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

2. SINGLE-DOF SYSTEM UNDAMPED FORCED VIBRATION - 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. m k
F(t) = F0sin(wt) x, . ..

3. SINGLE-DOF SYSTEM UNDAMPED FORCED VIBRATION - HARMONIC EXCITING FORCE
The equation of motion
The solution of equation
The particular solution xp
The constant Cp is determined for

4. SINGLE-DOF SYSTEM UNDAMPED FORCED VIBRATION - HARMONIC EXCITING FORCE
The solution
resp.
The constants A and B (C and j) are determined from the initial conditions

5. SINGLE-DOF SYSTEM UNDAMPED FORCED VIBRATION - HARMONIC EXCITING FORCE
The constants
The derivative with respect to time
The solution is

6. SINGLE-DOF SYSTEM UNDAMPED FORCED VIBRATION - 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 SYSTEM UNDAMPED FORCED VIBRATION - HARMONIC EXCITING FORCE
Resonance - excitating frequency w is equal to the natural angular frequency w0 - the resonance phenomenon appears.
Curve of resonance
Diagram of resonance phenomenon

8. SINGLE-DOF SYSTEM UNDAMPED FORCED VIBRATION - 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 an unbalanced 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 SYSTEM UNDAMPED FORCED VIBRATION - CENTRIFUGAL EXCITING FORCE
The amplitude of the combined vibration
where m = m1 + m0.
The magnification factor

10. SINGLE-DOF SYSTEM UNDAMPED FORCED VIBRATION - CENTRIFUGAL EXCITING FORCE
Variation of the magnification factor

11. SINGLE-DOF SYSTEM UNDAMPED FORCED VIBRATION - ARBITRARY EXCITING FORCE
The general case of exciting force is an arbitrary function of time

12. SINGLE-DOF SYSTEM UNDAMPED FORCED VIBRATION - 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. SINGLE-DOF SYSTEM DAMPED FORCED VIBRATION - HARMONIC EXCITING FORCE
The mechanical model m b k
F(t) = F0sin(wt) x, . .. x
The equation of motion
The following notation is used:

14. SINGLE-DOF SYSTEM DAMPED FORCED VIBRATION - 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 SYSTEM DAMPED FORCED VIBRATION - 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 SYSTEM DAMPED FORCED VIBRATION - HARMONIC EXCITING FORCE
The linear system of algebraic equation
D1, D2 are obtained

17. SINGLE-DOF SYSTEM DAMPED FORCED VIBRATION - HARMONIC EXCITING FORCE
The forced vibration x2 or
The motion of the system

18. SINGLE-DOF SYSTEM DAMPED FORCED VIBRATION - HARMONIC EXCITING FORCE
The amplitude of forced vibration
The magnification factor and phase delay
- damping ratio

19. SINGLE-DOF SYSTEM DAMPED FORCED VIBRATION - HARMONIC EXCITING FORCE
The graphic of the vibration

20. SINGLE-DOF SYSTEM DAMPED FORCED VIBRATION - HARMONIC EXCITING FORCE
Resonance
A-F characteristics
Phase delay