Advanced Symbolic and Numeric Techniques for Machine Vibration Analysis

Maplesoft Illustrates Advantages to Using Sophisticated Lumped Mass Models With Experimental Measurements

By Dr. Thanh-Son Dao, Maplesoft

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In this article submitted to us from Maplesoft, Application Engineer Dr. Thanh-Son Dao illustrates some of the advantages of using sophisticated lumped mass models combined with experimental measurements for vibration analysis. By applying advanced symbolic and numeric techniques using a tool such as MapleSim, this approach can be used to diagnose and solve a wide variety of vibration issues, including gear box rattling, driveshaft breakdowns, shaft bends, driver's comfort, machine noise, and many others. Overcoming these problems can significantly improve the customer's satisfaction and hence the manufacturer's success.

SEE ALSO: How to Build Models, Analyze With MapleSim

The operational reliability of a rotary equipment train is dependent on the vibration of its components. All rotating machines experience some degree of vibration during all phases of operation: startup, shutdown, and normal continuous operation. Often, the only evidence of this vibration is gear noise or coupling wear. However, these early indicators of vibration might eventually develop into high-amplitude vibration, resulting in gear wear, gear tooth failures, or broken shafts. The torsional response characteristics of rotating and reciprocating equipment should therefore be analyzed and evaluated to ensure system reliability.

In industrial environments, the most common cause of machine breakdowns is bearing failure. This is due to mechanical problems that can affect the bearings but cannot be seen directly, such as unbalance, misalignment, mechanical looseness, bent shafts, resonance, belt problems, gear problems, hydraulic/aerodynamic problems, electrical problems, and others. Among these, the main source for vibration of a driveline system is the engine fluctuating torque. Other vibration sources in the driveline which are important to study are the forced response to the periodic excitation fluctuating torque from the engine, transient response of the driveline to any sudden change of the gear shifting, clutch judder phenomena which may occur during clutch engagement, or natural frequencies of the driveline in order to avoid coincidence of engine torque harmonics. The vibrations of machines are also coupled with other motions of the machine structure like vertical and horizontal motions, or a vehicle might be subjected to frequent stop and start. All of these make the analysis of vibration behavior of mechanical structures complicated and challenging to understand.

In general, there are two approaches to vibration analysis of mechanical structures. The first technique is finite element analysis (FEA) which describes the mechanical structure as a system of thousands of discretized and idealized elementary elements. This results in solving a huge system of thousands of linear equations and therefore remains very time-consuming and expensive. As an alternative, it has become increasingly common to use lumped mass models combined with experimental measurements for vibration analysis. This approach can be expanded to develop more sophisticated models to allow the creation of a highly refined, realistic vibration analysis environment. This article illustrates the latter approach, using the advanced system-level modeling and simulation platform MapleSim, from Maplesoft, as the modeling and analysis tool.

Developing Models for Vibration Analysis
MapleSim gives users the ability to create physical models with any desired level of complexity. There are several ways to develop a model for vibration analysis in MapleSim. The engineer can start with vibration analysis in a one-dimensional setting using the 1D mechanical components. This allows the engineer to build models consisting of springs, dampers, masses/rotational inertias, and force/moment components for either torsional or translational domain while ignoring the other motions to keep the models simple.

Engineers can then use table-lookup components to incorporate experimental data describing characteristics of springs and dampers into the model to describe non-linearity of spring, damper, and shaft components. In order to do this, MapleSim provides a custom component worksheet that calculates spring and damping forces/moments based on an Excel spreadsheet attached to the model. This capability also gives users the freedom to control the component equations to fit their own needs.

Engine Excitation Model
The most predominant cause of vibration in the engine is the gas pressure torque related to the cylinder gas pressure by geometry of the slider-crank mechanism. This gas pressure torque repeats after every complete working cycle. For a four-stroke, single acting engine, the interval of the repetition is two revolutions of the crankshaft (4 ), and for a two-stroke engine, the interval is 2. Due to this harmonic nature, the gas torque is usually represented as a Fourier series as follows:

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In engine vibration theory, the number of excitations received during one revolution of the crankshaft is usually called the harmonic order. As an example, the 4th order repeats itself 4 times during each revolution in a two stroke, and 2 times in a four-stroke engine. For a four-stroke engine, the harmonic orders are half multiples (1/2, 1, 3/2). For a two-stroke engine, there are no half-orders and the orders are integer multiples. The harmonic orders and half-orders are represented by the sine and cosine terms in the above gas torque equation.

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