Strand7:  Web notes:  Harmonic

Single degree of freedom system with harmonic base displacement

Harmonic.zip
A single degree of freedom system is subjected to a harmonic base displacement. We are interested in finding out the displacements at the top.
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
  • 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.
  • 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:
The natural frequency of the system is given by:
rad/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:
where,

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

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

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

Numerical Results

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

The Models




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


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


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)
  • 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:
with the following initial conditions:
and
where is the displacement at the top node because of the point load applied.
If we assign c as
the solution has the following expression:
where:
is the natural frequency of the damped system =

Numerical Results

The following graph is obtained: