This report investigates resonance phenomena of driven damped simple harmonic oscillations by measuring accelerations and frequencies of an oscillator. Since it turned out that oscillations with low damping have smaller errors than those with high damping, small damping cases are employed to examine the resonance phenomena. Based on the Newton's 2nd Law \cite{book}, the motion of oscillations can be modeled by the following differential equation and its solution.

\begin{equation} \label{eq1}

\ddot x+2\gamma \dot x+\omega_{0}^{2} x=\left(\frac{F_{0}}{m}\right)e^{i\omega t}

\end{equation}

\begin{equation} \label{eq2} x(\omega)=\frac{F_{0}/m}{\omega_{0}^{2}-\omega^{2}+2i\gamma\omega}=A(\omega)cos(\omega t-\phi)

\end{equation}

Acceleration
*…show more content…*

In total, 16 experimental data points have been taken after settling to each steady-state for various driven frequencies. The voltage data obtained in Lab View is then converted to the acceleration data by multiplying its amplitudes by the slope of the two calibration points. Here, the voltage data is calibrated with gravity only and with no acceleration. The slope is called calibration factor. Errors for accelerations are the calibration factor times the error of the cursor measurements of Lab View, and that is the minimum difference Lab View can measure. Errors for the phase shifts are obtained by equation \ref{eq5} where \(\Delta f¥\) is an error for the driven frequency derived from the variation in frequency displayed on the oscilloscope \cite{oscilloscope}, \(t\) is a time shift between the peak of the acceleration of the oscillator and the driven force, \(\Delta t\) is an error for the time derived from the grid square size in the Lab View waveform graph.

\begin{equation} \label{eq5}

\Delta \phi = \phi \sqrt{(\frac{\Delta f}{f})^{2}+(\frac{\Delta t}{t})^{2}}

\end{equation}

At the beginning of the experiment, the natural frequency of the oscillator with neither the eddy currents nor driving force is observed as \(\omega_{0} = 75.7 \pm 0.1 \) rad/s or equivalently \(f_{0} = 12.05 \pm 0.01 \) Hz.

The result of the experiment is summarized in Figure ~\ref{fg:A} and ~\ref{fg:AandPhi}. The overall

Related

- Decent Essays
## Physics : The Back And Forth Motion

- 1168 Words
- 5 Pages

The pendulum was pulled to about 15 cm from the motion detector. In case of the mass on a spring, the mass was pulled till just a few inches away from the motion detector.

- 1168 Words
- 5 Pages

Decent Essays - Good Essays
## Sir Isaac Newton: Standing on the Shoulders of Galileo and Aristotle

- 1454 Words
- 6 Pages

The general and widespread acceptance of Sir Isaac Newton’s models and laws may often be taken for granted, but this has not always been so. Throughout history, scientists and philosophers have built on each other’s theories to create improved and often revolutionary models. Although Newton was neither the first nor the last to bring major innovations to society, he was one of the most notable ones; many of his contributions are still in use today. With the formulation of his laws of motion, Sir Isaac Newton contributed to the downfall of Aristotelianism and provided a universal quantitative system for approximating and explaining a wide range of phenomena of space and the physics of motion, revolutionizing the study and understanding

- 1454 Words
- 6 Pages

Good Essays - Decent Essays
## Physics Mastery Lab

- 830 Words
- 4 Pages

After the positions were recorded for frequencies 1,803 Hz, 2,402 Hz, 3,002 Hz, 3,600 Hz, and 4,201 Hz, the wavelength was determined for each. This was done by subtracting the initial position from the final position (position final–position initial=wavelength). Using the calculated wavelength, the speed of sound in air at each frequency was determined by multiplying the wavelength by the frequency (speed of sound=wavelength x frequency). By adding the five speed values and dividing by the number of speeds, the average speed of sound was calculated. Then 344 m/s was used as the accepted

- 830 Words
- 4 Pages

Decent Essays - Satisfactory Essays
## Lab 5 B Friction Lab

- 297 Words
- 2 Pages

In addition to , the data of the major finds in the experiment were the control speeds or no sail had an average speed of 138.3 sail had an average speed of 127.3.Sliding friction data average was -252.4. Rolling friction data had an average was –30.21.

- 297 Words
- 2 Pages

Satisfactory Essays - Decent Essays
## Nt1310 Unit 7 Exercise 1

- 581 Words
- 3 Pages

-“The wavelength is 0.20 s.” This is wrong because wavelength’s unit is meter and it measures the distance between two adjacent locations in the disturbance, not the time one particle takes to finish one vibration.

- 581 Words
- 3 Pages

Decent Essays - Decent Essays
## Nt1310 Unit 6 Lab Report

- 1074 Words
- 5 Pages

Loosen the retaining screw and adjust the horizontal arm to the dimple that produce the 2nd largest radius between the spinning mass and the rotating shift. Allow the spinning mass to hang straight down, without being connected to the spring. Position the pointer directly below the tip of spinning mass and secure the pointer to the base. Use the scale located on the base to determine the radius, r, of the pointer from the rotating shaft. Record this value onto Data Sheet A. The experimental uncertainty in r is estimated to be the width of the spinning mass tip – approximately 0.2 cm.

