Why is damping generally ignored in modal analysis

It is an induced force that is represented in the equation of motion using the [ B ] matrix and velocity vector. The structural damping force is a displacement-dependent damping.

The structural damping force is a function of a damping coefficient and a complex component of the structural stiffness matrix. Assuming constant amplitude oscillatory response for a single degree of freedom system, the two damping forces are identical if:. Therefore, if structural damping is to be modeled using equivalent viscous damping b , then the equality holds at only one frequency see image below.

Two parameters are used to convert structural damping to equivalent viscous damping. An overall structural damping coefficient can be applied to the entire system stiffness matrix using PARAM, W3, r where r is the circular frequency at which damping is made equivalent.

The default for W3 is zero, which results in damping from this source to be ignored in transient analysis. PARAM, W4 is an alternate parameter used to convert element structural damping to equivalent viscous damping.

The default for W4 is zero, which results in damping from this source to be ignored in transient analysis. The choice of W3 or W4 is typically the dominant frequency at which damping is active. Often, the first natural frequency is selected, but isolated individual element damping can occur at different frequencies and can be handled by the appropriate data entries.

Modal transient response analysis uses the mode shapes of the structure to reduce the size, uncouple the equations of motion, and make numerical integration more efficient. To outline the procedure we first look at the general equation of equilibrium for a finite element system in motion:.

The transformation from physical coordinates to modal coordinates is given by:. As long as the system is linear and is represented correctly by the modes being used which are generally only a small subset of the total modes of the finite element model , the method is also very accurate because the integration operator used is exact whenever the forcing functions vary piecewise linearly with time.

You should ensure that the forcing function definition and the choice of time increment are consistent for this purpose. For example, if the forcing is a seismic record in which acceleration values are given every millisecond and it is assumed that the acceleration varies linearly between these values, the time increment used in the modal dynamic procedure should be a millisecond. The user-specified maximum number of increments is ignored in a modal dynamic step.

The number of increments is based on both the time increment and the total time chosen for the step. While the response in this procedure is for linear vibrations, the prior response can be nonlinear and stress stiffening initial stress effects will be included in the response if nonlinear geometric effects were included in the step definition for the base state of the eigenfrequency extraction procedure, as explained in Natural frequency extraction.

You can select the modes to be used in modal superposition and specify damping values for all selected modes. If you do not select the modes, all modes extracted in the prior eigenfrequency extraction step, including residual modes if they were activated, are used in the modal superposition. Damping is almost always specified for a mode-based procedure; see Material damping.

You can define a damping coefficient for all or some of the modes used in the response calculation. The damping coefficient can be given for a specified mode number or for a specified frequency range. When damping is defined by specifying a frequency range, the damping coefficient for a mode is interpolated linearly between the specified frequencies.

The frequency range can be discontinuous; the average damping value will be applied for an eigenfrequency at a discontinuity. The damping coefficients are assumed to be constant outside the range of specified frequencies.

Figure 1 illustrates how the damping coefficients at different eigenfrequencies are determined for the following input:. Mode selection and modal damping must be specified in the same way, using either mode numbers or a frequency range. If you do not select any modes, all modes extracted in the prior frequency analysis, including residual modes if they were activated, will be used in the superposition.

If you do not specify damping coefficients for modes that you have selected, zero damping values will be used for these modes. Damping coefficients for selected modes that are beyond the specified frequency range are constant and equal to the damping coefficient specified for the first or the last frequency depending which one is closer. This is consistent with the way Abaqus interprets amplitude definitions.

For convenience you can specify constant global damping factors for all selected eigenmodes for mass and stiffness proportional viscous factors, as well as stiffness proportional structural damping.

Structural damping is a commonly used damping model that represents damping as complex stiffness. This representation causes no difficulty for frequency domain analysis such as steady-state dynamics for which the solution is already complex.

However, the solution must remain real-valued in the time domain. To allow users to apply their structural damping model in the time domain, a method has been developed to convert structural damping to an equivalent viscous damping.

This technique was designed so that the viscous damping applied in the frequency domain is identical to the structural damping if the projected damping matrix is diagonal. For further details, see Modal dynamic analysis. Structural and viscous material damping see Material damping is taken into account in a SIM -based transient modal analysis.

Since the projection of damping onto the mode shapes is performed only one time during the frequency extraction step, significant performance advantages can be achieved by using the SIM-based transient modal procedure see Using the SIM architecture for modal superposition dynamic analyses. If the damping operators depend on frequency, they will be evaluated at the frequency specified for property evaluation during the frequency extraction procedure.

You can also control damping of the low frequency eigenmodes in transient modal analyses. This control is useful for free structures and models with secondary base motions, and it controls all sources of damping including the modal damping. To include low frequency eigenmodes, set the low frequency cutoff value to a small negative value.

