Dynamics of structures

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1 Dynamics of structures 2.Vibrations: single degree of freedom system Arnaud Deraemaeker 1 One degree of freedom systems in real life 2 1

2 Reduction of a system to a one dof system Example 1: 3 Reduction of a system to a one dof system Example 2: 4 2

3 Reduction of a system to a one dof system Example 3: 5 Reduction of a system to a one dof system Example 4: Offshore platform 6 3

4 Harmonic motion 7 Simple Harmonic Motion 8 4

5 Harmonic signals A periodic vibration of which the amplitude can be described by a sinusoidal function: is called an harmonic vibration with: amplitude a angular frequency frequency f period T = 1/f or f = 1/T phase angle at t=0 total phase angle 9 Harmonic signals Representation in the complex plane: Independent of time Projection of the rotating vector on the real axis is a cosine Projection of the rotating vector on the imaginary axis is a sine 10 5

6 Harmonic signals Displacement Velocity Acceleration the phase angle of u(t) is 90 behind v(t) the phase angle of v(t) is 90 behind a(t) 11 Harmonic signals 12 6

7 Undamped vibrations under free and forced harmonic excitations 13 Conservative system: equation of motion Newton s law: 14 7

8 Conservative system: equation of motion What about the effect of gravity? The displacement x is defined with respect to the equilibrium position of the mass subjected to gravity. The effect of gravity should therefore not be taken into account in the equation of motion of the system. 15 Equation of motion: general solution Characteristic equation: In the absence of external excitation force, the motion is oscillatory. The natural angular frequency n is defined by the values of k and m The motion is initialized by imposing initial conditions on the displacement and the velocity 16 8

9 Equation of motion: general solution 17 Equation of motion: general solution Alternative representation: The motion can be described by a cosine function with a phase. The phase is a function of the initial conditions. 18 9

10 Equation of motion: particular solution for harmonic excitation 19 Equation of motion: particular solution for harmonic excitation 20 10

11 Equation of motion: particular solution for harmonic excitation Dynamic amplification factor Bode diagram (linear frequency axis) 21 Resonance 22 11

12 How resonance can lead to failure 23 How resonance can lead to failure Wine glass 12

13 Is stiffer stronger? 25 Forced excitation videos 26 13

14 Building resonance 27 Damped vibrations under free and forced harmonic excitation 28 14

15 Effect of damping on a building 29 Damped equation of motion Damping force : viscous damping 30 15

16 Damped equation of motion: general solution damping coefficient General solution Characteristic equation: 31 Damped equation of motion: general solution Initial conditions: 32 16

17 Damped equation of motion: general solution 33 Damped equation of motion: general solution Number of oscillations after which the vibration amplitude is reduced by one half 34 17

18 Damped equation of motion: general solution Critical damping Critical damping 35 Impulse response Equivalent to initial velocity at t 36 18

19 Impulse response For an initial velocity, the response of the system is: with For a unit impulse, we define the impulse response h(t) 37 Damped equation of motion : particular solution 38 19

20 Damped equation of motion : particular solution 39 Damped equation of motion : particular solution Maximum Max ampl

21 Bode diagram Displacement Velocity Acceleration 41 Nyquist plot The plot enhances strongly the frequencies around resonance 42 21

22 Dynamic response for arbitrary force input 43 Time domain response using Duhamel s integral f(t) is decomposed into a series of short impulses at time The contribution of one impulse the response of the system is given by : to (h(t) is the impulse response) The total contribution is therefore: We have and for so that we can write 44 22

23 Convolution integral The convolution integral of two time functions x(t) and h(t) yields a new time function y(t) defined as: Take the two functions x(t) and h(t) and replace t by the dummy variable Mirror the function h( ) against the ordinate, this yields h(- ) Shift the function h(- ) with a quantity t Determine for each value of t the product of x( ) with h(t- ) Compute the integral of the product y(t) let t vary from (or a value small enough to make the product zero) to (or a value of t that is big enough) 45 Convolution integral 1st function Mirrored 2nd function Shift h( ): 46 23

24 Convolution integral Property: 47 Harmonic excitation below resonance Transient regime Steady-state regime 48 24

25 Harmonic excitation near resonance Transient regime Steady-state regime Beating 49 Harmonic excitation at resonance Transient regime Steady-state regime 50 25

26 Harmonic excitation at resonace Transient regime Steady-state regime Transient regime Steady-state regime 51 Harmonic excitation above resonance Transient regime Steady-state regime 52 26

27 Sine sweep excitation Resonance 53 Base excitation 54 27

28 Base excitation Excitation 55 Illustration 56 28

29 Application example: the accelerometer 57 Application example: the accelerometer In the frequency domain : 58 29

30 Application example: the accelerometer Piezoelectric accelerometer 59 Application example : the accelerometer High resonant frequency Low sensitivity Low resonant frequency High sensitivity 60 30

Dynamics of structures

Dynamics of structures 2.Vibrations: single degree of freedom system Arnaud Deraemaeker (aderaema@ulb.ac.be) 1 Outline of the chapter *One degree of freedom systems in real life Hypothesis Examples *Response