Mechanical Engineering - Mechanical Systems Dynamics

Laboratory notes - Report 1

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MECHANICAL SYSTEM DYNAMICS Assignment 1 Part 1A – Model of a cantilever beam I. La Paglia Compute the natural frequencies and the mode shapes of a cantilever beam, by making reference to the standing wave solution of a slender beam in bending vibration . Study the forced response of the system, by computing its Frequency Response Function (for assigned input and output positions) . Contents : ▪ Data of the reference structure (geometry and material properties) ▪ Vibration modes of the cantilever beam ▪ Frequency Response Function Target 2 I. La Paglia Mechanical Systems Dynamics Data of the reference structure 3 L b h Aluminum beam with rectangular cross -section I. La Paglia Mechanical Systems Dynamics 4 I. La Paglia Mechanical Systems Dynamics Vibration modes of the cantilever beam 1. Standing wave solution S T
�P L ># cos �T E $ sin �T E % Cosh �T E & Sinh :�T ;?cos :�P E � ; 2. Boundary conditions 3. Matrix formulation * :� ; V L r (V vector of unknown coefficients) 4. Solution of the characteristic equation det * :� ; L r \ � � (numerical solution in Matlab ) 5. Mode shapes computation � � \ * :� �; V:�; L r \ 0 � T 6. Plot the mode shapes with the associated natural frequencies L b h 5 I. La Paglia Mechanical Systems Dynamics A normalization is recommended for the visualization of the mode shapes. Vibration modes of the cantilever beam Mode shapes 6 I. La Paglia Mechanical Systems Dynamics Frequency Response Function w (x i , t) r o � L r o � i n�� 1. Frequency Response Function ) :F3 ; L � �@5 � 0 � T� 0 �:T�;I � F 3 6 E Ft��� �3 E � � 6 2. Choose input and output positions ( T� and T�respectively) 3. Define proper damping values (in the order of 1 %) �� 4. Compute the modal mass (hint: trapz.m Matlab function) I � L �4 �I 0 � 6 T @T 5. Plot the FRF 7 I. La Paglia Mechanical Systems Dynamics Frequency Response Function 8 I. La Paglia Mechanical Systems Dynamics Assignment 1 – Part 1A Work out the following items and include the corresponding results in the report of Assignment 1. 1. Briefly describe the procedure followed for computing natural frequencies and mode shapes . Plot the mode shapes of the first four modes with the indication of the associated natural frequencies and provide comments to the results . 2. Compute the FRFs for one or more combinations of input and output positions . Comment the results . MECHANICAL SYSTEM DYNAMICS Assignment 1 Part 1B – Experimental Modal Analysis I. La Paglia Target 2 I. La Paglia Mechanical Systems Dynamics Assuming the FRFs computed in Part 1A to be representative of an experimental test carried out on a cantilever beam, apply modal parameters identification according to the single mode procedure . Contents ▪ Data of the reference structure ▪ Frequency Response Function ▪ Modal parameters identification – simplified single mode procedure Data of the reference structure 3 L b h Aluminum beam with rectangular cross -section I. La Paglia Mechanical Systems Dynamics 4 I. La Paglia Mechanical Systems Dynamics Frequency Response Function 1. Consider the N FRFs numerically computed according to the procedure developed in Part 1A (for various input locations T� and measuring positions T�) to be representative of an experimental test 2. Identify the modal parameters according to the simplified single mode procedure ▪ Natural frequency ▪ Damping ratio ▪ Mode shape w (x 1 , t) r � � L r � � � ��� w ( xj, t) w ( xN , t) 5 I. La Paglia Mechanical Systems Dynamics Maximum of the FRF Phase change of  6 Modal parameters identification – simplified single mode procedure Natural frequency 6 Half -power points Slope of the phase diagram I. La Paglia Mechanical Systems Dynamics Modal parameters identification – simplified single mode procedure Damping ratio : N F u dB ; 7 I. La Paglia Mechanical Systems Dynamics )�� L :�4 (� L � �@5 � :� � : � � F I �3 6 E F3 ?� E G� L � �@5 � :� � : � � I � F 3 6 E F3 t� ��� E � � 6 According to modal superposition approach , the FRF can be written as The modal vector representing mode E is defined based on a common scaling factor (for variable F). Modal parameters identification – simplified single mode procedure Mode shape At resonance 3 L � �: )�� � �  F F : � � � �?