Response to deterministic and random input present together

Hi,

I am solving a mechanical vibration problem that has both deterministic and
random signals simultaneously as an input. I would like to know if I can
assume the deterministic signal as random signal and solve the problem as a
system subjected to random input. I mean, can I use PSD of the deerministic
signal and add it to the PSD of random signal and consider the resultant
PSD as the random input signal?
To make myself more clear, let's say I want to find response of a single
degree of freedom system(1D-spring and mass) subjected to 
1) force = F x sin wt
2) random force with psd = P, then, can I assume input PSD = P + F^2/2 and
then the output psd = (P + F^2/2)* H^2 where H is the transfer function.

Is this approach correct mathematically? would the resultant motion of the
suspended mass be random? or do I need to solve for deterministic signal
separately and random signal separately? How would the two outcome differ?

Thanks in advance
Rahul

rsk <kalerahul@n_o_s_p_a_m.yahoo.com> wrote:

> I am solving a mechanical vibration problem that has both deterministic and
> random signals simultaneously as an input. I would like to know if I can
> assume the deterministic signal as random signal and solve the problem as a
> system subjected to random input. I mean, can I use PSD of the deerministic
> signal and add it to the PSD of random signal and consider the resultant
> PSD as the random input signal?

Sometime you might be able to, but most often not.

> To make myself more clear, let's say I want to find response of a single
> degree of freedom system(1D-spring and mass) subjected to 
> 1) force = F x sin wt
> 2) random force with psd = P, then, can I assume input PSD = P + F^2/2 and
> then the output psd = (P + F^2/2)* H^2 where H is the transfer function.

Well, say the system has a high-Q resonance at w. If you ignore that,
and assume that the spectrum is all random, you will miss it.

If your deterministic signal has a spectrum similar to a random
signal, though, you might be able to do that. Consider that a
pseudo-random-number-generator is deterministic, but still looks random.

-- glen

On 1/28/12 12:18 PM, glen herrmannsfeldt wrote:
> rsk<kalerahul@n_o_s_p_a_m.yahoo.com>  wrote:
>
>> I am solving a mechanical vibration problem that has both deterministic and
>> random signals simultaneously as an input. I would like to know if I can
>> assume the deterministic signal as random signal and solve the problem as a
>> system subjected to random input. I mean, can I use PSD of the deerministic
>> signal and add it to the PSD of random signal and consider the resultant
>> PSD as the random input signal?
>
> Sometime you might be able to, but most often not.
>
>> To make myself more clear, let's say I want to find response of a single
>> degree of freedom system(1D-spring and mass) subjected to
>> 1) force = F x sin wt
>> 2) random force with psd = P, then, can I assume input PSD = P + F^2/2 and
>> then the output psd = (P + F^2/2)* H^2 where H is the transfer function.
>

the mechanical system is linear and time-invariant.  no?

the response of the sum is the sum of the responses.  for the 
deterministic component, you get a closed-form deterministic response. 
for the random component, you can do it with the PSD or, if you like 
time-domain, the autocorrelation of the random component.

maybe i'm missing something obvious.

-- 

r b-j                  rbj@audioimagination.com

"Imagination is more important than knowledge."

On Sat, 28 Jan 2012 10:51:56 -0600, rsk wrote:

> Hi,
> 
> I am solving a mechanical vibration problem that has both deterministic
> and random signals simultaneously as an input. I would like to know if I
> can assume the deterministic signal as random signal and solve the
> problem as a system subjected to random input. I mean, can I use PSD of
> the deerministic signal and add it to the PSD of random signal and
> consider the resultant PSD as the random input signal?
> To make myself more clear, let's say I want to find response of a single
> degree of freedom system(1D-spring and mass) subjected to 1) force = F x
> sin wt
> 2) random force with psd = P, then, can I assume input PSD = P + F^2/2
> and then the output psd = (P + F^2/2)* H^2 where H is the transfer
> function.
> 
> Is this approach correct mathematically? would the resultant motion of
> the suspended mass be random? or do I need to solve for deterministic
> signal separately and random signal separately? How would the two
> outcome differ?

It depends on what answer you're looking for.  You can find the PSD of 
the deterministic + random signal, keeping in mind that it'll have an 
impulse at the sine wave frequency.  Then you can run that through the 
transfer function of the system to find the result.  But depending on how 
you do the calculations you might lose the phase relationship between the 
sine wave at the output and the driving sine wave.

But because the character of the impulse is so different from the 
(presumably spread out) PSD of the random signal, you won't be saving 
yourself any work over just working out the two answers individually.

