Equation Case

Graph plot

The graph below shows the plot of graph of Y/rβ against Ω, for different values of ζ. The equation used here is:

Here, the ranges of value are:   for Y/rβ,  for Ω, and the values of ζ vary from 01 to 1.1, in increments of 0.2.

            Y/rβ

                                    Ω

                        Graph of Y/rβ against Ω, for different values of ζ

The table below shows the value calculated and used for plotting the graph

                                    Ω                                                                                Y/rβ

                                    Table of Values used for plotting the graph

Significance of equation with frequency

The equation (1) is:

This is the equation of motion for a system of mass (M + m), which is constrained to move in the vertical direction, when mass m placed at a distance r from its centre of gravity is rotating with an angular velocity ω. Here, k is the stiffness of spring, c is the damping coefficient for a system that has viscous damping, and  is the phase lag.

The natural frequency of the system depends only on the stiffness and the mass.. This can be written in equation form as:

                  (www.lds-group.com/docs/site_documents/App%2011.pdf)

That is to say if the spring used in the system was of greater stiffness, the natural frequency would increase, while if the mass of the systems was increased by adding more mass to the spring, the natural frequency would decrease.

At low speeds, where r is small, the amplitude of the motion entire mass of the system is almost 0, while at very large values of r, the value of the amplitude becomes constant, at a value equal to rm/M.  The value of damping in the system is not necessary in this case; also there is no need of the actual speed of the mass m (Grieve 2004).

Effect of rotational imbalance

Rotational imbalance exists in various types of rotating equipment for e.g. a rotating machine or a rotor. As seen above it is caused by when there is an uneven distribution of the mass around the designated axis of motion. This rotation when occurs in a machine i.e,. When the unbalanced part of a machine rotates, it can cause the entire structure to vibrate. This imbalance then generates forcing function which in turn that affects the structure. This is even seen in cars. Sometimes when a car reaches a, the car will shake, sometimes quite violently. What happens is that the rotational speed of the wheels tires to be near to the natural frequency of the car on its suspension, so that the amplitude becomes a maximum (Grieve 2004).

The rotating machines mentioned above include – turbines, electric motors and electric generators, fans, or rotating shafts, washing machine, steam/gas turbines, computer disk drives etc. (www.lds-group.com/docs/site_documents/App%2011.pdf)

Rotation vibration is usually an unwanted effect. It can be eliminated by properly balancing the machine. For example, a rotor can be balanced by placing a balancing mass, m1, on the rotor diametrically opposite m, and at a distance, h, such that m1h = mr. This balancing mass m1, then produces an excitation force that is exactly equal and opposite to the force that is produced by the out-of-balance, so that there is no resultant excitation force (Grieve 2004).

                                               References

“Basics of Structural Vibration: Testing and Analysis”, LDS group, 2003

www.lds-group.com/docs/site_documents/App%2011.pdf

Grieve D J, “Engineering Structures: STRC21 Home Page”, 16th February 2004,

http://www.tech.plym.ac.uk/sme/strc201/struchome.htm