Applied Analytical Methods In Engineering MODELLING
Applied Analytical Methods In Engineering MODELLING
- Referring to Handout paper “Filtering properties…”
Consider Eq. (1) with
have the initial amplitude of the solution Eq. (5) to be
- 2. A linear single-degree-of-freedom system with natural frequency and viscous damping factor respectively is excited by a stationary zero-mean random force Y(t) with a spectral density
where are constants (here are respectively mean or expected excitation frequency and excitation bandwidth). Thus, the equation of motion may be written as
Find stationary mean square of the steady-state system’s response using the spectral approach. You may use either direct numerical integration of the response PSD for the given data or rely upon available Table of the relevant integrals (which would require writing equation for shaping filter of Y(t) by expanding denominator of into two complex conjugated co-factors). Four different cases altogether:
The parameter may be
(arbitrarily) assumed to have value equal to unity.
Comment to Problem 2: for the case where its analytical solution as obtained from the Table may be written as
The final expression clearly shows that approximate representation of a narrow-band excitation Y(t) by a sinusoid may be adequate – at least as long as just mean-square response is considered – but only if it is narrow-band compared with the system excited (i.e.) For example, if Y(t) represent forcing due to ocean waves then is usually of the order 0.1 so that its approximation by a single harmonic may be inaccurate. Actually, the latter extreme case can be obtained directly from the general excitation/response PSDs relation by asymptotic approximation for transfer function being slowly varying compared with excitation PSD having a sharp peak at ; thus
- Referring to Handout “Intro Rot”. Whirl of a Jeffcott rotor with both external and internal damping is governed by two ODEs (6a) for transverse displacements X, Y of its disk (along nonrotating axes). This set is reproduced here as a set (*) with two extensions:
- Added white-noise random excitation along X as applied through supports
- Angular velocity of rotation (the fourth terms in each ODE) is renamed as ν to avoid confusion with ω which is argument of the PSDs. This imply renaming stability threshold (12) as
The updated set is
Denote constant PSD of the random force f X(t) as 2πW. Use basic theorem relating PSDs of excitation and response to derive expression for sum of PSDs of X(t) and Y(t) as functions of ν/ν* Integration by table of this sum yield
Verify the solution (**) by numerical integration of the total PSD for
Hint: this basic theorem is considered in detail in Handout “FFT” for the case of single excitation/single response. Using the same approach as based on algebraic relations for the FTs with finite limits it can be easily extended to multiple excitation/responses. Solution for the case of two ODEs with random RHSs is presented in detail in Section 4 of the Handout “Noncons”
Comment. Availability of measurable subcritical response of any
dynamic system may be used with advantage for on-line Mechanical Signature Analysis. In Section 4 of the Handout “Noncons” the coherence function of two displacements has been indicated as a potential index for stability margin on the system. In the present problem with only single external excitation available that index will not work – coherence should be always unity (implying permanent “false alarm”. Your solution to the present problem may provide alternative index – ratio of mean square responses <Y2>/<X2> should be monotonously increasing with ν/ν*from zero at ν/ν* = 0 to unity at ν/ν* = 1.
