MECHANICS LAB AM 37 EXP 8 FREE VIBRATIN F CUPLED PENDULUMS
I. BJECTIVES I. To observe the normal modes of oscillation of a two degree-of-freedom system. I. To determine the natural frequencies and mode shapes of the system from solution of the eigenvalue problem. I.3 To compare experimental and theoretical natural frequencies of the system. II. THERY The two simple pendulums shown in Fig. are coupled together by means of a light spring which has a spring constant k. The spring is unstrained when the two pendulums are in the vertical position. Figure a represents the pendulums oscillating and Fig. b is the corresponding free-body diagram assuming >. R y R x a k ka( - ) a sin L (a) Figure Double Pendulum (b) mg For a two degree-of-freedom system there are two coupled differential equations that govern the motion of the system. They are determined from application of Newton s Second Law - Equation of Motion. R. Ehgott (Created) / 4/7/ T. Hao (Revised) 8/7/6
Σ M & I 8. where: I mass moment of inertia of one pendulum about the pivot point M moment produced by gravitational force and spring force about point & angular acceleration of the pendulum Summing moments about point for the pendulum shown in Fig. b and assuming is small (i.e., sin and cos ): I & m gl ( ) or & + m gl + ka ( ) I 8. A similar differential equation can be obtained for the other pendulum I & + m gl + ka ( ) 8.3 Equations (8.) and (8.3) can be expressed in matrix form: I && + I && 8.4 A two degree of freedom system has two natural frequencies which can be determined by solving the eigenvalue problem. Assuming harmonic motion and making the following substitutions for and Eq. (8.4), we get & in A sin( ω ), && A ω sin( ω ) t t A sin( ω ), && A ω sin( ω ) t Equation (8.4) can be expressed as: t I A I A ω sin( ω t) + ω sin( ω t) A A sin( ω t) sin( ω t) R. Ehgott (Created) 3/ 4/7/ T. Hao (Revised) 8/7/6
or I A sin( ω t) ω 8.5 I A sin( ω t) Equation (8.5) is the eigenvalue problem discussed in Appendix A. A two degreeof-freedom system will have two roots or eigenvalues which physically represent the natural angular frequencies of the system. Taking the determinant and equating the determinant to zero, the natural angular frequencies ( ω and ω ) cab be solved for as follows. I ω I ω 8.6 ( I ω ) ( ka ) I ω ± ka Solving for the two natural frequencies ω and ω, we have ω mgl I mgl + ka ω 8.7 I Substituting the first natural frequency into Eq. (8.6) and taking the adjoint of the resulting matrix, the first eigenvector can be determined: I ω m gl + ka mgl ka ka ka I ω mgl Factoring out ka and taking the adjoint of the results in: We select one column of the above matrix as the first eigenvector. nly one R. Ehgott (Created) 4/ 4/7/ T. Hao (Revised) 8/7/6
column is needed since the other column will produce the same result. { φ } Following the same procedure and substituting the second eigenvalue into Eq. (8.6) we obtain the second eivenvector: { φ } Eigenvectors are sometimes referred to as the mode shapes of the system and give important information regarding the motion of each mass when the system is set into motion. The first mode shape indicates that both pendulums move together at the same amplitude (in phase) and the second mode shape indicates out of phase motion (shown in Fig. ). R. Ehgott (Created) 5/ 4/7/ T. Hao (Revised) 8/7/6
mgl ω, I { φ } mgl + ka ω, I { φ } k k.5 -.5 -.5 -.5 -.5 -.5 -.5 -.5 - If each pendulum is initially displaced amount and released, the two pendulums will vibrate at the first natural angular frequency ω. The frequency in Hz can be calculated from f ω /π. If one pendulum is initially displaced amount and the other, when released, the two pendulums will vibrate at the second natural angular frequency ω. The frequency in Hz can be calculated from f ω /π. Figure Natural Angular Frequencies and Mode Shapes of the Coupled Pendulums R. Ehgott (Created) 6/ 4/7/ T. Hao (Revised) 8/7/6
The pendulums discussed so far consider a single lumped mass at length L from the support. The pendulums used in this experiment have additional mass that needs to be considered in the inertia and gravity force calculations. L s Weight of spring holder.4 lb Spring constant.39 lb/in. L r L w Weight of rod.44 lb Added weight lb Weight of holder.36 lb Figure 3 Pendulum Data The inertia about the pivot point can be calculated from: I m L + m L + m L + m L 8.8 s s 3 r r w w h r where: m s mass of the spring holder m r mass of the rod m w mass of the added weight m h mass of the weight holder The moments produced by gravity forces are given by: mgl m gl m gl + m gl + m gl 8.9 Σ s s + r r w w h r R. Ehgott (Created) 7/ 4/7/ T. Hao (Revised) 8/7/6