Accordingly, the identification and model updating of such models are more complex. As most existing identification and updating methods are based on the assumption of viscous damping model, some corresponding improved methods have to be proposed for these kinds of damping models.

Adhikari and Woodhouse [ 43 ] identified the exponential damping by modal parameters based on the first-order perturbation method. Arora et al. Pan and Wang [ 45 ] updated the exponentially damped systems with the assumption of the Rayleigh damping model; however, the damping ratios of higher modes are obviously high in that model, which may not be valid for practical engineering structures. Consequently, it is necessary to propose a model updating method for systems with more general proportional damping.

As the derivation of modal data may lead to extra errors, especially for the imaginary parts of the mode-shapes, to overcome the drawback, this paper will focus on the model updating based on complex FRFs.

The main contributions of this paper is to derive the sensitivity of the dynamic stiffness matrix with respect to the physical and damping parameters for systems with nonviscous proportional damping, which can been seen in Section 2. To characterize the damping energy dissipation of the structure, in addition to the material and geometry parameters, the diagonal elements of the modal damping matrix and the relaxation parameters are chosen as the updating parameters.

In this way, without the assumption of Rayleigh or Cauchy damping models, the arbitrary variation of modal damping ratios with respect to frequency can be captured within the interested frequency range. The proposed method is validated by numerical and experimental examples.

The results show that the dynamic response can be predicted accurately by the updated analytical model. For the proportionally damped system, the damping matrix can be diagonalized as where is the diagonal modal damping matrix, is the k th diagonal element, and is the normalized undamped mode-shape matrix of the analytical model. With the mass normalization condition, the inverse of and can be calculated as follows:. Once the modal damping matrix is updated using experimental results, the damping matrix in the physical coordinate can be obtained accurately by the inverse transformation:.

For most real-life structures, the modes of high frequency are hard to be excited; hence, the modal information of the experimental data is often incomplete.

With the assumption that only the first N c elements of the modal damping matrix are considered in the updating procedure, then the damping matrix in the physical coordinate can be expressed as follows: where. In this paper, the updating parameters are divided into two categories: the physical parameter geometry, material, boundary condition, etc. The detailed updating formulas are given in the following sections.

Firstly, the mode-shape sensitivity with respect to the physical parameter is derived [ 46 ], which will be utilized later in Section 2. The characteristic equations of the undamped dynamic system are represented as follows: where and are the corresponding i th eigenvalue and eigenvector. Taking the derivative of equation 10 with respect to the k th physical updating parameter , we obtain where is the number of physical parameters that are chosen to be updated.

Generally, the derivative of the mode shape can be expressed by the linear combination of the mode-shape matrix: where is the coefficient to be solved. The superscript k indicates that the coefficient is related to the k th physical updating parameter. Substituting equation 12 into equation 11 yields:. Premultiplying equation 13 by and considering the orthogonality of the eigenvector with respect to the mass and stiffness matrices, one has where can be calculated by.

Combining equation 6 and equation 14 , the derivative of the inverse of mode-shape matrix with respect to the physical parameters can be calculated by. Based on the assumption of proportional damping, the mode shapes are the same in either damped or undamped systems. Regarding the mode-shape sensitivity with respect to the damping parameters, one can obtain the following relationship:.

From equation 3 , the sensitivity of dynamic stiffness with respect to the k th physical parameter can be expressed as follows:. The mass and stiffness sensitivity in equation 20 can be calculated easily while constructing the global mass and stiffness matrix by the FE method.

As the updating of physical parameters will lead to the changes of mode shapes, from equation 7 , it can be found that the damping matrix in the physical coordinate is also dependent on the physical parameters. The damping matrix sensitivity with respect to physical parameters can be calculated as follows:. Since the modal damping parameters are independent updating variables, regarding the sensitivity of the modal damping matrix with respect to the physical parameter, one can obtain the following relationship:.

Substituting equation 6 , 16 , 17 , and 22 into equation 21 , we can obtain. For modal incomplete case, using equation 8 , equation 23 can be expressed as follows:. Substituting equation 24 into equation 20 , we can obtain the dynamic stiffness sensitivity with respect to the physical parameter. For the exponential damping model, the damping parameters that need to be updated include the diagonal elements of the modal damping matrix and the relaxation parameter.

Since the mass and stiffness sensitivity with respect to the damping parameter are equal to zero, the dynamic stiffness sensitivity with respect to the k th element of the modal damping matrix can be calculated by. For the proportional damping model, substituting equation 18 into equation 25 , one has. Since the modal damping matrix is the diagonal matrix constructed by , one can obtain where is a zero matrix except for the unit element in the k th diagonal.

