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Structural Dynamics
Lecture 9
Outline of Lecture 9
� Continuous Systems.
� Introduction to Continuous Systems.
� Continuous Systems. Strings, Torsional Rods and Beams.
� Vibrations of Flexible Strings.
� Torsional Vibration of Rods.
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� Torsional Vibration of Rods.
� Bernoulli-Euler Beams.
� Undamped Eigenvibrations.
� Orthogonality Property of Eigenmodes.
� Forced Vibrations of a Bernoulli-Euler Beam Element.
Structural Dynamics
Lecture 9
� Continuous Systems
: Bending stiffnessaround -axis.: Mass per unitlength.
� Introduction to Continuous Systems
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length.
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� Discrete system (SDOF, MDOF) : Discrete distribution of mass or mass moment of inertia. Finite many dofs.
� Continuous system ( ) : Continuous distribution of mass. Infinite many dofs.
Elasticity may be continuously distributed in all cases. The discretization
approach illustrated in Figs. 1a and 1b are referred to as the lumped mass method.
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method.
Mass distributing functions:
� One-dimensional structures (strings, bars, beams):: Mass per unit length, [kg/m].
� Two-dimensional structures (slabs, plates, shells):
: Mass per unit area, [kg/m2].
� Three-dimensional continua:: Mass per unit volume (mass density), [kg/m3].
Structural Dynamics
Lecture 9
� Continuous Systems. Strings, Torsional Rods and Beams
� Vibrations of Flexible Strings
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� : Constant pre-stressing force of string.
� : Dynamic load per unit length in the -direction.
� : Constant mass per unit length.
� : Displacement in the -direction.
� : Rotation angle of cross-section in the -direction.
� : Length of string.
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Lecture 9
D’Alembert’s principle:
The inertial load per unit length is added to the external load on a differential string element of the length .
“Static” equilibrium in the -direction of the free string element (mass particle):
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Boundary conditions:
Initial conditions:
must be known for all differential string elements (mass
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must be known for all differential string elements (mass particles).
Eigenvibrations ( ):
(5) is known as the wave equation. is the phase velocity.
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Lecture 9
General solution to (5) :
� : Wave propagating in the positive -direction (same displacement at positions and times, where ).
� : Wave propagating in the negative -direction.
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� : Wave propagating in the negative -direction.
(6) is due to d’Alembert. The relation follows from the following identities:
where denotes the 2nd derivative of with respect to the
argument .
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Lecture 9
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The shape of the wave is preserved during the wave propagation. This is referred to as non-dispersive wave propagation.
All harmonic components in the Fourier series expansion of and travel with same velocity , i.e. does not depend on the frequency.
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Solution by separation of variables:
Product solutions to the homogeneous wave equation are searched on the form:
Insertion in Eq. (5):
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The left-hand side of Eq. (9) is a function of , and the right-hand side is a function of . This can only be true, if the left- and right-hand sides are equal to the same constant, which is chosen as . Hence, product solutions of the type (8) are only solutions to (5), if the following equations are fulfilled by the functions and :
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Lecture 9
The solutions to (10) and (11) are given as:
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Boundary conditions:
The displacement vanishes at the end of the string at all times:
Insertion of (14) in (12):
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Lecture 9
The 2nd equation of (15) may be fulfilled for . However, with this leads to , and hence to the trivial solution
. Non-trivial solutions implies that , leading to the condition:
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� : Angular eigenfrequency of the string. (i.e. the frequency is increased one octave, when the length of the string is halved).
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The modal coordinate differential equations of a MDOF system are given as, cf. Lecture 5, Eq. (70):
Then, Eq. (11) may be interpreted as the equation for undamped eigenvibrations of a modal coordinate:
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This motivates the following designations.
� Time function in the separation method : Modal coordinate.
� Spatial function in the separation method : Eigenmode function.
Infinite many modal coordinates exist for a continuous system.
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Superposition principle:
� : Determined from the initial value functions , .
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� Example 1 : Eigenvibrations of a flexible string
Let the initial conditions be given as:
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From Eq. (19):
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The 1st equation in Eq. (21) is multiplied with , followed by an integration over the interval :
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Above, the following identities have been used:
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The final solution becomes:
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� Torsional Vibration of Rods
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� : Torsional moment, [Nm].
� : Torsional moment load per unit length, [Nm/m].
� : Rotational angle in the -direction of a cross-section, [rad].
� : Mass moment of inertia per unit length, [kgm2/m].
� : Mass density, [kg/m3].
� : Torsional constant of a circular cylindrical bar, [m4].
� : Shear modulus, [N/m2].
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D’Alembert’s principle:
Constitutive relation for St. Venant torsion:
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(27) is a wave equation with the phase velocity .
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� Example 2 : Torsional undamped eigenvibrations of a fixed-free circular cylindrical bar
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Determine the undamped eigenfrequencies and eigenmodes of the clamped bar with the length sketched on Fig. 5 are determined.
Boundary conditions:
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The eigenvalue problem for the spatial function becomes, cf. Eq. (10):
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Insertion of into the solution given by Eq. (12) implies that. Hence:
. Then, undamped angular
eigenfrequencies are given as:
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eigenfrequencies are given as:
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� Bernoulli-Euler Beams
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� : Dynamic displacement in the -direction.
� : Dynamic load per unit length in the -direction. No dynamic load in the -direction.
� : Mass per unit length.
� : Static axial force.
� : Shear force from dynamic loads.
� : Bending moment from dynamic loads.
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� : Bending moment from dynamic loads.
� : Bending stiffness around the -axis.
� : Length of beam element.
