**4. Dynamic damping—Comparison of different concepts from the point of view of their physical nature and effects on civil engineering structures**

There are a variety of sources of dynamical damping of vibration in civil engineering structures. Generally, damping can be caused either by the external environment or by energy dissipation due to structural deformation. The most common way to account for the damping in the motion equation is Rayleigh damping (see [13–15]). The Rayleigh damping is proportional to a linear combination of the stiffness and mass matrix.

This subchapter deals with the justification for the use of Rayleigh damping and discusses another solution because there is no direct physical interpretation of the mass damping parameter and the stiffness damping parameter. The damping given by internal resistance of structure occurs in the case of using inelastic materials when the loading and unloading parts of the strain–stress diagram differ, and therefore dissipation occurs in loading cycles, as described in [1–3, 5, 6, 10–12, 16–20]. Another source of damping is friction in structural connections.

### **4.1 Finite element analysis of the equation of motion**

The virtual work of external forces has to be equal to the virtual work of internal forces. For one element with the volume Ω*<sup>e</sup>* and the surface boundary Γ*<sup>e</sup>* we can write this equilibrium of virtual work as follows:

$$\int\_{\Omega\_{\epsilon}} \delta \mathbf{u}^{T} \mathbf{b} d\Omega\_{\epsilon} + \int\_{\Gamma\_{\epsilon}} \delta \mathbf{u}^{T} \mathbf{t} d\Gamma\_{\epsilon} + \sum\_{i=1}^{n} \delta \mathbf{u}\_{i}^{T} \mathbf{f}\_{i} = \int\_{\Omega\_{\epsilon}} (\delta \mathbf{u}^{T} \rho \ddot{\mathbf{u}} + \delta \mathbf{u}^{T} c \dot{\mathbf{u}} + \delta \mathbf{e}^{T} \boldsymbol{\sigma}) d\Omega\_{\epsilon} \tag{19}$$

where *δ***u** and *δ***ε** represent the virtual displacement and pertinent strain, respectively, *ρ* is the material density, *c* is the viscous damping parameter, **b** and **t** are the volume and surface forces respectively, **f***<sup>i</sup>* and *δ***u***<sup>i</sup>* represent the concentrated forces and pertinent generalized displacement, respectively. The damping parameter *c* does not take into account material (internal) or external resistance. The material viscosity is not taken into account because it has an impact only in the case of a nonzero rate of deformation in the given mass point (strain of the body). The influence of the external environment is not taken into account because the stress vector *c***u**\_ , which acts against the motion **u**, arises even in the internal points of the bodies.

When discretizing for the finite element method we obtain the following relations:

$$\dot{\mathbf{u}} = \mathbf{N}\dot{\mathbf{d}} \qquad \dot{\mathbf{u}} = \mathbf{N}\dot{\mathbf{d}} \qquad \ddot{\mathbf{u}} = \mathbf{N}\ddot{\mathbf{d}} \qquad e = \mathbf{B}\mathbf{d} \tag{20}$$

Combining eqs. (22) and (23) we obtain:

$$\delta \mathbf{d}^T \left( \int\_{\Omega} \rho \mathbf{N}^T \mathbf{N} d\Omega\_\epsilon \ddot{\mathbf{d}} + \int\_{\Omega} c \mathbf{N}^T \mathbf{N} d\Omega\_\epsilon \dot{\mathbf{d}} + \int\_{\Omega\_\epsilon} \mathbf{B}^T \mathbf{o} d\Omega\_\epsilon - \int\_{\Omega\_\epsilon} \mathbf{N}^T \mathbf{b} d\Omega\_\epsilon - \int\_{\Gamma\_\epsilon} \mathbf{N}^T \mathbf{t} d\Gamma\_\epsilon - \sum\_{i=1}^n \mathbf{f}\_i \right) = \mathbf{0} \tag{21}$$

where it is assumed that the concentrated forces **f***<sup>i</sup>* act in nodes. Let us denote the first two integrals in the equation as the consistent mass matrix and the damping matrix.

$$\mathbf{M}\_{\epsilon} = \int\_{\Omega\_{\epsilon}} \rho \mathbf{N}^{T} \mathbf{N} d\Omega\_{\epsilon} \tag{22}$$

$$\mathbf{C}\_{\epsilon} = \int\_{\Omega\_{\epsilon}} c \mathbf{N}^{T} \mathbf{N} d\Omega\_{\epsilon} \tag{23}$$

The word consistent means that the matrix follows directly from the discretization of a finite element with corresponding shape functions **N**. Let us define the vectors of the internal and external nodal forces:

$$\mathbf{f}\_{\epsilon}^{\text{int}} = \int\_{\tilde{\Omega}} \mathbf{B}^{T} \boldsymbol{\sigma} d \Omega\_{\epsilon} \tag{24}$$

$$\mathbf{f}\_{\epsilon}^{\text{ext}} = \int\_{\tilde{\Omega}\_{\epsilon}} \mathbf{N}^{T} \mathbf{b} d\Omega\_{\epsilon} + \int\_{\tilde{\Gamma}\_{\epsilon}} \mathbf{N}^{T} \mathbf{t} d\Gamma\_{\epsilon} + \sum\_{i=1}^{n} \mathbf{f}\_{i} \tag{25}$$

Substituting from eqs. (25), (26), (27), and (28) into eq. (24), and taking into account the fact that variation *δ***d** can be arbitrary, and hence the second form of the product must be equal to zero, we obtain:

$$\mathbf{M}\_{\epsilon}\ddot{\mathbf{d}} + \mathbf{C}\_{\epsilon}\dot{\mathbf{d}} + \mathbf{f}\_{\epsilon}^{\text{int}} = \mathbf{f}\_{\epsilon}^{\text{ext}} \tag{26}$$

For a linear elastic material without viscosity, we can write the following relation for the internal nodal forces:

$$\mathbf{f}\_{\epsilon}^{\text{int}} = \mathbf{K}\_{\epsilon} \mathbf{d} \tag{27}$$

where

$$\mathbf{K}\_{\epsilon} = \int\_{\Omega\_{\epsilon}} \mathbf{B}^{T} \mathbf{D} \mathbf{B} d\Omega\_{\epsilon} \tag{28}$$

is the stiffness matrix of the element, with **D** being the constitutive matrix of the material. Then eq. (29) can be rewritten as:

$$\mathbf{M}\_{\epsilon}\ddot{\mathbf{d}} + \mathbf{C}\_{\epsilon}\dot{\mathbf{d}} + \mathbf{K}\_{\epsilon}\mathbf{d} = \mathbf{f}\_{\epsilon}^{\text{ext}} \tag{29}$$

Eqs. (30) and (32) express in discretized form Newton's second law, or, more generally, the law of conservation of momentum. When writing these equations in the global form, i.e., for all degrees of freedom of the analyzed structure, we obtain the following equation:

$$\mathbf{M}\ddot{\mathbf{d}} + \mathbf{C}\dot{\mathbf{d}} + \mathbf{f}^{\text{int}} = \mathbf{f}^{\text{ext}} \tag{30}$$

$$\mathbf{M}\ddot{\mathbf{d}} + \mathbf{C}\dot{\mathbf{d}} + \mathbf{K}\mathbf{d} = \mathbf{f}^{\text{ext}} \tag{31}$$

#### **4.2 Rayleigh damping**

For Rayleigh damping, the damping matrix **C***<sup>R</sup> <sup>e</sup>* is defined as the linear combination of the consistent mass matrix **M***<sup>e</sup>* and the stiffness matrix **K***e*.

$$\mathbf{C}\_{\epsilon}^{\mathbb{R}} = a\mathbf{M}\_{\epsilon} + \beta \mathbf{K}\_{\epsilon} \tag{32}$$

When substituting to eq. (35) **M***<sup>e</sup>* from eq. (25) and **K***<sup>e</sup>* from (31), we obtain the relation for the damping matrix in the following form:

$$\mathbf{C}\_{\epsilon}^{R} = a \int\_{\Omega\_{\epsilon}} \rho \mathbf{N}^{T} \mathbf{N} d\Omega\_{\epsilon} + \beta \int\_{\Omega\_{\epsilon}} \mathbf{B}^{T} \mathbf{D} \mathbf{B} d\Omega\_{\epsilon} \tag{33}$$

The damping matrix has two parts and the first one is identical with **C***<sup>e</sup>* as defined in (26), where *c* ¼ *αρ* and its physical nature is unclear. The second part of the expression does not correspond with relation (26) but is proportional to the stiffness matrix. The damping for *α* ¼ 0 and *β* >0 is thus proportional to the rate of deformation of the body. The internal nodal forces arise only when the body deforms over time and no internal forces arise when an element moves as a rigid body.

#### *4.2.1 Damping caused by material viscosity*

As we mentioned before the internal damping caused by material arises only when a body is strained and is proportional to the strain rate. We have chosen the three most well-known viscoelastic models for this article for the damping caused by material viscosity and they are described below.

### *4.2.2 The Maxwell material model*

For the Maxwell (see **Figure 10**) model the following relations are valid:

$$
\sigma(t) = \sigma^\varepsilon(t) = \sigma^v(t), \; \varepsilon(t) = \varepsilon^\varepsilon(t) + \varepsilon^v(t), \; \varepsilon^\varepsilon(t) = \frac{\sigma(t)}{E}, \; \dot{\varepsilon}^v(t) = \frac{\sigma(t)}{\eta} \tag{34}
$$

As a result, we obtain the following linear nonhomogenous ordinary differential equation which describes the constitutive relation between stress and strain:

**Figure 10.** *Scheme of the Maxwell model.*

$$
\sigma(t) + \frac{\eta}{E}\dot{\sigma}(t) = \eta \dot{\epsilon}(t) \tag{35}
$$

### *4.2.3 The kelvin-Voigt material model*

The Kelvin-Voigt model (see **Figure 11**) is based on the following relations:

$$
\varepsilon(t) = \varepsilon'(t) = \varepsilon''(t), \ \sigma(t) = \sigma'(t) + \sigma''(t), \ \varepsilon'(t) = \frac{\sigma(t)}{E}, \ \dot{\varepsilon}^p(t) = \frac{\sigma(t)}{\eta} \tag{36}
$$

Using these relations, we obtain as a result the following linear non-homogenous ordinary differential equation which describes the constitutive relation between stress and strain:

$$
\sigma(t) = E\epsilon(t) + \eta \dot{\epsilon}(t) \tag{37}
$$

*4.2.4 The standard linear solid (SLS) material model*

For the SLS model (see **Figure 12**) the following relations are valid:

**Figure 11.** *Scheme of the kelvin-Voigt model.*

**Figure 12.** *Scheme of the standard linear solid model.*

*On the Nonlinear Transient Analysis of Structures DOI: http://dx.doi.org/10.5772/intechopen.108446*

$$\varepsilon(t) = \varepsilon\_1(t) = \varepsilon\_2(t), \ \varepsilon\_1(t) = \varepsilon\_1^\varepsilon(t) = \frac{\sigma(t)}{E\_1}, \ \varepsilon\_2(t) = \varepsilon\_2^\varepsilon(t) + \varepsilon\_2^\nu(t), \ \sigma(t) = \sigma\_1(t) + \sigma\_2(t),$$

$$\sigma\_1(t) = E\_1\varepsilon(t) \text{ and } \sigma\_2(t) = E\_2\left(\varepsilon(t) - \varepsilon\_2^\nu(t)\right) \tag{38}$$

Then, for the resulting stress the following relations can be written:

$$
\sigma(t) = E\_1 \varepsilon(t) + E\_2 \left( \varepsilon(t) - \varepsilon\_2^v(t) \right) \tag{39}
$$

After several modifications, we obtain the final relation for expressing strain rate as a function of stress rate, stress, and actual strain (it is a differential equation describing the constitutive relation between stress and strain).

$$
\sigma(t) + \frac{\eta}{E\_2}\dot{\sigma}(t) = \frac{\eta(E\_1 + E\_2)}{E\_2}\dot{\epsilon}(t) + E\_1\varepsilon(t) \tag{40}
$$

#### *4.2.5 Damping caused by the kelvin-Voigt model*

Let us now demonstrate how the damping caused by the Kelvin-Voigt model will manifest itself in the equation of motion. Let us substitute for **σ**ð Þ*t* from (41) into eq. (27). For the vector of internal nodal forces, we obtain:

$$\mathbf{f}\_{\epsilon}^{\text{int}} = \int\_{\hat{\Omega}} \mathbf{B}^{T} (\sigma^{\epsilon} + \eta \dot{\mathbf{e}}) d\Omega\_{\epsilon} = \int\_{\hat{\Omega}} \mathbf{B}^{T} \left( \sigma^{\epsilon} + \eta \mathbf{B} \dot{\mathbf{d}} \right) = \int\_{\hat{\Omega}} \mathbf{B}^{T} \sigma^{\epsilon} + \eta \int\_{\hat{\Omega}} \mathbf{B}^{T} \mathbf{B} \dot{\mathbf{d}} \tag{41}$$

Taking into consideration the fact that the damping will be caused by the material, we will not expect damping to take place with the help of the matrix **C** here. The equation of motion shall then read:

$$\mathbf{M}\_{\epsilon}\ddot{\mathbf{d}} + \int\_{\Omega\_{\epsilon}} \mathbf{B}^{T} \boldsymbol{\sigma}^{\epsilon} + \eta \int\_{\Omega\_{\epsilon}} \mathbf{B}^{T} \mathbf{B} \dot{\mathbf{d}} = \mathbf{f}\_{\epsilon}^{\text{ext}} \tag{42}$$

The equation can then be rewritten for a linear elastic material as:

$$\mathbf{f}\_{\epsilon}^{\text{int}} = \mathbf{K}\_{\epsilon}\mathbf{d} + \eta \int\_{\Omega\_{\epsilon}} \mathbf{B}^{T} \mathbf{B} \dot{\mathbf{d}} d\Omega\_{\epsilon} \tag{43}$$

$$\mathbf{M}\_{\epsilon}\ddot{\mathbf{d}} + \mathbf{f}\_{\epsilon}^{\text{int}} = \mathbf{f}\_{\epsilon}^{\text{ext}} \tag{44}$$

and after the substitution for **f**int *<sup>e</sup>* , we can rewrite the equation of motion in the form

$$\mathbf{M}\_{\epsilon}\ddot{\mathbf{d}} + \mathbf{K}\_{\epsilon}\mathbf{d} + \eta \int\_{\Omega} \mathbf{B}^{T}\mathbf{B}\dot{\mathbf{d}}d\Omega\_{\epsilon} = \mathbf{f}\_{\epsilon}^{\text{ext}} \tag{45}$$