-- 
Tim Wescott
Control system and signal processing consulting
www.wescottdesign.com

On 1/28/2012 11:51 AM, rsk wrote:
> Hi,
>
> I am solving a mechanical vibration problem that has both deterministic and
> random signals simultaneously as an input. I would like to know if I can
> assume the deterministic signal as random signal and solve the problem as a
> system subjected to random input. I mean, can I use PSD of the deerministic
> signal and add it to the PSD of random signal and consider the resultant
> PSD as the random input signal?
> To make myself more clear, let's say I want to find response of a single
> degree of freedom system(1D-spring and mass) subjected to
> 1) force = F x sin wt
> 2) random force with psd = P, then, can I assume input PSD = P + F^2/2 and
> then the output psd = (P + F^2/2)* H^2 where H is the transfer function.
>
> Is this approach correct mathematically? would the resultant motion of the
> suspended mass be random? or do I need to solve for deterministic signal
> separately and random signal separately? How would the two outcome differ?
>
> Thanks in advance
> Rahul

Superposition applies if the system is linear, which in your case, 
probably means that it neither breaks nor undergoes plastic deformation. 
(If the system doesn't behave elastically, the actual signal, not merely 
the PSD, will determine the outcome. In that case, you will need 
something like a Monte Carlo simulation.) If superposition applies, it 
provides the simplest way to proceed.

Jerry
-- 
Engineering is the art of making what you want from things you can get.
�����������������������������������������������������������������������

On Jan 29, 5:51&#4294967295;am, "rsk" <kalerahul@n_o_s_p_a_m.yahoo.com> wrote:
> Hi,
>
> I am solving a mechanical vibration problem that has both deterministic and
> random signals simultaneously as an input. I would like to know if I can
> assume the deterministic signal as random signal and solve the problem as a
> system subjected to random input. I mean, can I use PSD of the deerministic
> signal and add it to the PSD of random signal and consider the resultant
> PSD as the random input signal?
> To make myself more clear, let's say I want to find response of a single
> degree of freedom system(1D-spring and mass) subjected to
> 1) force = F x sin wt
> 2) random force with psd = P, then, can I assume input PSD = P + F^2/2 and
> then the output psd = (P + F^2/2)* H^2 where H is the transfer function.
>
> Is this approach correct mathematically? would the resultant motion of the
> suspended mass be random? or do I need to solve for deterministic signal
> separately and random signal separately? How would the two outcome differ?
>
> Thanks in advance
> Rahul

Yes, superposition!

>On Jan 29, 5:51=A0am, "rsk" <kalerahul@n_o_s_p_a_m.yahoo.com> wrote:
>> Hi,
>>
>> I am solving a mechanical vibration problem that has both deterministic
a=
>nd
>> random signals simultaneously as an input. I would like to know if I
can
>> assume the deterministic signal as random signal and solve the problem
as=
> a
>> system subjected to random input. I mean, can I use PSD of the
deerminist=
>ic
>> signal and add it to the PSD of random signal and consider the
resultant
>> PSD as the random input signal?
>> To make myself more clear, let's say I want to find response of a
single
>> degree of freedom system(1D-spring and mass) subjected to
>> 1) force =3D F x sin wt
>> 2) random force with psd =3D P, then, can I assume input PSD =3D P +
F^2/=
>2 and
>> then the output psd =3D (P + F^2/2)* H^2 where H is the transfer
function=
>.
>>
>> Is this approach correct mathematically? would the resultant motion of
th=
>e
>> suspended mass be random? or do I need to solve for deterministic
signal
>> separately and random signal separately? How would the two outcome
differ=
>?
>>
>> Thanks in advance
>> Rahul
>
>Yes, superposition!
>


When we do testing of mechanical systems that are excited by deterministic
signals such as engine sinusoidal force, we need to be careful about
collecting the phase information along with the magnitude. For example,
forces at the engine mounts need to be known in magnitude and phase from
the test so that they can be applied to the vehicle to know the vehicle
response. If we need to use mathematical tools such as simulation using
Finite Element Method, we take these measured forecs in magnitude and phase
and carry out frequency response analysis. We can then find the response at
a particular location such as steering or seat. This response will also be
output in magnitude and phase by the software.
My question is, can we not use PSD analysis using just the magnitude of
these forces so that the efforts needed to accurately capture the phase
would not be necssary? We can then just use the auto and cross PSDs of
these forces at the mounts and carry out response PSD analysis for the
vehicle using multiple input-single output methods of signal analysis.
If we can, then why is this generally not done? would there be some
mathematical reason for not carrying out analysis using PSDs of these
deterministic signals? Generally, we use PSDs for random signal analysis
only. I have not seen people using PSDs for deterministic signals. Is it
because they do not want to deal with RMS output quantities? if so, I guess
if we do take auto and cross PSDs of input, we can find peak values from
RMS output by multiplying by sqrt(2). This would make the use of "Phase
information" unnecessary.

On Jan 28, 9:36&#4294967295;pm, "rsk" <kalerahul@n_o_s_p_a_m.yahoo.com> wrote:
...
> My question is, can we not use PSD analysis using just the magnitude of
> these forces so that the efforts needed to accurately capture the phase
> would not be necssary? We can then just use the auto and cross PSDs of
> these forces at the mounts and carry out response PSD analysis for the
> vehicle using multiple input-single output methods of signal analysis.
> If we can, then why is this generally not done? would there be some
> mathematical reason for not carrying out analysis using PSDs of these
> deterministic signals? Generally, we use PSDs for random signal analysis
> only. I have not seen people using PSDs for deterministic signals. Is it
> because they do not want to deal with RMS output quantities?
...

