„Micromechatronics and MEMS” – Lab2 2014/2015
The model of the accelerometer with one degree of freedom is shown in Fig.1.
Fig. 1 Model of accelerometer
The equation of motion for the relative displacement x is given by the following expression:
0 2
x x 0x a
Q
, (1.1)
where 0 denotes the resonance frequency in rad/s; Q is the mechanical quality factor and a is the measured acceleration. The equation (1.1) can be written in the form of the 1st order system of equation, which is required for the MATLAB’s algorithms:
0 2 0
x v
v a v x
Q
. (1.2)
The mechanical quality factor Q is given by:
Q0m , (1.3)
where is the damping coefficient and m is the proof mass. The eigenfrequency (the resonance frequency) may be calculated by:
0
ktot
m , (1.4)
where the total stiffness of the proof mass suspension is denoted by ktot. The proof mass may be suspended by various designs of springs. The fundamental form of the spring is the elastic beam, which is shown in Fig. 2.
Fig. 2 Elastic beam as a spring
The second moment of inertia of the beam cross-section is given by (1.5), where the z-axis determines the bending moment direction; a and b are dimensions as shown in Fig. 2.
3 z 12
I ab . (1.5)
Complex springs can often be broken into series and parallel connections of basic beams. The total spring constant for n parallel (the same displacement) springs are given by:
1 n
tot i
i
k k
, (1.6)where ki is the stiffness of the i-th spring. When spring are connected in series, the same force acts on each spring, then:
1
1 n 1
tot i i
k
k . (1.7)The stiffness of the beam depends on the boundary conditions, dimensions and material properties of the spring. The each segment of the serpentine spring, which is denoted by view W in Fig. 3, may be modeled by the so called guided beam spring.
Fig. 3 The serpentine spring and its segment (top view)
If the displacement of the proof mass is perpendicular to the Fig. 3, the stiffness of the single segment can be expressed by equation (1.8):
3
12EIz
k L , (1.8)
where E denotes the Young’s modulus of the beam material and Iz is given by (1.5); the length of the segment is denoted by L.
The accelerometer is a sensor and in sensors, the noise sets the limit for the smallest acceptable size.
For small enough devices, the signal level falls below the thermal noise floor. Hence, the understanding of the noise limitations is crucial for optimal sensor design.
The noise induced displacement spectral density is given by:
4 B
n
tot
x k T k
, (1.9)
where kB=1.38·10-23 J/K is the Boltzmann’s constant and T denotes the temperature conditions. The rms-vibration amplitude is given by:
B rms
tot
x k T
k (1.10)
and the equivalent acceleration noise spectral density is defined by:
4 B 0 n
a k T
mQ
. (1.11)
The rms-acceleration, which describes the accelerometer quality is shown by (1.12).
2 0 B rms
a k T
m
. (1.12)
To describe the quality of the accelerometer other measures may be applied. The typical step response of the accelerometer is shown in Fig. 4.
Fig. 4 Accelerometer typical step response
Settling time ts is the time required for an output to reach and remain within a given error band following some input stimulus. For example, the settling ts95%
denotes that the output remains in the interval 0.95xfinal;1.05xfinal , where xfinal is the steady-state displacement.
The percentage overshoot PO may be defined as:
max 1 100%
final
PO x x
, (1.13)
where the maximum value of the output is shown in Fig. 4.
To obtain the accelerometer response, the equation of motion (1.1) or (1.2) must be integrated. The part of the MATLAB m-function, which allows to integrate the equation of motion may have the following structure:
T=[0 Tmax]; % time interval of analysis
ic=[0 0]; % initial conditions: displacement and velocity
% the integration algorithm
% the following data must be defined earlier
% omega_0*10^-6 - the resonance frequency in rad/s
% Q – the mechanical quality factor
% a_max*10^-6 - the measured acceleration step in m/s
[ts,xs]=ode45('motion_acc',T,ic,[],omega_0*10^-6,Q,a_max*10^-6);
% ts are moments of time in s
% xs denotes the matrix of the solution
x_acc=xs(:,1)*10^6; % the 1st column contains the displacement output in pm
The m-function ‘motion_acc’ has the following form:
function xdot=motion_acc(t,x,flag,omega,q,a);
% the right side of the equation (1.2)
xdot=[x(2); a-(omega/q)*x(2)-omega^2*x(1)];
