Outline of 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. Analytical Dynamics. Properties of, and. Undamped Eigenvibrations. Generalized Eigenvalue Problem. 1
Multi-Degree-of-Freedom Systems Formulation of Equations of Motion Newton s 2 nd Law Applied to Free Masses 2
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. : 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 2 nd law of motion is applied to each of the free masses: 3
Equivalent matrix formulation: : Displacement vector. : Dynamic load vector. : Mass matrix. : Stiffness matrix. : Damping matrix. 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. Vector formulation of the initial conditions: (2) is solved with the initial conditions (4). 5
Example 1 : Shear building exposed to a horizontal earthquake 6
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 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. 7
Basic Equations of Motion for Forced Vibrations of Linear Viscous Damped Systems : 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. 8
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: : Flexibility coefficient for displacement or rotational degree of freedom due to a unit force or unit moment in degree of freedom. 9
Maxwell-Betti s reciprocal theorem implies the symmetry property: (7) may be written in the following matrix form: 10
: Mass matrix. : Flexibility matrix. Due to (8), becomes a symmetric matrix. (9) is multiplied by providing: : Stiffness matrix. 11
Power balance: Scalar multiplication of (13) with : : Kinetic energy. : Strain energy. : Supplied power. : Dissipated power. 12
Dissipative damping force vector: Constitutive equation for damping forces: Linear viscous damping model: A dissipative linear viscous damping model vector requires that: 13
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
Analytical Dynamics The method is explained with reference to the system defined on Fig. 1. Conservative dynamic loads:. Kinetic energy: Potential energy (strain energy plus potential energy of and ): 15
Non-conservative forces: Lagrange function (Lagrangian): Notice that the kinetic energy in a more general case may depend on both and. 16
Lagrange equations of motion: (24) holds for linear as well for non-linear MDOF systems. Proof by verification of Eq. (2):,. 17
Then, the identity of (2) and (29) follows from (27) and (30). Vector formulation of Eq. (29): Convention: : Row vector. : Column vector. 18
: 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. Shear force between th and th story beam: The shear forces and are applied to the i th story mass in the opposite direction and the direction of, respectively. Next Newton s 2 nd law is applied to all free story masses: 19
20
Matrix formulation: where: 21
Example 2 : Torsional eigenvibrations of multi-degree-of-freedom system. : Rotation of flywheel, [rad]. : Mass moment of inertia of flywheels, [kgm 2 ]. : St. Venant torsional stiffness of shaft, [Nm 2 ] : Distance between flywheels. 22
Lagrange equations of motion: Kinetic energy: Potential energy: where: 23
At first the following partial derivative of the Lagrangian are evaluated ( ): 24
Then: 25
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
Properties of, and Strain energy: Non-negative scalar. 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. 27
Rigid body motions fulfill: Existence of non-trivial solutions to (49) requires that: Kinetic energy: Positive scalar. is a symmetric positive definite matrix, so exist (i.e. ). 28
Linear viscous damping forces: Mechanical energy must be dissipated during any motion with non-zero velocity vector : Hence, the damping matrix must be positive definite. The damping matrix may be decomposed into a symmetric part and an anti-symmetric part : where: 29
Since,, it follows that: It follows that only the symmetric part of dissipates energy. Antisymmetric viscous damping matrices occur, when the equations of motion 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. 30
Undamped Eigenvibrations Undamped vibrations : Eigenvibrations : From (2) and (4): 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. 31
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: Insertion of (59) and (60) into (58): 32
(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
: Invariants.,. Roots: : i th undamped angular eigenfrequency. For a solution exists to (61). : i th eigenmode. The phase angle in (59) is undetermined, and may be chosen arbitrarily. Any value of produces a useful solution. 34
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
Let: Characteristic equation: 36
Determination of : 37
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
Hence, the following eigenmodes are obtained: The eigenmodes (70) have been illustrated in Fig. 7. 39
Box 1 : Determination of the j th eigenmode It is assumed that the j th eigenvalue equation is fulfilled: is known, so the characteristic Then, is determined as indicated below, so the following linear homogeneous equations are fulfilled: 1. Choose an arbitrary component of to, e.g. the first component, so has the structure: 40
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 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. 41
Example 4 : Matematical double pendulum The equations of motion for the indicated double pendulum is formulated using d Alembert s principle. 42
Moment equilibrium around the support points provides: Next, the undamped eigenfrequency and eigenmodes will be determined for small vibrations around the equilibrium state. Linearization around :, 43
Eigenvalues and eigenmodes become: (This follows also immediately from symmetry!) 44
Example 5 : Eigenfrequencies and mode shapes for shear building The generalized eigenvalue value problem reads, cf. Eq. (34): where is given by (36). (78) may be restated into the following component equations: 45
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: (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 : 46
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: where (82) has been used to eliminate. (83) has the solutions: 47
The first solution implies that, and hence provides trivial solutions. The second solution determines the non-trivial solutions to the problem. Then, the components of the th eigenmode become: The undamped angular eigenfrequencies follow from (78), (82) and (84): 48
Summary of Multi-Degrees-of-Freedom Systems. Formulation of Equations of Motion. Newton s 2 nd law of motion for free masses. D Alembert s principle. Basic Equations of Motion for Forced Vibrations of Linear Viscous Damped Systems. Analytical dynamics. Properties of, and. Undamped Eigenvibrations. Generalized eigenvalue problem. Eigenvalues, eigenmodes. Analytical solution for plane shear frame. 49