Institute of Mechatronics and Information Systems

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1 EXERCISE 2 Free vibrations of a beam arget Getting familiar with the fundamental issues of free vibrations analysis of elastic medium, with the use of a finite element computation system ANSYS. Program 2.1 Determine the first natural frequency and the associated deflection shape of a steel cantilever beam. 2.2 Determine the first ten natural frequencies and the associated deflection shapes of a steel beam supported on both ends by articulated joints. ask 2.1 Determine the first natural frequency of a steel bar, with a rectangular cross-section, fixed at one end (cantilever beam from Exercise 1). Dimensions of the bar: - length L = 1000 mm = 1. 0 m; - width b = 60 mm = m; - height h = 20 mm = m. Material parameters of steel: - Young s modulus 11 E = 2 10 Pa; - Poisson s coefficient ν = 0. ; - density ρ = kg/m ; Note: use a DO as the decimal point heoretical solution Formulas he literature about this topic gives us the theoretical solution of free vibrations frequency for some, simple, elastic constructions. For example cantilever beam. he period of the natural vibrations of the first deflection shape (mode shape) is equal: + mb L + mb L [s] (2.1.1) E I E where: m - concentrated mass fixed on the end f the rod, [kg], 0 m - mass of the beam, [kg], b Andrzej Młotkowski, Ph.D. M.Sc. Eng.

2 I - inertia moment of the beam s cross-section: I = [m 4 ] (2.1.2) he frequency of the free vibrations: 1 f = [Hz] (2.1.) Calculations Mass of the beam: m = ρ L = kg b = Concentrated mass fixed on the end of the rod: m 0 = 0 Inertia moment of the beam s cross-section: I = = = 4 10 m 4 Period of the natural vibrations: + mb L = s 11 8 E I Frequency of the free vibrations: 1 1 f = = = 16.6 Hz Finite Element Method solution Model Use the numerical model of the beam defined in Exercise 1 (shape, material, and meshing into finite elements). Open the existing database with the command from the top menu: File Resume from, Pick the appropriate folder and file Deletion of boundary conditions and loads he model contains data from the previous static calculations. Delete them with the command: Delete All Load Data Boundary conditions Repeat the procedure of fixing one end of the beam to the Y-Z surface of the coordinate system, identical as in task 1.2. Apply Structural Displacement On Areas. Click with the mouse on the appropriate surface. Andrzej Młotkowski, Ph.D. M.Sc. Eng. 1

3 In the window Apply U, Rot on Areas confirm the choice of the surface OK, and then: All DOF OK. he software automatically selects the nodes belonging to the chosen surface and sets the displacements UX=UY=UZ=0, which means that these nodes can not move in all directions - of the X, Y, and Z axis Solution Determination of the result of calculations is performed in the Solution segment of the ANSYS software. Choose: analysis type: modal analysis, calculation method (option): Lanczos, number of deflection shapes (modes) to be considered: 10, range of frequencies to be considered: from 0 to Hz. Main Menu Solution Analysis ype New Analysis choose Modal OK. Next select the analysis options: Main Menu Solution Analysis ype Analysis Options In the window Modal Analysis, in section Mode extraction method, mark Block Lanczos and in the window No. of modes to extract enter 10. In the lower section: Expand mode shapes mark the box Yes, and No. of modes to expand enter 10. Click OK to save the entries. In the window Block Lanczos Method, in the box Start Freq enter 0, in the box End Frequency enter Click OK to save the entries. he computer performs the calculations after clicking consecutively: Main Menu Solution Solve Current LS. he end of the computation process is announced by a window with the text: Solution is done. Click Close Results he visualisation and analysis of the results is possible in the segment General Postproc. Numerical results he numerical results of all calculated free vibration frequencies are summarised in: Main Menu General Postproc Result Summary. See Fig he ordinal number of the deflection shape (mode), of natural vibrations, is given in the first column of the results table SE. he second column IME/FREQ presents the value of the calculated frequency of the vibration for the particular mode. he first computed frequency slightly differs from the one calculated from the theoretical formula (16.6 Hz). he reasons of the difference are the initial assumptions and simplifications in both methods. Andrzej Młotkowski, Ph.D. M.Sc. Eng. 14

Fig. 2.1.1 Results summary display window.

