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Slovak University of Technology Faculty of Material Science and Technology in Trnava APPLIED MECHANICS Lecture 04 SINGLE-DOF SYSTEM FREE VIBRATION, WITH DAMPING Model of single-DOF system with a viscous damper . .. x, x, x b k m b - coefficient of viscous damping, k - stiffness of spring, m - mass. SINGLE-DOF SYSTEM FREE VIBRATION, WITH DAMPING Equation of motion mx bx kx 0 b 2m x b k x x 0 m m x 2x 02 x 0 - damping ratio, k 0 - natural angular frequency of undamped system m . The characteristic equation r 2 2r 02 0 with the solutions r1,2 2 02 SINGLE-DOF SYSTEM FREE VIBRATION, WITH DAMPING r1,2 2 02 Case 1 2 02 0 the roots r1 and r2 are complex conjugate (r1, r2 C, where C - the set of complex numbers ) Case 2 2 02 0 the roots r1 and r2 real and distinct (r1, r2 R , where R - the set of real numbers) Case 3 2 02 0 the roots r1 = r2 are real and identical 2 (r1, r2 R , where R - rthe set of real numbers) SINGLE-DOF SYSTEM FREE VIBRATION, WITH DAMPING For case 3 bcr k 0 2m m b = bcr is called the critical damping coefficient bcr 2 km 2m0 One can classify the vibrations with respect to critical damping coefficient as follows: Case 1: b bcr - complex conjugate roots (low damping and oscillatory motion). Case 2: b bcr - real and distinct roots (great damping and aperiodic motion). Case 3: b bcr - real and identical roots (critical damping and aperiodic motion). SINGLE-DOF SYSTEM FREE VIBRATION, WITH DAMPING Case 1: Complex conjugate roots 2 2 The term d 0 2 02 0 is the quasicircular frequency. The roots r1,2 2 02 i 02 2 id The solution x e t ( Ad sin( d t ) Bd cos(d t )) Cd e t sin( d t d ) where Ad, Bd and Cd, d are integration constants. SINGLE-DOF SYSTEM FREE VIBRATION, WITH DAMPING Using the initial condition x x0 t 0 x v0 and the derivative of equation x with respect to time x et ( Ad cos(d t ) Bd sin( d t )) d ( Bd cos(d t ) Ad sin( d t )) the integration constants are v x0 Ad 0 Bd x0 2 v x0 C d Ad2 Bd2 x02 0 , B x0 tan d d . Ad v0 x0 SINGLE-DOF SYSTEM FREE VIBRATION, WITH DAMPING The solution is xe t v0 x0 sin( d t ) x0 cos(d t ) or in amplitude form x x02 2 x0 v0 x0 t e sin d t arctan v0 x0 2 b d 0 1 bcr 2 2 Td d 2 2 0 - the quasicircular (or quasiangular) frequency - the quasiperiod of motion SINGLE-DOF SYSTEM FREE VIBRATION, WITH DAMPING The displacement, velocity and acceleration for parameters: x0 0 m v0 0,2 m/s 0 0,8 rad/s SINGLE-DOF SYSTEM FREE VIBRATION, WITH DAMPING The rate of decay of oscillation: Cd e t sin( d t d ) x(t ) 1 Td e x(t Td ) Cd e (t Td ) sin[ d (t Td ) d ] e Td The logarithmic decrement L is defined: L ln x(t ) 2 b ln e Td Td x(t Td ) 02 2 md SINGLE-DOF SYSTEM FREE VIBRATION, WITH DAMPING Case 2: Real and Distinct Roots 2 02 0 The roots of the characteristic equation r1 2 02 1 , r2 2 02 2 , The solution x C1e 1t C2e 2t Aperiodic motion - tends asymptotically to the rest position t 0, x 0 SINGLE-DOF SYSTEM FREE VIBRATION, WITH DAMPING Case 3: Real identical roots 2 02 0 The roots of the characteristic equation r1 r2 The solution xe t (C1t C2 ) C1t C2 e t The diagrams of motion are similar to the previous case, with the same initial conditions. Critical damping represents the limit of periodic motion; hence, the displaced body is restored to equilibrium in the shortest possible time, and without oscillation. Many devices, particularly electrical instruments, are critically damped to take advantage of this property.