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EXP8.pdf
MECHANICS LAB AM 317
EXP 8 FREE VIBRATION
OF COUPLED PENDULUMS
I. OBJECTIVES
I.1 To observe the normal modes of oscillation of a two degree-of-freedom system.
I.2 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. THEORY
The two simple pendulums shown in Fig. 1 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 1a represents the pendulums oscillating and Fig. 1b is the corresponding free-body diagram assuming 12 θθ > .
θ 1
θ 2
ka
L θ 2
ka (θ 2 - θ 1 )
O
R y
R x
a sin θ 2
mg (a) (b)
Figure 1 Double Pendulum
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) 2/12 04/07/01 T. Hao (Revised) 08/07/16
θ&&OO IM =Σ 8.1
where:
IO = mass moment of inertia of one pendulum about the pivot point O MO = moment produced by gravitational force and spring force about point
O = angular acceleration of the pendulum θ&&
Summing moments about point O for the pendulum shown in Fig. 1b and assuming θ is small (i.e., θθ ≅sin and 1cos ≅θ ):
)( 12 2
22 θθθθ −−−= akLgmI O &&
or 0)( 12 2
22 =−++ θθθθ akLgmI O && 8.2
A similar differential equation can be obtained for the other pendulum
0)( 21 2
11 =−++ θθθθ akLgmI O && 8.3
Equations (8.2) and (8.3) can be expressed in matrix form:
⎭ ⎬ ⎫
⎩ ⎨ ⎧
= ⎪⎭
⎪ ⎬ ⎫
⎪⎩
⎪ ⎨ ⎧
⎥ ⎦
⎤ ⎢ ⎣
⎡
+− −+
+ ⎪⎭
⎪ ⎬ ⎫
⎪⎩
⎪ ⎨ ⎧
⎥ ⎦
⎤ ⎢ ⎣
⎡ 0 0
0 0
2
1
22
22
2
1
θ
θ
θ
θ
akLgmak akakLgm
I I
O
O
&&
&& 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 in Eq. (8.4), we get
θ&&
)sin(11 tA ωθ = , )sin( 2
11 tA ωωθ −=&&
)sin(22 tA ωθ = , )sin( 2
22 tA ωωθ −=&&
Equation (8.4) can be expressed as:
⎭ ⎬ ⎫
⎩ ⎨ ⎧
= ⎪⎭
⎪ ⎬ ⎫
⎪⎩
⎪ ⎨ ⎧
⎥ ⎦
⎤ ⎢ ⎣
⎡
+− −+
+ ⎪⎭
⎪ ⎬ ⎫
⎪⎩
⎪ ⎨ ⎧
−
− ⎥ ⎦
⎤ ⎢ ⎣
⎡ 0 0
)sin(
)sin(
)sin(
)sin( 0
0
2
1
22
22
2 2
2 1
tA
tA
akLgmak akakLgm
tA
tA I
I
O
O
ω
ω
ωω
ωω
R. Ehgott (Created) 3/12 04/07/01 T. Hao (Revised) 08/07/16
or ⎭ ⎬ ⎫
⎩ ⎨ ⎧
= ⎪⎭
⎪ ⎬ ⎫
⎪⎩
⎪ ⎨ ⎧
⎟ ⎟ ⎠
⎞ ⎜ ⎜ ⎝
⎛ ⎥ ⎦
⎤ ⎢ ⎣
⎡ −⎥
⎦
⎤ ⎢ ⎣
⎡
+− −+
0 0
)sin(
)sin( 0
0
2
12 22
22
tA
tA I
I
akLgmak akakLgm
O
O
ω
ω ω 8.5
Equation (8.5) is the eigenvalue problem discussed in Appendix A. A two degree- of-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 ( 1ω and 2ω ) cab be solved for as follows.
