Outline of Stochastic Dynamics of SDOF Systems (cont.). Weakly Stationary Response Processes. Equivalent White Noise Approximations. Gaussian Response Processes as Conditional Normal Distributions. Stochastic Dynamics of MDOF Systems. Introduction to MDOF Systems. 1
Stochastic Dynamics of SDOF Systems (cont.) The frequency response function of a SDOF oscillator is given as, cf. Lecture 2, Eq. (58): 2
Weakly Stationary Response Processes If the load process is weakly or strictly stationary, and has been applied to the structure in infinite long time, the displacement process becomes weakly or strictly stationary. Mean value function: From Lecture 2, Eq. (54): From (1), cf. Lecture 2, Eq. (56): Auto-spectral density function: From Lecture 2, Eq. (57): 3
SDOF system exposed to Gaussian white noise: 4
is a Gaussian white noise process with the autospectral density function. Then, in the stationary state the autospectral density function of the displacement process becomes, cf. (1), (4): (5) is identical to Lecture 2, Eq. (25). The related auto-covariance function is given as, cf. Lecture 2, Eqs. (30), (31): 5
The velocity process and the acceleration process have the auto-spectral density function, cf. Lecture 3, Eqs. (27), (28): Hence, the auto-spectral density function of the acceleration process is not integrable, so. The acceleration process resembles the load process. Neither have finite or continous realizations. Typical realizations of the displacement and velocity processes are depicted in Fig. 2. 6
SDOF system exposed to filtered Gaussian white noise: The load process is obtained by a filtration of a Gaussian unit intensity white noise process defined by the auto-covariance and auto-spectral density functions: 7
The filter is defined by the rational frequency response function of the order : Then, the auto-spectral density function of the load process becomes: given target load spectrum. are determined, so (13) at best fits a 8
The auto-spectral density function of the displacement process becomes, cf. (4), (13): The resulting frequency response function is a rational function of the order, obtained as a product of the components and : where, and: 9
and form a series connection of frequency response functions, which is known as a cascade. It follows from (14), (15), (16) that has the asymptotic behavior for : Hence, the displacement process exists with a finite variance, if: 10
Example 1: Kanai-Tajimi filter 11
The equation of motion for the free storey beam: The sediment layer is modeled as a SDOF oscillator with the mass, the stiffness and the linear viscous damping constant. The mass of the storey beam is assumed to be ignorable compared to. Then, the reaction force from the frame can be ignored in the equation of motion for the subsoil: (19) may be written as: 12
where and are the angular eigenfrequency and damping ratio of the frame: (20) and (22) determines the earthquake load on the frame as the output of a rational filter of the order with the filter constants, cf. (12): The indicated earthquake model is known as a Kanai-Tajimi filter. The input to the filter is the bedrock acceleration process. The primary energy drain in the subsoil is due to energy transport carried by the elastic stress waves, and not due to mechanical dissipation in the soil. For this reason the damping ratio of the subsoil in the model need to be chosen relatively large,.. 13
Example 2: Single-degree-of-freedom system exposed to an indirectly acting dynamic load and damping force. : Point mass at point 1. : Damper constant. Damper is acting at point 2. : Dynamic load. Load is acting at point 3. : Degree of freedom of point mass. : Auxiliary degree of freedom of support point of damper. : Auxiliary degree of freedom of attack point of load. 14
Equations of motion is formulated by means of d Alembert s principle: j th component 15
The frequency response matrix becomes: The component denotes the frequency response function for the displacement due to a harmonically varying load at the degree of freedom. This is given as: where: 16
Hence, is a rational function of the order. The load process and the displacement process are related by the stochastic differential equations, cf. Lecture 4, Eqs. (1), (2): The double-sided auto-spectral density function of is given as, cf. Lecture 2, Eq. (57): 17
If is a Gaussian white noise process with the autospectral density function, (34) reduces to: Hence, and have continous realizations. does not exist with a finite variance. The realizations of and resemble those shown in Fig. 2. 18
Equivalent White Noise Approximations 19
The load process is assumed to be weakly stationary with a broad-banded auto-spectral density function without any marked peaks. The oscillator is assumed to be lightly damped, i.e.. Then, has a marked peak at. Actually, cf. Eq. (1): Then, the following approximation for the variance of the displacement process applies: 20
Hence, the approximations leading to (37) is equivalent to the replacement of the actual broad-banded load process with an equivalent Gaussian white noise process with the auto-spectral density function, see Fig. 4a. This is so, because only harmonic load components with angular frequencies close to the angular eigenfrequency contributes significantly to the variance of the reponse. 21
22
Wave and wind gust (turbulence) processes have much higher spectral densities at lower angular frequencies than at the angular eigenfrequency of the structure. Hence, the approximate result (37) is not valid, and a modified approximation is needed. The peak angular frequency of the excitation is typically of the magnitude, and is placed well below the resonance frequency interval of the oscillator. Then, the variance of the response may be approximately calculated as: 23
