1.1 To observe the normal modes of oscillation of a two degree of freedom system.

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2 I. BJECTIVES. To observe the normal modes of oscillation of a two degree of freedom system.. To determine the natural frequencies and mode shapes of the system from solution of the Eigenvalue problem..3 To compare experimental and theoretical natural frequencies of the system. II. THERY The two simple pendulums shown below 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 Figure b is the corresponding free body diagram assuming θ > θ. Rx Ry a Κ Ka(θ θ ) a sin(θ ) L θ θ θ mg (a) Figure Double Pendulum (b) 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 of Motion (Equation 8.). R. Ehrgott / 3//

3 Where: M & θ 8. I o Mass moments of inertia of one pendulum about the pivot point o. Mo Moment produced by gravity forces and spring forces about o. & θ The angular acceleration of the pendulum. Summing moments about point o for the pendulum shown in Figure b and assuming θ is small (sin(θ) θ and cos(θ) ): & θ mglθ ( θ θ) or & θ + mglθ + ka ( θ θ) 8. A similar differential equation can be obtained for the other pendulum & θ + mglθ + ka ( θ θ ) 8.3 Equations 8. and 8.3 can be expressed in matrix form: && θ && θ + mgl + ka θ θ 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 θ and & θ in Equation 8.4: θ A, & θ Aω θ A, & θ Aω Equation 8.4 can be expressed as R. Ehrgott 3/ 3//

4 Aω Aω mgl + ka + A A or mgl + ka ω A A 8.5 Equation 8.5 is the Eigenvalue problem discussed in Appendix A. By equating the determinant mgl + ka ω 8.6 to zero, the natural frequencies (ω, ) can be solved for. A two degree of freedom system will have two roots or eigenvalues which physically represent the natural frequencies of the system. Taking the determinant yields: ( ω ) (ka ) Taking the square root of both sides, ω ± ka Solving for the two natural frequencies ω and ω : ω mgl ω 8.7 R. Ehrgott 4/ 3//

5 Substituting the first natural frequency into Equation 8.6 and taking the adjoint of the resulting matrix, the first eigenvector can be determined: mgl + ka mgl o mgl mgl ka ka 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 column is needed since the other column will produce the same result. { φ } Following the same procedure and substituting the second eigenvalue into Equation 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 (Figure ). R. Ehrgott 5/ 3//

6 mgl ω, { φ} ω, { φ} Κ Κ θ θ θ θ If each pendulum is displaced amount +θ and released, the two pendulums will vibrate at the first natural frequency ω. The frequency in Hertz can be calculated from f ω /π. If one pendulum is displaced amount +θ and the other - θ, when released, the two pendulums will vibrate at the second natural frequency ω. The frequency in Hertz can be calculated from f ω /π. Figure Natural Frequencies and Mode Shapes of the Coupled Pendulums R. Ehrgott 6/ 3//

7 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. Ls Weight spring holder.4 lb Spring constant.39 lb/in Lr Lw Weight of rod.44 lb Added weight. lb Weight holder.36 lb Figure 3 Pendulum Data The inertia about the pivot point o can be calculated from: Where: mrlr m sls + + mwl w + mhl 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 r The moments produced by gravity forces are given by: m gl r r mgl msgl s + + mwgl w + mhglr 8.9 III. EQUIPMENT 3. Double pendulum system 3. Tape measure 3.3 Stopwatch R. Ehrgott 7/ 3//

8 IV. PRCEDURE 4. The natural frequency calculations were based on an assumption that the value for the angle θ was small and that sin(θ) θ. Determine the accuracy of this assumption by calculating sin(θ) and comparing it to θ and record the calculations in Table I. 4. 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 acceleration of gravity in inch units (386.4 in/s ). Record these values in Table. 4.3 Calculate the mass moment of Inertia I o from Equation 8.8 and the moments due to gravity forces from Equation Calculate the two circular natural frequencies from Equation 8.7 and convert the result to Hz. 4.5 Set the two pendulums in motion with both pendulums moving together in the same direction (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. 4.6 Set the two pendulums in motion with both pendulums moving in opposite directions direction (mode ). Experimentally determine the second natural frequency using the procedure above. V. REPRT 5. Give all dimensions and calculated values. 5. Report the error in the sin(θ) θ approximation. 5.3 Report the theoretical and experimental natural frequency. 5.4 Determine the natural frequencies and mode shapes if one of the pendulums has a 4 lb added weight instead of lb. The numerical solution of the Eigenvalue problem must be obtained in this case (see Appendix A for example). VI. REFERENCES Vierck, R.K., Vibration Analysis, Harper & Row, 979 R. Ehrgott 8/ 3//

9 θ in Degrees θ in Radians Sin(θ) % Error referenced to sin(θ). 5 3 Table Accuracy of Sin(θ) θ m s m r m w m h L s L r L w Pendulum data (in) (in) (in) (lb s /in) (lb s /in) (lb s /in) (lb s /in) Lr Lw Ls m r m s Table Pendulum Data m w m h Time (sec) Number of cycles Experimental Freq. (Hz) Theoretical Freq. (Hz) % Error Ref. to Exp. f f Table 3 Natural Frequency Comparison R. Ehrgott 9/ 3//

10 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 ]. This determinant can be expanded into an n th order polynomial in λ. λ n + c λ n- + c n- λ + c n The n roots (λ i ) 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 ] and then computing any column of the Adjoint of matrix [B]. Example Determine the eigenvalues and eigenvectors of matrix [A] below. [A] 5 det ([A] λ [ I ]) det λ 5 λ ( - λ) (5 - λ) - 4 λ 7λ + 6 Giving two roots or eigenvalues for λ, λ, λ 6. R. Ehrgott / 3//

11 For vibration related problems, the roots correspond to the circular natural frequencies of the system. In this case, the circular natural frequency equals the square root of λ. To obtain the first eigenvector {φ } we substitute λ into matrix [B] and determine the Adjoint [B] a. [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 determine the Adjoint [B] a. [B] [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 frequency ω λ R. Ehrgott / 3//

MECHANICS LAB AM 317 EXP 8 FREE VIBRATION OF COUPLED PENDULUMS

MECHANICS LAB AM 317 EXP 8 FREE VIBRATION OF COUPLED PENDULUMS 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