Characterization of Varying Pendulum Mass

In this lab project, we characterize an experimental system and utilize that characterization to develop a theoretical model to understand the relationship between the physical properties involved. Specifically, we practiced modeling a pendulum system with varying mass of the bob to understand how the varying mass will affect the drag and air resistance of the system. The data acquired from the experimental setup was processed and plotted to verify the relation between varying mass, damping coefficient and resistance coefficient 

Introduction:

This experiment emphasizes the theory of drag on the mass of a pendulum. In a conventional pendulum model, metrics such as max velocity and frequency are not governed by the end mass under the assumption of a friction-less, drag-less system. However, as part of this experiment, we want to account for drag to see if there is correlation between mass and the dampening of the system Our hypothesis is that we will still see an increase in dampening with an increase is mass, which we will explore through our experiments. 

Another aspect that we want to emphasize as part of this experiment is the use of a variety of sensors to inform our data. While the data used to find the dampening coefficient is only reliant on angular velocity as measured by the encoder, we also wanted to verify the accuracy of our encoder readings through acceleration data from an IMU (Interial Measurment Unit) and camera which can plot the position of the mass center through Matlab image-processing. Through this project, we hoped to be able to implement Matlab experiment-related tools such as image processing, FFT techniques, noise removal, plotting and regression to determine if mass plays a significant role in the dampening of the system. 

Theory:

A basic pendulum motion is a classic example in physics to demonstrate the behavior of an oscillation system and to understand Newtonian mechanics, including the concepts of periodic motion, gravity, oscillation, and friction. In vacuum with no air resistance, such a pendulum will oscillate indefinitely with constant amplitude. However, in the real world, there is always some air resistance that will damp down the oscillation and the amplitude of the pendulum will continuously decrease over time as the system is gradually losing mechanical energy to the air resistance. 

The model we will be using in this experiment will be used to establish a relation between mass of the pendulum bob (m), resistance coefficient (γ ) and damping coefficient(c). 

According to fluid dynamics, the frictional force (or drag) for a spherical object in motion depends on the properties of fluid and on the size, shape, and speed of the object. This relationship is described by the drag equation: FD = 1/2*CD*ρ*A*v^2 (1), where FD, CD, ρ, A and v are the drag force, drag coefficient, density of the fluid, cross-sectional area, and velocity of the object relative to the fluid respectively.

The equation of motion of a pendulum is given by: Iα = Στ (2), where all torques τ are calculated relative to the support point, α is the angular acceleration of the pendulum about this point, and I = mL^2 is the moment of inertia of the pendulum about the support point. 

The damping coefficient is related to the resistance coefficient and mass as γ = c/2m (3). 

After detailed derivation with these three equations which is showed below in the lab report, we extract an expression for the linear velocity of the pendulum at the lowest point which is given as follows:  |v(tlow)| = vm exp(−γtlow) (4), where vm is the maximum velocity,  γ is the damping coefficient that resulted from the drag and v(tlow) is velocity of the bob at the lowest point of oscillation.

This expression tells us that the velocity of the pendulum bob decreases over time exponentially due to the drag force experienced in the system, starting with the maximum velocity of vm. 

For easy plotting of this equation, it is ideal to take natural log on both side of the equation which gives: ln|v(tlow)| = −γtlow + ln(vm) (5). When ln|v(tlow)| is plotted against tlow, the corresponding slope of the plot should give us the damping coefficient and we can use Eq. (3) to understand and verify the relation between mass, damping coefficient and resistance coefficient. Therefore, we have derived a fundamental relation of three important physical parameters using the maximum velocity at the lowest point over time of a pendulum system. 

Sensors Used:

For our project, we mainly used 4 sensors as explained below:

1. Accelerometer: They are a class of sensors which measure acceleration and vibration. We use a 9-axis MEMS accelerometer to measure the acceleration of the pendulum end and compare it the to the measurements of the photo-encoder and the camera tracking. 

