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Unformatted text preview: HIGHERORDER LINEAR ORDINARY DIFFERENTIAL EQUATIONS III: Mechanical Vibrations David Levermore Department of Mathematics University of Maryland 26 October 2009 Because the presentation of this material in class will differ from that in the book, I felt that notes that closely follow the class presentation might be appreciated. 7. Mechanical Vibrations 7.1. SpringMass Systems 2 7.2. Unforced, Undamped Motion 3 7.3. Unforced, Damped Motion 6 7.4. Forced, Undamped Motion 8 7.5. Forced, Damped Motion 9 1 2 7. Mechanical Vibrations 7.1. SpringMass Systems. Consider a spring hanging from a support. When an object of mass m is attached to the free end of the spring, the object will eventually come to rest at a lower position. Let y o and y r be the vertical rest positions of the free end of the spring without and with the mass attached. We will assume that the mass is constrained to only move vertically and want to describe the vertical postition y ( t ) of the mass as a function of time t when the mass is initially displaced from y r , or is given some initial velocity, or is driven by an external force F ext ( t ). The forces acting on the mass that we will consider are the gravitational force F grav , the spring force F spr , the damping or drag force F damp , and the external or driving force F ext . Newtons law of motion then states that m d 2 y dt 2 = F grav + F spr + F damp + F ext . (7.1) Always be sure you are working in one of the standard systems of units. In MKS units length is given in meters (m), time in seconds (sec), mass in kilograms (kg), and force in Newtons (1 Newton = 1 kg m/sec 2 ). In CGS units length is given in centimeters (cm), time in seconds (sec), mass in grams (g), and force in dynes (1 dyne = 1 g cm/sec 2 ). In British units length is given in feet (ft), time in seconds (sec), mass in slugs (sl), and force in pounds (1 lb = 1 sl ft/sec 2 ). The gravitational force F grav is simply the downward weight of the mass. If we assume a uniform gravitational acceleration g then F grav = mg , (7.2) where g = 9 . 8 m/sec 2 in MKS units, g = 980 cm/sec 2 in CGS units, and g = 32 ft/sec 2 in British units. The spring force is modeled by Hookes law F spr = k ( y y o ) , (7.3) where k is the socalled spring constant or spring coefficient. This is a fairly good model provided y y o does not get too big. When there is no external driving force, the mass has a rest position y r < y o that satisfies 0 = F grav + F spr at y = y r . Hence, we have mg = k ( y r y o ) = k ( y o y r ) = k  y r y o  . (7.4) Sometimes you will be given  y r y o  and have to figure out k from this relation. 3 The damping force is modeled by F damp = dy dt , (7.5) where 0 is the socalled damping coefficient. This is not as good a model for damping force as Hookes Law was for the spring force, but we will use it because of its simplicity....
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