On the Interaction of an Electro-dynamic On the Interaction of an - PowerPoint PPT Presentation

On the Interaction of an Electro-dynamic On the Interaction of an Electro-dynamic Shaker and a Beam with Stiffness Nonlinearity B. Tang 1 , M.J. Brennan 2 , G. Gatti 3 1 Institute of Internal Combustion Engine, Dalian University of Technology, China 2 Departamento de Engenharia Mecânica, UNESP, Ilha Solteira, Brazil 3 Department of Mechanical, Energy and Management Engineering, University of Calabria, Italy

Dalian Dalian Dalian is a small city in China……, Only 6 million people live in the city  Only 6 million people live in the city 

Dalian Dalian Dalian University of Technology (DUT) Dalian University of Technology (DUT)

2010, ISVR, Dalian Univ Soton, UK 2013-14, UNESP, Brasil x m O k h , δ k h , δ c v k v y l

Motivation 40 40 softening 35 de t Amplitud 30 30 linear 25 hardening 20 20 placement Nonlinear Isolators 15 10 10 Disp 5 0 0 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 Non-dimensional frequency Vibration Jumps Energy Harvesting Devices

Objective Objective • To identify nonlinear stiffness characteristics of a • To identify nonlinear stiffness characteristics of a structure experimentally (b) (b) f ( t ) Input f ( t ) Output x ( t ) Input Output m O O x ( t ) c k 1 , k 3 Grey Box – we know the type of nonlinear stiffness expected

Example Structure 1 Example Structure 1 A Vibration Absorber

A Vibration Absorber Deformed brass plate m = 7 5 g m 7.5 g Brass plate (0.15 mm) Support structure (52 mm diameter) Support structure (52 mm diameter)

Vibration Test x x 1 m c x 2 k 1 , k 3 Shaker mass m shaker Shaker c shaker k shaker suspension Bin Tang, M.J. Brennan, G. Gatti, N.S. Ferguson. Experimental characterization of a nonlinear vibration absorber using free vibration. Journal of Sound and Vibration 367 (2016) 159-169.

Vibration Test    shaker absorber m m  shaker m EQ Q   m m m m shaker x m EQ c k 1 , k 3 1 , 3

Experimental result Time range over which analysis is conducted -4 x 10 3 0 s 0.3 s 2 nt (m) nt (m) 1 placemen placemen 148 Hz 0 Disp -1 Disp Low frequency -2 High frequency -3 0 0.2 0.4 0.6 0.8 Ti Time (s) ( ) t (s) Free vibration Steady state CS

System Characteristics     x t ( ) A t ( ) cos ( ) t Create analytic signal Create analytic signal Hilbert transform   w t ( ) x t ( ) j H[ ( )] x t         A t ( ( ) ) w t ( ( ) ) ( t ) angle w ( t ) ( t ) a g e w ( t )   d   Envelope Envelope d t ( ) ( ) t B Backbone curve kb dt

System Characteristics Backbone curve Envelope -4 -4 2 5 x 10 3 x 10 2.5 3 2 m) m) acement (m ) elope (m) cement ( 2 velope (m) 1.5 Displac 1 1 Displa Enve Env 1 0.5 0 0 80 90 100 110 120 130 140 0 0.05 0.1 0.15 0.2 0.25 0.3 f (Hz) t (s) Frequency (Hz) Frequency (Hz) Time (s) Time (s)

Backbone Curve 2      1 3 k     dd n 2 2 2     3 t A t d dd    k k m m 4 4 m m n n 1 1 EQ EQ EQ     c 2 m k EQ 1 Backbone curve Envelope 5 7.5 x 10 -8 7 Slope -8.5 og(m)) pe) -9 9 6 6 nvelope) (lo g(Envelo 2  2 (rad/s) 2  d -9.5 5  log(En Log -10 4 -10.5 3 -11 0 1 2 3 4 5 6 6.5 0 0.05 0.1 0.15 0.2 0.25 0.3 Displacement 2 (m 2 ) -8 t (s) Displacement 2 (m) 2 x 10 Time (s)

Estimated Parameters Linear Linear Nonlinear Nonlinear Damping D i Mass (g) stiffness stiffness Ratio (N/m) (N/m 3 ) Shaker and 351 0.04 17000 - support support Vibration Current 7 55 7.55 0 02 0.02 2380 2380 5 93 × 10 10 5.93 × 10 10 absorber Method RSFM 2730 5.17 × 10 10

Estimated Parameters Restoring Force Surface Method (RFSM) 3          cx cx k x k x k x k x m m x x • Set Set 1 3 EQ   ( , , x x m x ) • Plot 3D surface EQ • • Extract a section of the surface between Extract a section of the surface between     0.1 m/s x 0.1 m/s 2 rce (N) 1 e (N) oring For oring force 0 Resto Resto -1 -2 -3 -2 -1 0 1 2 3 Displacement (m) -4 Displacement (m) x 10

Estimated Parameters Restoring Force Surface Method (RFSM) 3          cx cx k x k x k x k x m m x x   • Set Set   1 3 EQ x t ( ) A t ( ) cos ( ) t   ( , , x x m x ) • Plot 3D surface EQ • • Extract a section of the surface between Extract a section of the surface between     0.1 m/s x 0.1 m/s -4 3 x 10 2 2 rce (N) t (m) 1 e (N) (m) 1 oring For oring force acement placement 0 0 Disp Resto Displa -1 Resto -1 -2 -3 -2 0 0.05 0.1 0.15 0.2 0.25 0.3 -3 -2 -1 0 1 2 3 t (s) Displacement (m) -4 Displacement (m) Time (s) x 10

Example Structure 2 Example Structure 2 A Beam with a Compressive Load Compressive Load Bin Tang, M.J. Brennan, V. Lopes Jr., S. da Silva, R. Ramlan. Using nonlinear jumps to estimate cubic stiffness nonlinearity: An experimental study Proceedings of the to estimate cubic stiffness nonlinearity: An experimental study, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 230(19) (2016) 3575-3581.

