ME 3456 Technical memorandum Report to: | TA | from: | HN, 3.36 | subject: | Vibration due to rotational Unbalance | date: | 3/27/2014 | Date Submitted: | 4/20/2014 | | |

Introduction
This experiment was conducted in order to study the relationship between rotational unbalance and , the basic attributes of a rotational system. This involves determining equivalent rotational quantities such as deflection, natural frequency, stiffness, moment of inertia, equivalent damping constant, logarithmic decrement and other attributes of this rotational system to determine the effect of rotational unbalance on a larger mass system. The experiment involved use of a bar fixed at one end, with suspension by a spring in the vicinity of the opposite end, as well as a dashpot to remove energy from the system.
The aim of the investigation is to determine the relationship between the voltage, frequency, and amplitude in relation to the natural and damped frequencies: especially in terms of approaching and leaving the resonance.

Equipment Description
Equipment used included: I. Vibrating beam apparatus (in diagram, A-D) II. Power supply (2) III. Motor with power supply (C, G) IV. Contact tachometer (to measure rpm) V. LVDT (linear variable differential transformer) with voltage source from power supply (F, I) VI. Oscilloscope (H) VII. Fixed Mass of 500g (E)

The LVDT measured the motion of the beam end with a range of ±2 inches. The voltage source power supply must be set to 10 V as it is proportional to the signal out, per inch. LVDT sensitivity is ~0.365V.

The oscilloscope should be calibrated such that the horizontal scale which refers to time between major division (“major division” is scaled as per centimeter) should be 250ms or 500ms. The vertical scale refers to the magnitude of voltage per major division (cm) and should be set to about 500mV.

Calibration of the system involved ensuring an accurate relation between the oscilloscope voltage and the beam angle, which meant the LVDT vertical sensitivity was set to 200mV/division and evaluating the voltage change on the oscilloscope as well as the displacement at the tip of the beam as the 500g mass is added.

Procedure

Table 1: Measured Values Static | Attribute | Value | Unit | Fixed mass | 500 | g | Weight of Mass | 4.905 | N | Change in LVDT | 1.07 | V | Displacement of beam | 8 | mm | Beam Length | 825.5 | mm | beam pivot to fixed mass | 145.26 | mm | Free Vibration | Eq. LVDT (for 8mm displ.) | 1.07 | V | mm/Volt | 7.4766 | mm/V | Rad/Volt | 0.009057 | rad/V | | | |
Static
The change in LVDT signal on the oscilloscope was 1.07V and displacement amplitude was 8mm, which gives the rate of deflection per voltage change 7.4766mm/V.

This was then converted to angular deflection, as the beam is a torsional system. The rate of radians per volt is used throughout the report.

Weight and moment should be calculated to determine the equivalent stiffness of the system. Weight multiplied by distance to pivot supplies the moment, which is 0.7125N-m. This is divided by the angle change to produce the equivalent stiffness.

Angular displacement is determined by the displacement divided by beam distance, according to small angle approximation. The deflection in radians is 0.00969, making Keq = 360.29 N-m/rad. The equation is noted later on. Weight of the static mass was used for force, and the moment from this was determined by its distance along the beam.
Equation 1
Keq=M∙lbeamδang=F∙lmasslbeamδang
Table 2 : Forced Vibration Measurements Forced Vibration | Flywheel Diameter | 124.37 | mm | Hole diameter | 24.77 | mm | Hole depth | 8.71 | mm | Motor shaft from beam pivot | 368.3 | mm | Hole center from wheel center | 43.6 | mm |
The equation for static deflection:
Equation 2 δ=lnθ1θN+1N This was used to determine the damping ratio ζ. It was calculated when the beam was displaced: the time for oscillations to decay by half of the original magnitude was used. The static deflection was applied to equation 3 in order to determine the damping ratio:

Equation 3 ζ=δ2π2+δ2 Results
Table 3: Computed Values Attribute | Value | | δ | 0.181 | Per cycle | ζ | 0.0287815 | - | Keq | 360.29 | N-m/rad | Ceq | 1.07991 | - | J0 / Meq | 1.025 | kg | ωd | 18.756 | Rad/s | ωn | 18.748 | Rad/s |
After determining the damped natural frequency from visual analysis of the oscillation graph output from the LVDT onto the oscillator, it was possible to determine the natural frequency:
Equation 4 ωn=ωd1-ζ2 The values found allow us to compute the equivalent damping of the system using the formula:
Equation 5
Ceq=2ζKeqMeq

We can also solve for Meq, which is the same as J0:
Equation 6
Meq=Keqω2
And using these values we can apply them to the equation:
Equation 7
J0θ+Ceqθ+Keqθ=Rmeω2sinωt
Where R is the distance from pivor to motor shaft, m is the mass of the missing hole (determined by multiplying it’s volume dimensions by the density of steel), e is the radius of the small hole. Using previously calculated values, we can evaluate calculated displacement amplitude for each point of data using equations 7 and 8. These values are listed in Table 5.

