Vibration suppression in MEMS devices using electrostatic forces H. - PowerPoint PPT Presentation

Introduction Las Vegas, Nevada, United States, 20-24 March 2016 Theory Application Vibration suppression in MEMS devices using electrostatic forces H. Khodaparast, H. Madinei, M.I. Friswell & S. Adhikari College of Engineering, Swansea University, UK March 23, 2016 H. Khodaparast, H. Madinei, M.I. Friswell & S. Adhikari Passive and Active Vibration Isolation Systems 1/24

Introduction Las Vegas, Nevada, United States, 20-24 March 2016 Theory Application Overview Introduction and motivation Theory of Incremental Non-linear Control Parameters (INCP) Application to vibration suppression in Microelectromechanical Systems (MEMS) device Conclusions and Future Works H. Khodaparast, H. Madinei, M.I. Friswell & S. Adhikari Passive and Active Vibration Isolation Systems 2/24

Introduction Las Vegas, Nevada, United States, 20-24 March 2016 Theory Application Application of MEMS devices Automotive (MEMS pressure sensors) Biomedical (smart pills) Wireless and optical communications Optical displays Chemical H. Khodaparast, H. Madinei, M.I. Friswell & S. Adhikari Passive and Active Vibration Isolation Systems 3/24

Introduction Las Vegas, Nevada, United States, 20-24 March 2016 Theory Application Types of actuation mechanism in MEMS Electrostatic Thermal Pneumatic Piezoelectric Pull-in: the voltage at which the system becomes unstable H. Khodaparast, H. Madinei, M.I. Friswell & S. Adhikari Passive and Active Vibration Isolation Systems 4/24

Introduction Las Vegas, Nevada, United States, 20-24 March 2016 Theory Application Motivation- the objective of this work The objective of this study is: to minimize the vibration amplitude of a MEMS device by controlling the resonance frequency of the system. To this end, DC voltages are applied to the electrodes to change the resonance frequency of the system. Applying DC voltages to the system makes the system non- linear. To solve the non-linear system of equations, the non-linearity is parametrised by a set of ’non-linear control parameters’ such that the dynamic system is effectively linear for zero values of these parameters and non-linearity increases with increasing values of these parameters. ’non-linear control parameters’ are the applied DC voltages in this problem as when they are zeros, the system is linear. H. Khodaparast, H. Madinei, M.I. Friswell & S. Adhikari Passive and Active Vibration Isolation Systems 5/24

Introduction Las Vegas, Nevada, United States, 20-24 March 2016 Theory Application Incremental non-linear control parameters (1) The idea is to develop an extended harmonic balance method for the steady-state solution of non-linear multiple-degree-of- freedom dynamic problems based on incremental non-linear con- trol parameters. The method only requires the solution of linear equations for the non-linear problem It also provides the sensitivities of the solution with respect to non-linear control parameters. The non-linear control parameters are those with which the non-linearity in the model is triggered. H. Khodaparast, H. Madinei, M.I. Friswell & S. Adhikari Passive and Active Vibration Isolation Systems 6/24

Introduction Las Vegas, Nevada, United States, 20-24 March 2016 Theory Application Incremental non-linear control parameters (2) This property of the non-linear control parameters can be ex- ploited in the solution of a non-linear problem. They are incremented from zero to one (note that the parame- ters are normalised so that the maximum values are unity) and a linear equation giving the sensitivities of the responses with respect to the parameters is obtained at each increment. Using these sensitivities, the solution at each step can be cal- culated through the solution at the previous increment. The method starts from the linear system and continues until all non-linear control parameters reach unity. H. Khodaparast, H. Madinei, M.I. Friswell & S. Adhikari Passive and Active Vibration Isolation Systems 7/24

Introduction Las Vegas, Nevada, United States, 20-24 March 2016 Theory Application Description of the method Consider the model of a MEMS cantilever beam with electrodes (shown in the figure below) z d 1 1 d 3 3 V 1 V 2 x Ω z(t)=z 0 cos( t ) Micro-beam g 2 V 2 g 1 V 1 4 d 2 2 H. Khodaparast, H. Madinei, M.I. Friswell & S. Adhikari Passive and Active Vibration Isolation Systems 8/24

Introduction Las Vegas, Nevada, United States, 20-24 March 2016 Theory Application Mathematical model The equation of motion of the beam can be expressed as: EI ∂ 4 w ( x , t ) + ρ A ∂ 2 w ( x , t ) ∂ w ( x , t ) + c a = ∂ x 4 ∂ t 2 ∂ t � V 2 � ǫ 0 aH ( x − d 1 ) 1 − ( g 1 − w ( x , t )) 2 2 � V 2 � ǫ 0 aH ( x − d 1 ) 1 + ( g 1 + w ( x , t )) 2 2 � V 2 � ǫ 0 a ( H ( x − d 2 ) − H ( x − d 3 )) 2 − ( g 2 − w ( x , t )) 2 2 � � V 2 − ρ A ∂ 2 z ( t ) ǫ 0 a ( H ( x − d 2 ) − H ( x − d 3 )) 2 ( g 2 + w ( x , t )) 2 ∂ t 2 2 (1) H. Khodaparast, H. Madinei, M.I. Friswell & S. Adhikari Passive and Active Vibration Isolation Systems 9/24

