Transformation of Gear Inter-Teeth Forces into Acceleration and Velocity *

The paper deals with mathematical modelling and computer simulation of a gearbox system. Results of computer simulation show new possibilities of extended interpretation of a diagnostic acceleration signal if signal is obtained by synchronous summation. Four groups of factors: design, production technology, operation, change of gear condition are discussed. Results of computer simulations give the relation between inter-teeth forces and vibration (acceleration, velocity). Some results of computer simulations are referred to the results obtained in rig measurements and in field practice. The paper shows a way of increasing the expert’s knowledge on the diagnostic signal, which is generated by a gearbox system, on a base of mathematical modelling and computer simulation.


INTRODUCTION
Mathematical modelling and computer simulation give new possibilities of investigating the vibration diagnostic signals generated by a gearing of gearbox system for aiding diagnostic inference. In Bartelmus (1994Bartelmus ( , 1996Bartelmus ( , 1997, relations between condition of the gearing and inter-teeth forces are given. It comes from that the results of computer simulations were referred to results obtained by measurements presented in Rettig (1977). Because the diagnostic assessment of the gearing condition is taken from vibration measurements, the This paper was originally presented at ISROMAC-7. Tel.: 4871441201. Ext.: 406. Fax: 48714481 23. 203 presented paper is mainly concentrated on relation between inter-teeth forces and vibration parameters like acceleration and velocity. Simulations may be done for any conditions which may change the vibration diagnostic signal. Gear conditions under which investigations were done by Rettig were limited to the change of the operation caused by the change of rotation speed of gears. The gear conditions may be described, as stated in Bartelmus (1992), by four groups of factors: design, production technology, operation, change of gear condition. Many factors will be taken into consideration and results of computer simulations given by the 204 W. BARTELMUS relation between inter-teeth forces and vibration (acceleration, velocity) will be presented. First mathematical model of a gear system taken into consideration design, production technology, operation and change of condition was given in Bartelmus (1994). The mathematical model is constantly developed and by the use of computer simulations different aspect of gear system dynamics are investigated (Bartelmus, 1996;1997). The results of computer simulations are referred to the results obtained by a rig investigation (Rettig, 1977) and a field investigation (Bartelmus, 1988;. The results obtained by Rettig (1977) are given in Fig. 1. A dynamic factor giving as a ratio Kd F(t)/F against a length of line of action, expressed by % of the length, is given. A gear system may run under resonance and over-resonance. The gear system operates at resonance when a meshing frequency equals to a natural frequency of a gearing. Results of computer simulations are given in Figs. Fig. 2 the result of computer simulation is given for the gearing operation under resonance run. It is easy to see similarities obtained by measurements and computer simulations, compare
FIGURE 4 Function of gearing dynamic factor Kd for unstable run of gearing (Bartelmus, 1996), Tmeshing period, Gearing dynamic factor Kd as function of inter-teeth error and shape of error (Bartelmus, 1996), (a) error mode (a; e; r) (parameter of error function; maximum value of error; coefficient of error change), (0.1; e; 0), (b) error mode for (0.5; e; 0), (c) error mode for (0.5;-e; 0). distinct pick for one period of meshing the same as for a result taken from measurements ( Fig. 1).
Figures 1-3 describe the influence of operation factors to a signal generated by a gearing. The cooperation of a gearing depends also on the design and the change of condition factors given by errors of teeth. Figure 4 gives a result of computer simulation when a gearing runs at unstable conditions caused by tooth errors (Bartelmus, 1996). Figure 4 shows influence of a natural frequency to a course of inter-teeth forces. In Fig. 4 a period of a meshing (0.0-100%) and period Tn of a natural frequency is given. Figure 5(a) and (b) shows courses of Kd against error e (l.tm) and as the function of error shape, defined later. Presented results of computer simulations are also referred to the results presented in Penter (1991) where diagnostic signal is obtained by synchronous summation. An example of a vibration signal is given in Fig. 6 (Penter, 1991). The signal is given as a function of time (s). It is the signal that shows a broken tooth in the gearing.
Presented results of computer simulations may be considered as idealised results of synchronous summation.

