S.S.E Al Akhawayn University

Dynamics (Egr2301) - Spring 2015 ProjectVideo Analysis and Modeling of Dynamic Systems

Project done by:Ilias LAROUSSIAnass MILOUDI

Supervised by : Dr Hassan DarhmaouiObjective: Modeling of a real life dynamic system (mainly related to vibrations). Then define and modify the force expressions, parameter values and initial conditions, in a way that our dynamic model simulation will synchronize with and draws itself on the video.

A. Introduction:The dynamic behavior of structures is an important topic in many fields. Aerospace engineers must understand dynamics to simulate space vehicles and airplanes, while mechanical engineers must understand dynamics to isolate or control the vibration of machinery. In civil engineering, an understanding of structural dynamics is important in the design and retrofit of structures to withstand severe dynamic loading from earthquakes, hurricanes, and strong winds, or to identify the occurrence and location of damage within an existing structure.In this project we recorded four videos related to mechanical vibrations. Then we tried to analyze and model those videos using tracker. Tracker is a video analysis and modeling program that enables the creation of particle model simulations based on Newton's laws and to compare their behavior directly with that of real world objects captured on a video.B. Theoretical background about the motions studied: 1) Free vibrations on rigid bodies:Considering first the free vibration of the undamped system .Newtons equation is written for the mass m. The force mx exerted by the mass on the spring is equal and opposite to the force kx applied by the spring on the mass:Equation 1 mx + kx = 0 Where x = 0 defines the equilibrium position of the mass. The solution of Eq. 1 is x = A sin ( t) + B cos ( t )where the term is the angular natural frequency defined by n = rad/sec

2) Damped forced vibrations :

The behavior of the spring mass damper model varies with the addition of a harmonic force. A force of this type could, for example, be generated by a rotating imbalance.

Summing the forces on the mass results in the following ordinary differential equation:

Thesteady statesolution of this problem can be written as:

The result states that the mass will oscillate at the same frequency,f, of the applied force, but with a phase shiftThe amplitude of the vibration X is defined by the following formula.

Where r is defined as the ratio of the harmonic force frequency over the undamped natural frequency of the massspringdamper model.

The phase shift,is defined by the following formula.

3) Torsional pendulum: Letbe the angle of rotation of the disk, and letcorrespond to the case in which the wire is untwisted.

A torsional pendulum.

Any twisting of the wire is inevitably associated with mechanical deformation. The wire resists such deformation by developing arestoring torque, which acts to restore the wire to its untwisted state. For relatively small angles of twist, the magnitude of this torque is directly proportional to the twist angle. Hence, we can write(1)

Whereis thetorque constantof the wire. The above equation is essentially a torsional equivalent to Hooke's law. The rotational equation of motion of the system is written(2)

Whereis the moment of inertia of the disk (about a perpendicular axis through its Centre). The moment of inertia of the wire is assumed to be negligible. Combining the previous two equations, we obtain(3)

Equation(3) is clearly a simple harmonic equation. Hence, we can immediately write the standard solution is(4)

Where (5)

We conclude that when a torsion pendulum is perturbed from its equilibrium state (i.e.,), it executes torsional oscillations about this state at a fixed frequency,, which depends only on the torque constant of the wire and the moment of inertia of the disk. Note, in particular, that the frequency is independent of the amplitude of the oscillation [providedremains small enough that Eq.(1) still applies]. Torsion pendulums are often used for time-keeping purposes. For instance, the balance wheel in a mechanical wristwatch is a torsion pendulum in which the restoring torque is provided by a coiled spring.

