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Wind Turbine Vibrations and Their Effects

Introduction

Oil is the single most important source of non-renewable energy driving the economies of the world. However, with increased prices due to diminishing resources, governments are opting for alternative sources of energy that are renewable in nature. Among these alternative sources is the wind energy that is harnessed through the wind turbines, generating power that is vital for both industrial and domestic use. Wind energy faces competition from its substitutes that include solar and nuclear energy among others. To this end, there is need to enhance its efficacy in order for it to be above its substitutes (Wylie 2007). The modifications required to enhance the efficacy of these equipment (wind turbines) are reserved to the engineers and designers of the technology. Contemporary designs are characterized by turbines that are light-weight and mounted on flexible towers to enhance their performances. One important element that engineers grapple with in an effort to enhance wind turbine performance is vibrations. To some extent, engineers have managed to decimate vibration effects by using dampers as well as using control strategies. Therefore, in this paper, our main focus is on wind turbine vibrations and their effects. To many, the effect of vibrations on the efficacy of wind turbines is undecipherable, but its influence is seen when exclusion zones are created. Ideally, in these zones the energy output is zero. Exclusion zone happens when the natural frequencies of the turbine system overlap with the frequencies due to excitations emanating from the vibrating elements of the system (e.g. a rotor blade). One important “cause of vibrations is the changing circumstances” (Serway & Jewett 2003). This is owed to the changing structure of the seabed, rendering the wind turbine tower to the no-energy zone. Nonetheless, with advanced damping devices, engineers have managed to minimize vibrations to a greater extent. Contemporary damper designs (e.g. Tuned Liquid Column Dampers (TLCD)) have managed to reduce dumping without increasing the overall weight of the system (Serway & Jewett 2003).

The main objective of this report is to analyze the damping effects in a given system. The vibration behavior of an ‘undamped’ system is different from a damped one. This behavior can be exhibited by plotting the amplitude (x (t)) as a function of time (t). This is represented in the figure below:

the x-t plot for ‘undamped’ vibration
Figure 1: showing the x-t plot for ‘undamped’ vibration.

When we consider working with a spring-type system, the below expressions can be derived.

Formula

Formula

Formula

The parameters Formula represent the angular velocity, frequency, periodic time, spring constant, and mass respectively.

For a free damped vibration, the x-t plot assumes the trend below:

the x-t plot for a free damped vibration
Figure 2: showing the x-t plot for a free damped vibration

For the same system (spring-type system), the following equations are derived:

Formula

Formula

Formula

Formula

Where; Formula and m represent the logarithmic decrement, cycles apart, damping force per velocity, the critical damping, damped angular velocity, normal angular velocity, and the mass.

When damping is achieved via a damping device incorporated into a system, the system is said to be ‘forced’ damped. One important formula to note is:

Formula

Where; r and Formula represent the resonant frequency and the angular velocity emanating from the external force applied respectively.

To understand the behavior of a ‘forced’ damped vibration, a plot of Q (magnification factor) and α (phase angle) are plotted against r. These trends are exhibited in the figures below:

the trends of Q and α against r respectively
Figure 3: showing the trends of Q and α against r respectively

Some of the important equations associated with these plots are shown below.

Formula

Formula

Where; R is the dumping ratio.

Formula

Experimental setup

In this experiment, the experimental setup was as shown in the figure below. The values L1, L2 and L3 represented 0.066, 0.658 and 0.769 m respectively. Importantly, the LDVT calibration was such that 1 volt represented 1.25mm.

Experimental setup

The equipment above was used to analyze different damping behaviors that are displayed on a PC that is connected to it. The analysis was divided into two tasks (A and B). For task A, the damping properties were analyzed and this was labeled A1. Measurements such as spring extension (Δx), spring stiffness (k), and the weight of the bob mw were calculated and recorded. Forced vibration was analyzed as task A2 where the RPM was adjusted momentarily at intervals of 100 from 500 to 1600 RPM. The parameters of interest i.e. Xmax (maximum amplitude) and T (periodic time) were recorded for analysis. For task B, the idea was to increase the weight of the damper. Akin to task A1, the damping properties were identified and recorded. Just like task A2, the RPM was adjusted momentarily at intervals of 100 from 500 to 1600 RPM. Finally, Xmax and T were recorded for analysis. With these parameters such as the critical damping as well as the logarithmic decrement equation for both air and oil were determined. Moreover, the values of c, ωd, R, r, Xmax and Q for both air and oil were also determined. A graph of Xmax against r was plotted. Also, a graph of Q against r was plotted.

