Driven Harmonic Oscillator free essay sample

The value of k of the second string by calculating the mean is (9. 79  ± 0. 001) N/m and the value of k calculated using linear regression is (9. 1  ± 0. 5) N/m. The damping coefficient caused by magnet is (0. 021  ± 0. 001) Ns/m. The value of resonant frequency using spring 2 with no mass added is (2. 1  ± 0. 1) Hz, this value is obtained from looking at the maximum amplitude, the other values obtained from calculation and looking at the phase are 2. 05 Hz and (2. 5  ± 0. 1) Hz. Experiment Conducted: 27th April 2004 AIM General: Investigate various aspects of harmonic motion using a driven harmonic motion analyser Specific: Measure the spring constant k for two springs Measure the resonant frequency Investigate the effect of damping on the period of the oscillation, and to determine the damping coefficient b To investigate the relationship between the phase and resonant frequency. EQUIPMENT Driven harmonic motion analyser Computer interface Additional masses (0. 01 kg each) Magnetic damping unit 2 Springs Meter ruler INTRODUCTION Consider a mass attached to a spring, damped and oscillated with external force as seen on the following diagram: [pic] Figure 1. Mass attached to a spring with constant k and it is damped with a damping factor of b. F is force that is changing with time. The equation of motion for the mass is: [pic] Now consider the following cases: 1. Undamped (b = 0), Undriven (F0 = 0) harmonic oscillator The equation of motion become: [pic] The general solution for this second order differential equation is: [pic] Where [pic] 2. Underdamped, undriven (F0 = 0) harmonic oscillator The equation of motion for this case is: [pic] The general solution to this equation is: [pic] The new frequency of the oscillation is ? ’, where: [pic] 3. Underdamped, driven harmonic oscillator The equation of motion for this case is: [pic] The solution to this differential equation can be divided into two parts, which are the complementary and the particular solution. The complementary solution will decay with time and will become very small compared with the particular solution which is the steady state solution, which is: [pic] [pic] The resonant frequency is when the mechanical impedance is at the minimum value, that is when m? = k/?. EXPERIMENTAL SETUP [pic] Figure 2. The experimental setup. This is the overview of DHMA assembly EXPERIMENTAL PROCEDURE Taken from the lab sheet 1. Measuring the Spring Constant †¢ Adjust the length of the Drive Cord so that the 17 cm mark on the mass bar is aligned with the top edge of the Upper Mass Guide. †¢ Add 10g mass to the hanger, and record the position of the Mass Bar scale. Repeat the procedure for each additional mass added to the hanger. Using the balance check that the accuracy of the nominal values on the added masses. †¢ Determine the displacement caused by the addition of each mass. †¢ Two approaches can be taken to calculate the spring constant 1. Calculate the spring constant using Hooke’s law: F = kx 2. A better approach is to plot the added mass against the total displacement. Least square fit the data to a straight line and determine the spring constant from the gradient of the line. †¢ Compare the two values and account for any differences. †¢ Repeat experiment using other spring. 2. Measuring resonant frequency †¢ Reposition the bar. †¢ Set the DRIVE switch to off and the FUNCTION to PERIOD †¢ Record the period of oscillation for different masses. Compare with the calculated values. †¢ Do the same for the other spring 3. Extraneous Damping †¢ Record the mass of the mass bar. Set the drive switch to off and the FUNCTION to AMPLITUDE †¢ Pull the mass bar down and release it. The mass will oscillate freely with decreasing amplitude. Record it in a table. †¢ Plot the natural logarithm of the amplitude as a function of time and use least square line fitting technique to determine the damping factor b from the slope of the line. 4. Amplitud e versus driving frequency †¢ Set the driver amplitude at the back of the control unit to 1 mm and carefully realign the mass bar so that it does not contact the edge of the slot †¢ Add the magnetic damping to stabilise the oscillations †¢ Set the DRIVE to ON. Starting with 0. 5 Hz, increase the frequency in 0. 2 Hz increments, recording the amplitude of oscillation for each setting. When the amplitude starts to increase dramatically, reduce the increments to 0. 05 Hz for better resolution. For each reading wait until the amplitude stabilises, increasing the magnetic damping if it takes too long to stabilise (by decreasing the distance between the magnets). †¢ Using the computer interface to record the data. †¢ Plot the amplitude as a function of frequency. 5. Phase as a function of time †¢ Add the magnetic damping unit. †¢ Switch DRIVE to ON. Measure the resonant frequency. †¢ Select a driving amplitude and a magnetic damping setting that pro vide a reasonably large amplitude and stable phase and amplitude. †¢ Realign the bar (very important! ). †¢ Using the computer interface to record the data. †¢ Beginning with the lowest driving frequency, record the phase for the full frequency range of the DHMA. For each setting, wait for the transients to die out for an accurate reading. †¢ Plot the phase as a function of frequency, noting the location resonance. RESULTS 1. Measuring spring constant First Spring Table 1.