Choose the following options: Main Menu General Postproc Read Results First Set.

4 Fig Results summary display window. Visualisation and animation of results It is more convenient to view the results as animated pictures of the deflections. Choose the following options: Main Menu General Postproc Read Results First Set. In the menu bar at the top of the screen select PlotCtrls, in the next windows choose: Animate Mode Shape... DOF solution Deformed Shape OK. Fig Example of deflection shape of free vibration ninth mode. Andrzej Młotkowski, Ph.D. M.Sc. Eng. 15

5 Animation of the first mode of free vibrations of the beam should appear on the screen. he (apparent) speed of the beam s movement can be adjusted by the delay slide, in the Animation Control window. Stop and close the animation: click Stop Close. he other deflection shapes (modes) of the natural vibrations can be inspected by loading next sets of data: Main Menu General Postproc Read Results Next Set. In the menu bar at the top of the screen select PlotCtrls, in the next windows choose: Animate Mode Shape... DOF solution Deformed Shape OK. ask 2.2 Determine the first, ten natural frequencies of a steel bar, with a rectangular cross-section, supported on both ends by articulated joints. Geometry and material properties the same as in ask heoretical solution Formulas For a beam supported on both ends by articulated joints the period of the natural vibrations, of the first deflection shape (mode shape), is equal: mb L + mb L 5 5 [s] (2.2.1) E I E where: m - concentrated mass fixed on he end f the rod, [kg], 0 m - mass of the beam, [kg], b I - inertia moment of the beam s cross-section: I = [m 4 ] (2.2.2) he frequency of the free vibrations: 1 f = [Hz] (2.2.) Calculations Use the results of previous calculations, from ask 2.1. he period of natural vibrations of the first deflection shape: mb L = s 11 E I Frequency of the free vibrations: 1 1 f = = = 46. Hz Andrzej Młotkowski, Ph.D. M.Sc. Eng. 16

6 2.2.2 Finite Element Method solution Model Use the numerical model of the beam defined in ask 2.1 (shape, material, and meshing into finite elements). Open the existing database with the command from the top menu: File Resume from, Pick the appropriate folder and file Deletion of boundary conditions and loads he model contains data from the previous calculations. Delete them with the command: Delete All Load Data Boundary conditions Carry out the following operations: In the menu bar at the top of the screen select PlotCtrls, in the next windows choose: Numbering, then Plot Numbering Control KP Line Area OK. Plot Areas. he boundary conditions are given as loads and constraints applied to chosen nodes lying on certain lines. An articulated joint is simulated by blocking the displacement of nodes of the chosen line without blocking the possibility of rotation of the model around this line as an axis. Apply Structural Displacement On Lines. Click with the mouse on the appropriate line (L9). In the window Apply U, Rot on Lines confirm the choice of the line OK, and then: UY OK. Proceed the same way with line L10, on the other side of the beam. Movement in the direction of Z axis should be shut off in points 2 and 4: Apply Structural Displacement On KPs. Mark with the cursor points number 2 and 4, in the window Apply U, Rot on KPs confirm the choice of the points OK, and then: UZ OK. Movement in the direction of X axis should be blocked in point 5: Apply Structural Displacement On KPs. Mark with the cursor point number 5, in the window Apply U, Rot on KPs confirm the choice OK, and then: UX OK. Approve the entered data and close the Preprocessor Solution Solution according to point Results Results of the calculations according to point Last edited 4 December 201 Andrzej Młotkowski, Ph.D. M.Sc. Eng. 17

Institute of Mechatronics and Information Systems

Institute of Mechatronics and Information Systems EXERCISE 4 Free vibrations of an electrical machine model Target Getting familiar with the fundamental issues of free vibrations analysis of a simplified model of an electrical machine, with the use of