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 of a single degree of freedom (sdof) system Free response Forced response Influence of the damping *Reduction to a sdof system Equivalent stiffness Equivalent mass Equivalent damping Validity of the reduction 2 1
One degree of freedom systems in real life 3 Reduction of a system to a one dof system Hypothesis : -Rigid body -Motion in one direction -Linked to a reference point through a flexible element (with negligible mass) Valid under certains conditions -Specific direction of excitation and motion -Limited frequency band 4 2
Rigid vs flexible body In statics: flexibility = ability to deform under applied load, function of the stiffness In dynamics, flexibility is a function of: - the stiffness - the mass - the frequency of excitation - the damping in the system Rigid vs flexible body : illustration Freq < 20 Hz (earthquake) 3
Rigid vs flexible body : illustration Freq > 100 Hz Not a rigid body Rigid vs flexible body : illustration Massless spring Can be reduced to a one dof system Cannot be reduced to a one dof system 4
Reduction of a system to a one dof system Example 1: 9 Reduction of a system to a one dof system Example 2: 10 5
Reduction of a system to a one dof system Example 3: 11 Reduction of a system to a one dof system Example 4: The motion is the rotation instead of the displacement 12 6
Reduction of a system to a one dof system Example 5: Offshore platform 13 Response of a single degree of freedom (sdof) system 14 7
Conservative system: equation of motion Newton law: 15 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. 16 8
Equation of motion: general solution 0 Characteristic equation: In the absence of external excitation force, the motion is oscillatory. The natural angular frequency is defined by the values of k and m The motion is initialized by imposing initial conditions on the displacement and the velocity 17 Equation of motion: general solution * 18 9
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. 19 Equation of motion: particular solution Consider the continuous inverse Fourier transform of the excitation f and isolate a single frequency component of the form We now compute the particular solution of the equation of motion for this single frequency harmonic excitation 20 10
Equation of motion: particular solution 21 Equation of motion: particular solution 22 11
Equation of motion: particular solution Positive if < n Infinite if n Negative if n 23 Equation of motion: particular solution Dynamic amplification factor Bode diagram (linear frequency axis) 24 12
Equation of motion: particular solution * Bode diagram (logarithmic frequency axis) see MATH-H-201 (ULB) 25 Equation of motion: back to the time domain Continuous Fourier transform Inverse continuous Fourier transform Displacement in the time domain 26 13
What is resonance? Amplification of given frequencies How resonance can lead to failure 28 14
How resonance can lead to failure Wine glass Damped equation of motion Damping force : viscous damping 30 15
Damped equation of motion: general solution * General solution damping coefficient Characteristic equation: 31 Damped equation of motion: general solution Initial conditions: * 32 16
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
Damped equation of motion: general solution Critical damping Critical damping 35 Impulse response Equivalent to initial velocity at t 36 18
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
Damped equation of motion : particular solution * 39 Damped equation of motion : particular solution Maximum Max ampl. * For small 40 20
Bode diagram Displacement Velocity Acceleration 41 Nyquist plot The plot enhances strongly the frequencies around resonance 42 21
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 43 Time domain response using Duhamel s integral For a single degree of freedom, we have : Special case : harmonic excitation The Fourier transform of h(t) is the transfer function 44 22
Base excitation Excitation 45 Illustration 46 23
Reduction to a sdof system 47 Reduction to a sdof system -Equivalent stiffness? -Equivalent mass? -Validity (frequency band)? 48 24
Equivalent stiffness General principle : F in the direction of excitation (and motion) of the mass Particular case: the system is a beam subject to - traction - bending - torsion 49 Equivalent stiffness: bar in traction Direction of excitation Attachment point Constitutive equation: Stress is constant Static equilibrium Displacement of the attachment point (in the direction of excitation): 50 25
Equivalent stiffness: bar in torsion Direction of excitation J=rotational intertia Attachment point Constitutive equation : Circular section Static equilibrium Rotation at the attachment point: 51 Equivalent stiffness : cantilever beam in bending Direction of excitation Attachment point From tables 52 26
Equivalent stiffness : of beams in bending Equivalent stiffness of beams in bending can be computed using formulas from resistance of materials or using tables : 53 Equivalent stiffness: portal frame 54 27
Equivalent stiffness: portal frame From tables (internet) 55 Equivalent stiffness: energy approach Strain energy in a mass-spring system : Bar in traction: Bar in torsion: Beam in bending: 56 28
Equivalent stiffness : general case General case:- simplified model - numerical approximation (finite elements) 57 Equivalent mass : energy method Kinetic energy of a mass-spring system Example 1 : Kinetic energy of the spring Additional mass 58 29
Equivalent mass : energy method Kinetic energy of a mass-spring system Example 2 : Kinetic energy of the bar Additional mass m a 59 Equivalent damping : Types of damping Damping = dissipation of energy [Vibration Problems in Structures, CEB, 1991] 60 30
Hysteresis loop due to damping When dissipation is present, the stress is not in phase with the strain, which results in a hysteresis loop [Vibration Problems in Structures, CEB, 1991] The mechanical energy dissipated in one cycle per unit volume is given by the area inside the loop T=period 61 Damping factor The damping factor of a material is proportional to the ratio of energy dissipated in one cycle to the maximum strain potential energy The damping factor of the structure is given by (V is the volume of the structure) : For a homogeneous structure, we have 62 31
Damping factor of a SDOF system with viscous damping * 63 Damping factor of a SDOF system with viscous damping At resonance : at resonance 64 32
Equivalent viscous damping in real structures Example of energy dissipation in a real structure Frequency dependent damping coefficient -> difficult to use for time domain computations 65 Equivalent viscous damping in real structures The real damping is replaced by an equivalent viscous damping The energy dissipated for this equivalent system is equal to the real dissipation only for n, where, for lightly damped systems, the effect of damping is the most important 66 33
Hysteretic damping : equivalent damping In many engineering applications, S is found to be constant with the frequency Hysteretic damping -> depends on frequency 67 Coulomb friction damping : equivalent damping Coulomb friction damping -> Depends on frequency and amplitude 68 34
Measurement of damping Logarithmic decrement method Half-power bandwidth Estimation of in the time domain Estimation of in the frequency domain 69 Logarithmic decrement method 70 35
Half-power bandwidth 71 Typical values of damping Contributions to damping - bare structure - materials - joints -non-structural elements -Footbridges : pavement, railings -Building floors : partition walls, flooring, furniture, ceilings.. -energy radiation in soil Damping coefficients are usually derived from practice if no measurement is possible 72 36
Frequency limits for one dof systems m=100 kg, E =210 GPa, =7800 kg/m 3, L=1m; A = 0.0004 m 2 Continuous model SDOF model 1 SDOF model 2 73 Frequency limits of SDOF systems for real continuous systems k =EA/L = 8.4 10 7 N/m M = 100 kg m a = AL/3 = 1.04 kg <<M f n =145.87 Hz (mass M) f n =145.11 Hz (mass M+m a ) SDOF hypothesis not valid for frequencies > 2000 Hz 74 37
Frequency limits for one dof systems m=100 kg, E =210 GPa, =78000 kg/m 3, L=1m; A = 0.0004 m 2 Continuous model SDOF model 1 SDOF model 2 75 Frequency limits of SDOF systems for real continuous systems k =EA/L = 8.4 10 7 N/m M = 100 kg m a = AL/3 = 10.4 kg <M f n =145.87 Hz (mass M) f n =138.83 Hz (mass M+m a ) m a should be taken into account SDOF hypothesis not valid for frequencies > 500 Hz 76 38
Summary In particular cases, a real system can be reduced to a sdof system. This requires computing the equivalent mass, stiffness and damping of the system The validity of the model is limited and it should not be used outside of a certain frequency band, and for motions and excitations in other directions. The tools to compute the response of single degree of freedom systems have been described in this chapter, both in the frequency and in the time domain 77 Application example: the accelerometer In the frequency domain : 78 39
Application example: the accelerometer Piezoelectric accelerometer 79 Application example : the accelerometer High resonant frequency Low sensitivity Low resonant frequency High sensitivity 80 40