Structural Dynamics Lecture 4 Outline of Lecture 4 Multi-Degree-of-Freedom Systems Formulation of Equations of Motions. Newton’s 2 nd Law Applied to Free Masses. D’Alembert’s Principle. Basic Equations of Motion for Forced Vibrations of Linear Viscous Damped Systems. 1 Basic Equations of Motion for Forced Vibrations of Linear Viscous Damped Systems. Analytical Dynamics. Properties of , and . Undamped Eigenvibrations. Generalized Eigenvalue Problem.
Transcript
Structural Dynamics
Lecture 4
Outline of Lecture 4
� Multi-Degree-of-Freedom Systems
� Formulation of Equations of Motions.
� Newton’s 2nd Law Applied to Free Masses.
� D’Alembert’s Principle.
� Basic Equations of Motion for Forced Vibrations of Linear Viscous Damped Systems.
1
� Basic Equations of Motion for Forced Vibrations of Linear Viscous Damped Systems.
� Analytical Dynamics.
� Properties of , and .
� Undamped Eigenvibrations.
� Generalized Eigenvalue Problem.
Structural Dynamics
Lecture 4
� Multi-Degree-of-Freedom Systems
� Formulation of Equations of Motion
� Newton’s 2nd Law Applied to Free Masses
2
Structural Dynamics
Lecture 4
MDOF Systems: coordinates used for the description of the system.
Fig. 1 shows two “roller scates” with the masses and , moving in the same direction on a smooth horizontal plane.
� : Degrees of freedom.
3
: Degrees of freedom.
� : Dynamic loads on masses.
and are measured from the statical equilibrium position of the masses. Springs and dampers are assumed to be linear. Then, any static force disappears from the dynamic equations of motion.
Masses are cut free from the springs and dampers. External dynamic loads and internal spring and damper forces are applied to the free masses with the signs shown on Fig. 1. Newton’s 2nd law of motion is applied to each of the free masses:
Structural Dynamics
Lecture 4
Equivalent matrix formulation:
4
� : Displacement vector.
� : Dynamic load vector.
� : Mass matrix.
� : Stiffness matrix.
� : Damping matrix.
Structural Dynamics
Lecture 4
(2) is the general matrix format for the equations of motion of a linear MDOF system. may include rotational components. If so, the conjugated components of are dynamic external moments.
Initial conditions:
� : Initial displacements.
� : Initial velocities.
5
� : Initial velocities.
Vector formulation of the initial conditions:
(2) is solved with the initial conditions (4).
Structural Dynamics
Lecture 4
� Example 1 : Shear building exposed to a horizontal earthquake
6
Structural Dynamics
Lecture 4
� D’Alembert’s Principle
(2) is written as an equivalent static equilibrium equation:
� : Inertial load vector.
The components of are applied on the free masses in the direction of
7
The components of are applied on the free masses in the direction of
the selected degrees of freedom along with the components of the dynamic load vector and the internal force vector .
The application of d’Alembert’s principle at the formulation of dynamic equations of motion will be extensively used in what follows.
Next, the equations of motion may be formulated by any of the two analysis methods of static structural mechanics:
� Force method.
� Deformation method.
Structural Dynamics
Lecture 4
� Basic Equations of Motion for Forced Vibrations of Linear Viscous Damped Systems
8
� : Translational or rotational degrees of freedom.
� : Mass or mass moment of inertia.
� : Dynamic load or moment.
� : Damping force or damping moment.
� : Inertial load or inertial moment.
Structural Dynamics
Lecture 4
D’Alembert’s principle is used, and the structure is analyzed statically with the loads , and .
Force method:
The displacement or rotation in all degrees of freedom are given as:
9
� : Flexibility coefficient for displacement or rotational degree of
freedom due to a unit force or unit moment in degree of
freedom .
Structural Dynamics
Lecture 4
Maxwell-Betti’s reciprocal theorem implies the symmetry property:
(7) may be written in the following matrix form:
10
Structural Dynamics
Lecture 4
� : Mass matrix.
� : Flexibility matrix.
11
� : Flexibility matrix.
Due to (8), becomes a symmetric matrix. (9) is multiplied by providing:
� : Stiffness matrix.
Structural Dynamics
Lecture 4
Power balance:
Scalar multiplication of (13) with :
12
: Kinetic energy.
: Strain energy.
: Supplied power.
: Dissipated power.
