A Digital Recursive Filtering Method for Calculating the Applied Force
for a Measured Response Acceleration
By Tom Irvine
Email: tomirvine@aol.com
March 29, 2012
_____________________________________________________________________________________
Introduction
Consider a single-degree-of-freedom (SDOF) system subjected to an applied force, as shown in
Figure 1.
There are certain cases where the response of a system is known, but applied force is unknown.
Note that response acceleration is usually easier to measure than applied force. The
corresponding force can be calculated via a deconvolution process in the form of a digital
recursive filtering relationship.
f(t)
x
m
k
c
Figure 1.
The variables are
m
is the mass
c
k
x
f(t)
is the viscous damping coefficient
is the stiffness
is the absolute displacement of the mass
is the applied force
1
Note that the double-dot denotes acceleration.
The free-body diagram is
f(t)

x
m
c x
kx
Figure 2.
Summation of forces in the vertical direction
F 

mx
(1)
   cx  kx  f ( t)
mx
(2)
  cx  kx  f ( t )
mx
(3)
 c
 k
 1

x    x    x    f ( t )
 m
 m
 m
(4)
Divide through by m,
By convention,
(c / m)  2 n
(5)
(k / m)  n 2
(6)
where
n

is the natural frequency in (radians/sec)
is the damping ratio.
2
By substitution,

x  2   n x   n2 x 
1
f (t)
m
(7)
Equation (7) does not have a closed-form solution for the general case in which f(t) is an
arbitrary function. A convolution integral approach must be used to solve the equation.
.
Absolute Acceleration
The acceleration response for a unit impulse from Reference 1 is
yt  
1
m





n 2

2  1 sin  t 







(
t
)

exp



t

2

cos

t

2



n
n
d
d
d









(8)
The corresponding Laplace transform for H a (s) (acceleration/force) is
H a (s) =

1
s2


m  s 2  2 s   2 
n 
n

(9)
The corresponding Laplace transform for H i (s) (force/acceleration) is
 s 2  2 s   2 
n 
n
H i (s) = m 
2


s
(10)
The Z-transform is found using the bilinear transform.
s
2 z 1
T z 1
(11)
3
  2 z  1 2

T z  1
H i (z) = m  



 2 z  1
2
 2 n 


n
 T z  1


2
 2 z  1

 T z  1



(12)
2
 2


2

  z  1  2 n  z  1z  1  n 2 z  12 
T

T


H i (z) = m  

2
2



z

1
T





(13)
2
 1


1

 4 z  1  4 n  z  1z  1  n 2 z  12 
T

T


H i (z) = m  

2
1



4  z  1


T


(14)
 4z  12  4 Tz  1z  1   2 T 2 z  12 
n
n

H i (z) = m 
2


4 z  1






(15)
 4 z 2  2z  1  4 T z 2  1   2 T 2 z 2  2z  1 
n
n

H i (z) = m 


4 z 2  2z  1
(16)
 4z 2  8z  4  4 Tz 2  4 T   2 T 2 z 2  2 2 T 2 z   2 T 2 
n
n
n
n
n

H i (z) = m 
2


4z  8z  4
(17)


4

 



(18)

(19)
 4  4 T   2 T 2 z 2   8  2 2 T 2 z  4  4 T   2 T 2 
n
n
n
n
n

H i ( z) = m 
2


4z  8z  4

 4  4 T   2 T 2 z 2  2  4   2 T 2 z  4  4 T   2 T 2 
n
n
n
n
n

H i ( z) = m 
2


4z  8z  4
Solve for the filter coefficients using the method in Reference 1.




 4  4 T   2 T 2 z 2  2  4   2 T 2 z  4  4 T   2 T 2 
c 0 z 2  c1 z  c 2
n
n
n
n
n

=m 
2
2


z  a1z  a 2
4z  8z  4

(20)
c 0 z 2  c1 z  c 2
z 2  a1z  a 2




2 2 2
2 2
2 2
m  4  4 n T  n T z  2  4  n T z  4  4 n T  n T 


=
2
4

z  2z  1

(21)
Solve for a1.
a1 = -2
(22)
a2 = 1
(23)
Solve for a2.
Solve for c0.
c0 

m
4  4 n T  n 2 T 2
4
5

(24)
Solve for c1.
c1 
Solve for c2.
c2 

m
 4  n 2 T 2
2

(25)

m
4  4 n T  n 2 T 2
4

(26)
The digital recursive filtering relationship is
fi
 a1 f i 1
c 0 x i
 a 2f i  2
 c1 x i 1  c 2 x i  2
(27)
The digital recursive filtering relationship is
fi
 2 f i 1


 f i2

m
4  4 n T  n 2 T 2 x i
4





m
m
 4  n 2 T 2 x i 1 
4  4 n T  n 2 x i  2
2
4
(28)
References
1. T. Irvine, The Time-Domain Response of a Single-degree-of-freedom System Subjected
to an Impulse Force, Revision B, Vibrationdata, 2012.
2. T. Irvine, An Introduction to the Shock Response Spectrum, Revision R, Vibrationdata,
2010.
3. T. Irvine, Modal Transient Analysis of a System Subjected to an Applied Force via a
Ramp Invariant Digital Recursive Filtering Relationship, Revision J, Vibrationdata, 2012
6
APPENDIX A
Example
INPUT: WAVELET 80 Hz 7 half-sines
SDOF RESPONSE (fn=80 Hz, Q=10)
40
Acceleration
Applied Force
FORCE (lbf)
30
30
20
20
10
10
0
0
-10
-10
-20
-20
-30
-30
-40
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
ACCEL (G)
40
-40
0.10
TIME (SEC)
Figure A-1.
An SDOF system is subjected to an applied force in the form of a wavelet pulse.
Both the input and response are shown in Figure A-1.
The response is calculated via Reference 3.
7
APPLIED FORCE
WAVELET 80 Hz 7 half-sines
15
Calculated
Original
10
ACCEL (G)
5
0
-5
-10
-15
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.10
TIME (SEC)
Figure A-2.
The Original and Calculated Applied Force curves are nearly identical.
The calculation was performed via equation (28) given the response in Figure A-1.
8