Vibration of a Cantilever Plate (Level 7) Lecturer: Dr Guowu Wei [email protected] School of Computing, Science and Engineering, University of Salford Notes: This assignment contributes 30% to the mark of the whole module, the report for this assignment should be no more than 30 pages. Report associated with this assignment is due at 4pm on Friday 6th January 2017. 1 Aims and Objectives This assignment aims to carry out the theoretical, experimental and FEA-based vibration analysis of a 6-DoF aluminium lumped mass plate. The objectives of this assignment are listed as follows: • By using the compliance matrix obtained (provided in Section 4) through experimental measurements and the MATLAB program provided, construct matrix A (see Sections 3 and 4) so as to find the eigenvalues and eigenvectors of A which correspond to the 6 theoretical values for λ (natural frequency squared) and the corresponding 6 modal vectors X that describe the mode shapes. Then plot the theoretical mode shapes so that you can compare them with the experimental mode shapes and the results obtained through FEA simulation. • Carry out experiments on 6-DoF aluminium plate and find the first 3 bending modes along with the first 3 torsional modes by sprinkling sand onto the plate. • Simulate the experiment in ANSYS or Solidworks software and carry out modal analysis for the 6-DoF aluminium plate. • Report the results achieved from theoretical analysis, experimental study and FEA simulation, providing comprehensive discussion and comparison. 2 Introduction for the Experiment In this experiment you will make observations of the natural frequencies and corresponding vibration modes of a flat cantilever plate. An electrodynamic shaker will be used to find the plate's first 9 resonances, which occur at approximately the natural frequencies. At a particular resonance the corresponding vibration mode dominates the motion of the plate and mode shape can be observed using sand which gathers along lines where the amplitude of plate vibration is zero or close to zero. You will also use a simplified six degree-of-freedom (DoF) lumped parameter model of the plate to obtain theoretical results for comparison with the experimental results and the FEA results. This model will be based on the assumption that the mass of the plate is lumped at 6 locations, the vertical displacement of those locations being the simplified model's 6 DoF. To make theory and experiment closer, 6 bolts have been placed at the chosen locations to increase the mass at those positions. 3 Theory Background 3.1 Normal mode vibration The simultaneous equations of motion of an n-DoF system can be written in matrix form as follows Mx ¨ + Kx = 0 (1) where M is the system's n × n mass matrix, K is the system's n × n stiffness matrix, and x is an n × 1 vector of the displacements of the n degrees-of-freedom. 1Assuming that the system undergoes normal mode vibration (synchronous harmonic motion), we can write x = X A sin (!t + φ) (2) e.g.  x x1 2  =  X X1 2  A sin (!t + φ) for a 2 degrees-of-freedom system. Where X is a constant modal vector which describes the relative vibration amplitudes of the degrees-of-freedom. By differentiating Eq. (2) twice with respect to time we get the following expression for acceleration x ¨ = −!2X A sin (!t + φ) or x ¨ = −!2x (3) Therefore, by substituting Eq. (3) into Eq. (1), the matrix form equation of motion becomes − M!2x + Kx = 0 (4) Rearranging, substituting λ = !2 into Eq. (4), and dividing through by A sin (!t + φ) gives (K − Mλ) X = 0 (5) 3.2 The eigen-problem approach Pre-multiplying Eq. (5) by the inverse of the mass matrix, it yields M−1K − Iλ
X = 0 (6) This can be written in the format of a standard eigenvalue problem as follows (A − λI) X = 0 or AX = λX (7) where A = M−1K and I is the identity matrix. Values of λ (natural frequency squared) that allow non-trivial solutions of the above are known as the eigenvalues of A. The corresponding solutions for the modal vectors X are A's eigenvectors. These concepts should be familiar to you from your mathematics lectures. There are many maths software packages which will calculate a matrix inverse and the eigenvalues and eigenvectors of a matrix. For this reason the eigen-problem formulation is preferred for computer solutions. It is recommended that any problem with more than 2-DoF should be solved using maths software such as MATLAB and Mathematica. In this experiment, you will be measuring the compliance matrix H, which is the inverse of the stiffness matrix; in other words, K = H−1. Therefore, the matrix A that can be used to find the natural frequencies ! = pλ and the modal vectors X , can also be expressed as A = M−1K = M−1H−1 (8) 3.3 The compliance (flexibility) matrix The compliance matrix H is defined by δ = Hf (9) In this assignment, for the plate H is a 6 × 6 matrix which leads to 26666664 δ1 δ2 δ3 δ4 δ5 δ6 37777775 = 26666664 h11 h12 h13 h14 h15 h16 h21 h22 h23 h24 h25 h26 h31 h32 h33 h34 h35 h36 h41 h42 h43 h44 h45 h46 h51 h52 h53 h54 h55 h56 h61 h62 h63 h64 h65 h66 37777775 26666664 f1 f2 f3 f4 f5 f6 37777775 (10) If each row is considered individually we can see that δi = 6X j =1 hijfj (11) 2where i is the row number and j is the column number. The contribution of a single force fj to deflection δi is given by δi(j) = hijfj (12) Rearranging Eq. (12) gives hij = δi(j) fj (13) Thus we can obtain hij by applying a known force fj (at degree of freedom j and measuring the resulting deflection δi(j) (at degree of freedom i). 4 Methods 4.1 Matrices for 6-DoF lumped parameter model Read Section 3.2 above and use the equipment provided to measure the following 8 of the hij terms in the compliance matrix: H = 26666664 − − − − − h16 − − − − h25 − − − h33 h34 − − − − h43 h44 − − − h52 − − − − h61 − − − − − 37777775 (14) Theoretically, reciprocity and plate symmetry mean that: h16 = h61 = h25 = h52, h34 = h43 and h33 = h44. So you should calculate the averages of: h16, h61, h25 and h52; h34 and h43, and h33 and h44. Then use these three experimentally obtained values to replace the corresponding values in the theoretical compliance matrix which is as follows: H = 26666664 11:5 2:7 28:7 16:0 46:0 36:5 2:7 11:5 16:0 28:7 36:5 46:0 28:7 16:0 171:0 113:0 317:0 240:0 16:0 28:7 113:0 171:0 240:0 317:0 46:0 36:5 317:0 240:0 714:0 575:0 36:5 46:0 240:0 317:0 575:0 714:0 37777775 × 10−6 (m=N) (15) Form the mass matrix by assuming it is diagonal and the 6 diagonal elements are all equal to 1/6 of the total mass (i.e. 1/6 mass of plate + mass of a bolt − mass of the bolt hole). Take the density of aluminium alloy to be 2770 kg=m3. 4.2 Normal mode vibration experiment versus theory and FEA simulation • Undertake resonance testing of the plate using the electrodynamic shaker as demonstrated during the laboratory session. Beginning at a frequency of 5Hz, increase frequency until the first 9 resonances have been identified. • At each resonance, record the resonant frequency (≈ a natural frequency) and use the sand provided to capture the mode shape. The sand gathers along lines where the amplitude of plate vibration is zero or close to zero. • Read Sections 3.1 and 3.2 above and use the MATLAB program provided to: 1) Form the matrix A = M−1H−1. 2) Find the eigenvalues and eigenvectors of A which correspond to the 6 theoretical values for λ (natural frequency squared) and the corresponding 6 modal vectors X that describe the mode shapes. 3) Plot the theoretical mode shapes so that you can compare them with the experimental mode shapes (sand patterns). 3• Use Solidworks or ANSYS to create the model of the 6-DoF aluminium lumped mass plate, then carry out modal analysis of the plate in ANSYS or Solidworks and plot the mode shapes so that you can compare them with the experimental and theoretical mode shapes. 5 Reporting • Properly describe the apparatus and methods used. Simply copying the text above is inadequate as too little detail is provided. • Briefly describe the modelling and modal analysis process of the 6-DoF cantilever plate in ANSYS or Solidworks. • The results section is an important contributor to the overall mark. So please take note of the following: a) All results and calculations must be properly explained. It is not good enough to simply present results without explanation. Also a printout of the MATLAB program does not count as an explanation and will attract no marks on its own. b) Present your experimental, theoretical, and FEA simulation results in such a way that they can be compared. • The discussion section is an important contributor to the overall mark. So please take note of the following: a) Summarise the main results and key points arising from them. b) Explain the approximations that have been made in the theoretical model and how these have contributed to the differences in your experimental, theoretical and FEA simulation results. c) Discuss any sources of experimental errors. • Please submit your report electronically via blackboard before the deadline. References [1] Rao, S. S., 2011, Mechanical Vibrations, Prentice Hall, Upper Saddle River, New Jersey. [2] Thomson, W. T. and Dahleh, M. D., 1998, Theory of Vibration with Applications, Prentice Hall, Upper Saddle River, New Jersey. Notes: The University of Salford values academic integrity. Therefore, all students must understand the meaning and consequences of cheating, plagiarism and other academic offences under the University Policy on the Conduct of Assessed Work. (see http://policies.salford.ac.uk/display.php?id=255 for more information). 4Appendix A brief instruction for the experiment is provided below. 1. Study the information sheet provided. 2. Form the flexibility matrix associated with the 6 stations defining the vertical deflections of the bolt head centre points, following the procedure below which refers to the station numbers shown in the diagram illustrated in Fig. 1. 1 2 3 4 5 6 Figure 1: Diagram of the cantilever plate (a) Position the dial gauge above station 1 and take a reading for the un-deflected plate. (b) Attach a 1.5 kg mass to the plate at station 1 and record the subsequent dial gauge reading. (c) Remove the mass from station 1 and take further dial gauge readings with masses attached to stations 2 to 5. It is recommended that the loading scheme indicated in Table 1 should be used. Table 1: Loading scheme Loading Station ! Measuring Station # 1 2 3 4 5 6 1 1.5 kg 1.5 kg 1 kg 1 kg 1 kg 1 kg 2 1.5 kg 1.5 kg 1 kg 1 kg 1 kg 1 kg 3 1 kg 1 kg 1 kg 1 kg 1 kg 1 kg 4 1 kg 1 kg 1 kg 1 kg 1 kg 1 kg 5 1 kg 1 kg 1 kg 1 kg 0.5 kg 0.5 kg 6 1 kg 1 kg 1 kg 1 kg 0.5 kg 0.5 kg (d) Repeat steps (a) to (c) with the dial gauge positioned at stations 2 to 5. (e) Evaluate the flexibility matrix elements hij using the relationship hij = δij mjg (m=N) where δij is the deflection measured at station i when a mass mj is attached at station j. (f) Check the reciprocity principle which states that hij = hji (i 6= j), implying that the flexibility matrix should be symmetric. If the matrix is not symmetric and the deviations are large, check the measurements. If there are small deviations then form a symmetric flexibility matrix in which the elements in the ij positions are given by (hij + hji)=2. 5