Forced Harmonic Oscillations

Analysis of forced oscillations is performed to predict the behavior of a structure under external actions which change in accordance with the harmonic law. These actions include force and/or kinematic excitation. In addition to it, the impact of the system damping may be taken into account.
The objective of the forced oscillation analysis is to obtain a dependence of the system’s response on frequency of compelling actions. As a result of calculations we get amplitudes of displacements, vibration acceleration, and vibration overload at the preset compelling frequency. According to these results, one can obtain, for a frequency range, the dependencies of amplitudes and vibration acceleration on frequency of compelling actions, which is important with the evaluation of vibration stability of the system in this preset frequency range.


The module “Forced oscillations” of the finite element modeling system, AutoFEM Analysis, can be used for analyzing the established forced oscillations of the following types:

Forced oscillations of the system with no account of damping under the action of harmonic compelling force. In the system with numerous degrees of freedom, these oscillations are described by the following system of linear differential equations:
where M is a symmetric square matrix of mass;
K is a symmetric square stiffness matrix of the system;
F0 is a vector of amplitudes of compelling action;
ω is frequency of compelling action;
u, u'' are vectors of coordinates of the system’s points, which change their position in time t, and their accelerations;
φ is the initial phase of the exciter.
Forced oscillations of the system with the account of damping under action of harmonic compelling force. These oscillations are described by the following system of linear differential equations:
where C is symmetric square matrix of damping and u' is vector of velocity of the system’s points. We assume that the system is damped in accordance with Rayleigh law, i.e. it is proportional to distribution of stiffnesses and masses of the system and is defined with the equation

where a is an aspect ratio of masses and b is an aspect ratio of stiffness (both ratios are scalars).
Forced oscillations of the system of both above types, arising from the displacement of supports under the harmonic law, i.e. bringing one ore more system supports to oscillatory displacement. Differential equations describing this type of oscillations are analogous to those set above but differ in the harmonic compelling force which is calculated via formula:.

Several compelling forces and/or displacements of supports may be applied to the system, but their frequencies must be equal.

The rotation of a shaft or spindle in the unbalanced state on elastic supports may serve an example of harmonic compelling force. Kinematic excitation is applied when values of compelling forces are not known, in contrast to amplitudes of oscillations of some structure elements which are known.

When considering forced oscillations, it is important to take into account the impact of damping forces. The process of energy dissipation of mechanic oscillations, leading to step-by-step attenuation of lump-sum produced oscillations of the system, is termed “damping”. Damping forces may have different origin, namely: friction between dry sliding surfaces, friction between lubricated surfaces, internal friction, air or liquid resistance, etc. It is usually presumed that the damping force is proportional to velocity (viscous damping). Resistance forces changing under the voluntary law are replaced for equivalent damping forces, proceeding from the condition that in one cycle, they dissipate the same amount of energy as real forces. The equation for forced oscillations with the account of damping for i th mass, which is the solution of the above differential equations, is given below:
where ω0 is angular frequency of damping, ω is angular frequency of the compelling force.
2n=c/m  ,
where ci is damping factor for i th mode; and mi is mass. It consists of two terms: expression in parentheses describes attenuating free oscillations at damping frequency which differs little from the frequency of free oscillations; the remainder corresponds to forced oscillations with the frequency of forced action ω.

In order to clarify the impact of damping, let us consider the figure which presents the graph of dependence of the gain factor of amplitudes

on the ratio of frequencies of forced and free oscillations ω/ω0 at differing values of the damping factor γ=n/ω=c/ccritical , where ccritical is critical factor of viscous damping, at which oscillations do not occur while the displacement of the system monotonously decays.

The figure shows that, when frequency of forced oscillations is low compared with the own frequency of free oscillations, the displacement of points of the system is approximately equal to the displacement at static load, driven by the compelling force. When the driving harmonic force has high frequency, than regardless of the damping factor, it does not cause forced oscillations of the system, which has low own frequency. In both cases ω<<ω0 and ω>>ω0, damping does not virtually impact the forced oscillations, albeit when the ratio of the above frequencies is about 1, than damping poses a significant impact on the gain factor. At small damping factor, the greatest impact of damping is observed near resonance frequencies, which is crucial to consider when analyzing the structures. For purposes of analyzing forced oscillations of a structure near own frequencies, import of values of own frequencies from the results of computation of the frequency analysis is provided (see below).

Damping factor γj for j th mode is tied to aspect ratios a, b by the equation

The value of the damping factor γ takes values from 0.01 for weakly damped systems (all-metal parts); 0.02-0.04 (metal structures with permanent joints, which are deformed below the yield strength); 0.03-0.07 (metal structures with in-cut connections); 0.05 for rubber; and up to 0.15 for strongly damped systems.

If damping factors are known for i th and j th modes, than aspect ratios will be calculated under formulas:
If the coefficient a is equal to zero, such damping will be called relative, and the damping factor for j th mode will be proportional to angular frequency of that mode without damping. Therefore, oscillations corresponding to higher modes will attenuate faster. If the coefficient b is equal to zero, such damping will be termed absolute, and the damping factor for j th mode will be inversely proportional to angular frequency of this mode without damping. Therefore, oscillations corresponding to lower modes will attenuate slower.

The main results of calculation in the module of forced oscillations are the following values:

amplitudes of displacement at finite element mesh nodes Um;
vibration accelerations at finite element mesh nodes expressed via amplitudes Um as U''m=Umω2 ;
vibration overloads, defined as the ratio of vibration acceleration to free fall acceleration U''m/g.

