Modelling Excercise - Assignment Example

Name Tutor Course Date Mass spring systems Objectives These study aims at studying the diverse behaviors of a linear mechanical system. This involves developing mathematical formulas that describe the models under different conditions. Further the physical implications of the models should be ascertained from the mathematical models. This helps in understanding various mathematical relationship of a system which can be used to describe case scenarios of spring motion Introduction The basic principal behind the operation of springs is the Hooke’s law which states that the length of a string containing a mass hang freely will always be directly proportional to the force due the mass attached to it unless the stretching limit is exceeded. When the spring is stretched to some point and then released the string will vibrate in a harmonic motion. Procedure A spring of negligible mass at its normal state has a length of a, the stiffness of the spring is given by k which is a value greater than zero as shown in figure 1a. Figure 1a A particle of mass m is then attached at the end of the spring as shown in figure 1b Figure 1b X is defined as the distance from the ceiling which is treated as the origin. Under this condition, the mass will exert a force on the spring. The string will therefore exert an opposite force on the mass which is directly proportional to constant k in an attempt to move to its natural length a. At the same time the mass will exert a downward force mg where g is the gravitational force approximated to be 9.8m/s When the air resistance is neglected; gravitational force and the force due to the spring will therefore be acting on the particle. Question one Derivation of equation of motion of the particle; The force exerted on a free hanging spring is dependent on the change in length and the spring constant as shown by the equation below. Where; is the change in length of the spring At equilibrium this force is equal to the force exerted by the mass m, mg but opposite Therefore; To change the coordinate system Then we define the following; Where ; V=velocity =the phase =angular frequency If the particle is pulled for a distance of X0 beyond the equilibrium position, and then released at time t = 0 with zero initial velocity. The forces acting on the particle will therefore be; The system will therefore be oscillating for a distance xo The point of reference can be taken as Taking the initial direction to be upward, then At time t=0, the following variables apply to define the equation in term of time t When the particle is in motion, it is governed by a second order differential equation since; The differential equation then is given as; M= the mass k = the spring constant. From the above equation, the velocity is given as The displacement is therefore found by getting the integrating of the above equation which becomes; Substituting this into the differential equation gives; Then an example of plot of the displacement time curve is given as shown in figure 2 Figure 2 The displacement therefore varies directly proportional to the stiffness of the spring k while it varies inversely to the mass attached to the spring m. This gives an explanation of the fact that the oscillations will remain constant at a particular absolute value. Question two When the air resistance is considered, it will introduce a third variable C which is the damping factor of the system. The differential equation then is given as; C= the damping factor. From the above equation, the velocity is given as The displacement is therefore found by getting the integral of the above equation which becomes; Substituting this into the equation gives; In order to plot the values we first determine assign the constants some arbitrary values say M=1 k=1 c=1/3 Then the plot of the displacement time curve is given as shown in figure 2 Figure 2 The displacement therefore varies directly proportional to the stiffness of the spring k while it varies inversely to the mass attached to the spring m. The resistance due to air friction exerted in real systems indicates a direct proportionally relationship with the magnitude of displacement of the particle. The air resistance will provide the damping factor to the system and therefore the oscillations will die away with time. At time t=∞ the damping effect will have completed will be over and the system will be now operate steadily at equilibrium point. This is because the oscillations die away with time. From the above function, the time when the system will change when the displacement is zero and thus the system is at equilibrium since there is no oscillation. The critical value of c can then be calculated by equating displacement to zero Therefore the value of c is given as Question three When an actuator of is added; The resultant force on the system can be represented by the function below; Therefore the differential equation describing the new function will change to The nature of the actuator or the added force is sinusoidal and in the direction of oscillation. The equation which describes the movement of the spring in this case is given below; This shows that for t>0 the oscillations will always be there since the forceful impact of the actuator will enhance the oscillations. The behavior of this system is that the amplitude of the oscillations will be increased by an equivalent force to the actuator. The damping of factor due to air resistance will remain the same however these might not bring the system back to equilibrium depending on the magnitude of the actuating force (Kelly 46). Question four. The behavior of the system when the air resistance is neglected that is c=0 The equation of motion will then change to = When This means that the actuating force is the same as the natural frequency of the system. Depending on the phase of the two motions the actuating force can be used to enhance dumping or pump energy to the oscillations. For this case the oscillations will keep on increasing since the actuating force is in phase with the actuating the system oscillation When The natural frequency is not the same then that means there will be distortion of the oscillation of the system. The smooth harmonic motion of the particle will be interfered since there will be more than one frequencies of oscillation in the system. Conclusion Mechanical systems behave uniquely when the conditions are changed. The harmonic oscillation of a system can be achieved in two situations where there is no damping and in situations where an actuating force with the same frequency as the natural frequency is applied Work cited Kelly, Graham. Fundamentals of Mechanical Vibrations. 1st ed. New York: McGraw Hill, 1993. Print. Read More