Next, the following approximations are applied: 24
The first term on the right hand side of (39) represents the variance from the quasi-static response. The second term indicates the variance from the narrow-banded dynamic response. The variance contributions add linearly, which indicates that the response components are uncorrelated. The interaction of the two response components has been illustrated by the realization shown in Fig. 6. The approximation (39) has been applied in the Danish Code of Practice for wind gust loadings. Gaussian Response Processes as Conditional Normal Distributions A SDOF oscillator with deterministic initial values is subjected to a stationary Gaussian load process : 25
Due to the linearity the response process becomes Gaussian as well. Next, consider the stationary displacement process, obtained as solution to the stochastic differential equation (40), when the load process has been acting in infinite long time. The process is defined by the mean value function and the auto-covariance function, where is the stationary variance, and signifies the auto-correlation coefficient function. The displacement and the velocity at the time are in this case random variables. Consider the 4-dimensional normal distributed stochastic vector: where and denotes the state vector at the times and. The joint probability density function of becomes, cf. Lecture 1, Eqs. (20), (21), (22): 26
The mean value vector and the covariance matrix become: At the evaluation of (44) the following results have been used, cf. Lecture 3, Eqs. (14), (15): 27
Next, the distribution of on condition of is determined. Any marginal or conditional distribution of is normal. Hence, on condition of is jointly normal distributed with the conditional mean value vector and the conditional covariance matrix. With the definitions in (43) and (44) these are given as: 28
where has been introduced in (46) and (47). (40) may be formulated on the state vector form, cf. Lecture 4, Eq. (42): 29
The considered conditional mean value vector and the conditional covariance matrix are per definition the mean value vector function and the zero time-lag covariance matrix obtained from (48) at the time. Next, assume that is a Gaussian white noise process with the auto-spectral density function.then, is given by (7) and is given by (8). Further, the components of and are determined from the following differential equations, cf. Lecture 4, Eqs. (28) and (36): 30
The differential equations (50) follows by explicit calculation of the right-hand side of Lecture 4, Eq. (39), cf. Lecture 4, Eq. (44). From the above argumentation it follows that the solution to (49) and (50) are given by (46) and (47) for : 31
Stochastic Dynamic of MDOF Systems Introduction to MDOF Systems MDOF systems are brought forward by discrete distribution of mass and damping. Elasticity (the bending stiffness in beam theory) may be continuously distributed. 32
The number of degrees of freedom specifies the number of unconstrained displacement degrees of each point mass (up to 3 dofs) and the number of unconstrained displacement and rotational degrees of freedom of each rigid, distributed mass (up to 6 dofs). : -dimensional load vector process. : -dimensional displacement vector process. : Index time interval. may contain both displacement and rotational component processes. 33
Stochastic vector differential equation: : Initial value vectors. Stochastic vectors of dimension. : Mass matrix. : Damping matrix.. : Stiffness matrix.. 34
Stochastic integral equation:. Dimension:.. Dimension:. : Initial value response vector. : Impulse response matrix. : Frequency response matrix. 35
Modal analysis: Undamped eigenvibrations are assumed on the form: The amplitude vector, the angular frequency and the common phase are determined by insertion in (53) for,. This leads to the generalized eigenvalue problem: Nontrivial solutions for exist for: Eigensolutions are real, due to the symmetry properties,, and because or are positive definite. : Undamped angular eigenfrequency of the j th mode. : Eigenmode. 36
Solutions indicate rigid body motions. The related eigenmodes neither induce elastic forces nor damping forces: The linear independent eigenmodes form an N-dimensional vector basis. Hence, the displacement process may be written: : -dimensional modal coordinate vector process. : Modal matrix. 37
Insertion of (64) into (53), and premultiplication with provides: 38
: Modal mass matrix. : Modal mass. : Modal damping matrix. : Modal damping. : Modal stiffness matrix. : Modal stiffness. : -dimensional modal load vector process. and are always diagonal due to the orthogonality properties of the eigenmodes. may be assumed to be diagonal for lightly damped systems with well-separated eigenfrequencies. Notice that for rigid-body modes. The diagonal structure of,, implies that the j th component process of is determined from the uncoupled SDOF equation: 39
Alternatively, (71) may be written on the form: where: : Modal damping ratio. The solution of (72) becomes, cf. Lecture 4, Eqs. (47), (48): 40
: Modal impulse response function of the j th mode. : Damped angular eigenfrequency of the j th mode. : Frequency response function of the j th mode. 41
Summary of Weakly Stationary Response Processes. : Frequency response function. 42
Equivalent White Noise Approximations. is stationary and broad-banded. Gaussian Response Processes as Conditional Normal Distributions. Given a zero mean stationary Gaussian process the auto-covariance function: with Let: We search the joint probability density function of on condition of. 43
on condition of is normal distributed, where: 44