0222 222
= ⎥ ⎥ ⎦
⎤
⎢ ⎢ ⎣
⎡
−+− −−+
ω ω
O
O
IakLgmak akIakLgm
8.6
0)()( 22222 =−−+ akIakLgm Oω
222 akIakLgm O ±=−+ ω
Solving for the two natural frequencies 1ω and 2ω , we have
OI mgL
=1ω OI
akmgL 2 2
2+ =ω 8.7
Substituting the first natural frequency into Eq. (8.6) and taking the adjoint of the resulting matrix, the first eigenvector can be determined:
⎥ ⎥ ⎦
⎤
⎢ ⎢ ⎣
⎡
−+− −−+
2 1
22
22 1
2
ω ω
O
O
IakLgmak akIakLgm
⎥ ⎦
⎤ ⎢ ⎣
⎡
−+− −−+
mgLakLgmak akmgLakLgm
22
22
⎥ ⎦
⎤ ⎢ ⎣
⎡
− −
22
22
akak akak
Factoring out and taking the adjoint of the results in: 2ak
⎥ ⎦
⎤ ⎢ ⎣
⎡ 11 11
We select one column of the above matrix as the first eigenvector. Only one
R. Ehgott (Created) 4/12 04/07/01 T. Hao (Revised) 08/07/16
column is needed since the other column will produce the same result.
⎭ ⎬ ⎫
⎩ ⎨ ⎧
= 1 1
}{ 1φ
Following the same procedure and substituting the second eigenvalue into Eq. (8.6) we obtain the second eivenvector:
⎭ ⎬ ⎫
⎩ ⎨ ⎧
− =
1 1
}{ 2φ
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. 2).
R. Ehgott (Created) 5/12 04/07/01 T. Hao (Revised) 08/07/16
OI mgL
=1ω , ⎭ ⎬ ⎫
⎩ ⎨ ⎧
= 1 1
}{ 1φ OI
akmgL 2 2
2+ =ω ,
⎭ ⎬ ⎫
⎩ ⎨ ⎧
− =
1 1
}{ 2φ
θ 0
θ 0
k
0
0 .5 1
-0 .5-1
−θ 0
θ 0
k
0
0 .5 1
-0 .5-1 0
0 .5 1
-0 .5-10
0 .5 1
-0 .5-1
R. Eh T. Ha
If each pendulum is initially displaced amount θ 0 and released, the two pendulums will vibrate at the first natural angular frequency ω 1. The frequency in Hz can be calculated from f 1 = ω 1/2π.
Figure Natural Angular Frequencies and Mode
gott (Created) 6/12 o (Revised)
If one pendulum is initially displaced amount θ 0 and the other −θ 0, when released, the two pendulums will vibrate at the second natural angular frequency ω 2. The frequency in Hz can be calculated from f 2 = ω 2/2π.
2 Shapes of the Coupled Pendulums
04/07/01 08/07/16
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
LwLr
Weight of spring holder = 0.141 lb
Spring constant = 0.39 lb/in.
Weight of rod = 1.44 lb
Added weight = 2 lb
Weight of holder = 0.326 lb
Figure 3 Pendulum Data
The inertia about the pivot point O can be calculated from:
222 3 12
rhwwrrssO LmLmLmLmI +++= 8.8
where:
ms = mass of the spring holder mr = mass of the rod mw = mass of the added weight mh = mass of the weight holder
The moments produced by gravity forces are given by:
rhwwrrss LgmLgmLgmLgmmgL +++=Σ 2 1 8.9
R. Ehgott (Created) 7/12 04/07/01 T. Hao (Revised) 08/07/16
III. EQUIPMENT
III.1 Double pendulum system
III.2 Tape measure
III.3 Stopwatch
IV. PROCEDURE
IV.1 The natural angular frequency calculations were based on an assumption that the value for the angle θ was small and that sinθ = θ. Determine the accuracy of this assumption by calculating sinθ and comparing it to θ and record the calculations in Table I.
