Introduction to Vibration Mike Brennan UNESP, Ilha Solteira São Paulo Brazil
Vibration Most vibrations are undesirable, but there are many instances where vibrations are useful Ultrasonic (very high frequency) vibrations Tooth cleaning Imaging of internal organs Welding Structural Health Monitoring Vibration conveyers Time-keeping instruments Impactors Music Heartbeat
Introduction to Vibration Nature of vibration of mechanical systems Free and forced vibrations Frequency response functions
Fundamentals For free vibration to occur we need mass m stiffness k The other vibration quantity is damping c
Fundamentals - potential and kinetic energy energy.mov
Fundamentals - damping
Fundamental definitions A x( t) x = Asin( ωt) t T T ω = 2π f (radians/second) Period T = 2π ω Frequency f = 1 T (seconds) (cycles/second) (Hz)
Phase A x( t) t x = A sin( ω t ) x = Asin( ωt + φ) φ ω Green curve lags the blue curve by radians 2 π
Harmonic motion A x( ωt) ω angular displacement φ = ωt φ = ωt One cycle of motion 2π radians
+ imaginary - real + imaginary e Complex number representation b φ of harmonic motion A a Euler s Equation ± jφ = cosφ ± j x = a + sinφ jb + real So x x = Acosφ + jasinφ ( cosφ j sinφ ) = A + x = j Ae φ magnitude phase magnitude 2 2 1 x = A = a + b phase φ = tan ( b a)
Relationship between circular motion in the complex plane with harmonic motion Imaginary part sine wave Real part cosine wave
Sinusoidal signals other descriptions Average value x( t) T 1 xav = Asinωt T 0 dt T t For a sine wave x av = 0 For a rectified sine wave xav = 0.637A
Sinusoidal signals other descriptions Average value x( t) DC t Average value of a signal = DC component of signal
x( t) Sinusoidal signals other descriptions Mean square value T 2 1 xmean = A t T 0 ( sinω ) 2 For a sine wave x = 0.5A 2 2 mean dt T t Root Mean Square (rms) x = x = A rms 2 mean 2 Many measuring devices, for example a digital voltmeter, record the rms value
Sinusoidal signals Example A vibration signal is described by: x = 0.15sin200t Amplitude (or peak value) = 0.15 m Average value = 0 Mean square value = 0.01125 m 2 Root mean square value = 0.10607 m Peak-to-peak value = 0.3 m Frequency = 31.83 Hz
x( t) Vibration signals Periodic or deterministic (not sinusoidal) Heartbeat IC Engine t T T T is the fundamental period
Fourier Analysis (Jean Baptiste Fourier 1830) Representation of a signal by sines and cosine waves x( t) + t + + :
Fourier Composition of a Square wave frequency
Vibration signals x( t) Transient Gunshot Earthquake Impact t
Vibration signals x( t) Random Uneven Road Wind Turbulence t
Free Vibration System vibrates at its natural frequency x( t) t x = Asin( ωnt) Natural frequency
Forced Vibration System vibrates at the forcing frequency x( t) f ( t) x( t) t x = Asin( ωf t) Forcing frequency
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
Linear system Mechanical Systems Linear system Same frequency as input Magnitude change Phase change Output proportional to input
Linear system Mechanical Systems input excitation b a M system output response, y y = Ma + Mb = M( a + b)
Nonlinear system Mechanical Systems Nonlinear system output comprises frequencies other than the input frequency output not proportional to input
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
Mechanical Systems Nonlinear systems example: nonlinear spring f k hardening spring x For a linear system f = kx force f linear softening spring displacement x
Mechanical Systems Nonlinear systems example: nonlinear spring Peak-to-peak vibration (approximately linear) force f Peak-to-peak vibration (nonlinear) Static displacement displacement x stiffness f = x
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 θ Independent coordinate
Degrees of Freedom Single-degree-of-freedom systems x Independent coordinate k m
Spring Idealised Elements k f1 f2 x1 x2 no mass k is the spring constant with units N/m ( ) f = k x x 1 1 2 ( ) f = k x x f 2 2 1 = f 1 2
Idealised Elements Addition of Spring Elements Series k 1 k 2 k = total 1 1 1 + k k 1 2 k total is smaller than the smallest stiffness k 1 Parallel k total 1 2 2 k = k + k k total is larger than the largest stiffness
Idealised Elements Addition of Spring Elements - example f k R x k T stiffness = f x Is k T in parallel or series with k R? Series!!
Viscous damper c Idealised Elements f1 f2 xɺ 1 xɺ 2 f = c xɺ xɺ ( ) 1 1 2 f = c xɺ xɺ ( ) 2 2 1 no mass no elasticity c is the damping constant with units Ns/m f = f 1 2 Rules for addition of dampers is as for springs
Viscous damper Idealised Elements f1 f2 m xɺɺ f1 + f2 = mxɺɺ f = mxɺɺ f 2 1 rigid m is mass with units of kg Forces do not pass unattenuated through a mass
Free vibration of an undamped SDOF system Undeformed spring System equilibrium position k k m System vibrates about its equilibrium position
Free vibration of an undamped SDOF system System at equilibrium position Extended position m m mxɺɺ k m k mxɺɺ kx + kx = 0 inertia force stiffness force
Simple harmonic motion The equation of motion is: k where m ω = 2 n k m x mxɺɺ + kx = k x ɺɺ + x = m xɺɺ + ω x = is the natural frequency of the system 2 n 0 0 0 The motion of the mass is given by x = X sin( ω t ) o n
Simple harmonic motion Real Notation Complex Notation k m x Displacement x X sin( ( ω t ) j n = o Velocity Acceleration n xɺ = ω X cos( ω t ) n o n 2 n o n x ɺ = Xe ω j nt x = jωn Xe ω xɺɺ = ω X sin( ω t ) 2 j n xɺ = ω Xe ω n t t
Simple harmonic motion Imag xɺɺ ω x Real xɺ ωt
Free vibration effect of damping k m c x The equation of motion is mxɺɺ + cxɺ + kx = 0 inertia force damping force stiffness force
Free vibration effect of damping x = Xe ζω nt φ time ω x = Xe ζωnt sin( ω t + φ ) d d T d = 2π ω d ζ = Damping ratio T d = Damping period φ = Phase angle
Free vibration - effect of damping The underdamped displacement of the mass is given by x = Xe ζωnt sin( ω t + φ ) d Exponential decay term Oscillatory term ζ = Damping ratio = c ( 2mω ) ( 0 < ζ < 1) ω n = Undamped natural frequency = k m ω d = Damped natural frequency = φ = Phase angle n ωn = 1 ζ 2
Free vibration - effect of damping
Free vibration - effect of damping ( ) x t t Undamped ζ=0 Underdamped ζ<1 Critically damped ζ=1 Overdamped ζ>1
Variation of natural frequency with damping ωd ω n 1 0 1 ζ
Degrees-of-freedom Single-degree-of-freedom system k m x 1 Multi-degree-of-freedom (lumped parameter systems) N modes, N natural frequencies k m k m k m k m x 1 x 2 x 3 x 4
Degrees-of-freedom Infinite number of degrees-of-freedom (Systems having distributed mass and stiffness) beams, plates etc. Example - beam Mode 1
Degrees-of-freedom Infinite number of degrees-of-freedom (Systems having distributed mass and stiffness) beams, plates etc. Example - beam Mode 1 Mode 2
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
Free response of multi-degree-of-freedom systems Example - Cantilever ω 1 X + ω 2 x( t) + + ω 3 t ω 4
k m c Response of a SDOF system to harmonic excitation F sin( ωt ) x x ( t) f x ( p t ) Steady-state Forced vibration t t x ( t) + x ( t) p f t
Steady-state response of a SDOF system to harmonic excitation k m c F sin( ωt ) x The equation of motion is mxɺɺ + cxɺ + kx = F sin( ωt ) The displacement is given by x = X sin( ωt + φ ) o where X is the amplitude φ is the phase angle between the response and the force
Frequency response of a SDOF system k m c F sin( ωt ) x The amplitude of the response is given by X o = F 2 2 ( 2 k ω m) + ( ωc) Applied force Inertia force F φ 2 ω mx o Stiffness force Damping force ωcx o kxo The phase angle is given by 1 ωc φ = tan 2 k ω m
Frequency response of a SDOF system k m c j t Fe ω x The equation of motion is mxɺɺ + cxɺ + kx = Fe jωt The displacement is given by x = j t Xe ω This leads to the complex amplitude given by X 1 = X 1 1 2 or = F k ω m + jωc F k 1 ( ω ω ) 2 n + j2 Where 2 ω n = k m and ζ = c ( 2 mk ) ζ ω ω Complex notation allows the amplitude and phase information to be combined into one equation n
Frequency response functions Receptance X = ω 1 ω 2 F k m + j c Other frequency response functions (FRFs) are Accelerance = Mobility = Acceleration Force Velocity Force Force Apparent Mass = Acceleration Force Impedance = Velocity Force Dynamic Stiffness = Displacement
Representation of frequency response data Log receptance 1 k Increasing damping ω n Log frequency phase -90 Increasing damping
k Vibration control of a SDOF system j t Fe ω X o 1 = F 2 2 2 ( k ω m) + ( ωc) m x c Frequency Regions Low frequency ω 0 X F= 1 k o Stiffness controlled Resonance X F= ωc 2 ω = k m o 1 Damping controlled High frequency 2 ω >> ω n = 2 Xo F 1 ω m Mass controlled Log X o F 1 k Stiffness controlled Damping controlled Mass controlled Log frequency
Representation of frequency response data Recall X 1 1 = F k 1 ( ω ω ) 2 n + j2ζ ω ωn This includes amplitude and phase information. It is possible to write this in terms of real and imaginary components. 2 X 1 1 ( ω ωn ) 1 2ζ ω ωn = j ( ( ) ) 2 ( ) ( ( ) ) 2 F k 2 2 + 2 2 1 ω ωn 2ζ ω ω k + n 1 ω ωn + ( 2ζ ω ωn ) real part imaginary part
Real and Imaginary parts of FRF Re X F ω n frequency Im X F
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 k φ Re X F Increasing frequency ω n Im X F
3D Plot of Real and Imaginary parts of FRF Im X F Re X F ζ = 0 ζ = 0.1 frequency
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