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A PPLIED M ECHANICS

Slovak University of Technology Faculty of Material Science and Technology in Trnava. A PPLIED M ECHANICS. Lecture 0 5. x,. x,. x,. F ( t ) = F 0 sin( w t ). k. m. SINGLE- DOF S YSTEM UNDAMPED F ORCED V IBRATION - HARMONIC EXCITING FORCE.

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A PPLIED M ECHANICS

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  1. Slovak University of Technology Faculty of Material Science and Technology in Trnava APPLIED MECHANICS Lecture 05

  2. .. . x, x, x, F(t) = F0sin(wt) k m SINGLE-DOF SYSTEMUNDAMPED FORCEDVIBRATION - HARMONIC EXCITING FORCE • The system is excited by a harmonic force of the form where F0- amplitude of the forced vibration, w - the forced angular frequencies.

  3. SINGLE-DOF SYSTEMUNDAMPED FORCEDVIBRATION - HARMONIC EXCITING FORCE • The equation of motion • The solution of equation • The particular solution xp • The constant Cpis determined for

  4. SINGLE-DOF SYSTEMUNDAMPED FORCEDVIBRATION - HARMONIC EXCITING FORCE • The solution resp. • The constants A and B (C and j) are determined from the initial conditions

  5. SINGLE-DOF SYSTEMUNDAMPED FORCEDVIBRATION - HARMONIC EXCITING FORCE • The constants • The derivative with respect to time • The solution is

  6. SINGLE-DOF SYSTEMUNDAMPED FORCEDVIBRATION - HARMONIC EXCITING FORCE • The displacement is a combined motion of two vibrations: • one with the natural frequency w0, • one with the forced frequency w • The resultant is a nonharmonic vibration • The amplitude is: where

  7. SINGLE-DOF SYSTEMUNDAMPED FORCEDVIBRATION - HARMONIC EXCITING FORCE • Resonance - excitating frequency w is equal to the natural angular frequencyw0- the resonance phenomenon appears. Curve of resonance Diagram of resonance phenomenon

  8. SINGLE-DOF SYSTEMUNDAMPED FORCEDVIBRATION - CENTRIFUGAL EXCITING FORCE • Unbalance in rotating machines is a common source of vibration excitation. Frequently, the excited harmonic force came from an unbalanced mass that is in a rotating motion that generates a centrifugal force m0 is anunbalanced mass connected to the mass m1 with a massless crank of lengths r, the mass m0 rotates with a constant angular frequency w.

  9. SINGLE-DOF SYSTEMUNDAMPED FORCEDVIBRATION - CENTRIFUGAL EXCITING FORCE • The amplitude of the combined vibration wherem = m1+ m0. • The magnification factor

  10. SINGLE-DOF SYSTEMUNDAMPED FORCEDVIBRATION - CENTRIFUGAL EXCITING FORCE Variation of the magnification factor

  11. SINGLE-DOF SYSTEMUNDAMPED FORCEDVIBRATION - ARBITRARY EXCITING FORCE • The general case of exciting force is an arbitrary function of time

  12. SINGLE-DOF SYSTEMUNDAMPED FORCEDVIBRATION - ARBITRARY EXCITING FORCE • The differential equation of motion • The vibration in this case is described where t is presented in Figure; A, B are constants. The integral in equation is called the Duhamel integral.

  13. . x, .. b x, x F(t) = F0sin(wt) k m SINGLE-DOF SYSTEMDAMPED FORCEDVIBRATION - HARMONIC EXCITING FORCE • The mechanical model • The equation of motion The following notation is used:

  14. SINGLE-DOF SYSTEMDAMPED FORCEDVIBRATION - HARMONIC EXCITING FORCE • The equation of motion becomes • Case 1: or • The characteristic equation with the roots • The general solution of differential equation x1 - solution of the differential homogenous equation, x2 - particular solution of the differential nonhomogeneous equation

  15. SINGLE-DOF SYSTEMDAMPED FORCEDVIBRATION - HARMONIC EXCITING FORCE • The solution of the free damped system • The solution of the forced (excited) vibration • D1, D2 are determined by the identification method. • Solution of the forced vibration is introduced into equation of motion

  16. SINGLE-DOF SYSTEMDAMPED FORCEDVIBRATION - HARMONIC EXCITING FORCE • The linear system of algebraic equation • D1, D2 are obtained

  17. SINGLE-DOF SYSTEMDAMPED FORCEDVIBRATION - HARMONIC EXCITING FORCE • The forced vibration x2 or • The motion of the system

  18. SINGLE-DOF SYSTEMDAMPED FORCEDVIBRATION - HARMONIC EXCITING FORCE • The amplitude of forced vibration • The magnification factor and phase delay - damping ratio

  19. SINGLE-DOF SYSTEMDAMPED FORCEDVIBRATION - HARMONIC EXCITING FORCE • The graphic of the vibration

  20. SINGLE-DOF SYSTEMDAMPED FORCEDVIBRATION - HARMONIC EXCITING FORCE • Resonance A-F characteristics Phase delay

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