Report

Viscously Damped Free Vibration • Viscous damping force is expressed by the equation where c is a constant of proportionality. • Symbolically. it is designated by a dashpot • From the free body diagram, the equation of motion is .seen to be • The solution of this equation has two parts. • If F(t) = 0, we have the homogeneous differential equation whose solution corresponds physically to that of freedamped vibration. • With F(t) ≠ 0, we obtain the particular solution that is due to the excitation irrespective of the homogeneous solution. • → Today we will discuss the first condition • With the homogeneous equation : the traditional approach is to assume a solution of the form : where s is a constant. • Upon substitution into the differential equation, we obtain : • which is satisfied for all values of t when • Above equation, which is known as the characteristic equation, has two roots : Hence, the general solution is given by the equation: where A and B are constants to be evaluated from the initial conditions and • Substitution characteristic equation into general solution gives • The first term, , is simply an exponentially decaying function of time. • The behavior of the terms in the parentheses, however, depends on whether the numerical value within the radical is positive, zero, or negative. Positive → Real number Negative → Imaginary number • When the damping term (c/2m)2 is larger than k/m, the exponents in the previous equation are real numbers and no oscillations are possible. • We refer to this case as overdamped. • When the damping term (c/2m)2 is less than k/m, the exponent becomes an imaginary number, . • Because • the terms within the parentheses are oscillatory. • We refer to this case as underdamped. • In the limiting case between the oscillatory and non oscillatory motion , and the radical is zero. • The damping corresponding to this case is called critical damping, cc. • Any damping can then be expressed in terms of the critical damping by a non dimensional number ζ , called the damping ratio: and s1, 2 1 n 2 • The three condition of damping depend on the value of ζ i. ζ < 1 (underdamped) ii. ζ > 1 (overdamped) iii ζ = 1 (criticaldamped) See Blackboard i. ζ < 1 (underdamped) s1, 2 1 n 2 • The frequency of damped oscillation is equal to : i. ζ < 1 (underdamped) • the general nature of the oscillatory motion. ii. ζ > 1 (overdamped) • The motion is an exponentially decreasing function of time iii ζ = 1 (criticaldamped) • Three types of response with initial displacement x(0). STABILITY AND SPEED OF RESPONSE • The free response of a dynamic system (particularly a vibrating system) can provide valuable information concerning the natural characteristics of the system. • The free (unforced) excitation can be obtained, for example, by giving an initial-condition excitation to the system and then allowing it to respond freely. • Two important characteristics that can be determined in this manner are: 1. Stability 2. Speed of response STABILITY AND SPEED OF RESPONSE • The stability of a system implies that the response will not grow without bounds when the excitation force itself is finite. This is known as bounded-input-bounded-output (BIBO) stability. • In particular, if the free response eventually decays to zero, in the absence of a forcing input, the system is said to be asymptotically stable. • It was shown that a damped simple oscillator is asymptotically stable. • But an undamped oscillator, while being stable in a general (BIBO) sense, is not asymptotically stable. It is marginally stable. STABILITY AND SPEED OF RESPONSE • Speed of response of a system indicates how fast the system responds to an excitation force. • It is also a measure of how fast the free response (1) rises or falls if the system is oscillatory; or (2) decays, if the system is non-oscillatory. • Hence, the two characteristics — stability and speed of response — are not completely independent. • In particular, for non-oscillatory (overdamped) systems, these two properties are very closely related. • It is clear then, that stability and speed of response are important considerations in the analysis, design, and control of vibrating systems. STABILITY AND SPEED OF RESPONSE • Level of stability: Depends on decay rate of free response • Speed of response: Depends on natural frequency and damping for oscillatory systems and decay rate for non-oscillatory systems Decrement Logarithmic