Date: March 2025 - May 2025

Role: Data Acquisition and Analysis, Team Member

My team of 3 peers and myself were tasked with instrumenting the response of a 2nd-order mechanical oscillator to various inputs and determining the systems characteristics such as natural frequency, damping ratio, and spring constant. The system consisted of two masses connected by a spring with damping due to friction as the masses move along the rod as shown in the diagrams above and below. Our team devised 3 distinct measurement schemes to determine the system characteristics: static measurements of the spring constant and dynamic measurements of the system’s response to sinusoidal and step inputs. Through each scheme, the overall results could be determined and compared. We then explained our methods, results, and their significance in a 70 page technical report found at this link. My role in this project was to assist in determining the measurement procedures and perform the data acquisition and analysis for each measurement scheme. This required creating MATLAB scripts to efficiently collect and analyze data, resulting in an overall time to obtain results for each measurement of 30 seconds per sample for our over 50 input signals. Then using these intermediate results, I created an extensive Excel spreadsheet to convert these measurements into values for the overall system characteristics and their uncertainties.

For the static spring constant measurements, the mass of the top mass and the length of the spring were measured. Then additional weights were placed on the top mass and the displacement of the spring was measured. These weights and corresponding displacements were fit to a line whose slope was the spring constant of the spring. Lastly, using the measured mass and spring constant, the natural frequency of the system was estimated

For the dynamic response of the system, a PCB Piezotronics accelerometer was placed on the top mass and used to convert the acceleration of this mass into a measurable voltage. To associate a voltage to an acceleration, a standard 3-point accelerometer calibration was used. A sinusoidal input of an unknown frequency was created by turning on the motor which oscillated the bottom mass, resulting in a response of the connected top mass. By running the motor at a range of speeds, the top mass’s response to various input frequencies was determined. Voltage data was collected from the accelerometer, filtered using a bandpass filter, converted to acceleration using the sensor calibration, integrated twice, and filtered using a high pass filter to remove any accumulated constants to obtain the measured displacement of the top mass over time. The amplitude of this displacement was then compared to the amplitude of the displacement of the bottom mass to obtain the magnitude ratio. The frequency at which the magnitude ratio was maximized was the resonance frequency of the system and the value of the magnitude ratio at this frequency was the quality factor. These values could be used to determine the natural frequency, damping ratio, spring constant, and damping coefficient of the system. To validate our results we compared the frequency measured by the accelerometer at a certain motor power to the frequency measured at the same motor power by a stroboscope. Additionally, we measured the difference in phase between the top and bottom masses at different input frequencies.

Lastly, a step input was created by displacing the top mass and suddenly releasing it, resulting in damped oscillations. The peak voltages and times at which they occurred were determined and used to calculate the damped natural frequency, time constant, natural frequency, damping ratio, spring constant, and damping coefficient of the system.

When compared, the results from each measurement method had uncertainty which overlapped with each other for each characteristic except for the damping ratio, providing significant confidence in the accuracy of our results. The measurements were also compared to theoretical predictions, exposing the non-idealities of our system such as friction’s magnified effects at low speeds and slight variation of the spring constant over the range of the spring’s displacement.