Introduction Las Vegas, Nevada, United States, 20-24 March 2016 Theory Application The nondimensionalized equation of the micro-beam The electrostatic force functions in Eq.(1) may be expressed in terms of its Taylor series. Therefore the nondimensionalised form of Eq.(1) with the trun- cated cubic terms of electrostatic force becomes ∂ 4 ˆ + ∂ 2 ˆ � ˆ x , ˆ � � ˆ x , ˆ � � ˆ x , ˆ � w t + c ∂ ˆ w t w t � w 5 � w 3 + O + α 1 ˆ w + α 3 ˆ ˆ = x 4 ∂ ˆ ∂ ˆ t 2 ∂ ˆ t � � i ˆ Ωˆ γ exp t + cc . (2) H. Khodaparast, H. Madinei, M.I. Friswell & S. Adhikari Passive and Active Vibration Isolation Systems 10/24

Introduction Las Vegas, Nevada, United States, 20-24 March 2016 Theory Application Is cubic order accurate enough? This depends on the amplitude of excitation frequency and damping If the beam is excited at its first non-dimensionalized resonance frequency and V 1 = V 2 = 7 V and z 0 = 0 . 1 µ m, the electro- static force can be estimated by its third-order Taylor series with a reasonable degree of accuracy. This is shown in the figure below V DC = 7 V V DC = 7 V 0.335 Nondimensionalized tip displacement Numerical integration (True electrostatic forces) Nondimensionalized tip displacement Numerical integration (True electrostatic forces) 0.4 Harmonic Balance (Electrostatic forces up to order 3) Harmonic Balance (Electrostatic forces up to order 3) 0.33 Error= 2.7 % 0.3 0.325 0.2 0.1 0.32 0 0.315 -0.1 0.31 -0.2 -0.3 0.305 -0.4 0 20 40 60 80 100 65 70 75 Nondimensionalized time ˆ Nondimensionalized time ˆ t t H. Khodaparast, H. Madinei, M.I. Friswell & S. Adhikari Passive and Active Vibration Isolation Systems 11/24

Introduction Las Vegas, Nevada, United States, 20-24 March 2016 Theory Application How the method works When the voltages are zeros, the system is linear and therefore the solution of linear system can be assumed as N � � � � i ˆ � Ωˆ w 0 = ˆ Y j ( x ) + cc . (3) Q 0 j exp t j =1 where Q 0 j , the components of vector q 0 ∈ R N , are obtained from the following equation � − 1 F � − ˆ Ω 2 M + i ˆ q 0 = Ω C + K (4) H. Khodaparast, H. Madinei, M.I. Friswell & S. Adhikari Passive and Active Vibration Isolation Systems 12/24

Introduction Las Vegas, Nevada, United States, 20-24 March 2016 Theory Application Linear system The non-linear control parameters are normalised in which θ i = V i V pi ( V pi is pull-in voltage (the maximum voltage that can be applied to the system) If all the normalised non-linear parameters are perturbed by δθ , the steady state solution of weakly non-linear system may be expressed by � ∂ ˆ w 0 + ∂ ˆ w 0 � � δθ (2) � w 0 + ´ w 1 = ˆ ˆ w 0 + δθ + O ≈ ˆ w 1 δθ (5) ˆ ∂θ 1 ∂θ 2 where ´ � � ∂ ˆ ∂θ 1 + ∂ ˆ w 0 w 0 w 1 = ˆ ∂θ 2 H. Khodaparast, H. Madinei, M.I. Friswell & S. Adhikari Passive and Active Vibration Isolation Systems 13/24

Introduction Las Vegas, Nevada, United States, 20-24 March 2016 Theory Application Perturbation w 0 + ´ Substituting ˆ w 1 = ˆ w 1 δθ into the governing equation of ˆ the beam and neglecting the higher order terms of δθ yield ∂ 4 ´ t + ∂ 2 ´ � ∂ x 4 + c ∂ ´ � ˆ ˆ ˆ w 1 w 1 w 1 t 2 + α 1 ( θ ) ´ w 1 ˆ δθ + ∂ ˆ ∂ ˆ w 03 + 3 ˆ w 02 ´ � � α 1 ( θ ) ˆ w 0 + α 3 ( θ ) ˆ w 1 δθ ˆ = 0 (6) The above partial differential equation is a linear function in terms of ´ w 1 and standard discretization methods (such as Galerkin) ˆ can be used to obtain the solution of ´ w 1 . ˆ H. Khodaparast, H. Madinei, M.I. Friswell & S. Adhikari Passive and Active Vibration Isolation Systems 14/24

Introduction Las Vegas, Nevada, United States, 20-24 March 2016 Theory Application Non-linear solution (1) The steady state solution of ´ w 1 includes primary and higher ˆ harmonics of the excitation frequency. One may ignore the higher harmonics and assume � ´ m ´ � i ˆ � � � Ωˆ w 1 = ˆ Y j ( x ) + cc . (7) Q 1 j exp t j =1 Balancing the harmonic terms and applying standard Galerkin projection gives A 1 ´ q 1 = b 1 (8) � ´ � ∈ R N where ´ q 1 = Q 1 j H. Khodaparast, H. Madinei, M.I. Friswell & S. Adhikari Passive and Active Vibration Isolation Systems 15/24