Mass Spring Damper

Existing mass spring damper or vibration theory is directly applicable to point absorbers devices. Using standard notation, the governing equation is given by

𝑚𝑧̈ + 𝑏𝑧̇ + 𝑐𝑧 = 0 (2.87)

Dividing by 𝑚 yields,

𝑧̈ + 2𝜁𝜔𝑛𝑧̇ + 𝜔𝑛2𝑧 = 0 (2.88)

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𝜁 = 𝑏 2𝑚𝜔𝑛 (2.89) 𝜔𝑛= √ 𝑐 𝑚 (2.90)

Here 𝜔𝑛 is the natural frequency (in rad/s) of the system and 𝜁 is a measure of damping

present in the system relative to the critical level of damping. In an actual vibration, there will always be some damping present. The solution of equation (2.88) must be a function which has the property that repeated differentiations do not change its form since the function and its first and second derivatives must be added together to give zero (Housner & Hudson, 1950). Assume a solution of the form;

𝑧(𝑡) = 𝐴𝑒𝛼𝑡 _(2.91)

Substituting equation (2.91) into equation (2.88) yields the characteristic equation,

𝛼2+ 2𝜁𝜔𝑛𝛼 + 𝜔𝑛2= 0 (2.92)

An equation, whose roots are

𝛼1 = 𝜔𝑛(−𝜁 + √𝜁2− 1) , 𝛼2= 𝜔𝑛(−𝜁 − √𝜁2− 1) (2.93)

The general solution of the governing equation is therefore

𝑧(𝑡) = 𝐴1𝑒𝛼1𝑡+ 𝐴2𝑒𝛼2𝑡 =𝐴1𝑒(−𝜁+√𝜁 2_−1)𝜔 𝑛𝑡_{+ 𝐴} 2𝑒(−𝜁−√𝜁 2_−1)𝜔 𝑛𝑡 (2.94)

The physical significance of the solution depends on the amount of damping present. There are three distinct scenarios to be considered:

𝜁 > 1: The system is said to be overdamped; the roots 𝛼1, 𝛼2 are real negative numbers.

When an overdamped system is given an initial displacement, the damping is sufficiently large to ensure the mass never oscillates past the static equilibrium position but rather exponentially subsides.

𝜁 = 0: The system is said to be critically damped with 𝛼1 = 𝛼2 = −𝜔𝑛. A critically damped

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𝜁 < 0: The system is underdamped. Such a system, when given a displacement, will oscillate about the equilibrium before eventually coming to rest. The frequency of oscillation is known as the damped angular frequency 𝜔𝑑 , and is given by

𝜔𝑑= 𝜔𝑛(√1 − 𝜁2) (2.95)

Rearranging equation (2.94) gives

z(t) = { 𝐴1𝑒𝑖(√1−𝜁 2_)𝜔 𝑛𝑡_{+ 𝐴} 2𝑒−𝑖(√1−𝜁 2_)𝜔 𝑛𝑡_{} 𝑒}−𝜁𝜔𝑛𝑡 = { 𝐴1𝑒𝑖𝜔𝑑𝑡+ 𝐴2𝑒−𝑖𝜔𝑑𝑡}𝑒−𝜁𝜔𝑛𝑡 (2.96)

Euler’s identity 𝑒𝑖𝜃= cos(𝜃) + 𝑖 sin(𝜃), allows the displacement to be written

𝑧(𝑡) = 𝑒−𝜁𝜔𝑛𝑡{ (𝐴 1+ 𝐴2) cos(𝜔𝑑𝑡) + 𝑖( 𝐴1− 𝐴2)𝑠𝑖𝑛(𝜔𝑑𝑡)} =𝑒−𝜁𝜔𝑛𝑡{𝐴 1 ′ _cos(𝜔 𝑑𝑡) + 𝐴2′𝑠𝑖𝑛(𝜔𝑑𝑡)} (2.97)

Alternatively, equation (2.97) can be written in amplitude-phase form as

𝑧(𝑡) = 𝐶𝑒−𝜁𝜔𝑛𝑡_cos(𝜔

𝑑𝑡 + 𝜙) (2.98)

In an effort to determine the hydrodynamic parameters of added mass and damping, it is usual to carry out a decay test. A decay test or extinction test is one where the buoy is initially displaced a distance 𝑧0 from its equilibrium position and then released. The buoy then

undergoes a damped free oscillation; parameters such as the natural frequency and hydrodynamic coefficients of added mass and damping can then be inferred from the recorded response of the buoy as it comes to rest. The unforced solution to a pure mass spring damper system is that of Equation (2.98) where the parameters 𝐶 and 𝜙 are determined by the initial conditions. If the buoy is initially displaced a distance 𝑧0, with an

accompanying initial velocity 𝑣0= 0 , then

𝐶 = 𝑧0

√1 − 𝜁2 (2.99)

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𝜙 = tan−1( −𝜁

√1 − 𝜁2) (2.100)

A typical response of a damped free oscillation is shown in Figure 2:16(a). The envelope of decay is determined by the 𝐶𝑒−𝜁𝜔𝑛𝑡 _{term, whilst the trigonometric term determines the}

frequency of oscillation.

Figure 2:16: Typical free decay curve of (a) a viscously damped system; (b) Coulomb damping

According to (Feeny & Liang, 1996):

“In his Theory of Sound, Lord Rayleigh noted that, for the free vibration of a linear damped oscillator, ‘the difference in the logarithms of successive extreme excursions is nearly constant, and is called the logarithmic decrement’. In fact, the idea goes back to Hermann Helmhotz, who in 1863 applied the logarithmic decrement to determine frequency information in musical tones given a known damping coefficient”

Therefore, the logarithmic ratio of successive peaks (called the logarithmic decrement δ) can be used to estimate the damping experimentally

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𝛿 = ln (𝑧1 𝑧2 ) = 𝜁𝜔𝑛𝑇𝑑= 2𝜋𝜁 √1 − 𝜁2 (2.101)

The influence of mechanical friction in any test set up cannot be ignored. If the energy dissipation in a system is achieved purely by friction, as is the case in the free response of a mass and spring with Coulomb friction, then the amplitude of successive peaks decays linearly with time as shown in Figure 2:16(b). Thus, to truly encapsulate all the dynamics of the experimental system both viscous and Coulomb damping must be considered. A method of identifying Coulomb and viscous friction from free vibration decrements is described in (Liang and Feeny 1998).

The equation of motion of a single degree of freedom SDOF mass spring damper with dry friction is

𝑚𝑧̈ + 𝑏𝑧̇ + 𝑐𝑧 + 𝑓(𝑧̇) = 0 (2.102)

The dry friction will always oppose the direction of motion. Hence

𝑓(𝑧̇) = 𝑓𝑘𝑠𝑖𝑔𝑛(𝑧̇)

𝑧̇ ≠ 0, −𝑓𝑠≤ 𝑓(0) ≤ 𝑓𝑠

(2.103)

Where 𝑓𝑘, is the kinetic friction force and 𝑓𝑠 is the static friction force. The equilibrium

solution of Equation (2.102) can be obtained by letting 𝑧̈ = 𝑧̇ = 0. A physical consequence of the friction is that the system doesn’t have a single equilibrium point but rather a locus of equilibria −𝑧𝑠≤ 𝑧 ≤ 𝑧𝑠 where 𝑧𝑠= 𝑓𝑠⁄ . 𝑐

If follows from the above that if the buoy where to be submerged by some small distance 𝑧 < 𝑧𝑠, then the restoring buoyancy force would not be able to overcome the stiction force

so the buoy would remain stationary.

Equation (2.102) is piecewise solvable, when written as

𝑧̈ + 2𝜁𝜔𝑛𝑧̇ + 𝜔𝑛2𝑧 = −𝜔𝑛2𝑧𝑘 𝑧̇ > 0 (2.104)

𝑧̈ + 2𝜁𝜔𝑛𝑧̇ + 𝜔𝑛2𝑧 = +𝜔𝑛2𝑧𝑘 𝑧̇ < 0 (2.105)

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In a decay test the buoy is initially displaced 𝑧(𝑡0) = 𝑍0< −𝑧𝑠 with 𝑧̇(𝑡0) = 0. Then the

motion starts with 𝑧̇ > 0, and the response to equation (2.104) has the form

𝑧(𝑡) = (𝑍0+ 𝑧𝑘)𝑒−𝜁𝜔𝑛(𝑡−𝑡0){𝑐𝑜𝑠(𝜔𝑑(𝑡 − 𝑡0)) + 𝛽𝑠𝑖𝑛(𝜔𝑑(𝑡 − 𝑡0))} − 𝑧𝑘 (2.106)

Where 𝛽 = 𝜁

√1−𝜁2.

This equation is valid until the velocity vanishes after time 𝑡 = 𝑡1= 𝑡0+ 𝜋 𝜔⁄ 𝑑. If the next

peak excursion point is denoted 𝑍1, then

𝑍1= 𝑧(𝑡1) = −𝑒−𝛽𝜋𝑍0+ (𝑒−𝛽𝜋+ 1)𝑧𝑘 (2.107)

If 𝑍1> 𝑧𝑠 then the mass will reverse direction and continue moving only with 𝑧̇ < 0, so that

the governing equation is now that given in equation (2.105). The solution for this interval is

𝑧(𝑡) = (𝑍1− 𝑧𝑘)𝑒−𝜁𝜔𝑛(𝑡−𝑡1){𝑐𝑜𝑠(𝜔𝑑(𝑡 − 𝑡1)) + 𝛽𝑠𝑖𝑛(𝜔𝑑(𝑡 − 𝑡1))} + 𝑧𝑘 (2.108)

This is valid until 𝑧̇ = 0 after time 𝑡 = 𝑡2= 𝑡1+ 𝜋 𝜔⁄ 𝑑. The next peak excursion point 𝑍2 =

𝑧(𝑡2) = −𝑒−𝛽𝜋𝑍1− (𝑒−𝛽𝜋+ 1)𝑧𝑘. If 𝑍2< −𝑧𝑠, motion will continue again. This process will

repeat itself until −𝑧𝑠≤ 𝑍𝑛≤ 𝑧𝑠, at which time the motion stops. This iterated process leads

to a recursive relation for the successive peaks and valleys in the oscillatory response:

𝑍𝑖 = −𝑒−𝛽𝜋𝑍𝑖−1+ (−1)𝑖(𝑒−𝛽𝜋+ 1)𝑧𝑘, i=1,2,... ,n. (2.109)

It is therefore possible to utilise successive maxima and minima points to isolate the viscous effect and then calculate the Coulomb effect. Taking the sum of extreme displacement values cancels out the friction contribution

𝑍𝑖+ 𝑍𝑖+1

𝑍𝑖−1+ 𝑍𝑖

= −𝑒−𝛽𝜋 (2.110)

Taking a logarithmic decrement gives the viscous dependence

ln (𝑍𝑖+ 𝑍𝑖+1 𝑍𝑖−1+ 𝑍𝑖

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In document Laboratory scale modelling of a point absorber wave energy converter (Page 100-106)