Beam Experiment P Axial force First Mode Beam        3 mq cq k q k q F cos t 1 3   2  4 P l EI  k = 1 Accelerometer   1  2 3 EI 2 l      4 EA   k k Shaker Shaker 3 3 8 l   Al Al  m 2

Frequency Sweep (0.05 Hz/s) leration Acce 0 0 50 50 100 100 150 150 Time (s)

Frequency Sweep (0.05 Hz/s) Increasing excitation amplitude 0 0 0 0 0 leration 0 0 0 0 0 0 0 0 Acce 0 0 0 0 0 0 0 0 150 0 0 0 50 50 100 100 150 150 0 0 50 50 100 100 150 0 50 50 100 100 150 0 150 0 50 50 100 100 150 150 180 180 Time (s) Increasing excitation amplitude Increasing excitation amplitude 0 0 0 0 on 0 0 0 0 Accelerati 0 0 0 0 0 0 0 0 0 50 100 150 200 230 0 50 100 150 200 230 0 50 100 150 200 230 0 50 100 150 200 Time (s)

Frequency Sweep (0.05 Hz/s)   k 3 Amplitude at 2 2 2 2       3 1 2 Y Jump frequency dd n d jump frequency 4 4 m m 4 7 x 10 7 2 nd Mode 2 nd Mode Slope p 6 5 6.5 ad/s) 2 2  dd 6 6 2 (ra 2 (rad/s )  d 5 5 5.5 Shaker 5 0 0.2 0.4 0.6 0.8 1 2 (mm) -5 Y d x 10

Comparison with RFSM 30 20 e (N) k  k 7 43 kN/m 7.43 kN/m g force 10 1 0 0 storing 3 k  4.78e8 N/m 3 -10 Res -20 -30 -3 -2 -1 0 1 2 3 Displacement (mm) ( )

Frequency Sweep (0.05 Hz/s) Increasing excitation amplitude 5 6 6 isp. 4 4 4 4 4 en Point Di 2 2 2 0 . 0 Drive 2 2 2 2 2 4 4 4 5 6 6 0 50 100 150 0 50 100 150 0 50 100 150 0 50 100 150 180 Time (s) i ( ) Why are there peaks and bulge? 0 2.5 2 0 2 0 Disp. 1 0 1 0 0 dle Point D 0 0 0 0 -1 0 -1 0 -2 0 -2 -2.5 Mid 0 0 50 100 150 0 50 100 150 180 150 0 50 100 150 0 50 100 Time (s) Increasing excitation amplitude

Beam Model (a) Axial force Beam Accelerometer Shaker Shaker

Measured and estimated FRF 10 1 10 0 10 10 -1 1 10 -2 10 -3 20 30 40 50 60 70 80 90     X X Y Y X X Y Y X X n n      A 11 , B 21 Y A Ar Ar 11        2 2   F j 2 F 1 Y Z F 1 Y Z  r 1 b r r r 11 sh 11 sh   X n  2      Z k m j c   Y B Ar Br sh sh sh sh 21        2 2 F j 2  r 1 b r r r

Measured displacement transmissibility 5 4.5 4 5 4 3.5 T 3 2.5 2 32 34 36 38 40 42 44 f (H ) f (Hz) X T  T B Beam model shape changing? Beam model shape changing? X A

Measured relative displacement -3 2 x 10 x 10 1.5 mm) - X A ) (m 1 ( X B 0.5 0 32 32 34 34 36 36 38 38 40 40 42 42 44 44 f (Hz) Relative displacements follow the similar trend when Relative displacements follow the similar trend when the nonlinearity in the beam is severity

Dynamic stiffness of the whole system Linear case:   Z Z Z cs 11 sh F F 1 1   Z b 11 Y X 11 11 A A Nonlinear case: 3 3     2            2 2 Z Z k k k k X X k k X X X X m m + + jc jc cs cs sh3 A b 3 B A cs cs 4 4

Nelder-Mead simplex algorithm X 1 X X T 3 3   A = , B = A 2        2 2 Z k k X k X X m + jc F Z F F cs cs sh3 A b 3 B A cs cs 4 4 cs 2.5 x 10 -3 2 x 10 -3 2 x 10 -3 x 10 -3 2.5 x 10 3 10 -3 2.5 x 10 2 1.5 1.5 2 2 ) / F | (m/N) / F | (m/N) 1.5 F | (m/N) 1.5 1.5 1 1 1 1 | X / 1 1 | X / | X / 1 0.5 0.5 0.5 0.5 0.5 0 0 0 0 32 34 36 38 40 42 32 34 36 38 40 42 32 34 36 38 40 42 0 32 34 36 38 40 42 32 34 36 38 40 42 f (Hz) f (Hz) f (Hz) f (Hz)