Applied Dynamics II Applied Dynamics II

2013 2013

Student

Student Name: Name: Steven Steven GiffneyGiffney Student

Student Number: Number: 1032216510322165 Date: 28

Date: 28ththJanuaryJanuary

Laboratory Title:

Laboratory Title: To Investigate the vibration motion of To Investigate the vibration motion of a damped mechanicala damped mechanical system having a single degree of freedom

Abstract:

A mechanical system consisting of a vibrating beam and various add-ons such as dampers, springs etc. was analysed during the laboratory session. Theoretical values for the damped natural frequencies were calculated and then compared with values taken from the experiment. The values for our theoretical (free vibration) and experimental (forced vibration) damped natural frequencies were 5.1586Hz and 4.808 respectively. The difference is attributed to human/experimental error. Zeta was found to be 0.02307 meaning that the system was to be underdamped.

Introduction:

Vibration is described as oscillation of the parts of a fluid or an elastic solid about equilibrium as a result of some disturbance. This is a largely utilised

phenomenon and it’s applications in our world can be seen in many forms e.g. the strumming of a guitar string to produce sound. However excessive vibration levels cause problems especially when it comes to material

limitations and the results of high levels of vibrations on a material. Vibration is manipulated to give us such creations as the loudspeaker

however when vibrations occur in structures that are not catered for them such as bridges or building, the results can be disastrous. So the importance of being able to quantify vibration is clear especially in the case of large frequently used structures as it is crucial to ensure structural stability and being able to measure vibration and account for it is a huge factor that needs to be dealt with in many projects. So, there are cases in which it is desired to dampen vibrations so as to reduce their effect on whatever they are encountering.

The objective of this laboratory session is to investigate the motion as a result of vibration, of a damped mechanical system with one degree of freedom.

Free Vibration:

Free Vibration occurs when some impulse applied to a system results in an initial displacement which causes the system to vibrate at its natural

frequency (Fn). In a vacuum, the system would continue to vibrate at this frequency, however in real life applications there is some inherent damping that occurs as a results of real world factors such as friction with the air being displaced by the vibrating system will eventually r educe the

Forced Vibration:

An alternative case in which vibration occurs is that in which a sinusoidal force is applied to the system. This causes the system to vibrate at a specific frequency dependant on the force applied until that force is changed or removed. Systems vibrating as a result of sinusoidal forces are particularly susceptible to resonance phenomena should the induced vibration

frequency correspond to the natural frequency of the system. For this reason the natural frequency of a system is good to know as engineers can take it into account and take measures to avoid or aim for such amplitudes depending on the specific case.

Apparatus:

Note: mmotor= mtotal – mbar

Fig 1: Diagram representation of Apparatus

Theory:

Uniform bar: weight act through centre of gravity at Length/2

### Kx

_{s}

### m

motor### g

### m

_{bar}

### g

## F

Fi 3: Schematic of Forces resent on s stem

Using this, we can convert standard general system equations to particular equations for the system in question and arrive at the following:

Spring Masses

A m p l i t u d e Damping:

The logarithmic decrement δ was used to deduce the damping ratio ζ.

Time

And from the below equations the logarithmic decrement can be calculated.

Hence we get the following expression for the logarithmic decrement

Values for X_{o},X_{n :}

Trial 1 Trial 2 Average

X_{o} 32 38 35

X_{n} 8 10 9

We can use the relationship

Rearranging we get the following:

Once a value for ζ had been established , the type of damping present in the system could be identified.

ζ = 1: Critically Damped ζ > 1: Overdamped

ζ < 1:Underdamped

Method:

• First weight was placed on bar suspended and the resultant deflection was recorded.

• Process repeated with 3 more weights being added.

• Process then carried out with the weights being removedand the deflection at each of the 4 loadings again recorded. This is done so as to allow an average deflection for each loading to be calculated and so reduce the experimental error present in our readings.