Structural Dynamics
Lecture 4
Dissipative damping force vector:
Constitutive equation for damping forces:
13
Linear viscous damping model:
A dissipative linear viscous damping model vector requires that:
Structural Dynamics
Lecture 4
Hence, the damping matrix must be positive definite.
From (13) and (22) follows the general equation of motion for forced vibrations of a linear viscous damped system of degrees of freedom:
14
Structural Dynamics
Lecture 4
� Analytical Dynamics
The method is explained with reference to the system defined on Fig. 1.
Conservative dynamic loads: .
Kinetic energy:
15
Potential energy (strain energy plus potential energy of and ):
Structural Dynamics
Lecture 4
Non-conservative forces:
16
Lagrange function (Lagrangian):
Notice that the kinetic energy in a more general case may depend on both and .
Structural Dynamics
Lecture 4
Lagrange equations of motion:
(24) holds for linear as well for non-linear MDOF systems.
Proof by verification of Eq. (2): , .
17
Proof by verification of Eq. (2): , .
Structural Dynamics
Lecture 4
Then, the identity of (2) and (29) follows from (27) and (30).
Vector formulation of Eq. (29):
18
Convention:
� : Row vector.
� : Column vector.
Structural Dynamics
Lecture 4
� : Story mass of infinite stiff story beams.
� : Shear stiffness of linear elastic story columns.
� : Damping constant in story columns.
� : Horizontal displacement of support.
� : Displacement of the th story beam relative to the ground surface.
19
surface.
Shear force between th and th story beam:
The shear forces and are applied to the ith story mass in the opposite direction and the direction of , respectively. Next Newton’s 2nd law is applied to all free story masses:
Structural Dynamics
Lecture 4
20
Structural Dynamics
Lecture 4
Matrix formulation:
where:
21
Structural Dynamics
Lecture 4
� Example 2 : Torsional eigenvibrations of multi-degree-of-freedom system.
22
� : Rotation of flywheel , [rad].� : Mass moment of inertia of flywheels, [kgm2]. � : St. Venant torsional stiffness of shaft, [Nm2]� : Distance between flywheels.
Structural Dynamics
Lecture 4
Lagrange equations of motion:
Kinetic energy:
Potential energy:
23
Potential energy:
where:
Structural Dynamics
Lecture 4
At first the following partial derivative of the Lagrangianare evaluated ( ):
24
Structural Dynamics
Lecture 4
Then:
25
Structural Dynamics
Lecture 4
The mass- and stiffness matrices have the same structure as those of the shear frame, cf. Eqs. (34) and (36). In this case the parameter is given as:
26
Structural Dynamics
Lecture 4
� Properties of , and
Strain energy: Non-negative scalar.
27
must be symmetric.
Supported system:
is positive definite.
Unsupported system:
In this case so-called rigid body motions exist, which are related with zero strain energy. is positive semi-definite.
Structural Dynamics
Lecture 4
Rigid body motions fulfill:
Existence of non-trivial solutions to (49) requires that:
Kinetic energy: Positive scalar.
28
is a symmetric positive definite matrix, so exist (i.e. ).
Structural Dynamics
Lecture 4
Linear viscous damping forces:
Mechanical energy must be dissipated during any motion with non-zero velocity vector :
29
Hence, the damping matrix must be positive definite. The damping matrix may be decomposed into a symmetric part and an anti-symmetric part
:
where:
Structural Dynamics
Lecture 4
Since, , it follows that:
It follows that only the symmetric part of dissipates energy. Anti-symmetric viscous damping matrices occur, when the equations of motion are formulated in a rotating coordinate system, where they are denoted
30
are formulated in a rotating coordinate system, where they are denoted gyroscopic damping matrices.
The eigenvalues of are real, because is a symmetric matrix. is positive definite, if all eigenvalues are positive.
Structural Dynamics
Lecture 4
� Undamped Eigenvibrations
� Undamped vibrations :
� Eigenvibrations :
From (2) and (4):
31
� Generalized Eigenvalue Problem:
Linear independent solutions to the homogeneous matrix differential equation (58) are searched.
� : Trivial solution. Only solutions are of interest.
Guess: Solutions are harmonic motions with the amplitude vector , the angular frequency and the common phase angle .
Structural Dynamics
Lecture 4
Next, the amplitude vector , the angular frequency and the phase angle are determined, so the homogeneous matrix differential equation (58) is fulfilled.
