MEASUREMENT OF MOMENT OF INERTIA


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1 1. measurement MESUREMENT OF MOMENT OF INERTI The am of ths measurement s to determne the moment of nerta of the rotor of an electrc motor. 1. General relatons Rotatng moton and moment of nerta Let us consder the case when a body of mass m moves on a crcular path wth acceleraton at due to the tangental component Ft of force F. By vrtue of Newton's second law we have Fr = m ar, or n scalar form Ft = m a t. Let us multply both sdes by the radus r of the crcular path: r F = r m t a t Usng the relaton at = rε (n whch ε s the angular acceleraton) and denotng the product r Ft by moment M we obtan M = m r ε. The rght sde multpler mr s the moment of nerta Θ[kg m ]. Thus Newton's second law for rotary moton s M = ε. The moment of nerta Θ = mr depends on the rotatng masses, as well on the dstances of the masses from the centre of rotaton. In Fg. 1 depcts a sold dsk (a dsk of constant thckness).
2 Fg. 1 The mass m s n dfferent dstances from the O axs of rotaton. If we cut a mass m, on r radus, the moment of nerta of ths thn rng wll be = m r. By summng up these thn rngs over the whole radus we obtan the moment of nerta of the dsc as = = m r To determne the moment of nerta n such way, we must know the dstrbuton of mass along the radus,.e. the functon, from whch one can see what m mass belongs to radus r. For example, let us calculate the moment of nerta of a dsc of constant densty and wdth b. Let us dvde the radus to N peces, wth r=r/n, =1 N. Than we have R R m = b ρ = r { π r b b { { ρ = N π N ρ crcumference wdth thckness Thus the moment of nerta s R R R R = = π ρ = π ρ N N 4 N 3 m r b b 4 = 1 = 1 N N N N = 1 R 1 1 R 1 = π bρ N (1 + N) πbρr = R πbρ = mr 4 N For practcal purposes, we ntroduce the concept of reduced mass: mr. The moment of nerta of a body s expressed wth the moment of nerta of the mass beng on a sngle radus tmes a reducton factor λ: = m r = { λm r. The reducton factor λ s 0.5 n the case of a sold dsc (as seen above). The value of reducng factor depends on the form. mr
3 Physcal pendulum Makng use of the wellknown relaton of the mathematcal pendulum let us examne the perod of oscllaton of such a rgd body whch  due to gravtatonal force  can turn round a fxed horzontal axle. On the lefthand sde of Fg. the mathematcal pendulum and on the rghthand sde the physcal pendulum can be seen. Fg. By contnuously measurng the moton of the physcal pendulum let us change the length of the mathematcal pendulum untl the two pendulums swng together. If we denote wth s the dstance of the centre of gravty of the physcal pendulum from the pont of suspenson, ts moment of nerta to the pont of suspenson wth Θ, the mass wth m, the angular acceleraton of the physcal pendulum εph wll be M m g s snφ ε ph = = (The negatve sgn means the moment effectng n the drecton opposte to the deflecton.) The εmat angular acceleraton of the mathematcal pendulum deflected wth α = φ and swngng together wth the physcal pendulum: M m g l snφ snφ ε mat = = = g m l l From the equalty of the two angular acceleratons (εph = εmat) we can get the reduced length of that mathematcal pendulum whch swngs together wth the physcal pendulum. We may wrte m g s snα g snφ = l From ths the reduced length l = m s nd wth ths the perod of oscllaton of the physcal pendulum becomes l T = π = π g m g s
4 . Determnaton of moment of nerta wth measurement of the perod of oscllaton Descrpton of the measurement technque Ths method s based on the measurement T perod of oscllaton of a physcal pendulum. ccordng to Fg.3 we turn our rotor nto a physcal pendulum by mountng a cylnder (of unform massdstrbuton) to the rotatng part at a dstance of e from the rotaton axs. Ths physcal pendulum bult n a measurng apparatus can be seen n Fg 4. e d = r Fg. 3 The perod of oscllaton of the physcal pendulum s: T = π, ( M + m) g s where Θ s the moment of nerta of the swngng (M+m) mass to the axs whch s composed of the moment of nerta of the body wth mass M and the addtonal m mass fxed on t. From the torque equlbrum at pont : M g 0 + m g e = ( M + m) g s. Therefore T = π m g e T = π m g e
5 Θ s the moment of nerta of the whole swngng (M+m) mass, that can be wrtten as Θ = ΘM+ Θm = Θ+Θm, where ΘM= Θ s the moment of nerta of the rotatng part of the motor and Θm s the moment of nerta of the addtonal mass (a cylnder of mass m and wth radus r) takng also nto account that t s fxed at an offset dstance of e from pont. Thus: 1 m = m r + m e Therefore: = + = 1 1 m r m r + m e m e The formula of the perod of oscllaton gves a precse value only for lttle angle deflectons. The deflecton of the pendulum should not be greater than 10 durng the measurements. The measurement In the equpment seen n Fg 4., we mount a mass m (marked by 1) to the mass of the rotor M. Then we devate ths physcal pendulum tll the mark (10 ) scratched to the plate (3) and then we let t go to swng freely. The plate cuts the way of the lght and through the photoelectrc mpulse sender () t starts and after a whole swngng t stops the electronc chronometer (4). Knowng the T perod of oscllaton the moment of nerta can be calculated. Ths wll be done for dfferent masses (m) mounted at dfferent offset dstances (e). For each of them the moment of nertal of the rotatng part of the motor wll be determned, and ther average wll be calculated. Durng the measurement we note the three perods of the oscllaton, 1 then after takng the arthmetc mean of the tme values T av = ( T1 + T + T3 ) 3 we calculate the value of the moment of nerta wth the followng formulae: Tav = m g e π 1 = m r m e The dameter (d) of the addtonal mass can be measured by a calper from that the radus r=d/ can be determned. The mass (m) of the addtonal mass of cylnder should be noted at each measurement. lso the dstance e must be measured, snce the addtonal mass can be mounted to 3 dfferent places.
6 Fg Preparaton questons 1. Make a sketch of a mathematcal and a physcal pendulum and defne the quanttes nfluencng ther perod of swngng.. Explan the tran of thought wth whch the moment of nerta wll be measured. 3. Defne reduced mass and gve the reducton factor for a homogenous dsc. 4. Make a sketch of the test rg on whch the measurement wll be carred out. Gve a short descrpton of the man parts.
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