When the continuous systems such as string/beam vibrate in an viscous environment, we know that the natural frequencies of the system become damped natural frequencies and there exists a decay constant associated with each of the mode.
Are these decay constants (or damping factors) different for every mode or same? i.e. given an initial condition corresponding to specific modes, will the system decay down in same amount of period?
The question is raised due to this:
When string vibrates in viscous environment, the equation can be written as,
rho*w,tt+c*w,t-Tw,xx=0.
In this case when we discretize the equation using Galerkin's approach, the damping matrix becomes diagonal with every element same for standard boundary conditions.
This is an excellent question.
A good introduction to damping modes of beams/strings in terms of the finite element analsis of a tennis racquet is given in Section 3 in
Vibration modeling and supression in tennis racquets
https://www.researchgate.net/publication/268253872_Vibration_modeling_and_supression_in_tennis_racquets
More to the point, a time damping model is included in the auspicious article available on RG:
https://www.researchgate.net/profile/Guillaume_Vermot_Des_Roches/publication/278636108_Frequency_and_time_simulation_of_squeal_instabilities_Application_to_the_design_of_industrial_automotive_brakes/links/5609387308ae576ce63ded6c/Frequency-and-time-simulation-of-squeal-instabilities-Application-to-the-design-of-industrial-automotive-brakes.pdf
See Section 4.4.1 for common damping modells and Section 4.5.1 on modal damping.
Article Vibration modeling and supression in tennis racquets
Article Frequency and time simulation of squeal instabilities. Appli...
My experience is that if you assume a damping model and discretize the system using Galerkin (or something else), the eigenvalues you get using the full damping matrix and the damping matrix with the off-diagonal terms are neglected, are very close to each other, as long as damping is low (5% or less).
The important question that is asked is what type of damping models to use. There is no good answer to this. Probably the best thing is to do, as the previous answer recommends, is an experimental study and see what happens. I assume the answer will vary greatly from structure to structure.
Hello Pranav Lad
It is better to answer your question from the assessment.of free vibration of a sdof system governed by the ode
MUtt + CUt + KU =0 The solution to this equation is U(t) = A*exp(-ωdt)Sin(ωdt +α) where ωd = ωn*sqrt(1-D2) and ωn = circular natural frequency and D is the damping ratio = C/Cc. The decay factor is exp(-ωdt). The higher the frequency the faster the decay and the more the damping. Thus, the decay rate will vary with frequency. As is well known the viscous damping coefficient for a sdof system is defined by c = 2DMωn This idea can be extended to strings and beams.
I hope this answer will be useful to you.
Passive structural damping in distributed-parameter structures has been studied scientifically since at least the middle 1800s, but it is still an unsolved mystery. Vast numbers of research papers, book chapters, even entire books have dealt with the subject, so a simple answer to your general question does not exist.
In general, there is no single correct passive-damping model or single correct method for analyzing theoretically the influence of passive damping on structural dynamics. The best we can do experimentally is to measure the approximate influence, and the best we can do theoretically is to predict the order-of-magnitude energy dissipation.
Consider "small" passive damping. Suppose that you can magically deform statically a linear distributed-parameter structure into one of its theoretical pure undamped mode shapes, say mode i, and then you release the structure to vibrate freely. If the motion decays exponentially in time to half the original static amplitude after two (or more) of the undamped modal periods, then the modal damping is generally considered linear and "small" or “light,” and it is characterized by modal viscous damping factor zeta_i (ratio relative to critical damping of mode i with undamped natural frequency omega_i) being 0.055 or smaller.
The following is my experience analyzing theoretically the dynamic response of lightly damped, linear, distributed-parameter structures. Measure or derive theoretically your best estimates of the undamped normal modes (natural frequencies and mode shapes), under the good assumption that light damping has negligible practical influence over these modes. Obtain from measurements the general level of light structural damping, for example, just from the decay with time of the dominant fundamental mode. Then use modal analysis with order-of-magnitude modal viscous damping to analyze general response. The orders of magnitude for zeta_i that I prefer are 0.05, 0.01, 0.005, and 0.001. For examples, I would usually use 0.05 or 0.01 for a complex, built-up structure such as an airplane, but 0.005 for a simpler laboratory structures without fastened joints, and perhaps 0.001 for a truly free-free simple laboratory structure.
Now, I address your specific question. If you discretize your string equation using the modes of the undamped uniform string, then each of the resulting free-vibration modal equations has complex conjugate roots with real part -sigma= –c/(2*rho). This means that the exponential decay envelope of each mode has the same damping time constant tau=1/sigma, but that the modal damping factor zeta_i=sigma/omega_i varies inversely with omega_i. This uniformly decreasing zeta_i for higher-frequency modes is contrary to the usual experimental observation for lightly damped distributed-parameter structures, namely, that most modes have damping factors of roughly the same magnitude. (For examples of this, see my articles about experimentally inferred modes of a taut strap for two different cases of tension, “MATLAB Curve Fitting for Estimation of Structural Dynamic Parameters” and “Curve-Fitting Free Vibration Time Response for Estimation of Structural Dynamic Parameters.”) So the damping model in your PDE is physically unrealistic.
However, there is a simple damping model for beams, strings, bars, and shafts that can conform more closely with experimental observations, as described in the textbook Dynamics of Structures by R.W. Clough and J. Penzien, 1975, pp. 301-302. For example, suppose a taut string has two types of light viscous damping, the first proportional to velocity of string deflection (as in your model) and the second proportional to velocity of string curvature. Let the free-vibration PDE be
rho*w,tt+c1*rho*w,t-T*w,xx-c2*T*w,xxt=0.
Going through a discretization procedure will lead to modal damping factor for mode i
zeta_i=(c1/omega_i+c2*omega_i)/2
The two types of damping vary oppositely with increasing omega_i, so that, over a range of omega_i around the minimum zeta_i, the zeta_i can remain roughly the same order of magnitude, as is usually observed experimentally.
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