Vibration and analysis of the gear systems

Many investigations on vibration and analysis of gear systems were conducted over time. August (1986) used the three planet transmission technique to replicate vibration characteristics. Pike (1987) and Choy (1988b) examine the gear system dynamics using the gear structure relationships reported by Cornell (1981). Mitchell (1987) uses matrix transfer to simulate gear vibrations and determine the link between experimental and theoretical results. Furthermore, Ozguve and Houser (1988) and Kahraman (1990) employ the finite element model to estimate multi-gear mesh systems on gears. Choy (1991) on the other hand used the modal synthesis technique alongside the finite elements and the matrix transfer of both time and frequency. This is seen in calculating the casing motions and transient rotor. These studies unravel dynamic modelling as powerful tool for prediction of forces, vibrations and frequencies from the machine operations. The study in this case shall employ Fast Fourier Transform (FFT) to measure both the magnitude and the frequencies of the vibrations.

Aim of the experiment

To investigate whether dynamic modelling can simulate gear mesh problem

Literature Review

Differentials are considered as devices that transmit torque and rotation by the use of shafts, in one of the two ways: receiving input from one end, that is directed as output to the other end in the form of input. This is the principle applied in gears. This is mainly seen in automobiles in the creation of two inputs as the sum, difference or the mean of the inputs as outputs on other gear systems (Maitra, 2004)

In automobiles, differential principle is used on the wheels, with some form of gears attached to it, allowing the vehicles to rotate at different speeds with the supply of equal torque on each of the wheels (Zeping, 2004).

Nordiana, Ogbeide, Ehigiamusoe and F.I.Anyasi, (2007) indicated that by observing the structural analysis of different material components of the gears, Aluminum offers stress within the acceptable limits on the gears made. Aluminum alloys are better for the differential gear. This is evident when varying the stress of three metals, Aluminum, Steel and Cast Iron alloys. Comparing the stress values at different speeds of 2400rpm. 5000rpm and 6400rpm, Aluminum alloy indicates less vibrations as compared to the other alloys., making aluminum best material for the differential on the gears.

Naik (2015) indicates that there have been a myriad of developments in the field of engineering with the demand of more refined gear teeth based on the loading capacities and their operating speeds. It indicates that the dynamic and static analysis of the spur gear helps in the determination of maximum displacement, highest induced stress with the causative effects of variation of stress dependent on time. It is seen that the loading capacity and operating speed of the gears can be increased by the reduction of highest induced speed.

Method

The study involved the measurement of vibration, dynamic load and noise of the transmission using the gear noise rig. The measuring instrument was made up of a simple gearbox with pairs of axis as rolling elements. The 150Kw was the power supplied, with speed of 5000rpm used in obtaining the vibrations of the helical and the spur gears.

Figure 1: Lab Setups and the Gears used

Figure 2: Gearbox vibrations, frequencies and dynamic load apparatus

Results

Data for Helical Gear

Experimental, Hz

Analytical, Hz

650

650

1048

1006

1709

1763

2002

2049

2270

2342

2530

2530

2722

2750

2960

3010

Table 1: Data for Helical Gear

Data for Spur Gear

Experimental, Hz

Analytical, Hz

500

540

1000

990

1650

1600

2000

2000

2150

2100

2350

2200

2540

2450

3000

3050

Table 1: Table for Data for Spur Gear

Descriptive statistics

Helical Gear

Experimental Frequency

Analytical Frequency

Mean

1986.375

Mean

2012.5

Standard Error

286.8762761

Standard Error

294.2399

Median

2136

Median

2195.5

Standard Deviation

811.4086407

Standard Deviation

832.2362

Sample Variance

658383.9821

Sample Variance

692617.1

Kurtosis

-0.742743038

Kurtosis

-0.72201

Skewness

-0.616746154

Skewness

-0.65972

Range

2310

Range

2360

Minimum

650

Minimum

650

Maximum

2960

Maximum

3010

Sum

15891

Sum

16100

Count

8

Count

8

Table 3: Descriptive statistics for the Helical Gear

From the helical gear, the mean frequency for the experimental frequency was 1986.4 with a standard deviation (SD=286.876). The data was positively skewed with a total of 8 observations (N=8). The analytical frequency on the other hand had a mean of 2012.5 with a standard deviation (SD=294.2399). It was also made up of 8 observations (N=8).

Spur Gear

Experimental Frequency

Analytical Frequency

Mean

1897.5

Mean

1866.25

Standard Error

290.2631196

Standard Error

284.2153

Median

2075

Median

2050

Standard Deviation

820.9880807

Standard Deviation

803.8823

Sample Variance

674021.4286

Sample Variance

646226.8

Kurtosis

-0.253248306

Kurtosis

-0.1515

Skewness

-0.605228816

Skewness

-0.40863

Range

2500

Range

2510

Minimum

500

Minimum

540

Maximum

3000

Maximum

3050

Sum

15180

Sum

14930

Count

8

Count

8

Table 4: Descriptive statistics for the spur gear

On the spur gear, the experimental frequency indicated a mean of 1897.5 with a standard deviation (SD=290.263). The number of observations were 8 (N=8). The analytical frequency had a mean of 1866.25 with a standard deviation (SD=803.882). The number of observations (N=8).

Graphical representations

Helical Gear

Figure 3: Boxplot for the helical gear experimental frequency

From the helical gear experimental frequency boxplot, the median is 2000. The interquartile range lies between 1543.75 and 2578 from the graph.

Figure 4: Boxplot for the helical gear analytical frequency

From the helical gear analytical frequency boxplot, the median is 2000. The interquartile range lies between 1573.75 and 3010 from the graph.

Spur Gear

Figure 5: Boxplot for the spur gear experimental frequency

From the spur gear experimental frequency boxplot, the median is 2000. The interquartile range lies between 1487.5 and 3000 from the graph.

Figure 6: Boxplot for the spur gear analytical frequency

From the spur gear analytical frequency boxplot, the median is 2000. The interquartile range lies between 1447.5 and 3050 from the graph.

Correlation Analysis

Helical Gear

The hypothesis to be tested in this case involves:

Ho: There is no significance correlation between the experimental and analytical frequencies of the helical gear.

H1: There is significance correlation between the experimental and analytical frequencies of the helical gear.

Correlation matrix (Pearson):

Variables

Experimental Frequency

Analytical Frequency

Experimental Frequency

1

0.999

Analytical Frequency

0.999

1

Table 5: Correlation coefficients for helical gear

From the correlation between the frequencies of the experimental and analytical frequency of helical gear, the gear indicated a perfect linear relationship with each other (rho=0.999). This indicated that a change in the experimental frequency causes a great change in the analytical frequency.

p-values:

Variables

Experimental Frequency

Analytical Frequency

Experimental Frequency

0

0.000

Analytical Frequency

< 0.0001

0

Table 6: Correlation significance for helical gear

From the hypothesis test on the correlation, we have adequate evidence to reject the null hypothesis at 5% level of significance. This is because the correlation p-value is less than 0.05 (p-value