Split beam method for determining mode shapes of cantilever beams under multiple loading conditions

2021 ◽  
pp. 107754632110276
Author(s):  
Jun-Jie Li ◽  
Shuo-Feng Chiu ◽  
Sheng D Chao
Keyword(s):  
Operation Mode ◽  
Point Contact ◽  
Mode Shapes ◽  
Cantilever Beams ◽  
Loading Conditions ◽  
Mechanical Responses ◽  
Relative Contribution ◽  
Resonance Frequencies ◽  
Beam Method ◽  
Multiple Loading

We have developed a general method, dubbed the split beam method, to solve Euler–Bernoulli equations for cantilever beams under multiple loading conditions. This kind of problem is, in general, a difficult inhomogeneous eigenvalue problem. The new idea is to split the original beam into two (or more) effective beams, each of which corresponds to one specific load and bears its own Young’s modulus. The mode shape of the original beam can be obtained by linearly superposing those of the effective beams. We apply the split beam method to simulating mechanical responses of an atomic force microscope probe in the “dynamical” operation mode, under which there are a stabilizing force at the positioner and a point-contact force at the tip. Compared with traditional analytical or numerical methods, the split beam method uses only a few number of basis functions from each effective beam, so a very fast convergence rate is observed in solving both the resonance frequencies and the mode shapes at the same time. Moreover, by examining the superposition coefficients, the split beam method provides a physical insight into the relative contribution of an individual load on the beam.

scholarly journals A Multi-Mode Broadband Vibration Energy Harvester Composed of Symmetrically Distributed U-Shaped Cantilever Beams

Micromachines ◽  
2021 ◽  
Vol 12 (2) ◽  
pp. 203
Author(s):  
Xiaohua Huang ◽  
Cheng Zhang ◽  
Keren Dai
Keyword(s):  
Energy Harvester ◽  
Low Frequency ◽  
Experimental Results ◽  
Vibration Energy ◽  
Cantilever Beams ◽  
Resonant Frequencies ◽  
Promising Alternative ◽  
Piezoelectric Energy ◽  
Straight Beam ◽  
Resonance Frequencies

Using the piezoelectric effect to harvest energy from surrounding vibrations is a promising alternative solution for powering small electronic devices such as wireless sensors and portable devices. A conventional piezoelectric energy harvester (PEH) can only efficiently collect energy within a small range around the resonance frequency. To realize broadband vibration energy harvesting, the idea of multiple-degrees-of-freedom (DOF) PEH to realize multiple resonant frequencies within a certain range has been recently proposed and some preliminary research has validated its feasibility. Therefore, this paper proposed a multi-DOF wideband PEH based on the frequency interval shortening mechanism to realize five resonance frequencies close enough to each other. The PEH consists of five tip masses, two U-shaped cantilever beams and a straight beam, and tuning of the resonance frequencies is realized by specific parameter design. The electrical characteristics of the PEH are analyzed by simulation and experiment, validating that the PEH can effectively expand the operating bandwidth and collect vibration energy in the low frequency. Experimental results show that the PEH has five low-frequency resonant frequencies, which are 13, 15, 18, 21 and 24 Hz; under the action of 0.5 g acceleration, the maximum output power is 52.2, 49.4, 61.3, 39.2 and 32.1 μW, respectively. In view of the difference between the simulation and the experimental results, this paper conducted an error analysis and revealed that the material parameters and parasitic capacitance are important factors that affect the simulation results. Based on the analysis, the simulation is improved for better agreement with experiments.

