Fig. 4 bME 260W - Measurement Techniques6. Acquiring Test DataTaking Measurements:Department of Mechanical EngineeringME 260W - Measurement TechniquesSpring 2005Experiment #3: Motion Analysis of a Compound Pendulum1. OBJECTIVESThe objectives of the experiment are: i) To introduce a sensing device, Rotary Potentiometer.ii) To introduce the student to signal conditioning as applied to critical timing measurementsand motion analysis of a compound pendulum.iii) To apply basic statistical procedures for the purpose of data reduction, model fitting andcorrelation2. EQUIPMENT- Rotary Potentiometer, RP100 Series- Compound Pendulum - AC/DC Variable Elenco Power supply, Model XP800- Tektronix Digital Storage Oscilloscope; Model TDS 210 - Computer station with data acquisition A/D board3. DESCRIPTIONSTheoryConsidering the motion of a compound pendulum, which is illustrated in Fig. 1, the potentialenergy V and kinetic energy T at any arbitrary state of position and velocity are given as:    cos-1LgmV(1)2θI)θT(20(2)where - is the angle measured from the vertical equilibrium position, and dtdθθ  is the angularvelocity. The total energy of the pendulum at any instant in time is thus:    2Icos1LgmTV,E20(3)1Fig. 1 A Compound Pendulum.If the system is conservative (no energy dissipation), the total energy is constant with a valuedictated by the initial values of - andθ. In this case, the differential equation of motion of thependulum can be found by differentiating the energy with respect to time;Thus, differentiating equation (3) results in;0sinLgmI0(5)which can be simplified to:0sinImgL0(6)Equation (6) is the governing differential equation of motion for a pendulum. The second term isnon-linear, which means that equation (6) does not have a simple, straightforward solution. Wetreat below, small and large angle (-) cases separately.Small Angle Oscillations: Linearized Equation of MotionIf one considers small angles  (from the equilibrium position), the equation can belinearized by the approximationθsinθ (7)Yielding0ImgL0(8)Equation (8) describes a simple harmonic oscillator; for the case where the initial velocity iszero, the solution is given by:   tcostn(9)where - is the intial angle of the pendulum, and 0nImgL is the frequency of oscillation. The period of oscillation, or time needed for the pendulum to make a complete cycle, is related to the natural frequency by:nn2(10)It should be emphasized that this is the period for small oscillations, since the solution wasobtained by the assumption (Eq. 7), which is only valid for small angles.2LmgCGOAdLmgCGOAdLarge Angle Oscillations and Non-Linear EffectsConsider now the case where the restriction of small angles is removed. The governingequation is given by equation (6). Rather than solving this explicitly for the angle, -, as afunction of time, we wish merely to derive an expression for the period of the oscillation as afunction of initial angle. This expression will replace equation (10).Again, we start by noting that the total energy of the system is constant. The value of thisenergy is dictated by the initial angle of the pendulum; for free vibration, the initial angle is themaximum angle reached during oscillation. The energy of the pendulum system at any giveninstant is;   cos1Lgm0,E(11)Equating this to the right-hand side of equation (3) and re-arranging, we obtain: 20dtd2Icos-cosLgm(12)We can put this equation in a differential form in terms of - and t: dtdcoscosI2mgL0 or  dtdcoscos22n(13)where 0nImgL (rad/sec) is the angular frequency as before. This nonlinear ODE is in theform of  θfθ  and it can be solved numerically using many ODE solvers such as Runge-Kutta, Adams Bashfort etc. (Commercial codes, such as MATLAB, SIMULINK or MAPLE canhelp you substantially in this).Find the period for different initial conditions (-’s) and show that the curve in Fig. 2 holds.Notice that for linear case (i.e. small angle oscillation) 1n and this feature disappears asthe swing angle, -, increases.FrictionThe exact nature of the friction present in the experiment is indeterminate. Two types of frictionthat might be in effect are viscous friction and Coulomb (or dry) friction. The effect of these twotypes will be established by considering linear mass-spring systems with damping.Viscous frictionViscous friction is characterized by a damping force that is proportional to angular velocity. Itis a convenient assumption, since the governing differential equation of motion remains linear.Referring to Fig. 3a, the differential equation for the translating mass is given as:0kxxcxm (14)where the first term is the inertial term, the second term is the viscous damping term, and the lastterm is the force from the displaced spring. It is convenient to normalize in the following form:0xx2x2nn(15)3wheremkn is the natural frequency of the system, andkm2cis referred to as thedamping ratio. If the system is given an initial displacement x0 and released, the resulting motionwill take the form (see your vibration textbook):φ)tcos(e1xx(t)dt20n(16)where 2nd1is the damped natural frequency and the phase angle is given by:211tan(17)The response described by equation (16) has the basic form of an oscillating signal withdecaying amplitude. The amplitude shows an exponential decay. The response of a viscouslydamped mass-spring system is illustrated in Fig. 3b. Regardless of the value n, the relative sizeof the (n+1)th oscillation with respect to nth, xn+1/xn is given by (derive this equation from 16):2n1n12πxxlnδ(18)For small amounts of damping, - is small and the denominator in equation (18) is approximatelyequal to one. The quantity - is commonly referred to as the logarithmic decrement. It can benumerically calculated. For small damping, damping ratio parameter, -, can be approximatelyevaluated