Introduction to Vibration. Professor Mike Brennan

Transcription

1 Introduction to Vibration Professor Mie Brennan

2 Introduction to Vibration Nature of vibration of mechanical systems Free and forced vibrations Frequency response functions

3 Fundamentals For free vibration to occur we need mass m stiffness The other vibration quantity is damping c

4 Fundamentals - potential and inetic energy

5 Fundamentals - damping

6 Fundamental definitions A xt () x Asin( t) t T T 2f (radians/second) Period T 2 Frequency f 1 T (seconds) (cycles/second) (Hz)

7 Phase A xt () t x Asin( t) x Asin( t ) Green curve lags the blue curve by radians 2

8 Harmonic motion A x( t) angular displacement t t One cycle of motion 2π radians

9 + imaginary - real + imaginary e Complex number representation b of harmonic motion A a Euler s Equation j cos x a jsin jb + real So x Acos x A cos x j Ae magnitude jasin jsin phase magnitude x A a b phase tan ba

10 Relationship between circular motion in the complex plane with harmonic motion Imaginary part sine wave Real part cosine wave

11 Free Vibration System vibrates at its natural frequency xt () t x Asin( t) n Natural frequency

12 Forced Vibration System vibrates at the forcing frequency xt () ft () xt () t x Asin( t) f Forcing frequency

13 Mechanical Systems Systems maybe linear or nonlinear input excitation system output response Linear Systems 1. Output frequency = Input frequency 2. If the magnitude of the excitation is changed, the response will change by the same amount 3. Superposition applies

14 Linear system Mechanical Systems Linear system Same frequency as input Magnitude change Phase change Output proportional to input

15 Linear system Mechanical Systems input excitation b a M system output response, y y Ma Mb M( a b)

16 Nonlinear system Mechanical Systems Nonlinear system output comprises frequencies other than the input frequency output not proportional to input

17 Nonlinear systems Mechanical Systems Generally system dynamics are a function of frequency and displacement Contain nonlinear springs and dampers Do not follow the principle of superposition

18 Mechanical Systems Nonlinear systems example: nonlinear spring f hardening spring x For a linear system f x force f linear softening spring displacement x

19 Mechanical Systems Nonlinear systems example: nonlinear spring Pea-to-pea vibration (approximately linear) force f Pea-to-pea vibration (nonlinear) Static displacement displacement x stiffness f x

20 Degrees of Freedom The number of independent coordinates required to describe the motion is called the degrees-of-freedom (dof) of the system Single-degree-of-freedom systems Independent coordinate

21 Degrees of Freedom Single-degree-of-freedom systems x Independent coordinate m

22 Spring Idealised Elements f1 2 f x1 x2 no mass is the spring constant with units N/m f x x f x x f f 1 2

23 Idealised Elements Addition of Spring Elements Series 1 2 total total is smaller than the smallest stiffness 1 Parallel total total is larger than the largest stiffness

24 Idealised Elements Addition of Spring Elements - example f R x T stiffness f x Is T in parallel or series with R? Series!!

25 Viscous damper c Idealised Elements f1 2 f x1 x2 no mass no elasticity c is the damping constant with units Ns/m f c x x f c x x f f 1 2 Rules for addition of dampers is as for springs

26 Viscous damper Idealised Elements f1 2 m x f f f mx 1 2 f mx f 2 1 rigid m is mass with units of g Forces do not pass unattenuated through a mass

27 Free vibration of an undamped SDOF system Undeformed spring System equilibrium position m System vibrates about its equilibrium position

28 Free vibration of an undamped SDOF system System at equilibrium position Extended position m m mx m mx x x 0 inertia force stiffness force

29 Simple harmonic motion The equation of motion is: m where n m x mx x x x m x x is the natural frequency of the system 2 n The motion of the mass is given by x X sin t o n

30 Simple harmonic motion Real Notation Complex Notation m x Displacement x X sin t o Velocity x X cos t Acceleration n n o n 2 n o n x j nt Xe j nt x jn Xe 2 j x X sin t n x Xe n t

31 Simple harmonic motion Imag x x Real t x

32 Free vibration effect of damping m c x The equation of motion is mx cx x 0 inertia force damping force stiffness force

33 Free vibration effect of damping x Xe nt time x Xe nt sin t d d T d 2 d Damping ratio Td Damping period Phase angle

34 Free vibration effect of damping The underdamped displacement of the mass is given by x Xe nt sin t d Exponential decay term Oscillatory term = Damping ratio = c 2m 0 1 n = Undamped natural frequency = m n d = Damped natural frequency = = Phase angle n 1 2

35 Free vibration effect of damping x t t Undamped ζ=0 Underdamped ζ<1 Critically damped ζ=1 Overdamped ζ>1

36 Degrees-of-freedom Single-degree-of-freedom system m x 1 Multi-degree-of-freedom (lumped parameter systems) N modes, N natural frequencies m m m m x 1 x 2 x 3 x 4

37 Degrees-of-freedom Infinite number of degrees-of-freedom (Systems having distributed mass and stiffness) beams, plates etc. Example - beam Mode 1 Mode 2 Mode 3

38 Free response of multi-degree-of-freedom systems Example - Cantilever 1 X + 2 xt t 4

39 m c Response of a SDOF system to harmonic excitation Fsint x x () t f x () p t Steady-state Forced vibration t t x ( t) x ( t) p f t

40 Steady-state response of a SDOF system to harmonic excitation m c F sint x The equation of motion is mx cx x Fsint The displacement is given by x X sint o where X is the amplitude is the phase angle between the response and the force

41 Frequency response of a SDOF system m c Fsint x The amplitude of the response is given by X o F m c Applied force Inertia force F 2 mx o Stiffness force Damping force cx o Xo The phase angle is given by 1 c tan 2 m

42 Frequency response of a SDOF system j t Fe The equation of motion is m c x mx cx x Fe jt The displacement is given by x j t Xe This leads to the complex amplitude given by X 1 X or F m jc F 1 2 n j2 n Where 2 n m and c 2 m Complex notation allows the amplitude and phase information to be combined into one equation

43 Frequency response functions Receptance X 1 2 F m jc Other frequency response functions (FRFs) are Accelerance = Mobility = Acceleration Force Velocity Force Force Apparent Mass = Acceleration Force Impedance = Velocity Force Dynamic Stiffness = Displacement

44 Representation of frequency response data Log receptance 1 Increasing damping n Log frequency phase -90 Increasing damping

45 Vibration control of a SDOF system j t Fe X o 1 F m c m x c Low frequency 0 Frequency Regions X F 1 Stiffness controlled o Resonance X F c Damping controlled 2 m o 1 High frequency 2 n Mass controlled 2 Xo F 1 m Log X o F 1 Stiffness controlled Damping controlled Mass controlled Log frequency

46 Representation of frequency response data Recall X 1 1 F 1 2 n j2 n This includes amplitude and phase information. It is possible to write this in terms of real and imaginary components. 2 X 1 1 n 1 2 n j 2 2 F n n 1 n 2 n real part imaginary part

47 Real and Imaginary parts of FRF Re X F n frequency Im X F

48 Real and Imaginary parts of FRF Real and Imaginary components can be plotted on one diagram. This is called an Argand diagram or Nyquist plot 1 Re X F Increasing frequency n Im X F

49 3D Plot of Real and Imaginary parts of FRF Im X F Re X F frequency

50 Summary Basic concepts Mass, stiffness and damping Introduction to free and forced vibrations Role of damping Frequency response functions Stiffness, damping and mass controlled frequency regions