Equations between those coordinates for example a

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Unformatted text preview: coordinates. For example, a point in space has three degrees-of-freedom ( x, y, z ) and a rigid body in space has six ( x, y, z,θ x ,θ y ,θ z ) . Lecture Notes -5- 06/16/06 12:25 PM Mechanical Vibrations I A decibel is a log scale measure of relative power. Specifically, it is ten times the log P base ten of the power to a reference power, dB = 10log10 . If the quantity being P ref referenced is not power based, then the quantities are first squared, yielding V2 V dB = 10log10 2 or dB = 20log10 . V V ref ref An octave is a frequency range where the upper frequency is twice the lower frequency. A decade is a frequency range where the upper frequency is ten times the lower frequency. Bandwidth and frequency span both terms describe a range of frequencies from low to high. Bandwidth is particularly associated with the energy of the response of a system and in this course is used in reference to damping (see section eight.) Having established a small set of basic nomenclature, it is now possible to provide a brief overview of the direction of this course. In this course, Mechanical Vibrations I, much of the effort will be spent upon bringing together the many concepts and techniques developed in previous courses and applying them to the solution of vibrations problems. Note that, it is possible to solve simple, but representative, problems utilizing only information previously learned. To help clarify which courses and knowledge are used, the following introductory example has been developed. Lecture Notes -6- 06/16/06 12:25 PM Mechanical Vibrations I Introductory Example For the figure given, assume that the object rolls without slipping. The objective will be to develop the equation of motion and determine the natural frequency of the system. (A more complete example would involve a general multi-degree of freedom system, where the objective would be to determine; the equations of motion, the system pole, mode shapes, and steady-state response.) Rolls w/o slipping f (t ) θ (t ) k x(t ) r m, J cg Step 1: Draw the free-body diagram for each independent rigid body in the system. Be sure to include all external forces and moments and their points of application. (Recall that the internal force resulting from the spring is due to the actual relative motion of the ends of the spring. Therefore the coordinate x is measured from the free length of the spring.) [Mechanics II, Kinematics & Dynamics] Step 2: Write down all the force and moment balance equations. [Mechanics II, Kinematics & Dynamics] f ∑ F ) f − k x − f = mx ∑ F ) N − mg = 0 ∑M ) f r + f r = J θ x f y kx = J cg θ mx mg ff cg f cg N Step 3: Identify all relevant coordinate constraint equations. [Mechanics II, Kinematics & Dynamics] x = rθ and therefore x = rθ & x = rθ Step 4: Choose the desired independent coordinates and eliminate all the dependent coordinates by substituting the relevant constraint equations. [Mechanics II, Kinematics & Dynamics] ∑F )...
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