� :� � L F F : � � I � t� � 6�� :� :�; scaling factor 8 I. La Paglia Mechanical Systems Dynamics A common normalization is recommended for the visualization of the mode shapes comparison. MODE 1 Natural frequency (Hz) Damping ratio ( -) Model 4.50 Hz 0.01 Identified 4.46 Hz 0.0102 Modal parameters identification – simplified single mode procedure Results of the identification Target 9 I. La Paglia Mechanical Systems Dynamics Assuming the FRFs computed in Part 1A to be representative of an experimental test carried out on a cantilever beam, apply modal parameters identification according to the single mode procedure . Identify the natural frequencies and mode shapes of a real system through Experimental Modal Analysis . Contents ▪ Data of the reference structure ▪ Frequency Response Function ▪ Modal parameters identification – simplified single mode procedure ▪ The structure under test ▪ Experimental setup o Constraints → how to fix the system o Input → how to excite the system o Output → what to measure ▪ Signal processing (FRF and coherence function) Railway noise can be regarded as environmental noise, resulting from the operation of rail vehicles . In many railway noise problems (especially rolling noise and curve squeal noise), the wheel plays a fundamental role in terms of sound radiation . In particular, the axial vibration of the wheel surface results into efficient sound radiation (the surface vibration of the wheel induces a perturbation of the surrounding air and generates a consequent radiated sound field) . 10 I. La Paglia Mechanical Systems Dynamics Context and motivation 11 I. La Paglia Mechanical Systems Dynamics The structure under test Resilient wheel Wheelset Resilient wheel of a rail vehicle 12 I. La Paglia Mechanical Systems Dynamics Experimental setup Constraints Resilient wheel suspended through elastic support (free -free system) Input Dynamometric impact hammer applying an axial load Output Piezoelectric accelerometers to sense the axial vibration of the wheel rim . Due to the symmetry of the structure, 12 measurement positions have been considered, which are located only on half of the wheel, with a regular angular spacing of 15 ° Experimental setup 13 Dynamometric impact hammer ▪ Impulsive excitation excites all frequencies (theoretically) ▪ The bigger the hammer, the lower the frequency range ▪ The harder the tip, the higher the frequency range I. La Paglia Mechanical Systems Dynamics 14 I. La Paglia Mechanical Systems Dynamics Experimental setup Piezoelectric accelerometer 15 Time histories Frequency Response Function I. La Paglia Mechanical Systems Dynamics Signal processing 16 I. La Paglia Mechanical Systems Dynamics Axial modes to be identified Experimental Frequency Response Functions 17 I. La Paglia Mechanical Systems Dynamics Vibration modes experimentally identified N = 2 nodal diameters N = 3 nodal diameters FEM model of the resilient wheel, validated against the experimental data 18 I. La Paglia Mechanical Systems Dynamics Assignment 1 – Part 1B Work out the following items and include the corresponding results in the report of Assignment 1. Based on the cantilever beam model developed in Part 1A : 1. Briefly describe the procedure followed for identifying the natural frequencies, damping ratios (according to the half -power points and slope of the phase diagram methods) and mode shapes of the first four modes . 2. Compare the parameters defined at the simulation stage to the identified ones . Collect the results in table form and plot a diagram showing the comparison of the simulated and identified mode shapes . Considering the experimental data of the rail vehicle resilient wheel : 3. Apply the single mode identification procedure to the provided experimental data .Plot a diagram showing the identified mode shapes with the indication of the corresponding natural frequencies and damping ratios . Experimental data: ▪ freq frequency vector (resolution 0.333 Hz) ▪ frf Inertance ( I O60 ) frequency response functions (complex) collected by columns according to the measuring grid shown in slide 12 ▪ cohe coherence function, collected by columns Hints to plot the identified mode shapes : ▪ Measuring grid with regular angular spacing of 15 ° → define an angular spatial domain ▪ Polar symmetry of the system → polarplot.m Matlab function 19 I. La Paglia Mechanical Systems Dynamics Data provided