- 1074 Words
- 5 Pages

Decent Essays - Decent Essays
## Nt1310 Unit 4 Lab Report

- 1127 Words
- 5 Pages

We get that the percent error for method 1 is 19.3% and the percent error for method 2 is 1.2%. This affirms that the method 2 is the more accurate method to measure the resonant frequency. This may be due to the fact that method 1 involves using estimations by eye, which allows for human error and inaccurate estimations. The difference between the experimental and theoretical values for the resonant frequency could also result from faulty equipment, which may not function as the manufacturer claims or be worn down by natural decay.

- 1127 Words
- 5 Pages

Decent Essays - Decent Essays
## Nt1310 Unit 1 Lab Report

- 600 Words
- 3 Pages

The main mechanical structure is modified version of the structure that is described in the [1,3]. The device consists of a proof mass which is attached to two similar double-ended tuning fork resonators via a force amplifier like a mechanical lever with a special configuration. This schematic of this structure is depicted in Fig. 1. Each of the DETF resonators is actuated to their resonance electrostatically and they will be sustained in their resonance by a feedback loop[17]. When an external acceleration is applied to the structure in proper direction, the proof mass will deflect and as a result of this movement an axial force will transfer to the DETF resonators. This axial force will result a change in the stiffness of the resonators so there will be a shift in the resonance frequencies of these DETF resonators[3]. These shift in resonance

- 600 Words
- 3 Pages

Decent Essays - Better Essays
## How The Mass And Force Affect The Acceleration Of An Object

- 1691 Words
- 7 Pages

Newton’s laws of motion are three physical laws that describe the connection between a body and the different forces acting upon it, as well as its motion in response to those forces. Isaac Newton developed Galileo’s ideas further and developed three law of motions. Newton’s First Law of Motion states that an object at rest with remain this way unless if it affected by a force. Also if an object that is moving will continue at the same speed as well as the same direction until an unbalanced force acts upon it. An example of unbalance force is when a scooter is being driven, the friction and air resistance is going at it, the weight of the scooter is keeping the weight on the ground, the reaction force is going up and the thrust of the scooter going forward. The force’s tendency to resist any change in motion is called an object’s inertia. Newton’s Second Law of Motion states that an object will keep on accelerating in the direction of an unbalance force acting upon it. The mass of the object and the size of the force acting depends upon the size of the acceleration., F_net=m x a, is the formula to work out the total amount of force acting upon an object. This formula can be

- 1691 Words
- 7 Pages

Better Essays - Best Essays
## Torsional Mechanism System

- 2477 Words
- 10 Pages

ECP software runs on an IBM PC or compatible computer and corresponds with ECP's digital signal processor (DSP) based real-time controller. Its primary functions are supporting the downloading of various control algorithm parameters (gains), specifying command trajectories, selecting data to be obtained, and specifying how data should be plotted. ECP Model 205 torsion apparatus represents a broad and important class of practical plants including: rigid bodies, flexibility in drives, and coupled discrete vibrating systems. It easily transforms into second, fourth, and sixth (optional) order plants with collocated or no collocated sensor / actuator control. This laboratory dealt with a single simple degree of freedom.

- 2477 Words
- 10 Pages

Best Essays - Satisfactory Essays
## Spring Simulations

- 621 Words
- 3 Pages

Describe (sentences) the effect that manipulating each of the following variables has on the SIMPLE HARMONIC MOTION of the spring system. (Note, period, amplitude, speed max in description.

- 621 Words
- 3 Pages

Satisfactory Essays - Satisfactory Essays
## Nt1310 Lab 6 Hobbreviation

- 93 Words
- 1 Pages

Record the amplitude and corresponding period of vibration for one oscillation, followed by the sixth oscillation.

- 93 Words
- 1 Pages

Satisfactory Essays - Good Essays
## Sound Waves Essay

- 1607 Words
- 7 Pages

where smax is the maximum displacement or displacement amplitude, k is the angular wave number, and w is the angular frequency of the piston.

- 1607 Words
- 7 Pages

Good Essays - Decent Essays
## Physics Lab Report On Using A Simple Pendulum Model

- 932 Words
- 4 Pages

The objective of this practical is to determine acceleration due to gravity ‘g’ using the simple pendulum model. This is shown when a period of oscillation is seen to be independent of the mass of the mass ‘m’.

- 932 Words
- 4 Pages

Decent Essays - Decent Essays
## Pendulum Lab

- 923 Words
- 4 Pages

Theory states that T and d are related by the equation: T2 = kd3+ (4π2 l)/g where g is the acceleration of free fall and k is a constant.

- 923 Words
- 4 Pages

Decent Essays