedom system is subjected to a harmonic base displacement. We are interested in finding out the displacements at the top.
Freedom_1.gif
The following geometrical data are available:
  • m: lumped mass of the single degree of freedom system = 0.5 Tonne.
  • k: axial stiffness of the spring-type element = 200 N/mm.
  • l : spring length = 500 mm.
  • w : frequency of the external harmonic base displacement = 5 and then 20 rad/s.
  • Amplitude of the external harmonic base displacement = 20 mm.

S.d.o.f. with Support Excitation

Create elements and assign attributes
Create two nodes with a distance of y=500 mm (say), and a beam element between them. Assign the spring/damper property to the beam, entering the correct value for the axial stiffness. From the attribute menu select Node Translational Mass and apply a 0.5 T mass to the top node.

Assign the harmonic excitation to the base node
Assign a vertical unity displacement to the base node and then specify a Factor vs Time table which describes its harmonic variation.
Simply follow the next steps:
  • Select Attributes Node/Restraint
  • Check the Y value and specify 1 in the edit box.
  • Select the top node and press Apply
Freedom_2.gif
  • Now go to Table Factor vs. Time
  • To insert values by using a particular function press the f(x) button.
  • Now you can specify a function using the "x" unknown.
  • Enter the values range and the number of sample points (for example from 0 to 3 s and from 1 to 80 sample points).
  • Go to the Linear Transient Dynamic Solver.
  • Select Load Tables and choose the table name you have just specified for the freedom condition.
Freedom_3.gif
  • Set the time steps and any other information about the solver options.

Theoretical Solution - S.d.o.f. system without damping

The theoretical solution is obtained by solving the following ordinary linear, second order differential equation:
Freedom_4.jpg
The natural frequency of the system is given by:
Freedom_5.jpgrad/s
If the external and the natural frequency are equal we obtain the resonant response. In this case, the displacement of point B is harmonic but its amplitude increases linearly. Consider the case with an external frequency equal to 5 rad/s. The general solution for this case is:
Freedom_6.jpg
where, Freedom_7.jpg

Theoretical Solution - S.d.o.f. system with damping

In this case the equation describing the physical behaviour of the system is:
Freedom_9.jpg
with the following steady-state response:
Freedom_10.jpg
where, Freedom_11.jpg is the damping ratio.

To add the damping simply enter its value in the property dialog box.
Freedom_8.gif

Numerical Results

The Linear Transient Dynamic Solver was used to solve the models.

The Models


Freedom_12.gifFreedom_13.gif


Freedom_14.gif
S.d.o.f. without damping, external frequency = 5 rad/s


Freedom_15.gif
S.d.o.f. without damping, resonant response


Freedom_16.gif
S.d.o.f. with damping, resonant response

S.d.o.f. with non zero initial conditions

This example illustrates how to solve the same system of the previous example when a force is firstly applied to it and then realeased.

How to build the model

  • Create the Spring/Damper element and specify its properties.
  • Apply a Node Point Force at the top node.
  • Specify a Table with a 1 value before the release time and 0 after that.
  • Run the Linear Static Solver.
  • Go to Solver/Linear Transient Dynamic.
  • Specify as Initial Condition the previous linear static result file (*.lsa)
Freedom_17.gif
  • Specify the Time Steps and the other solver parameters
  • Press Solve

Theoretical Solution

The theoretical solution is obtained by solving the following ordinary second order differential equation:
Freedom_18.jpg
with the following initial conditions:
Freedom_19.jpg and Freedom_20.jpg
where Freedom_21.jpg is the displacement at the top node because of the point load applied.
If we assign c as
Freedom_22.jpg
the solution has the following expression:
Freedom_23.jpg
where:
Freedom_24.jpg is the natural frequency of the damped system = Freedom_25.jpg

Numerical Results

The following graph is obtained:
Freedom_26.gif

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