Lab #4 - Gyroscopic Motion of a Rigid BodyLast Updated: March 30, 2009INTRODUCTIONGyroscope is a word used to describe a rigid body, usually with symmetry about an axis,that has a large angular velocity (i.e. spin rate),˙ψ, about that axis. Some examples are aflywheel, symmetric top, football, navigational gyroscopes, and the Earth. The gyroscopediffers in some significant ways from the linear one and two degrees-of-freedom systems withwhich you have experimented so far . The governing equations are 3-dimensional equationsof motion and thus mathematical analysis of the gyroscope involves use of 3-dimensionalgeometry. The governing equations for the general motion of a gyroscope are non-linear.Non-linear equations are in general hard (or impossible) to solve. In this laboratory you willexperiment with some simple motions of a simple gyroscope. The purpose of the lab is foryou to learn the relation between applied moment, angular momentum, and rate of changeof angular momentum. You will learn this relation qualitatively by moving and feeling thegyroscope with your hands and quantitatively by experiments on the precession of the spinaxis.PRELAB QUESTIONSRead through the laboratory instructions and then answer the following questions:1. What is a gyroscope?2. Where is the fixed point of the lab gyroscope?3. How will moments (torques) be applied to the lab gyroscope?4. What angle in Figure 4.1 gives the gyroscope’s pitch? The rate of change of whichangle gives the precession rate? Spin rate?THE GYROSCOPEOur experiment uses a rotating sphere mounted on an air bearing (see Figure 4.2) so thatthe center of the sphere remains fixed in space (at least relative to the laboratory room).This is called a gyroscope with one fixed point.As the gyroscope rotates about its spin axis it is basically stable. That is, the spin axisremains pointing in the same direction in space. As you should see in the experiment, thelarger the spin rate the larger the applied moment needed to change the direction o f the spinaxis. When a moment is applied to a gyroscop e, the spin axis will itself rotate about a newaxis which is perpendicular to both the spin axis and to the axis of the applied moment.5960 Lab #4 - Gyroscopic Motion of a Rigid Bod yThis motion of the spin axis is called prece ssion, and comes from the vector form of AngularMomentum Balance:P−−→M/o=˙−→H/oDYNAMICS OF THE SYMMETRIC TOPWe will now use 3-dimensional rigid-body dynamics to determine the equations of motionfor a symmetric top under the influence of gravity. This is a famous mechanics prob-lem first solved by Lagrange in Mecanique Analytic, and is equivalent to our gyroscopesetup. Our analysis requires us to first define 2 different coordinate frames (see Figure 4.1).ThenˆX,ˆY,ˆZocoordinate system is our inertial frame that remains fixed in space. The{ ˆe1, ˆe2, ˆe3} coordinate system is rotating coordinate fra me that is semi-fixed to the rotatingtop. By semi-fixed, we mean that as the top spins about the ˆe3axes, the ˆe1and ˆe2axes willnot spin around with it.Figure 4.1: A free-body diagram of the symmetric top including both coordinate frames.The semi-fixed frame is produced in the following manner:• The semi-fixed coordinate a xis, ˆe3, is chosen to be the spin axis of the gyroscope. Theangle θ measures down from theˆZ axis to the ˆe3axis.• Next, we choose the ˆe1axis such that it is in the XY-plane and perpendicular to ˆe3.The angle in the XY-plane fromˆX to ˆe1is denoted φ.• Finally we choose ˆe2such that it is perpendicular to ˆe3and ˆe1forming a right handedcoordinate system. It will be rotated by an ang le φ in the XY-plane f r omˆY androtated up from the XY-plane by an angle θ.TAM 203 Lab Manual 61Because the semi-fixed coordinate system only tells us the direction of the spin-axis of ourgyroscope, we need one more angle to specify the orientation, i.e. the angle about the ˆe3axis thro ug h which the gyroscope has spun. That angle we denote as ψ (not drawn), andthus the spin rate is˙ψ.Using the aforementioned coordinate definitions, the frame rotation vector Ω which givesthe rotation rate of the semi-fixed frame { ˆe1, ˆe2, ˆe3} is given byΩ =˙φˆZ +˙θ ˆe1=˙θ ˆe1+˙φ sin θ ˆe2+˙φ cos θ ˆe3(4.1)If we add to the frame rotation vector Ω, the top’s rotation in the semi-fixed fra me, we getthe bo dy rotation vector ωω = Ω +˙ψ ˆe3=˙θ ˆe1+˙φ sin θ ˆe2+˙φ cos θ +˙ψˆe3(4.2)The angular momentum o f the to p about the fixed o r ig in, Ho, in the rotating coordinateframe, isHo= [Io]ω =I 0 00 I 00 0 Izzω1ω2ω3= Iω1ˆe1+ Iω2ˆe2+ Izzω3ˆe3(4.3)where Ixx= Iy y= I due to the symmetry of t he rigid body. Differentiating with respect totime, we find the time rate of change of the angular momentum to be˙Ho= I ˙ω1ˆe1+ I ˙ω2ˆe2+ Izz˙ω3ˆe3+ Ω × Ho(4.4)where the final term arises due to the use of a rotating coordinate frame (See the˙−→Q formulain Sec 15.2 of your book). Performing the required vector cross-product we getΩ × Ho=ˆe1ˆe2ˆe3ω1ω2Ω3Iω1Iω2Izzω3= (Izzω2ω3− Iω2Ω3) ˆe1+ (Iω1Ω3− Izzω1ω3) ˆe2+ 0 ˆe3(4.5)Using Figure 4.1 we find the total applied torque to beXMo= rcm× W = h ˆe3× −mgˆZ = hmg sin θ ˆe1(4.6)We now use angular momentum balance about the fixed origin, i.e.PMo=˙Ho. Substi-tuting (4.4), (4.5), and (4.6) into the angular momentum balance and “dotting” with all 3rotating unit vectors, we end up with 3 separate equations:I ˙ω1+ Izzω2ω3− Iω2Ω3= hmg sin θ (4.7a)I ˙ω2+ Iω1Ω3− Izzω1ω3= 0 (4.7b)I ˙ω3= 0 (4.7c)62 Lab #4 - Gyroscopic Motion of a Rigid Bod yEquation (4.7c) says that ω3=˙φ cos θ +˙ψ is constant. Physically, we interpret this as sayingthe “total spin” of the rigid body about the ˆe3-axis is constant.We simplify the analysis of the two remaining equations by restricting ourselves to “steady-precession”. Steady-precession occurs when we restrict the kinematics to constant spin rate˙ψo, constant precession˙φo, a nd constant pitch θo. With these restrictions, (4.7b) is triviallysatisfied and we are left with one equation˙φosin θohIzz˙φocos θo+˙ψo− I˙φocos θoi= hmg sin θo(4.8)There are 3 constants in (4.8), two of which can be independently fixed in order to solve forthe third. In this lab you will set t he spin rate˙ψoand the pitch angle θoand