4)Damped free vibration:To start the investigation of the massspringdamper assume the damping is negligible and that there is no external force applied to the mass (i.e. free vibration). The force applied to the mass by the spring is proportional to the amount the spring is stretched "x" (assuming the spring is already compressed due to the weight of the mass). The proportionality constant, k, is the stiffness of the spring and has units of force/distance (e.g. lbf/in or N/m). The negative sign indicates that the force is always opposing the motion of the mass attached to it:

The force generated by the mass is proportional to the acceleration of the mass as given byNewtons second law of motion:

The sum of the forces on the mass then generates thisordinary differential equation:Simple harmonic motion of the massspring systemAssuming that the initiation of vibration begins by stretching the spring by the distance ofAand releasing, the solution to the above equation that describes the motion of mass is:

This solution says that it will oscillate withsimple harmonic motionthat has anamplitudeofAand a frequency offn. The numberfnis called theundamped natural frequency. For the simple massspring system,fnis defined as:

C. Problem solving and Modeling: We used Newton second law of motion to model and analyze all the videos.Free vibrations on rigid bodies:Our video analysis and modeling

The motion of the rigid body can be mathematically determined through traditional video analysis of the y vs. t graph, v vs. t graph and a vs.t. In the Data Tool, select the region of data that corresponds to the motion, from t = 0.000 to 4 s and a Curve Fit allows us to determine a Fit Equation of y = A*sin (B*t+ C).The forces exerted are: the weight and the spring force

Damped free vibrations:Our video analysis and modeling:

Equation used:

The damped free vibration of the video can be mathematically determined through traditional video analysis of the y vs. t graph, v vs. t graph and a vs.t. In the Data Tool, select the region of data that corresponds to the motion, from t = 0.000 to 3 s and a vs. t graph. Curve Fit allows us to determine a Fit Equation of y = A*sin (B*t+ C). The forces exerted are: the weight and the spring force and the drag force

Torsional pendulum (bonus video):Our video analysis and modeling:

Equation used: we have chosen in this case Theta as Y The motion of the torsional pendulum can be mathematically determined through traditional video analysis of the y vs. t graph, v vs. t graph and a vs.t. In the Data Tool, select the region of data that corresponds to the motion, from t = 0.000 to 1.5 s and a Curve Fit allows us to determine a Fit Equation of y = A*sin (B*t+ C).The forces exerted are: the weight and the

Damped forced vibrations (bonus video):Our video analysis and modeling:

Equation used:

The motion of the ruler can be mathematically determined through traditional video analysis of the y vs. t graph, v vs. t graph and a vs.t. In the Data Tool, select the region of data that corresponds to the motion, from t = 0.000 to 7 s and a Curve Fit allows us to determine a Fit Equation of y = A*sin (B*t+ C).The forces exerted are: the weight and the drag force of the air, Initial force exerted on the ruler.

Simple pendulum:Our video analysis and modeling:

Equation used: The motion of the pendulum can be mathematically determined through traditional video analysis of the y vs. t graph, v vs. t graph and a vs.t. In the Data Tool, select the region of data that corresponds to the motion, from t = 0.000 to 2 s and a Curve Fit allows us to determine a Fit Equation of y = A*sin (B*t+ C) The forces exerted are: the weight and the tension of the rope.

Ball oil:Our video analysis and modeling:

Equation used:

The motion of the ball oil can be mathematically determined through traditional video analysis of the y vs. t graph, v vs. t graph and a vs.t. In the Data Tool, select the region of data that corresponds to the motion, from t = 0.000 to 6 s and a Curve Fit allows us to determine a Fit Equation of y = A*sin (B*t+ C)The forces exerted are: the weight and the drag force of oil, the spring force. SHMS scale:Our video analysis and modeling:

Equation used:

The motion of SHMscale can be mathematically determined through traditional video analysis of the y vs. t graph, v vs. t graph and a vs.t. In the Data Tool, select the region of data that corresponds to the motion, from t = 0.000 to 2 s and a Curve Fit allows us to determine a Fit Equation of y = A*sin (B*t+ C)The forces exerted are: the weight and the tension of the spring.

Discussion After modeling and analyzing the videos we found that the majority of the graphs are pretty much similar.

Conclusion:

2) References:

3) Appendix: