1. Dr-Ing Asrat Worku, AAIT
Soil Dynamics
Chapter 2
MDF Systems
October 2015
Dr-Ing Asrat Worku, AAIT

2. VIBRATION OF MDF SYSTEMS
Chapter 2
VIBRATION OF MDF SYSTEMS
October 2015
Dr-Ing Asrat Worku, AAIT

3. Simplest MDF System: Two DF Systems
A two-story shear building is idealized as a 2DF system as shown
Assumptions in the idealization:
Floor is rigid in flexure
Negligible axial deformations in columns and beams
Negligible effect of axial force on system stiffness
Newton’s 2nd law gives:
The restraining forces:
October 2015
Dr-Ing Asrat Worku, AAIT

4. Simplest MDF System: Two DF Systems
The damping forces follow similarly
The equation of motion becomes
These are coupled equations and can be written in matrix form
October 2015
Dr-Ing Asrat Worku, AAIT

5. Simplest MDF System: Two DF Systems
Or, in more compact form as
Where
An equivalent mass-spring-damper system is as shown in the figure
The same equations of motion evolve from this system
The above form of matrix equation applies to any MDF system
October 2015
Dr-Ing Asrat Worku, AAIT

6. Direct Stiffness Method to Determine [k]
The restoring force matrix may be written as
Or, generally, for n-DoF system
Determination of
E.g. k11 can be obtained as the restoring force, fs1, in the direction of u1 by giving u1 a unit displacement while keeping the rest to zero.
Similarly, k12 can be obtained as the restoring force in the direction of u1 by giving u2 a unit displacement and keeping the rest to zero.
October 2015
Dr-Ing Asrat Worku, AAIT

7. Simplest MDF Model: Example 1
Formulate the equation of motion for the shear frame shown without damping for a uniform floor height, h
For a shear frame:
The masses are:
Finally, the equation of motion becomes:
October 2015
Dr-Ing Asrat Worku, AAIT

8. Dr-Ing Asrat Worku, AAIT
MDF Systems: Example 2
The uniform rigid bar of total mass, m, subjected to the dynamic forces at its center is constrained to move in the vertical plane. It has two DFs. It is required to formulate the equation of motion without damping for the displacements as defined at the bar ends
Transfer external forces to DF locations as shown to obtain
Stiffness matrix: this is established by imposing unit displacements in turn as shown. It follows:
October 2015
Dr-Ing Asrat Worku, AAIT

9. Dr-Ing Asrat Worku, AAIT
MDF Systems: Example 2
Mass matrix: this is established by imposing unit accelerations in turn as shown. It follows:
Finally, the equation of motion:
If the DFs are defined as the vertical displacement at the center and the rotation of the bar as shown, the equation would become
October 2015
Dr-Ing Asrat Worku, AAIT

10. MDF Systems: Undamped Free Vibration
The undamped free vibration is expressed by
The solution is desired for a given initial condition of
The solution is of the form
Substituting back, one obtains
This leads to a standard mathematical problem called eigenvalue problem:
October 2015
Dr-Ing Asrat Worku, AAIT

11. MDF Systems: Undamped Free …
The eigenvalue problem can also be written as
A non-trivial solution is possible if
This condition results in a polynomial of order N (no. of DFs) with N different roots for λ
The roots of λ are positive real as far as the mass and stiffness matrices are positive definite. This condition is met by all civil engineering structures.
Positive Definiteness:
The stiffness matrix of a structure is positive definite as far as the structure does not undergo a rigid-body motion
The mass matrix is positive definite as far as the lumped mass is nonzero in all defined DFs
The N roots provide the N natural frequencies:
Substitution of each frequency in the above vector equation furnishes the corresponding natural mode shape.
The eigenvalue problem is normally dealt with numerically for a large number of DoFs.
October 2015
Dr-Ing Asrat Worku, AAIT

12. MDF Systems: Orthogonality Conditions
Subtracting (1) from (2), we obtain
Since it follows that
And from (1) follows
These two relations constitute the orthogonality conditions for MDF systems and play a pivotal role in uncoupling the coupled DEs
The eigenvalue problem may be written for the nth mode as
Pre-multiplying this by 𝜙 𝑟 𝑇 one obtains
Similarly follows for the rth mode of vibration
We transpose this latter matrix equation to obtain,
(2)
(1)
October 2015

13. MDF Systems: Orthogonality Conditions
For n=r, Eq. (1) yields,
Where are scalars given by
In matrix form Eq. (3) for all modes can be written as
In this equation all three matrices are diagonal matrices.
(3)
October 2015
Dr-Ing Asrat Worku, AAIT

14. MDF Systems: Orthogonality Conditions
These diagonal matrices are given by
The matrices and are called modal and spectral matrices.
October 2015
Dr-Ing Asrat Worku, AAIT

15. Normalization of Modes
Each natural mode satisfies the relation
This implies that any scalar product of each mode satisfies also this equation; thus, there is a need for normalization of modes
A number of normalization techniques exist
Selecting the largest element of a mode as unity;
Selecting an element along a particular DF as unity;
Normalization so that Mn is unity; i.e.
Modes normalized so that Mn=1 are called orthonormal modes.
October 2015
Dr-Ing Asrat Worku, AAIT