Stages of forced oscillations’ analysis

1.Creation of “Study”. When creating a study, you must state its type, “Forced oscillations” in the window of command properties. It is possible to create a copy of study type “Frequency analysis” and change the type; in this case, the mesh and border conditions would be transferred.
2.Setting materials. To set materials, it is necessary to click the right button of the mouse at the body from folder “Bodies” (or at the folder per se in order to set the same material for all bodies) and select item “Material”.
3.Imposing border conditions. In the analysis of forced oscillations, as in the static analysis, fixings and loads play the role of border conditions. In this type of analysis, all types of fixings and force loads can be used. Imposing the fixings and force loads is the mandatory condition of the calculation being correct. In total, the limitations imposed on the displacement of the body shall satisfy the following condition.
For the static analysis, the model must have a fixing excluding its freely moving in the space as the solid body. Failure to fulfill this condition would lead to incorrect results of finite element modeling or failure of computation process. Besides, kinematic load “oscillator” may replace partial or complete fixings.
4.Setting frequencies and damping. Performing calculations. Prior to the calculation, the user shall state in the properties of the study the values of frequencies of load actions, for which the analysis of forced oscillations will be performed, as well as the damping value.
5.Obtaining and analyzing results. To fine-tune the displayed results, you must click the right mouse at folder “Results” in the tree of the study and select item “Results”. The results of forced oscillation analysis: amplitudes of displacements, vibration accelerations or vibration overloads, and phases of oscillations. For each type of the “Result” the opportunity is provided to view the deformed state of a structure in different phases.


Installation-specific settings of the preprocessor in Forced Oscillation Analysis

1.Limitation setting. The model may be restricted both by complete and partial limitations. When partial limitations are set, the setting of displacements other than zero is prohibited.
2.Force load setting. To set up the amplitude of force loads, the following types of loads can be applied to facets, edges or vertices of a model (acceleration can be applied to bodies):
cylindrical load

A number of loads can action the system of bodies simultaneously, but all of them will have the same frequency. For each of permitted loads, its own initial phase of oscillations can be set.

3.Kinematic load setting (oscillator). To set the oscillation amplitude of the support, one must apply the oscillator load to elements of the model, which can be used instead of full or partial fixing. The load is applied to facets, edges or vertices of the bodies, as well as to single bodies of the assemblage. In order to specify the direction of oscillations LCS is selected in the command, and the direction itself is stated by a check mark, corresponding to the relevant LCS axis. The type of kinematic load is selected from the drop-down list and might be as follows:
amplitude of point displacements;

In the separate field, one can set the phase shift, measured either in degrees or radians.

At the combination of the oscillator load and partial fixings, directions of oscillations and limitations must be different for a single element of the model.


Settings of the processor of forced oscillations analysis

In tab [General], it is possible to determine or change the descriptive properties of the current study; that is: name, type of study, and comment.

In the tab [Parameters], it is possible to determine or change frequencies of external action, parameters of Rayleigh damping.

In the group of parameters “Frequencies of external action”, values of frequencies pertaining to actions of external loads are established. There several ways of adding new values:

1.Button [Add] allows the addition of both the single value of the frequency and a frequency range, for which the initial value, end value and gain are specified.

2.Button [Import] brings out the dialog window in which values of resonance frequencies can be imported out of results of the earlier performed frequency analysis.

3.When it is required to calculate the forced oscillations only at all resonance frequencies, determined via the earlier frequency analysis, you should activate the control element “Use frequency analysis results.” Here, the associative connection with the results of the selected frequency analysis is maintained, i.e. at the change of frequency analysis results, the values of updated own values will be automatically used for the forced oscillation analysis.

Over the existing list of frequency values, the following edit operations are available:

button [Change] permits to re-write the value of single specific value;
button [Remove] removes the selected frequency value from the list;
button [Clear] clears off the whole list of frequency values.

In the group of parameters “Rayleigh damping”, the values of damping factors of the structure a and b are defined.

Tab [Solve] permits to set the processor’s properties for the solution of a system of equations. Parameters, defining the settings of the processor, are similar to those in the Static Analysis.

Tab [Results] allows one to determine the types of results, displayed in the study tree upon the calculation.


Postprocessor settings and analysis of results for forced oscillations

In tab [Results], one can see the following types of results divided into eight groups:

The group “Loads” includes the following results: components applied to the finite element model and the module of loads reduced to nodes. This type of results is for reference.

The group “Displacements” includes:

Displacements of points of the finite element model with the account of the phase shift relative to the phase of exciter in the direction of axes of global coordinate system (CS): , , , as well as the displacement module .
Actual part of displacements in the direction of axes of global CS: Re(UX), Re(UY), Re(UZ), as well as the module of the actual part of displacements .
Virtual part of displacements in the direction of axes of global CS: Im(UX), Im(UY), Im(UZ), as well as the module of the virtual part of displacements .
Amplitude of displacements of points of the finite element model (with no account of the phase shift relative to the exciter phase) in the direction of axes of global CS: UXm, UYm, UZm, as well as the amplitude module .
Phase angles for displacement components of points of the finite element model in the direction of axes of global CS relative to the exciter phase , , , as well as the phase angle module .

In the case if the phase shift is preset, one can obtain separate epures for the actual and virtual components of displacements, as well as the absolute value of the phase in the points of the finite element model.

In the group “Vibration acceleration”, one may note the derivation of epures of amplitudes of vibration accelerations pertaining to the points of the finite element model U''m=Umω2 . The vibration acceleration phase differs by 180  (π rad) from the displacement phase.

In the group “Vibration overloads”, one may note the derivation of epures of vibration overloads U''m/g, measured in respect of the free fall acceleration.
Also the graph of the result can be displayed.

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