IV.2 Measure the dimensions Ls, Lr, and Lw and calculate the mass values ms, mr, mw, and mh. The mass is determined by dividing the weight by the gravitational acceleration in inch units (386.4 in/s2). Record these values in Table II.
IV.3 Calculate the mass moment of Inertia IO 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. REPORT
V.1 Give all dimensions and calculated values.
V.2 Report the error in the sinθ = θ approximation.
R. Ehgott (Created) 8/12 04/07/01 T. Hao (Revised) 08/07/16
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 2 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, 1997.
R. Ehgott (Created) 9/12 04/07/01 T. Hao (Revised) 08/07/16
Table I Accuracy of sinθ = θ
θ (º) θ (rad) sinθ % Error
referenced to sinθ 0.1 1 5
10 20 30
Table II Pendulum Data
Ls (in.) Lr (in.) Lw (in.) ms (lb-s
2/in.) mr (lb-s
2/in.) mw (lb-s
2/in.) mh (lb-s
2/in.)
Ls
LwLr
ms
mw
mr
mh
Table III Natural Frequency Comparison
f1 f2 Initial displacement,θ 0 (º) Time (sec) Number of cycles Experimental Freq. (Hz) Theoretical Freq. (Hz) % Error Ref. To Exp.
R. Ehgott (Created) 10/12 04/07/01 T. Hao (Revised) 08/07/16
Appendix A Eivenvalue Problem
Various problems in Mechanics require the solution of the eigenvalue problem. Eigenvalue problems are of the form: }0{}{)][][( =− xIA λ where is a square matrix that need not be symmetrical. A non trivial solution exists only if the determinant
][ A
0][][ =− IA λ 8.A1
This determinant can be expanded into an nth order polynomial in λ.
011 1
1 =++++ − −
nn nn ccc λλλ L 8.A2
The n roots ( ) obtained from this equation are the eigenvalues of matrix . The eigenvector
iλ ][ A }{ iφ corresponding to is obtained by substituting into: iλ iλ
][][][ IAB λ−= 8.A3
and then computing any column of the adjoint of matrix . ][B
Example
Determine the eigenvalues and eigenvectors of matrix below. ][ A
⎥ ⎦
⎤ ⎢ ⎣
⎡ −
− =
52 22
][ A
0)2)(2()5)(2( 52
22 )][][det( =−−−−−=
−− −−
→− λλ λ
λ λ IA
0672 =+−→ λλ
Giving two roots or eigenvalues for λ , 11 =λ , 62 =λ .
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) 11/12 04/07/01 T. Hao (Revised) 08/07/16
To obtain the first eigenvector }{ 1φ we substitute 11 =λ into matrix and
determine the adjoint .
][B aB][
⎥ ⎦
⎤ ⎢ ⎣
⎡ =⎥
⎦
⎤ ⎢ ⎣
⎡ −
− =⎥
⎦
⎤ ⎢ ⎣
⎡ −−
−− =
12 24
][, 42 21
152 212
][ aBB
The eigenvector }{ 1φ is then . ⎭ ⎬ ⎫
⎩ ⎨ ⎧
= 1 2
}{ 1φ
To obtain the second eigenvector }{ 2φ we substitute 62 =λ into matrix and
determine the adjoint .
][B aB][
⎥ ⎦
⎤ ⎢ ⎣
⎡ −
− =⎥
⎦
⎤ ⎢ ⎣
⎡ −− −−
=⎥ ⎦
⎤ ⎢ ⎣
⎡ −−
−− =
42 21
][, 12 24
652 262
][ aBB
The eigenvector }{ 2φ is then . ⎭ ⎬ ⎫
⎩ ⎨ ⎧−
= 2 1
}{ 2φ
Eigenvectors can be scaled for convenience. For example }{ 2φ could be expressed by:
⎭ ⎬ ⎫
⎩ ⎨ ⎧
− =
1 5.0
}{ 2φ
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 11 λω = .
R. Ehgott (Created) 12/12 04/07/01 T. Hao (Revised) 08/07/16