16. Simplest MDF Model: Example 2
Determine the natural frequencies and natural modes of the frame with rigid floors in the previous example if the frame is made of r.c. and a roof weight of 350 kN is shared by the frame. The columns are square in shape with a side length of 300 mm in the upper floor, and the floor height is 3 m
(To be done in class)
October 2015
Dr-Ing Asrat Worku, AAIT

17. Undamped Free Vibration: Example 1
Determine the natural frequencies and modes for the rigid bar treated earlier using both sets of coordinates
In the first case
Then
The characteristics equation:
October 2015
Dr-Ing Asrat Worku, AAIT

18. Undamped Free Vibration: Example …
From this follows:
The natural modes are found by substituting these in turn in
So that
The modes are given by the first set of vectors below:
The natural frequencies remain unchanged when using the second sets of DFs. The equivalent modes are given by the second set of vectors above. The two sets are equivalent.
October 2015
Dr-Ing Asrat Worku, AAIT

19. Free Vibration Response - Undamped
The time history of the free vibration is obtained from superposition of the individual mode contributions (see Slide 10):
An and Bn are obtained from the initial conditions as follows:
Let at t=0, and be known
After pre-multiplying the resulting two equations by and noting the orthogonality conditions, one obtains
With this, the time history of the free vibration is fully known.
October 2015
Dr-Ing Asrat Worku, AAIT

20. Free Vibration Response - Damped
The damped free vibration is given by
It is desired to find the solution for the initial conditions of and
We make the modal substitution of and pre-multiply by to obtain
Where
If [C] is a diagonal matrix, then we have a classically damped system with n-uncoupled equations for qn; if not, we deal with a non-classically damped system with coupled equations.
October 2015
Dr-Ing Asrat Worku, AAIT

21. Free Vibration Response - Damped
Cases of non-classically damped systems:
Structures made of different materials in distinctly different regions
Structures, whose foundations interact with soil: machine foundations, buildings, etc
Use of artificial dampers in selected regions of the structure
October 2015
Dr-Ing Asrat Worku, AAIT

22. Forced Vibration of Damped Systems: Modal Analysis
The forced vibration is expressed as
We make as before the modal substitution of and pre-multiply by to obtain
Where
For a classically damped system the elements Cnr are zero for n≠r so that we obtain the uncoupled equations of Or
October 2015
Dr-Ing Asrat Worku, AAIT

23. Dr-Ing Asrat Worku, AAIT
Forced Vibration …
Thus, the N coupled DEs in geometric coordinates are transformed into N uncoupled DEs in modal coordinates.
This is identical to solving the vibration problem of N independent single-mass oscillators!
A schematic of the problem in the nth mode is as sketched below.
The tools are already established!!
This method of solution is called Modal Analysis or Modal Superposition or modal transformation.
October 2015
Dr-Ing Asrat Worku, AAIT

24. Displacement Response
Once the modal coordinates are known, the displacement response is determined from
If the initial condition is different from at-rest condition, the free vibration solution should be included:
Restraining forces needed for design are determined from
Or, preferably from
October 2015
Dr-Ing Asrat Worku, AAIT

25. Construction of Proportional Damping Matrices: Rayleigh Damping
A proportional damping matrix can be constructed by setting the damping proportional to the mass and stiffness matrices. Such a damping is called Rayleigh damping.
Pre- and post-multiplying by the nth mode vector results in a diagonalized generalized damping matrix with:
Noting that , one obtains
October 2015
Dr-Ing Asrat Worku, AAIT

26. Construction of Classical Damping …
The plot of this expression as a function of ω is as shown
The first term is the mass proportional damping that decreases with increasing frequency.
The second term is the stiffness-proportional damping linearly increasing with increasing frequency.
October 2015
Dr-Ing Asrat Worku, AAIT

27. Construction of Classical Damping …
Specifying two values of ξn, say ξi and ξk, is sufficient to obtain the two unknowns α and β that can be easily shown to be
It is recommended, however, to use ξi = ξk= ξ for two modes one at around the lower end and one at around the upper end of the significant frequency spectrum (see figure on previous slide). In this case:
Once the two constants are known, the entire modal damping matrix is constructed from
October 2015
Dr-Ing Asrat Worku, AAIT

28. Construction of Classical Damping …
In the case of ξi = ξk= ξ , the damping ratios for all modes take the form
where
Note that the first term is the mass-proportional damping and the second one is the stiffness-proportional damping
If need be, the damping matrix in the geometric coordinate space can also be determined from
October 2015
Dr-Ing Asrat Worku, AAIT

29. Construction of Non-Classical Damping Matrices
In the case of non-classically damped systems, the damping matrix is constructed by combining proportional damping matrices established for different regions.
This process is shown schematically for the building frame given below
The case of soil-structure-interaction problems is more complicated, because the system is not amenable to ‘regionalization’.
October 2015
Dr-Ing Asrat Worku, AAIT

30. Dr-Ing Asrat Worku, AAIT
Example
Determine the displacement and story shears for the frame shown if it is subjected to a horizontal harmonic force of amplitude 60 kN and cyclic frequency of 4 Hz applied at the roof.
October 2015
Dr-Ing Asrat Worku, AAIT