2. H206 Photo-interrupter: The H206 photo-interrupter sensor consists of an infrared light-emitting diode (LED) and a phototransistor that are arranged in a package. The LED emits infrared light, which is then detected by the phototransistor when an object passes through the gap. The output signal will change when an object interrupts the IR beam and will return to its initial state when the phototransistor detects the beam again. This process will result in an increase of one "tick” which accumulates within the RAM and is reported over serial. We created a disk of 32 laser cut slots to utilize this sensor. 

3. OHAUS Adventurer Scale: The OHAUS Adventurer Scale is a high-precision laboratory grade scale for precise measurements. It has 95% confidence interval of 0.01 gram.

4. Phone 14 Pro Camera: The iPhone 14 Pro camera has recording capabilities of 4K resolution at 60 frames per second. However, for this lab, the video was recorded in 1080p and 30fps with continuous autofocusing 

 Methods:

Setup:

To conduct the experiment we designed and fabricated an instrumented pendulum setup which allowed us to vary the mass at the end of the link. The link was made out of a lightweight wood as to make its mass and moment of inertia negligible relative to the contributions of the end masses. Additionally, the axle was fixed relative to the stand while the link was allowed to freely rotate in order to reduce the rotating mass. To reduce frictional losses at the axle interface, brass bushings were incorporated to the pendulum arms and lubricant was applied during assembly 

The accelerometer was attached to the pendulum link and its distance from the pivot axis was recorded. The encoder was fixed relative to the base and the target was integrated with the pendulum arm. A 64, 48, and 32 tooth encoder target were tested and it was found the 32 tooth target yielded the fewest reading errors. 

To collect data, the sensors were connected to an ESP32 microcontroller running an Arduino script which collected and exported the data to a computer for post-processing in MATLAB. Additionally, a smartphone camera was placed on a stand and used to record the motion of the pendulum link from the side (with the camera pointed in the direction of the pivot axis). The uncertainty due to the sensor measurements is considered in section 

Procedure:

1. With no masses attached to the pendulum arm, displace the pendulum arm from the vertical. 

2. Start the data collection script and begin recording video, then immediately release the pendulum.

3. Allow the pendulum to freely oscillate until it settles at the vertical then stop data collection. 

4. Repeat the preceding steps four times to collect a total of 5 trials for a given link mass. 

5. Repeat the preceding steps after attaching calibrated masses to the link, adding up to four masses. 

Five trials each should be conducted for the five link masses, totaling 25 trials 

Results and Discussion:

After the end of the experiment, we were able to extract the maximum velocities of the pendulum bob at the lowest point for each set of masses with the help of encoder attached in our experimental setup. Using the set of maximum velocities and their corresponding times, a linear graph of ln(vm) against time(s) was plotted for each set according to Eq (5).

The damping and resistance coefficients measured for the various mass combination using Eq (5) and Eq (3) are presented in the table below: 

Based on our results from the table, we can observe that as the mass of in the pendulum system increases, the damping coefficient decreases. However, we find some inconsistency in the resistance coefficient with increase in mass for 3rd and 5th set of mass. This is because the decrease in 6 damping coefficient is more compared to the increase in mass going from 29.2g to 41.9g and 54.6g to 67.3g.  

To find the uncertainty of the resistance coefficient uc, we used Gaussian uncertainty analysis taking into account both the uncertainty of the mass and uncertainty of the damping coefficient γ. The 95% uncertainty of the mass balance is 0.01g. The 95% uncertainty of γ is determined using  Gauss’ uncertainty propagation relationship after uγ can be found based on the analytical expression. Detail derivation of the uncertainty is shown and explained in the lab report provided down below. 

Based on the detail propagation error analysis provided in the report, the calculated values and associated uncertainties of the damping coefficient and resistance coefficient are reported below:  

This table does not demonstrate a correlation between resistance coefficient, c and mass. This means that while the raw measurements did not show a consistent value, through the uncertainty of each measurement, the c value can still be a constant, with exception to the smallest mass. This inconsistency may have been caused by the bending of the system itself. For higher masses, the testing stand would also move, causing increased dampening. However, it is possible that for the lowest mass, the testing stand flex was significantly reduced, allowing the pendulum to oscillate longer and have a perceived lower c value. For future iterations of this experiment, a more rigid test stand can be used. 