III. EQUIPMENT III. Double pendulum system III. Tape measure III.3 Stopwatch IV. PRCEDURE IV. The natural angular frequency calculations were based on an assumption that the value for the angle was small and that sin. Determine the accuracy of this assumption by calculating sin and comparing it to and record the calculations in Table I. IV. Measure the dimensions L s, L r, and L w and calculate the mass values m s, m r, m w, and m h. The mass is determined by dividing the weight by the gravitational acceleration in inch units (386.4 in/s ). Record these values in Table II. IV.3 Calculate the mass moment of Inertia I from Eq. (8.8) and the moments due to gravity forces from Eq. (8.9). IV.4 Calculate the two angular natural frequencies from Eq. (8.7) and convert the result to Hz. IV.5 Set the two pendulums in motion with both pendulums moving together in the same direction (first mode). Experimentally determine the first natural frequency by measuring the time it takes for approximately ten cycles. Divide the number of cycles counted by the total time measured to obtain the frequency in Hz. IV.6 Set the two pendulums in motion with both pendulums moving in opposite directions direction (second mode). Experimentally determine the second natural frequency using the procedure above. V. REPRT V. Give all dimensions and calculated values. V. Report the error in the sin approximation. R. Ehgott (Created) 8/ 4/7/ T. Hao (Revised) 8/7/6
V.3 Report the theoretical and experimental natural frequency. V.4 Determine the natural frequencies and mode shapes if one of the pendulums has a 4 lb added weight instead of lb. The theoretical solution of the eigenvalue problem must be obtained in this case (see Appendix A for example). VI. SELECTED REFERENCES Thomson, W.T. and Dahleh, M.D., Theory of Vibration with Applications, 5th Edition, Pearson, 997. R. Ehgott (Created) 9/ 4/7/ T. Hao (Revised) 8/7/6
Table I Accuracy of sin (º) (rad) sin. 5 3 % Error referenced to sin Table II Pendulum Data L s (in.) L r (in.) L w (in.) m s (lb-s /in.) m r (lb-s /in.) m w (lb-s /in.) m h (lb-s /in.) L r L w L s m r m s m h m w Table III Natural Frequency Comparison Initial displacement, (º) Time (sec) Number of cycles Experimental Freq. (Hz) Theoretical Freq. (Hz) % Error Ref. To Exp. f f R. Ehgott (Created) / 4/7/ T. Hao (Revised) 8/7/6
Appendix A Eivenvalue Problem Various problems in Mechanics require the solution of the eigenvalue problem. Eigenvalue problems are of the form: ([ A] λ [ I]){ x} {} where [A] is a square matrix that need not be symmetrical. A non trivial solution exists only if the determinant [ A] λ [ I] 8.A This determinant can be expanded into an n th order polynomial in λ. λ i n n λ c λ + L + c λ + c 8.A + n n The n roots ( ) obtained from this equation are the eigenvalues of matrix [A]. The eigenvector { φ i } corresponding to λ i is obtained by substituting λ i into: [ B] [ A] λ [ I] 8.A3 and then computing any column of the adjoint of matrix [B]. Example Determine the eigenvalues and eigenvectors of matrix [A] below. [A] 5 λ det( [ A ] λ [ I]) ( λ)(5 λ ) ( )( ) 5 λ λ 7λ + 6 Giving two roots or eigenvalues for λ, λ, λ 6. For vibration related problems, the roots correspond to the natural angular frequencies of the system. In this case, the natural angular frequency equals the square root of λ. R. Ehgott (Created) / 4/7/ T. Hao (Revised) 8/7/6
To obtain the first eigenvector { φ } we substitute λ into matrix [B] and a determine the adjoint [B]. [ B ] 5, 4 [ B] a 4 The eigenvector { φ } is then { φ }. To obtain the second eigenvector { φ } we substitute λ 6 into matrix [B] and a determine the adjoint [B]. [ B ] 6 5 6 4, [ B] a 4 The eigenvector { φ } is then { φ }. Eigenvectors can be scaled for convenience. For example { φ } could be expressed by: { φ }.5 For vibration problems, the eigenvectors describe the relative displacement of each mass. The first mode shape states the first mass will have twice the displacement of the second mass when the system vibrating at the first natural angular frequency ω λ. R. Ehgott (Created) / 4/7/ T. Hao (Revised) 8/7/6