Vibrations of Spinning Rings With Radial and Circumferential Displacement Constraints

1991 ◽  
Author(s):  
Shyh-Chin Huang ◽  
Chen-Kai Su
Keyword(s):  
Natural Frequencies ◽  
Point Contact ◽  
Frequency Equation ◽  
Mode Shapes ◽  
Complex Conjugate ◽  
Constrained System ◽  
Form Complex ◽  
Circumferential Displacement ◽  
On The Road ◽  
Displacement Constraints

Abstract The frequencies and mode shapes of rolling rings with radial and circumferential displacement constraints are investigated. The displacement constraints practically come from the point contact, e.g., rolling tire on the road, or other applications. The proposed approach to analysis is calculating the natural frequencies and modes of a non-contacted spinning ring, then employing the receptance method for displacement constraints. The frequency equation for the constrained system is hence obtained, and it can be solved numerically or graphically. The receptance matrix developed for the spinning ring is surprisingly found not symmetric as usual. Moreover, the cross receptances are discovered to form complex conjugate pairs. That is a feature that has never been described in literature. The results show that the natural frequencies for the spinning ring in contact, as expected, higher than those for the non-contacted ring. The variance of frequencies to rotational speeds are then illustrated. The analytic forms of mode shapes are also derived and sketched. The traveling modes are then shown for cases.

Quantify Resonance Inspection With Finite-Element-Based Modal Analyses

2010 ◽  
Author(s):  
K. Lai ◽  
X. Sun ◽  
C. Dasch
Keyword(s):  
Finite Element ◽  
High Stress ◽  
Mesh Size ◽  
Element Model ◽  
Sensitivity Study ◽  
Production Level ◽  
Mode Shapes ◽  
Resonance Spectroscopy ◽  
Frequency Spectra ◽  
Resonance Frequencies

Resonance inspection uses the natural acoustic resonances of a part to identify anomalous parts. Modern instrumentation can measure the many resonant frequencies rapidly and accurately. Sophisticated sorting algorithms trained on sets of good and anomalous parts can rapidly and reliably inspect and sort parts. This paper aims at using finite-element-based modal analysis to put resonance inspection on a more quantitative basis. A production-level automotive steering knuckle is used as the example part for our study. First, the resonance frequency spectra for the knuckle are measured with two different experimental techniques. Next, scanning laser vibrometry is used to determine the mode shape corresponding to each resonance. The material properties including anisotropy are next measured to high accuracy using resonance spectroscopy on cuboids cut from the part. Then, finite element model (FEM) of the knuckle is generated by meshing the actual part geometry obtained with computed tomography (CT). The resonance frequencies and mode shapes are next predicted with a natural frequency extraction analysis after extensive mesh size sensitivity study. The good comparison between the predicted and the experimentally measured resonance spectra indicate that finite-element-based modal analyses have the potential to be a powerful tool in shortening the training process and improving the accuracy of the resonance inspection process for a complex, production level part. The finite element based analysis can also provide a means to computationally test the sensitivity of the frequencies to various possible defects such as porosity or oxide inclusions especially in the high stress regions that the part will experience in service.

METHODOLOGIES TO PREDICT HYDRODYNAMIC CHARACTERISTICS OF PUSHER AND PULLER PODDED PROPULSORS IN OBLIQUE FLOWS

2021 ◽  
Vol 158 (A4) ◽  
Author(s):  
M F Islam ◽  
F Jahra
Keyword(s):  
Computation Time ◽  
Performance Characteristics ◽  
Three Dimensions ◽  
Hydrodynamic Characteristics ◽  
Loading Conditions ◽  
Time Step ◽  
Podded Propulsor ◽  
Multiple Loading ◽  
Thrust And Torque ◽  
Straight Ahead

This paper presents the outcome of a numerical simulation based research program to evaluate the propulsive characteristics of puller and pusher podded propulsors in a straight course and at static azimuthing conditions while operating in open water. Methodologies to predict the propeller thrust and torque, and pod forces and moments in three dimensions using a Reynolds-Averaged Navier Stokes (RANS) solver at multiple azimuthing conditions and pod configurations are presented. To obtain insight into the reliability and accuracy of the results, grid and time step dependency studies are conducted for a podded propulsor in straight-ahead condition. The simulation techniques and results are first validated against measurements of a bare propeller and a podded propulsor in straight ahead condition for multiple loading scenarios and in both puller and pusher configurations. Next, simulations were carried out to model the podded propulsors in the two configurations at multiple loading conditions and at various azimuthing angles from +30° to –30° in 15° increments. The majority of the simulations are carried out using both steady state and unsteady state conditions, primarily to evaluate the effect of setup conditions on the computation time and prediction accuracy. The predicted performance characteristics of the pod unit using the unsteady RANS method were within 1% to 5% of the corresponding experimental measurements for all the loading conditions, azimuthing angles and pod configurations studied. The non-linear behaviour of the performance coefficients of the pod unit are well captured at various loading and azimuthing conditions in the predicted results. This study demonstrates that the RANS solver, with proper meshing arrangement, boundary conditions and setup techniques can predict the performance characteristics of the podded propulsor in multiple azimuthing angles, pod configurations and in the various loading conditions with a same level of accuracy as experimental results. Additionally, the velocity and pressure distributions on and around the pod-strut- propeller bodies are discussed as derived from the RANS predictions.

Crashworthiness performance of concentric structures with different cross-sectional shapes under multiple loading conditions

2020 ◽  
pp. 095440702096188
Author(s):  
Sadjad Pirmohammad
Keyword(s):  
Energy Absorption ◽  
Specific Energy ◽  
Absorption Maximum ◽  
Element Model ◽  
Hexagonal Structure ◽  
Loading Conditions ◽  
Cross Sectional ◽  
Specific Energy Absorption ◽  
Multiple Loading ◽  
Energy Absorbing

This paper evaluates the crashworthiness performance of concentric structures with different numbers of tubes (i.e. one to five) and cross-sectional shapes (i.e. hexagon, octagon, decagon and circle) under the multiple loadings of θ = 0, 10, 20 and 30°. An experimentally validated finite element model generated in LS-DYNA is employed to calculate the crashworthiness parameters including the specific energy absorption, maximum crush force and crush force efficiency. A total of 20 concentric structures are analyzed to explore the effects of number of tubes and cross-sectional shapes on the crushing performance. A multi-criteria decision-making method known as TOPSIS is also used to compare and rank the concentric structures in terms of crushing performance. Based on the results, the hexagonal structure including two tubes and octagonal, decagonal and circular structures including three tubes demonstrate the best results among their corresponding cross-sectional shapes. These structures show 9, 39, 38 and 39% higher specific energy absorption compared to their corresponding single tubal cases, respectively. However, in comparison to single tubal cases, they generate 4, 57, 57 and 58% higher maximum crush force, respectively. As such, the values for the improvement of the crush force efficiency are 3, 26, 25 and 21%, respectively. Furthermore, the decagonal structure including three tubes provides the highest energy absorbing characteristics as compared with all the other structures studied in this research. Meanwhile, taking into account all the multiple loading conditions, this structure shows 50% higher specific energy absorption than the hexagonal structure including single tube (as the weakest structure).

scholarly journals Dynamic Water Strider Algorithm for Optimal Design of Skeletal Structures

2020 ◽  
Author(s):  
Ali Kaveh ◽  
Armin Dadras Eslamlou ◽  
Nima Khodadadi
Keyword(s):  
Optimal Design ◽  
Structural Design ◽  
Territorial Behavior ◽  
Frame Structures ◽  
Loading Conditions ◽  
Water Strider ◽  
Skeletal Structures ◽  
Multiple Loading ◽  
Dynamic Version ◽  
Suitable Technique

In the present paper, a dynamic version of Water Strider Algorithm (WSA) is proposed. The WSA as well as the Dynamic Water Strider Algorithm (DWSA) are applied to minimize the weight of several skeletal structures. WSA is a nature-inspired metaheuristic that mimics the territorial behavior, intelligent ripple communication, mating style, feeding mechanisms, and succession of water strider insects. The efficiency of these algorithms is tested by optimizing different truss and frame structures subject to multiple loading conditions and constraints. Comparing the results obtained by DWSA with those of other methods it becomes evident that DWSA is a suitable technique for optimizing the structural design and minimizing the weight of structures while fulfilling all constraints.

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