Table 4: Experimental Data Relating Voltage and Motor Speed to Displacement Exact Voltage | Motor rpm[Tachometer] | Motor Hz[Oscilloscope] | Motor Rad/s[Tachometer] | Motor Rad/s[Oscilloscope] | Displacement Amplitude (V) | Angle Amplitude (Rad) | 7 | 124 | 2.53 | 12.98525 | 15.89646 | 0.08 | 0.000725 | 7.5 | 132 | 2.719 | 13.82301 | 17.08398 | 0.160 | 0.001449 | 8 | 137 | 2.89 | 14.34661 | 18.15841 | 0.560 | 0.005072 | 8.25 | 142 | 3.012 | 14.87021 | 18.92495 | 0.900 | 0.008151 | 8.5 | 148 | 3.096 | 15.49852 | 19.45274 | 0.540 | 0.004891 | 9 | 162 | 3.31 | 16.9646 | 20.79734 | 0.310 | 0.002808 | 10 | 181 | 3.704 | 18.95428 | 23.27292 | 0.190 | 0.001721 |

Figure 1: Graph representing the data presented in Table 4

The theoretical amplitude was calculated using the following equation:
Equation 8
X=Meqmer21-r222ζr2
Where X denotes the calculated amplitude in Volts, m refers to the mass of the missing hole (determined by multiplying it’s volume dimensions by the density of steel), e to the radius of the small hole, r is the ratio of the frequencies (“R Value” on table 5), Meq is the equivalent mass of the system determined above, and ζ is the damping ratio listed above.

Table 5: Experimental and Calculated Data for Displacement Amplitude Voltage | Motor rad/s Oscilloscope | Displacement Amplitude | Angle Amplitude | R Value | X- Amplitude Calc. | 7 | 15.89646 | 0.08 | 0.00072456 | 0.692612 | 0.042423 | 7.5 | 17.08398 | 0.16 | 0.00144912 | 0.911232 | 0.215769 | 8 | 18.15841 | 0.56 | 0.00507192 | 0.96854 | 0.519389 | 8.25 | 18.92495 | 0.9 | 0.0081513 | 1.009427 | 0.769056 | 8.5 | 19.45274 | 0.54 | 0.00489078 | 1.037578 | 0.511509 | 9 | 20.79734 | 0.31 | 0.00280767 | 1.109297 | 0.23737 | 10 | 23.27292 | 0.19 | 0.00172083 | 1.24134 | 0.130324 |

Figure 2: Graph depicting the data represented in Table 5
From Figure 2 it can be observed that the calculated amplitude fluctuates less than the experimentally derived values. The curves however only have slight variations in magnitude, although they maintain a similar shape. This resemblance shows that the calculations for predicting amplitude are consistent and accurate as well as that the use of speed from the oscilloscope proves to be more accurate. This is addressed below in the discussion.

Discussion
The volume of measurements and the need for attention to units gives a possibility for human and machine error. The values obtained by the tachometer are significantly lower than those determined from the oscilloscope, leading to the possibility of inaccuracy in one or both.

It is evident that there is a constant difference in the speeds calculated from the Tachometer and the oscilloscope. In calculating the R value for each data point, It became evident that the tachometer readings were significantly off, as altering the speed source for the R value created a nearly exponential graph in figure 2, which was a completely inaccurate representation of the data. Due to this misrepresentation, it may be possible that the Tachometer was broken or being used incorrectly. It may have been pushed too hard against the spinning motor.

Results from graphical representation of the data indicate that the oscilloscope was more accurate in collecting data. Other data relations and equations demonstrate accuracy due to the consistency with experimental values. This investigation proved successful in verifying the relationship between rotational unbalance characteristics, especially pertaining to amplitude and resonance to speed.