Hypotheses:

 The period of a pendulum is affected by both the length and the mass of the pendulum bob. The period is the time it takes for the pendulum to complete one full oscillation, and is given by the equation: 

                                                                                                                   T = 2π * sqr(L/g)

However, if the mass of the pendulum bob is increased while the length of the 8 pendulum remains constant, the period of the pendulum will also increase. This is because the added mass increases the inertia of the pendulum and reduces its ability to swing back and forth, which leads to a longer period. This new relationship can be expressed mathematically as follows: 

                                                                                            T2 = 2π s * sqrt(L/ g) ∗ sqrt(1 + (m2/ m1))

where T2 is the new period of the pendulum, m2 is the new mass of the bob, m1 is the original mass of the bob, and all other variables remain constant. 

Since the period is increasing, we hypothesize that the effective damping coefficient will decrease with increasing mass in order for the same power to be applied by the air/fluid on the surface area over a different length of time. 

Collected Data & Processing:

We collected data from 5 sets/iterations of 5 different point masses on the same pendulum apparatus. The values we sampled were the XYZ accelerations from the MPU9250 IMU sensor, the cumulative number of ticks from the photo-interrupter encoder, and the cumulative time in milliseconds. 

Unfortunately as shown in Figure above, the sampled acceleration data was extremely noisy and inconsistent that forced us to manually pick out "good” data from the entire batch. To summarize our data processing, we first isolated the data after we released the pendulum and reduced the noise of the accelerometer data by using FFT techniques to decompose and smooth the curves. Then we applied an appropriate offset to the raw data, removed outliers and missing sections of each set, and applied an offset to convert the data into SI units. 

Finally, to convert linear acceleration into radial acceleration we applied the following equation: 

                                                                                             α = sqrt( ax^2 + ay^2 + (az^2)/r ) 

For the encoder data, we also had the issue of missing samples. During processing, we first manually removed the repeat tick counts and their corresponding times, converted to SI units, then integrated in order to calculate the linear velocity. 

With use of a camera, we were also able to track the position of the end mass over the duration of oscillations. We drew a black circle at the center of the end mass so that the imfindcircles method could be used. Through visual examination of each of 5 runs from every mass, one run of each mass was selected. This was done as some runs had discrepencies in when the mass was released from rest causing offsets in the positional plots. 

Discrepancies:

We attempted to drastically reduce any impacts of friction and outside disturbances to the behavior of the pendulum by using bearings and lubricant within the axle, performing the experiment indoors, and tracking our movements with a camera. However, we still saw significant variations in our calculated resistance coefficients that is likely caused by these physical imperfections and disturbances from the outside world. Some of the discrepancies in the data can be accounted for by the limitations of the theoretical model, including the Stokes’ law drag proportionality FD ∝ v which neglects the advection terms in the Navier Stokes Equation for low Reynolds number flows [6] 

Additional assumptions include neglecting the mass of the pendulum arms, neglecting frictional losses, and ignoring the asymmetry of the drag area. The assumptions used to form the model were chosen as to not significantly detract from the model’s fidelity to the physical system. Other sources of error are listed in the Lab Report below. 

Conclusion:

Through our experiment, we were able to see through both the positional and velocity plots that the dampening decreased with increased mass, which was supported by the theory that more energy in the system will take longer to dissipate. The theory also provided a relationship c = 2γm. Since c is not a function of mass by basic drag theory, the c value should not change, ie. the dampening coefficient should change with the inverse rate of mass. Based on our calculations of γ and its uncertainty, we did not find a strong correlation between c and added mass, which is consistent with the theory. In general, through the use of the analysis techniques mentioned in the introduction such as FFT and image processing, we were able to determine the dampening of the system and its relationship with the coefficient of drag.