Understanding analytical methods to study vibration behavior of a mechanical system - Part 1

Vibration analysis of a mechanical system is carried out to study Free vibration response (i.e. Natural frequencies, Modes shapes) and Vibration amplitudes for a given 'Excitation Force'. In general purpose FEA software, Modal analysis provides natural frequencies and mode shapes, which is sufficient for most of practical purpose as objective is to keep system natural frequencies sufficiently away from excitation frequencies. Frequency response analysis helps predicting system steady state response (vibration amplitudes) under harmonic loads. Let us discuss how a detailed system vibration response study can be conducted through analytical methods.

Theoretical route for a vibration study of mechanical system through analytical method has following steps,

1. Building Spatial Model, which is a function of system response i.e amplitude of vibration and physical properties of the system i.e. Stiffness, Mass and Damping.
2. Derivation of a Modal Model from spatial model, which describes natural frequencies, mode shapes and damping factor of the system - Free vibration study.
3. Finally, calculating system response for a standard harmonic excitation, which is described as Response Model - Forced vibration study.

Let us consider a single degree of freedom system (SDOF) to understand these steps.

This system can be described by the following equation of motion, using system Mass, Stiffness and Damping values.

This equation is the Spatial Model of the system, a function relating system response with time and system physical properties i.e. x = f(t, K, M, C). Solution of differential equation of Spatial model provides system response behavior over time. However it doesn't provide an explicit way to, calculate natural frequencies and study mode shapes of the system. We are also unable to study system behavior under various forcing frequencies with this model. So we need some other form of mathematical model which provides an explicit way of calculation of natural frequencies and frequency response of the system. Let us consider a single degree of freedom (SDOF) system without damping.

SDOF system without damping

Free response study

The equation of motion without damping and under no external force reduces to the form :

Let us derive an analytical solution for this equation. From mathematics we know that solution of this equation is a harmonic function, which can be assumed as:

Which is a displacement function of the system with amplitude and phase information, being a complex function. Second derivative of the same provides acceleration function. Analytically solving equation of motion with displacement and acceleration (derived from displacement function) leads to:

Analytical solution of spatial model leads to the Modal Model of the SDOF system. If we further solve if for w, we get natural frequency of the system, w(n) = sqrt( k / m ). Note, X is the mode shape vector. So interesting thing to note here is that, we derived Modal model from analytical solution of equation of motion without external force. Natural frequencies and mode shapes are properties of a system, which do not depend on the external excitation.

Forced response study

Now let us assume an external harmonic force on our SDOF model.

Note, here w is external forcing frequency, we can also assume a trial displacement function, as we did earlier, with a frequency same as forcing frequency (but that's not natural frequency of the system).

Equation of motion with assumed x(t) and f(t) leads to:

H(w), which is the ratio of displacement amplitude of the system to external harmonic force amplitude, is called 'Receptance'. This function gives us the Response Model of the system as Frequency response function (FRF). Note that we started with a system differential equation in time domain (displacement as a function of time) which we called Spatial model and now we have system definition in terms of frequency (Displacement amplitude as a function of forcing frequency). With FRF we can calculate system behavior with varying frequencies of the forcing function. A pump with VFD (variable frequency drive) motor will have varying levels of vibration amplitudes with changing motor speeds. A response model let us study pump displacement amplitude behavior with varying motor speeds.

Note that H(w) becomes infinite (NaN) in absence of damping for w(n) = sqrt( k / m), i.e. when forcing frequency becomes equal to natural frequency of the system. So analytical solution from force response study leads to Response model of a system.

SDOF system with viscous damping

Free response study - Modal model

Let us consider SDOF with viscous damping and without external forces, to study free response of the system and to derive Modal model.

Spatial model of SDOF with viscous damping will have a more general solution like:

Where s is a complex number not just imaginary number, as in free response without damping. Solving equation of motion with this trial solution leads to:

Solution for x(t), which provide the Model model of the SDOF system, has some very useful information about system behavior under viscous damping conditions. It shows that amplitude of vibration A has decaying property and it keeps reducing with (omega*zeta) decay rate. It results in a single mode of vibration, with a complex natural frequency which has real and imaginary parts. The real part is the decaying part with (omega*zeta) as decay rate. The imaginary part is the oscillatory one with damped frequency:

A plot of real part of x(t), Free response plot, can be used to understand the physical significance of the derived modal model. The plot below shows that the vibration amplitude keeps reducing with (omega*zeta) decay rate and it has a oscillatory frequency given by w'.

Forced response study - Response model

Let us study forced vibration behavior of SDOF system with damping, for this assume a forcing function and a displacement function as we did earlier.

Equation of motion with these functions reduces to,

which gives us the Response Model for forced vibration with damping. H(w), as we defined earlier, is the Receptance FRF.

Receptance FRF for forced SDOF system with damping contains both amplitude and phase information. As discussed earlier we now have a function H(w) = f(K, M, C, w) which allows us to study vibration amplitudes with respect to change in forcing frequency i.e. in frequency domain. Let us plot Receptance FRF amplitude vs forcing frequency (w). Maximum value in the plot occurs at the natural frequency of the system.

Summary

We discussed analytical methods to study vibration response of a SDOF system with and without damping. Three models namely Spatial model, Modal model and Response model provides us different characteristics of a vibrating systems. In vibration studies Modal model is used to evaluate natural frequencies and mode shapes of a system i.e. free vibration characteristics. Response model on the other hand let us evaluate vibration amplitudes and phase angle w.r.t. forcing frequencies i.e. forced vibration characteristics.