29.1 VIBRATION MODELLING

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The most significant vibration in engineered systems is periodic. In these systems there is often an approximate spring-mass-damper system that gives us a second order response to disturbances.

In vibration modeling we typically assume that all components are linear. In a linear system the forcing (input) frequencies are directly related to response (output) frequencies.

In non-linear vibration systems we end up with the frequency of the forcing function being transformed to other frequencies. This tends to make the vibrations seem less clear, and appear more chaotic.

There are a few types of descriptive terms for these systems,

Damping Factor - The damping factor will indicate if vibrations will tend to die off. If the damping factor is too low the vibrations may build continually until failure.

Forced Vibration - When a periodic excitation is applied to these systems they will tend to show a steady state response

Free Vibration - When displaced/disturbed and released there is an oscillation at a natural frequency for any system. This is one measure of a system, and is typically induced by displacing a system and letting it go.

Natural Frequency - Each system will have one or more frequencies that it will prefer to vibrate at. When we excite a system at a natural frequency the system will resonate, and the response will become the greatest.

Response - This is a measure of how a system behaves when it is disturbed. For example, this could be measured by looking at the position of a point on a mechanism.

Steady State Response - After a system settles down it will assume a regular periodic response, this is steady state. The steady state excludes the transient.

Transient Response - When a forcing function on a system changes, there will be a short lived response that tends to be somewhat irregular. The transient will eventually die off, and the system will settle out to a steady state.

These systems can be modeled a number of ways, but we typically start with a differential equation.

29.1.1 Differential Equations

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In modeling any linear system we are best to start by developing a differential equation for the system components.

Consider the simple spring-mass-damper system shown below. A free body diagram can be drawn for the mass `M', and a sum of forces can be written, and expanded with the values for the mechanical components.

We may also consider a torsional vibration. We will assume that the vertical shaft has a stiffness of Ks and a damping coefficient of Kd. There is an applied torque `T' and a moment of inertia `I'.

29.1.2 Modeling Mechanical Systems with Laplace Transforms

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Before doing any sort of analysis of a vibrating system, a system model must be developed. The obvious traditional approach is with differential equations.

29.1.3 Second Order Systems

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Basically these systems tend to vibrate simply. This vibration will often decay naturally. The contrast is the first order system that tends to move towards new equilibrium points without any sort of resonance or vibration.

Some basic relations,

These generally have an effects on the Bode plot that are very evident.

Under the influence of damping, the natural frequency will shift slightly,

To continue the example with numerical values

29.1.4 Phase Plane Analysis

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When doing analysis of a system that has both a steady state and transient response it can be handy to do a phase plane analysis to help separate out the components.

To construct a phase plane graph we plot the value of a response variable against it's first or second derivative.

The shape of the graph exposes the phase between the displacement, and one of the derivatives. Here we see the system start to spiral out to an outer radius. The change in the radius of the spiral is the transient, the final radius is the steady state. If the forcing function changed, the path would then shift to a new steady state position.