• Plot results to graphically representfindings.

• Systemthen connected to oscilloscope. Vibration induced by bending and releasing the bar and Xo and Xn were determined.

• Using the derived equations, theoretical values are found

• Motor then turned on and period measured and from this frequency

calculated. (f = 1/T)

• Results plotted on graph from which the damped natural frequency may be found.

## Results

Experimental Results

Free Vibration

Mass Force Deflection Average

Deflection (kg) (N) ON OFF 0.453 4.44393 1.19 1.25 1.22 0.907 8.89767 2.28 2.33 2.305 1.358 13.32198 3.51 3.59 3.55 1.81 17.7561 4.83 4.79 4.81

As F=kx, k can be found by drawing a trend line with intercept at zero and calculating the slope of this line. The slope and subsequently the k value from arrived at from the results of our experiment were 3.687513 kN/m. 0 2 4 6 8 10 12 14 16 18 20 0 1 2 3 4 5 6 F o r c e Deflection

### Force Vs Deflection

Deflection on Deflection off Average Linear (Average)Forced Vibration Periodic Time a of Motor Frequency a (hertz) Amplitude a (Volts) 240 4.166666667 0.1 237 4.219409283 0.231 228 4.385964912 0.301 222 4.504504505 0.525 214 4.672897196 1.42 213 4.694835681 1.77 212 4.716981132 1.91 211 4.739336493 2.49 210 4.761904762 2.75 208 4.807692308 3.112 206 4.854368932 2.768 205 4.87804878 2.013 203 4.926108374 1.496 200 5 1.216 190 5.263157895 0.259 179 5.586592179 0.125

The damped natural frequency occurs at peak amplitude which in our experiment was 3.112 Volts. The frequency at this amplitude is 4.808Hz which is our damped natural frequency.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 A m p l i t u d e ( V o l t s ) Frequency(Hertz) Frequency(Hertz) Amax/Sqrt2 1 2 f f

Gra h 2: Am litude vs. Fre uenc

The bandwidth is calculated by taking the max Amplitude, dividing it by the square root of 2 and drawing a horizontal line across on the graph at this Amplitude. Where this cuts the curve of our graph we draw vertical lines down and the bandwidth is the difference between our 2 resultant

frequencies. In the case of our experiment the 2 values for the frequencies were 4.87 (fh) and 4.72 (fl) giving a bandwidth of 0.15Hz.

Discussion and Conclusion:

The values for our theoretical (free vibration) and experimental (forced vibration) damped natural frequencies were 5.1586Hz and 4.808

respectively. The difference is due to fact that the assumptions made to ease the complexity of analysing the system were too far from the reality. Some part of the discrepancy should also be attributed to human/

experimental error such as parallax, equipment error. The experimental equipment was quite old and the device for measuring the deflection in the beam was a bit temperamental.

The variable that’s value would have potentially suffered the most distortion due to these errors would likely have been that of the spring constant as it is the deflection measurements recorded. In the case of our experiment there was very little difference between the deflection values measured both in loading and unloading the beam which can be seen from graph 1 hence we can be quite confident our value of k. This is important as many of the other calculations used to analyse the beams vibration were based on this value.

Zeta was found to be 0.02307 meaning that the system was to be

underdamped. The graph of Amplitude vs. Frequency (graph 2) produced from our experimental results with its high peak amplitude and relatively small bandwidth of 15Hz confirms our theoretical deduction that the beam is underdamped. Because of the type of system that this is, frequencies that lie within the bandwidth should be avoided. If this was some type of audio equipment then some level of overdamping would be preferred as a large bandwidth would be ideal.

To conclude this lab was a very good demonstration of how vibrations are quantified and how they are seen in systems. Both theoretical and

experimental methods were used to calculate values for important system variables and then these values were compared. The small discrepancy between the two was explained. Overall our theoretical approach matched up very well with the experimental approach.

References: