Exercise 3 Identification of parameters of the vibrating system with one degree of freedom

Transcription

1 Exercise 3 Identificatin f parameters f the vibrating system with ne degree f freedm Gal T determine the value f the damping cefficient, the stiffness cefficient and the amplitude f the vibratin excitatin with ne degree f freedm. These are t be understd as parameters f vibrating system. Fig. 1 Scheme f the rig Drawing at Fig. 1 presents physical mdel f the vibrating system which pssesses the fllwing elements: - stiff beam beam cnnected t the supprt rtating nde O - set f cil springs with stiffness cefficient k, - il damper with damping cefficient c, - driving spring with stiffness cefficient k 1. A spring with a stiffness factr k 1 is cnnected t the eccentric pin n the driving mtr shaft. Rtatin f the shaft creates a kinematic frcing f the upper end f this springs, apprximately described as asinωt. Due t the frce, the beam swings ut f the balance psitin by an angle φ. The deflectin f the beam is measured by a linear displacement sensr, defining the displacement x f the pint f the beam, away frm the axis f rtatin. The euatin f the linear mtin f the physical mdel f the investigated system is as fllws: B ϕ+cl c ϕ+(k+k 1 )ϕ=k 1 asinωt (1) where B is mass mment f inertia relative t its rtatin axis. Dividing E. (1) by B and multiplying by we btain: 1 1

2 ϕ+ cl c l ϕ+ k+k 1 l ϕ= k 1 a sinωt () Intrducing: cl c k+k 1 k ϕ=ẍ, =h, l ϕ=ẋ, =α 1 a, ϕ=x, =, (3) we can rewrite E. (1) as: B ẍ+h ẋ+α x=sin ωt (4) where: h - represents damping, α - natural freuency f the system, - kinematic excitatin amplitude. Fig. Physical mdel f the rig Euatin (4) needs t be slved (remind yurself hw t determine such slutin?) Its specific slutin is functin (5) which describes scillatry mtin f the mdel: x=a sin(ωt+β), (5) where A is amplitude f the excited scillatins and β defines phase angle shift between actual value f the kinematic excitatin and the mdel scillatins. The value f A is defined as: A=. (6) (α ω ) +4h ω As ω is varying in mathematical sense frm 0 t infinity values f A can be understd as a functin A(ω). Its general frm takes graphical representatin shwn in the drawing belw.

3 Fig. 3 General frm f the functin A(ω) brken line verprinted n typical values cllected frm experimental measurements cntinuus line As E. (6) cntains yet unknwn values f α, h,, these values are t be treated as parameters which we are lking fr in the excercise. The way reaching their values can be as fllwing: Determine experimental measurements graph as in Fig. 3 by measuring amplitudes t the beam amplitudes at different excitatins at the rig (cntinuus line). Amplitude at very small (practically clse t zer) excitatin ω is marked as A R0. Maximal value f the beam mtin appears at resnance freuency ω m marked as A rm. Theretical resnance graph shwn with brken line in Fig. 3 shuld be a result f calculatins using frmula (6) with sme assumptins. We assume bth graphs have t fulfill three (why?) cnditins: 1. Fr excitatin freuency ω clse t zer bth amplitudes A and A R are t be the same: A(0)= = (α 0 ) +4h 0 α =A R0. (7). At excitatin eual t the natural (resnance) freuency ω m, bth amplitudes A and A R are t be the same: A(ω m )= =A (α ω m ) Rm. (8) +4h ω m 3. At excitatin eual t the natural (resnance) freuency ω m, bth amplitudes A and A R reach their maximum values: A ω ( ω=ω m ) = [ 4ω(α ω )+8h ω] =0 (9) ((α ω m ) +4h ω ) 3 3 3

4 When Es. (7), (8), (9) are understd as set f algebraic euatins they can be cnverted t the fllwing results: = ω m A R0, α = ω m, h=ω m (10) where = 1 A R 0 A Rm which allws t calculate numerical values f the unknwn parameters α, h and. Next, we can identify real rig parameters values as: c= h l c, k 1 a=, k= α k 1. (11) Earlier, sme physical values were measured r determined frm basic engineering frmulas: =1.38 kgm, a=0.003 m, l c =0.54 m, =0.54 m, =0.4 m. (1) Curse f the exerciser: 1. Measure the A R amplitude vibratins fr different angular velcity values ω; number f measurements shuld be abut 15. Determine the value f ω m, fr which the amplitude f vibratins reaches the maximum value f A Rm. Determine the amplitude value A R0 at clse t zer excitatin freuency. Cpy yur results int the table as belw.. Calculate the parameter values α, h, using the frmulas (10). 3. Calculate the amplitude values A f the theretical resnance plt using the frmula (6) fr thse ω values fr which the A R was measured. 4. Draw a real and theretical resnance graphs. 5. Calculate the values f the real system parameters k, c, k 1 using frmulas (11) and the values f the parameters given in (1). 4 4

5 Date:.. Grup:... Name and Family name:... Mark:.... Exercise 3 Reprt Identificatin f parameters f the vibrating system with ne degree f freedm Measurements ω A R A ω m = A Rm = A R0 = Calculatin f the physical mdel parameters: = 1 A R 0 = = ω m A Rm A R 0 = α = ω m = m h=ω = Frmula fr calculatin f the theretical mdel amplitude: A= = (α ω ) +4h ω Calculatin f the real mdel parameters: c= h l c = k 1 a= = k= α k 1 = 5 5