Results

Experiment A1 (light damping)

Table A1: showing results from task A1 (damping properties)

Parameter Value
Mass of bob (mw) (Kg) 7.5 Kg
Weight (w) (mw*g) N 75 N
Extension (Δx) x10-3*1.25/103M 4.2*10-5M
L2 0.769 M
L3 0.658 M
Spring stiffness (k) = L3/L2*w/Δx = (0.769/0.658)*(75/4.2*10-5) 2.08*10-4

Experiment A2 (Forced vibration results)

Table A2: showing results from task A2

RPM x 100 5 6 7 8 9 10
V (mV) 10.96 15.38 12.21 30.19 48.40 102.62
Xmax(Vx1.25 * 10-3/103) M 13.70 19.23 26.52 37.73 60.50 128.27
T (s) 0.39 0.32 0.28 0.23 0.22 0.19
RPM x 100 11 12 13 14 15 16
V (mV) 386.31 271.56 114.70 77.24 62.16 54.40
Xmax(Vx1.25 * 10-3/103) M 482.89 339.89 143.37 96.55 77.70 68.00
T (s) 0.15 0.15 0.13 0.125 0.12 0.11

RPM that coincide with the peak resonance = Experiment B1 (Forced Vibration)

Table A1: showing results from task B1 (forced damping properties)

Parameter Value
Mass of bob (mw) (Kg) 7.5 Kg
Weight (w) (mw*g) N 75 N
Spring stiffness (k) = L3/L2*w/Δx = (0.769/0.658)*(75/4.2*10-5) 2.08*10-4
Decaying oscillation ( Δx = x (t1) – x (t1 – T))

Experiment B2 (Forced vibration results)

Table A2: showing results from task B2

RPM x 100 5 6 7 8 9 10
V (mV) 9.68 15.34 19.03 28.21 44.77 91.25
Xmax(Vx1.25 * 10-3/103) M 12.10 19.18 23.78 35.26 55.97 114.07
T (s) 0.14 0.318 0.15 0.15 0.151 0.119
RPM x 100 11 12 13 14 15 16
V (mV) 416.20 230.54 110.56 74.28 59.67 54.43
Xmax(Vx1.25 * 10-3/103) M 520.25 288.18 138.20 92.85 74.59 68.04
T (s) 0.175 0.125 0.15 0.123 0.13 0.12

RPM that coincide with the peak resonance

Table C1: showing parameters used to obtain logarithmic decrement (Λ) as well as the critical damping (ωc). The parameters were obtained from Excel sheets at 500 RPM for both air and oil.

Component N X (x(t)) X (x(t+T.N)) t1(x(t)) t2 (x(t+TN)) T
Air 1 9.46 8.18 0.063 0.38 0.39
Oil 1 6.70 6.69 0.013 0.42 0.14

To obtain Λair and Λoil the formula above (4) is used. Therefore:

Formula

Table C2: showing a table of values vital in calculating c and ωd.

Component Λ ωc Formula Formula R=ωnc
Air 0.1454 0.1454 1.10 4.50 30.96
Oil 0.0015 0.0015 0.01 4.50 3001.23

The value of c is obtained using formula (5). But, ωc = Λ/T, ωn = ωcR and ωd is obtained using entry (6) formula.

Formula= √ (2.08*10-4 /7.5) = 4.50

Table C3: showing values of r (frequency ratio) as well as the resulting Xmax (maximum displacement).

For Air:

Air (RPM x 100) 5 6 7 8 9 10
Xmax(Vx1.25 * 10-3/103) M 13.70 19.23 26.52 37.73 60.50 128.27
ωf= 2πf 52.36 62.83 73.30 83.78 94.25 104.72
r = ωFn 11.64 13.96 16.29 18.62 20.94 23.27
RPM x 100 11 12 13 14 15 16
Xmax(Vx1.25 * 10-3/103) M 482.89 339.89 143.37 96.55 77.70 68.00
ωf= 2πf 115.19 125.66 136.14 146.61 157.08 167.55
r = ωFn 25.60 27.93 30.25 32.58 34.91 37.23

For oil:

Oil (RPM x 100) 5 6 7 8 9 10
Xmax(Vx1.25 * 10-3/103) M 12.10 19.18 23.78 35.26 55.97 114.07
ωf= 2πf 52.36 62.83 73.30 83.78 94.25 104.72
r = ωFn 11.64 13.96 16.29 18.62 20.94 23.27
RPM x 100 11 12 13 14 15 16
Xmax(Vx1.25 * 10-3/103) M 416.20 230.54 110.56 74.28 59.67 54.43
ωf= 2πf 115.19 125.66 136.14 146.61 157.08 167.55
r = ωFn 25.60 27.93 30.25 32.58 34.91 37.23
a plot of Xmax against r
Graph C3: showing a plot of Xmax against r

Table C4: showing the values for r (frequency ratio) as well as Q (the magnification factor)

Formula (9) above is used to obtain the values of Q.

the values for r (frequency ratio) as well as Q (the magnification factor)

the plot of Q against r
Graph C4: showing the plot of Q against r
the graph between Voltage and time
Graph D: showing the graph between Voltage and time

The graph above depicts a forced-damped vibration at 1600 RPM. There is a decrement witnessed on the wave motion.

a graph representing the Voltage against RPM
Graph E: a graph representing the Voltage against RPM

Discussion

The objective of this experiment was to analyze the damping behaviors of a vibrating system. Following the analysis, the damping behaviors of the system was analyzed and found to be in agreement with the theoretical expectations. Among the properties tested was the relationship between Xmax (maximum displacement) and r (frequency ratio). As expected, the trend reflected the theoretical behavior. The trend exhibited by the light damping plot (air) was similar to that one of a forced vibration (oil). However, the difference was witnessed on the peaks (see graph 3C above). Ideally, air peaked higher at a Vmax of approximately equal to 480 mm while its equivalent value for oil was approximately equal to 420 mm. This behavior was anticipated since oil acted as a damper and in effect it was expected to reduce the amplitude once the vibration commenced. Another property that was analyzed was the magnification properties of the two sets of experiment. To this end, a graph of Q (magnification factor) against r was plotted, and as such, the expected trends were portrayed. As is in the theory, graph 4C gave a similar trend. The trends for both air and oil were almost similar in shape, meaning that their damping effects are not much different. However, the inverse trend between the plotted parameters implies that the vibrating motion diminishes with an increase in r. On the other hand, the increase in r implies that the angular velocity of the entire system decreases with time until absolute damping is achieved.

The plot of voltage against time portrayed the behavior of the system. From the graph (D), it is evident that the motion exhibits an oscillatory type of motion. However, the decrement in amplitude is not significantly shown. Whatever was anticipated was a trend similar to the one exhibited by free-damped motion. The effect of air as a natural damper on a vibrating equipment is not shown here. To this end, we can point out to experimental errors that might have influence the results. Errors introduced while performing the experiment could be assumed in future by considering working with simulated experiment.

Another trend that exhibited the expected behavior of the system’s motion was the graph of voltage against RPM. Indeed, the trend shows that the increase in voltage increases the oscillations to an optimum point before decaying to a minimum value. This behavior proves that there is a presence of a damper in the system which functions to decrease vibration effectively (Tipler 2010).

In a conclusion, the objective of this experiment was to analyze the damping behaviors of a vibrating system. These properties confirmed that the behaviors of the systems echoed what is presented in theory. Among the properties tested was the relationship between Xmax (maximum displacement) and r (frequency ratio). As expected, the trend reflected the theoretical behavior. The trend exhibited by the light damping plot (air) was similar to that one of a forced vibration (oil). However, the difference was witnessed on the peaks (see graph 3C above). Also, a graph of Q against r portrayed the expected results, with oil giving the best damping ability vis-à-vis air. Furthermore, the plot of voltage against time portrayed the behavior of the system. From the graph (D), it is evident that the motion exhibits an oscillatory type of motion synonymous to systems experiencing vibrations. However, the decrement effect of a damper on the motion was not witnessed owing to experimental errors. Future experiments ought to avoid errors introduced by humans through designing a simulation-based experiment.

References

Serway, A. & Jewett, W 2003, Physics for Scientists and Engineers, New York University, New York.

Tipler, P 2010, Physics for Scientists and Engineers, McGraw-Hill, London.

Wylie, C. R 2007, Advanced Engineering Mathematics, McGraw-Hill, London.

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