Your continuous/periodic signals do not possess a Power Spectral
Density. They are correctly represented with the Power Spectrum. See
the application notes from a major supplier of vibration analysis
instrumentation below, particularly page 30 of Bv0031.pdf.

Bo0438.pdf
Choose your Units! (PWR, PSD, ESD)
www.bksv.com/doc/bo0438.pdf

Bv0031.pdf
Technical review No. 3 - 1987
http://bruel.ru/UserFiles/File/Review3_87.pdf
or
www.bksv.com/pdf/bv0031.pdf
(The bksv.com site seems to request registration.)
The article on page 29 this file is
"Signal and Units"
by Svend Gade and Henrik Herlufsen
see page 30

Dale B. Dalrymple

On 1/29/2012 12:36 AM, rsk wrote:

   ...

> When we do testing of mechanical systems that are excited by deterministic
> signals such as engine sinusoidal force, we need to be careful about
> collecting the phase information along with the magnitude. For example,
> forces at the engine mounts need to be known in magnitude and phase from
> the test so that they can be applied to the vehicle to know the vehicle
> response. If we need to use mathematical tools such as simulation using
> Finite Element Method, we take these measured forecs in magnitude and phase
> and carry out frequency response analysis. We can then find the response at
> a particular location such as steering or seat. This response will also be
> output in magnitude and phase by the software.
> My question is, can we not use PSD analysis using just the magnitude of
> these forces so that the efforts needed to accurately capture the phase
> would not be necssary? We can then just use the auto and cross PSDs of
> these forces at the mounts and carry out response PSD analysis for the
> vehicle using multiple input-single output methods of signal analysis.
> If we can, then why is this generally not done? would there be some
> mathematical reason for not carrying out analysis using PSDs of these
> deterministic signals? Generally, we use PSDs for random signal analysis
> only. I have not seen people using PSDs for deterministic signals. Is it
> because they do not want to deal with RMS output quantities? if so, I guess
> if we do take auto and cross PSDs of input, we can find peak values from
> RMS output by multiplying by sqrt(2). This would make the use of "Phase
> information" unnecessary.

PSD is a statistical measure, most appropriate, as far as I know, for 
signals about which only statistics are known. As an analogy, compare 
specifying a color by hue and saturation to giving a detailed spectrum.
I find it hard to see how truly random inputs with non-negligible 
magnitudes arise. Road irregularities possess the quality that only 
statistics are known, but that's not quite the same as truly random. 
There might be mathematical pitfalls if the distinction isn't at least 
kept in mind.
Automobile suspensions are rarely linear. Shock absorbers have 
directional qualities and are usually mounted so that they pivot at the 
ends when in motion. Often, spring "constants" aren't constant. Whether 
such nonlinearities make superposition useless can't be guessed from a 
distance. Good luck!

Jerry
-- 
Engineering is the art of making what you want from things you can get.
�����������������������������������������������������������������������

On Sat, 28 Jan 2012 23:26:33 -0800 (PST), dbd <dbd@ieee.org> wrote:

>On Jan 28, 9:36=A0pm, "rsk" <kalerahul@n_o_s_p_a_m.yahoo.com> wrote:
>...
>> My question is, can we not use PSD analysis using just the magnitude of
>> these forces so that the efforts needed to accurately capture the phase
>> would not be necssary? We can then just use the auto and cross PSDs of
>> these forces at the mounts and carry out response PSD analysis for the
>> vehicle using multiple input-single output methods of signal analysis.
>> If we can, then why is this generally not done? would there be some
>> mathematical reason for not carrying out analysis using PSDs of these
>> deterministic signals? Generally, we use PSDs for random signal analysis
>> only. I have not seen people using PSDs for deterministic signals. Is it
>> because they do not want to deal with RMS output quantities?
>...
>
>Your continuous/periodic signals do not possess a Power Spectral
>Density. They are correctly represented with the Power Spectrum. See
>the application notes from a major supplier of vibration analysis
>instrumentation below, particularly page 30 of Bv0031.pdf.
>
>Bo0438.pdf
>Choose your Units! (PWR, PSD, ESD)
>www.bksv.com/doc/bo0438.pdf
>
>Bv0031.pdf
>Technical review No. 3 - 1987
>http://bruel.ru/UserFiles/File/Review3_87.pdf
>or
>www.bksv.com/pdf/bv0031.pdf
>(The bksv.com site seems to request registration.)
>The article on page 29 this file is
>"Signal and Units"
>by Svend Gade and Henrik Herlufsen
>see page 30
>
>Dale B. Dalrymple

Nice references, thanks for posting those.   And I'd never heard of
that company before, but it looks like they know what they're doing!


Eric Jacobsen
Anchor Hill Communications
www.anchorhill.com