Are the resonant frequencies the same for a rigidly clamped beam and a free beam in space? While the modal frequencies may be equal, the mode shapes are expected to differ. Examples illustrating this comparison would be greatly appreciated. Thank you, Dave.
Hello, under the specified conditions, the natural frequencies of the beams will be the same. When both ends are free, they will have antinodes and the first natural frequency can be calculated as f=c/2L, where f is the frequency, c is the velocity of sound in the medium (steel), and L is the length of the tube. If both ends are fixed, they will have nodes and the first natural frequency will be f=c/2L. When one end is fixed and the other is free, the fixed end will have a node and the free end will have antinodes, resulting in the first natural frequency being f=c/4L. This scenario is different from the previous two cases.
The explanation provided above touches on an interesting concept. When considering the resonant frequencies of a clamped/clamped beam versus a free/free beam, it may seem counterintuitive that they can be the same. However, the key lies in understanding the role of the zero-frequency mode in the free/free beam. This mode, although not immediately apparent, actually sets the foundation for comparison with the modes of the clamped/clamped beam. Imagine a beam with variable-stiffness clamps on each end. Initially, with low stiffness, the beam behaves like a rigid body on a weak spring, echoing the free/free first mode at zero frequency. As stiffness is gradually increased, the frequency also rises, eventually reaching the resonant frequency of the clamped/clamped configuration. This mental experiment illustrates how the zero frequency first mode of the free/free beam transitions into the first frequency mode of the clamped/clamped beam as stiffness increases. Therefore, it is clear that the first mode of the stiffer and more constrained clamped/clamped case will indeed have a higher frequency than the first mode of the flexible and unconstrained free/free case. The apparent coincidence where the second mode of the free/free beam matches the first mode of the clamped/clamped beam can be understood through mathematical reasoning, but does not negate the expectation that the real modes of the free/free beam have lower resonant frequencies than their counterparts in the clamped/clamped beam. This phenomenon underscores the intricate relationship between stiffness, mode shapes, and resonant frequencies in beam configurations.
Looking for an "example"? See attached for the output of a transfer-matrix numerical solution for a 1-meter long cylindrical steel beam with a 0.1-meter diameter. The first slide displays the first three mode shapes and frequencies for the clamped/clamped scenario, while the second slide showcases the first three non-zero mode shapes and frequencies for the free/free case. Despite the varying mode shapes, the first three frequencies align with what was previously discussed. You can calculate these yourself using Den Hartog Appendix V (let me know if you need assistance with that). By examining the coefficients in the appendix, you can observe the resonance frequency match between the clamped/clamped and free/free scenarios (excluding the first zero-frequency mode of the free/free case).
Thank you for your responses and examples. I understand that clamped/clamped and free/free modal frequencies should theoretically be the same, with slight differences in mode shapes, as shown in Pete's file. However, during a recent modal test (data attached for the first mode only), I found that the frequencies were actually different from the predicted values. In this test, the beam was tested in both free/free and clamped/clamped modes. In the free/free mode, the beam was suspended horizontally and motion was measured in the horizontal direction. In the clamped/clamped mode, the beam was fixed to a solid foundation and measurements were taken in the vertical direction. I intentionally chose to conduct the test in this manner because vertical clamping was more effective and gravity was not a factor. Despite expectations, the modal frequencies varied significantly between the two modes. I will need to revisit the formulas to consider how the foundation stiffness impacts the results. It is possible that the theoretical calculation for the clamped/clamped frequency assumes an infinitely high foundation stiffness. For more information, please refer to the video "Fixed_beam_Movie_1.avi" provided by David.
Access free beam data in the video entitled Free_beam_1_Movie_1.avi. Get valuable information and analysis on beam properties at no cost.
Hi Dave, you're on the right track! The resonant frequencies of a rigidly clamped beam and a free beam in space will indeed differ due to boundary conditions. The clamped beam's frequencies are higher due to the constraint that exists at both ends, which is not the case with a free beam. Meanwhile, the mode shapes also depend on how the beam is fixed. When it's clamped, the mode shapes illustrate areas of zero displacement at both ends, whereas a free beam has its maximum displacement at the ends. In essence, the physical constraints of the system play an integral part in determining both resonant frequencies and mode shapes. Hope this helps!
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Answer: 1. Are the resonant frequencies the same for a rigidly clamped beam and a free beam in space? - The resonant frequencies may be the same, but the mode shapes are expected to differ between a rigidly clamped beam and a free beam in space.
Answer: - The modal frequencies may be equal between a rigidly clamped beam and a free beam, but the mode shapes are likely to be different.
Answer: - Examples that demonstrate the comparison of modal analysis between a rigidly clamped beam and a free beam would help in understanding how the mode shapes differ.
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