MODELLING OF GEARBOX SYSTEM
As it was stated for modelling of dynamic properties of gear system: design, production technology,  (Penter, 1991).
operation, change of gear condition factors ought to be considered. Design factors include specified flexibility/stiffness of the gear components, especially flexibility/stiffness of a meshing, and a specified machining tolerance and errors of components. Production technology factors include deviations from specified design factors obtained during machining and assembly of a gearbox.
Operational factors include peripheral speed (pitch line velocity) v (m/s) and its change Av (m/s) and load F and its change AF.
Change of condition includes influences of gear wear, pitting, fractured or broken tooth. 206 W. BARTELMUS From simulation point of view in the considered gear system the design factors can be divided into two groups: constant and controlled. The constant design factors are not controlled/changed for different simulation experiments. The controlled design factors are changed before a specified simulation experiment. The constant design factors are given by: Is, Ilp, I2p, I moments of inertia (kgm2) (Fig. 7); kl, k2 shaft stiffness coefficients (Nm/rad) ( Fig. 7); # coefficient of inter-teeth friction; Ch gearing damping coefficient (N s/m); rl, r2 base radii ofgears (m); a, b, c parameters ofgearing stiffness (0-1) ( Fig. 8(a)); 1inter-teeth backlash tm; C maximum value of gear stiffness (N/m), g changeability of gearing stiffness (0-0.4), 0.4 for spur-gear. The controlled design factors are given by: Cs clutch/coupler dumping coefficient (N m s/rad); a, e parameters of error function ( Fig. 8(b)); li random coefficient of error (0-1); r coefficient of error change (0-1), so a value of an error for a given tooth is expressed by where number of teeth pair; li is distributed randomly for z pair of teeth, number of teeth in the pinion of gears equals to Zl, el maximum value of teeth error (lam). The error of teeth may be described by error mode (a;el;r). Mathematical model for torsion vibration for the system ( Fig.   7(a)), is given by equations Values of forces and moments are given by

RESULTS OF COMPUTER SIMULATIONS
Controlled factors which were changed for investigating their influence to diagnostic signal are: Cs clutch/coupler dumping coefficient, a parameter of the error function ( Fig. 8(b)), and r coefficient of error change (0-1), Eq. (1). Obtained results of computer simulation are interpreted like results obtained by synchronous summation of a diagnostic signal. First set of results of computer simulations is given in Fig. 9. Figure 9(a) gives a picture of Kd function in four different periods: (1) acceleration of a gear system (Fig. 7), from 0 to 980 rpm; (2) free rotation; (3)  Mr. Inter-teeth forces are the reasons which cause the failure of a gearing. The inter-teeth force reveals all factors which have influence to vibration generated by a gearbox. The forces are transmitted through bearings to an outer housing. A direction of transmission of inter-teeth forces is given in Fig. 7(b) and lies along a line of action E1E2. We supposed that if we measure an acceleration on the gearbox housing we may infer on the inter-teeth force's change. It is supposed that the change of the inter-teeth forces is proportional to the difference of acceleration A a of co-operating gear wheels. The main aim of a computer investigation is presenting differences or similarities between force and acceleration Aa. Inter-teeth forces are presented as a ratio Kd. Figure 9(b) gives the acceleration difference Aa in four periods. In Fig. 9 results for the error mode (0.5; 10; 0) are given. The error mode function is given in Fig. 8(b). Figure 9(c) gives Kd function in 4th period of a gear system run. In Fig. 9(d) Aa acceleration is given. One can see similarities between Fig. 9(c) and (d). For the same period of time velocity difference Av is given in Fig. 9(e).
There is no direct similarity between inter-teeth force and velocity Av. Inter-teeth force function for st period ofgearing co-operation is given in Fig. 9(f).
The function shows the period of gearing and the period of natural vibration of gearing Tn. The function of acceleration Aa for the 1st period is given in Fig. 9(g). One can see that the error function given in Fig. 8(b) is a cause of two vibration impulses (increase and decrease), so a meshing period T is divided into two periods. It is better seen in acceleration function Aa than in the force function ( Fig. 9(f)). Figure 9(c) and (d) does not show full similarity (forces to acceleration), it is supposed that a cause of it is influence of damping moment in the clutch, Mh so for further investigation Cs 0 is taken. Figure 10 gives results of these simulations for condition of Cs=0. Figure 10(a) gives a course Kd function. Compare Fig. 9(a) with Fig. 10(a). In Fig. 9(a) influence of damping moment Ms is seen. Figure 10  Figs. 9(g) and 11 (f). The meshing period in Fig. 9(g) is divided into two equal parts (a-0.5) in the error mode, in Fig. 11 (f) the machine period is divided in different ways (a 0.1) in the error mode. Deterioration of a gearing causes random change of error function. A depth of error change for simulation of this condition is given by an error mode parameter r. Current error is given by Eq. (1). Error functions for r=0.1; 0.3; are given equivalently in Fig. 12(a), (d) and (h). In Bartelmus (1997) 0 where Ad normalised gearing condition function; A suitable constant to make the value positive; Aa acceleration; (M1 + Mh) moment on the first shaft; r radius of a pinion gear. Aa function is given for r--0.1;0.3; in Fig. 12(c), (g) and (j). An example of a function of Kd is given in Fig. 12(f) for r 0.3. Acceleration functions A a are given for r=0.1; 0.3; in Fig. 12(b), (e) and (i). One example of a course of velocity function is given in Fig.   12(k) for r 1. On a base of simulations given in Fig. 12, a conclusion is drawn that Ad function is very good parameter for condition change identification of a gearing. One of the most important thing in condition monitoring is identification of a fractured or broken tooth. Figure 6 shows a local change of the signal, for one broken tooth one local change of a diagnostic signal. So we may say there is one-to-one mapping. Another evidence for this is given in Fig. 9(g) where a decrease and an increase of a tooth error ( Fig. 8(b)) are identified by one impulse in a diagnostic signal. But for an error shape given in Fig. 11 (a) an identification of a decrease and an increase of a tooth error is not so clearly identified (Fig. l(e) and (f)). Taking in mind possibilities of one-to-one identification, further simulation experiments were undertaken. Figure 13 gives a set of results of simulation for one fractured tooth for which a stiffness fall to 0.68 of its normal stiffness. As it is seen from the results of simulation there is very little change of a diagnostic signal given by Kd; Aa; Kd ( Fig. 13(a) simulations for stiffness change to 0.25C. The results show that the change of stiffness to 0.25C gives change of diagnostic signal which may be easy to identify. For further stiffness change to 0.075C results are given in Fig. 15. In Fig. 15(a) and (b) one can see one-to-one mapping (one disturbance in the signal one fractured tooth). In Fig. 15(c) it is seen that one fractured tooth may cause disturbance on several teeth, so there is no one-to-one mapping. It may be stated that Ad function defined by Eq. (6) is very sensitive to condition change but for heavy fracture of a tooth function may be too sensitive. A set of simulation results for a broken tooth is given in Fig. 16. simulation when one of the tooth has a fault caused by pitting, model of error function is given in Fig. 17(a). Figure 17(c)-(e) shows that there is no one-to-one mapping if diagnostic signal is presented as functions of Kdl or Ad.