Velocity and acceleration vectors:
32
Velocity and acceleration vectors:
Insertion of (59) and (60) into (58):
Structural Dynamics
Lecture 4
(61) is called a generalized eigenvalue problem (GEVP). If , Eq. (61) has the solution:
, i.e. the trivial solution. Hence, non-trivial solutions requires that is singular. This implies that must fulfill the following characteristic equation:
33
following characteristic equation:
Structural Dynamics
Lecture 4
� : Invariants. , .
Roots:
� : ith undamped angular eigenfrequency.
34
� : ith undamped angular eigenfrequency.
For a solution exists to (61).
� : ith eigenmode.
The phase angle in (59) is undetermined, and may be chosen arbitrarily.Any value of produces a useful solution.
Structural Dynamics
Lecture 4
If is a solution to (61), then so is where is an arbitrary constant. Hence, the components of can only be determined within a common arbitrary factor. Then, we may choose one component of equal to . The remaining components are obtained from any of the homogeneous linear equations in (61) ( choices. Any of these choices give the same result).
� Example 3 : GEVP for 2DOF system
35
� Example 3 : GEVP for 2DOF system
Structural Dynamics
Lecture 4
Let:
Characteristic equation:
36
Characteristic equation:
Structural Dynamics
Lecture 4
Determination of :
37
Determination of :
Structural Dynamics
Lecture 4
The two solutions for given by (67) are identical, if:
This is indeed the case, if fulfils the characteristic equation, i.e. if is an eigenvalue.
38
is an eigenvalue.
Structural Dynamics
Lecture 4
Hence, the following eigenmodes are obtained:
The eigenmodes (70) have been illustrated in Fig. 7.
39
Structural Dynamics
Lecture 4
Box 1 : Determination of the jth eigenmode
It is assumed that the jth eigenvalue is known, so the characteristic equation is fulfilled:
Then, is determined as indicated below, so the following linear homogeneous equations are fulfilled:
40
homogeneous equations are fulfilled:
1. Choose an arbitrary component of to , e.g. the first component , so has the structure:
Structural Dynamics
Lecture 4
2. Choose any linear equations among the linear equations (72), and determine from these.
There are possible ways of choosing of linear equations from (72). All choices provides the same solution, if fulfils the characteristic equation (71).
Notice that the normalization used in (73) cannot be used, if . In
41
Notice that the normalization used in (73) cannot be used, if . In this case any of the possible systems of linear equations for the determination of becomes singular. Instead, one of the other components are set to .
Structural Dynamics
Lecture 4
� Example 4 : Matematical double pendulum
42
The equations of motion for the indicated double pendulum is formulated using d’Alembert’s principle.
Structural Dynamics
Lecture 4
Moment equilibrium around the support points provides:
Next, the undamped eigenfrequency and eigenmodes will be determined for small vibrations around the equilibrium state .
43
Linearization around : ,
Structural Dynamics
Lecture 4
Eigenvalues and eigenmodes become:
(This follows also immediately from symmetry!)
44
from symmetry!)
Structural Dynamics
Lecture 4
� Example 5 : Eigenfrequencies and mode shapes for shear building
The generalized eigenvalue value problem reads, cf. Eq. (34):
45
where is given by (36). (78) may be restated into the following
component equations:
Structural Dynamics
Lecture 4
A solution to (79) is searched on the form:
and are next determined, so all equations in (79) are fulfilled. Insertion into the equations for provides:
46
(81) is fulfilled for , which corresponds to the trivial solution. Hence, in order to fulfill (81) for non-trivial solutions
, the following relation must hold between and :
Structural Dynamics
Lecture 4
Insertion of (80) into the first equation in (79) again provides the relation (82) between and . Finally, insertion of (80) into the last equation of (79) provides:
47
where (82) has been used to eliminate . (83) has the solutions:
Structural Dynamics
Lecture 4
The first solution implies that , and hence provides trivial solutions. The second solution determines the non-trivialsolutions to the problem. Then, the components of the th eigenmodebecome:
48
The undamped angular eigenfrequencies follow from (78), (82) and (84):
Structural Dynamics
Lecture 4
Summary of Lecture 4
� Multi-Degrees-of-Freedom Systems.
� Formulation of Equations of Motion.
� Newton’s 2nd law of motion for free masses.
� D’Alembert’s principle.
49
� D’Alembert’s principle.
� Basic Equations of Motion for Forced Vibrations of Linear Viscous Damped Systems.