Substituting equation 27 into equation 26 , one has. For the k th relaxation parameter , the dynamic stiffness sensitivity can be computed easily by. For the analytical and experimental model, the following relationship always holds: where the superscripts A and E denote the analytical and experimental model, respectively. Express the experimental dynamic stiffness matrix as.

Substituting equations 32 and 30 into equation 31 , one has. Premultiply equation 33 with the analytical FRF ,. If only one column of the experimental FRF matrix is measured, then the above equation can be reduced as where j denotes the j th column of the experimental FRF. Linearizing with respect to the updating parameters , we have where and is the total number of updating parameters. The dynamic stiffness sensitivities with respect to the updating parameters are all given in Sections 2.

Substituting equation 36 into equation 35 , one obtains. If FRFs at several frequency points are considered, it is easy to construct the following algebraic equations: where. Since the updating parameters are all real variables, separating the real and imaginary parts of equation 38 , we can obtain. Once enough proper points are chosen within the interested frequency range, equation 41 can be solved by the SVD technique.

As equation 36 is just a first-order approximation, the iterative method of the above process can be applied to update the parameters. Assuming that at the k th iteration, the updating variable is , then can be calculated with the updated sensitivity matrix and residual vector by using equation 39 — The updated parameter can be obtained as.

The iteration procedure will not stop until the updating parameter satisfies either one of the following conditions: where denotes the norm of a vector and and are the given convergence threshold and the maximum number of iterations, respectively.

Usually, the incompleteness of experimental data is inevitable and some approximation has to be introduced during the updating procedure. For higher modes outside the interested frequency range, the corresponding modal damping is neglected when constructing the damping matrix in this paper. It should be pointed out that the number of the diagonal elements of the modal damping matrix which need to be updated should be larger than the number of modes considered in the interested frequency range, as the damping of each mode is described by the corresponding diagonal element.

The proposed model-updating method can be regarded as an application of the mode superposition. To solve the problem of spatial incompleteness, the unmeasured FRFs of in equation 37 are replaced by the analytical counterparts of [ 46 ]. Since that the damping effect is more pronounced around the resonance peaks, to have a better estimation of the damping parameter, the FRFs around the resonance peaks are used in the updating procedure. Besides, to improve the robustness of the updating procedure, some weighting techniques proposed by Lin [ 47 ] are adopted to balance the FRFs at different frequencies and locations.

In this section, as shown in Figure 1 , the cantilever beam is considered to evaluate the effectiveness of the proposed method.

The length of the beam is and the cross-sectional area is. The elastic modulus and density are and , respectively. The FE model of the Euler—Bernoulli beam consists of 10 elements, and each node has two degrees of freedom translational and rotational. Consequently, the size of the constructed mass and stiffness matrices is. Proportional exponential damping model is utilized to model the damping of the beam. For simplicity, only one exponential kernel function is considered here.

The degree of nonviscosity can be described by comparing the relaxation parameter with the maximum mode frequency within the interested frequency range. While the relaxation parameter is much larger than the interested frequency, the nonviscosity is so weak that the exponential damping model can be regarded as the traditional viscous damping model.

In this example, the damping coefficient matrix is assumed to have the following form as where is the constant variable, which denotes the damping level of the system. It is obvious that the given damping is proportional in the system. For viscous damping model, the damping matrix obtained from equation 44 ensures that the damping ratios of all the modes are the same.

Considering different levels of nonviscosity, two cases listed in Table 1 are discussed in this section. Then, random noise is added to the calculated time functions to simulate the experimental impulse response functions. Finally, the impulse responses are transformed back to frequency domain to simulate the FRFs contaminated with noise. Considering the spatial incompleteness in practical situations, only the FRFs of translational degrees of freedom are assumed to be predetermined in this example.

To simulate the errors between the real structure and the corresponding FE model, as shown in Table 2 , some discrepancies are added in the thicknesses of each beam element. Before the updating procedure, the initial values of the damping should be provided first. In this example, the initial relaxation parameter is set as the first natural frequency. That is to say, the nonviscous effect is assumed to be strong at first. With the modal damping ratios estimated from the half-power bandwidth, the initial diagonal elements of the modal damping matrix can be calculated easily by using the mass and stiffness matrices.

In this section, for the comparison with the proposed method, we also update the damping coefficient matrix which is expressed by the Rayleigh damping model. Usually, for the Rayleigh damping model, the damping coefficients can be estimated using just 2 modes.

If you have a large known damping that's unevenly spread across your structure you should use a direct solver for each frequency instead. For most structures of practical interest the error introduced to the natural frequency calculation by ignoring damping is small.

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