CONCLUSIONS
From the results of computer simulation which are considered as diagnostic signals obtained from synchronous summation detailed features of the signal may be drawn. For example on the basis of results presented in Fig. 2 one may draw a conclusion that an error shape of a tooth taken for investigation by Rettig (see Fig. 1) is as it is given in Fig. 8(b). The same conclusion may be drawn from Fig. 10(c) which gives a diagnostic signal in form of acceleration. From Figs. 9(d) and 10(c) one may draw a conclusion that not only properties of a gearing but also the damping properties of a coupling between an electric motor and a gearbox have influence on a diagnostic signal. One-to-one mapping (one fault one disturbance, in a signal, equivalent for one tooth) does not always hold (see Figs. 15(a) and 16(c)). It seems that this drawback may be eliminated by careful study of evolution/ development, of a diagnostic signal, as gearing condition changes to avoid misinterpretation. Deterioration of a gearing is described by a change of condition for all teeth in a gearing, models of error modes are given by Fig. 12(a), (d) and (h), and for a single fault, as pitting in one tooth ( Fig. 17(a)). The best results of diagnosing the gea.ring condition change are obtained for signal of acceleration which gives direct measure of inter-teeth forces when a gearbox system runs in steady condition, 4th period of Fig. 9(a). New gearing condition parameter is suggested, the parameter is denoted as Ad and is given by the formula (6) Economic and environmental factors are creating ever greater pressures for the efficient generation, transmission and use of energy. Materials developments are crucial to progress in all these areas: to innovation in design; to extending lifetime and maintenance intervals; and to successful operation in more demanding environments. Drawing together the broad community with interests in these areas, Energy Materials addresses materials needs in future energy generation, transmission, utilisation, conservation and storage. The journal covers thermal generation and gas turbines; renewable power (wind, wave, tidal, hydro, solar and geothermal); fuel cells (low and high temperature); materials issues relevant to biomass and biotechnology; nuclear power generation (fission and fusion); hydrogen generation and storage in the context of the 'hydrogen economy'; and the transmission and storage of the energy produced. As well as publishing high-quality peer-reviewed research, Energy Materials promotes discussion of issues common to all sectors, through commissioned reviews and commentaries. The journal includes coverage of energy economics and policy, and broader social issues, since the political and legislative context influence research and investment decisions.