advertisement

Periods and Frequencies Period the time that it takes for an event to repeat itself it takes about 365 days for your birthday to come around Frequency inverse of period your birthday comes once a year Frequency is a measure of how frequently an event or a process occurs. Period is the time that it takes for the event to complete one cycle. Periods and Frequency Consider a simple spring-mass system shown at the right. If the mass is pushed down and let go it will oscillate in a manner that manifests itself by an up-and-down motion. The natural undamped frequency of the system is given 1 fn 2 k m in Hz, cycles per second 1 fn 2 k m in Hz, cycles per second Where fn is the natural frequency of the system in cycles per second, or Hertz (Hz) k represents the stiffness of the spring or an elastic member (N/m) and m is the mass of the system (kg) The period of oscillation, T, for the given system – the time that it takes for the mass to complete on cycle – is given by 1 m T 2 fn k in seconds An Example of Period and Frequency Given: a simple spring-mass system Find: the natural frequency of the system Solution: 1 fn 2 k m 1 2 5000 N/m 8Hz 2 kg Angular (Rotational) Speeds Rotational motion is quite common in engineering applications. Examples of engineering components with rotational motion include shafts, wheels, gears, drills, helicopter blades, CD drives, Zip drives, and so on. The average angular speed of a line segment located on a rotating object is defined as the change in its angular position (angular displacement) over the time that it took the line to go through the angular displacement. Ω = angular speed (rad per sec) ∆θ = angular displacement (radians) ∆t = the time interval in seconds. 8-6 It is a common practice to express the angular speed of rotating objects in revolutions per minute (rpm) instead of radians per second (rad/s). Convert 1600 rpm to rad/s: 1600( revolutions 2π radians 1 minute rad )( )( ) 167.5 minutes 1 revolution 60 seconds s 8-7 There exists a relationship between linear and angular velocities of objects that not only rotate but also translate as well. For example, a car wheel, when not slipping, will not only rotate but also translate. S r and dividing both sides by t S r t t with t Then we have : V V = rω ω = r Determine the rotational speed of a car wheel if the car is translating along at a speed of 55 mph. The radius of the wheel is 12.5 in. miles 1h 5280 ft (55 )( )( ) V h 3600 s 1 mile 77.4 rad/s = 739 rpm 1 ft r (12.5 in.)( ) 12 in. Examples of Frequencies of Various Electrical and Electronic Systems Engineering Fundamentals, By Saeed Moaveni, Third Edition, Copyrighted 2007 8-10