Static equilibrium state:
Static loads in the - and -directions produce a static equilibrium configuration of the beam (drawn with a dashed signature in Fig. 6). Only the axial force is shown. Since there are no dynamic loads in the -direction, is unchanged during dynamic vibrations.
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Dynamically deformed state:
Force equilibrium in the -direction:
Moment equilibrium in the -direction around the bending centre at the
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Moment equilibrium in the -direction around the bending centre at the right-end section:
From Eqs. (33) and (34):
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Constitutive equation:
D’Alembert’s principle:
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� : External dynamic load per unit length.
� : Linear viscous damping coefficient per unit length.
From Eqs. (35), (36), (37):
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(38) must be solved with proper initial values at for all particles in the interval , and with boundary conditions at and for all times .
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For the beam in Fig. 7 the following quantities at the end-section are introduced:
� : Point masses.
� : Damper constant of linear viscous dampers.
� : Stiffness of linear elastic springs.
� : Mass moment of inertia of distributed masses, [ ].
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� : Mass moment of inertia of distributed masses, [ ].
� : Damper constants of linear viscous rotational dampers, [ ].
� : Stiffness of linear elastic rotational springs, [ ].
For each differential mass particle identified by the abscissa , an initial displacement and an initial velocitymust be formulated as a straightforward generalization of the discrete case.
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The boundary conditions are classified as either geometric or mechanical boundary conditions. At each end-section exactly 2 boundary conditions (geometric or mechanical) are specified.
Geometric boundary conditions are specified, whenever the end-section displacement or end-sections rotations are prescribed. In what follows only homogeneous geometric boundary conditions are considered.
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The following boundary and initial value problem may be stated for the beam shown in Fig. 7:
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Differential equation:
Initial values:
Geometric boundary conditions:
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Mechanical boundary conditions:
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Derivation of mechanical boundary conditions due to concentrated masses, dampers and springs:
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Mechanical boundary conditions specify that the bending moments, and the shear forces , immediately
to the right and the left of the end-sections must balance the inertial forces and the d’Alembert moments from the distributed masses, and the forces and moments in the concentrated dampers and springs, resulting in the following equations of equilibrium, see Fig. 8.
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The mechanical boundary conditions in (39) are obtained by insertion of (34) and (36) in (40).
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� Example 3: Boundary conditions for beam elements with constant cross section
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Special case , , and constant:
Harmonic wave propagating in the positive -direction:
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� : Amplitude, [ ].
� : Angular frequency, [ ].
� : Wave number, [ ].
: Phase velocity, [ ].
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Insertion of (42) into Eq. (41):
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Bending waves are dispersive. High-frequency components are moving faster than low-frequency components. This means that a displacement disturbancy is distorted during propagation in an infinite long Bernoulli-Euler beam.
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� Undamped Eigenvibrations
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Undamped vibrations:
Eigenvibrations:
Then, (39) attains the form:
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Differential equation:
Geometric boundary conditions:
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Mechanical boundary conditions:
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Guided by the experience with MDOF systems it can be anticipated that all mass particles are performing harmonic motions in phase during undamped eigenvibrations. Consequently, the solution of (48) is searched on the form, cf. Lecture 4, Eq. (41):
is the real amplitude of the mass particle , identified by the
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is the real amplitude of the mass particle , identified by the abscissa in the statical equilibrium state, and is the angular eigenfrequency. The phase can be selected arbitrarily. and are solutions to the following linear eigenvalue problem, obtained by insertion of (49) into (48):
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Differential equation:
Geometric boundary conditions:
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Mechanical boundary conditions:
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Homogeneous cross-section:
Solutions are determined to (50) for the special case of homogeneous cross-sections (constant value of , and ). The differential equation reduces to:
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The complete solution of (51) can be written as:
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, , , are integration constants, and and are the positive roots of the quadratic equations:
Especially, if , (52) and (53) reduces to:
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Especially, if , (52) and (53) reduces to:
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(52) or (54) are inserted into the 4 relevant boundary conditions in (50). Then, 4 homogeneous linear equations are obtained for the determination of the coefficients , , , , which can be formulated in the following way:
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and are functions of the angular frequency . Then,is a known function of . (56) always has the
solution , which implies the trivial solution. The necessary condition for non-trivial solutions is:
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Solutions to (57) determines non-trivial solutions to (56), and hence non-trivial solution to the amplitude function as given by (52) or (54).
� : Undamped angular eigenfrequency.
� : Eigenmode function.
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� Example 4: Boundary conditions of the eigenmode function for beam elements with constant cross-section
is inserted into the boundary conditions for , see Fig. 10.
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� Example 5 : Eigenfrequencies and eigenmodes of simply supported beam with a compressive axial force
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(52) is inserted into the boundary conditions shown on Fig. 11a:
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(58) has the non-trivial solution:
From (52), (53):
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In (62) the compressive axial force has been introduced. given by (63) signifies the angular eigenfrequency for . is the classical Euler buckling load.
For (62) provides:
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(65) provides a method for estimation by so-called non-destructive testing. Values of are measured for known values of (marked by a
on Fig. 13. The least-square fit determines as the intersection with the abscissa axis.
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� Example 6 : Eigenfrequencies and eigenmode functions of a cantilever beam
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The special case with no axial force, , is considered.
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(54) is inserted into the boundary conditions at as shown on Fig. 11b:
(54) may then be reduced to:
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(54) may then be reduced to:
(67) is inserted into the mechanical boundary conditions at :
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From (55):
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From the first equation of (68):
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� Example 7 : Eigenfrequencies and eigenmode functions of a free-free beam
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The special case with no axial force, , is considered. The eigenvalue problem follows from (51), (54), (55) and Fig. 11c:
(72) is fulfilled for the rigid body modes and
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(72) is fulfilled for the rigid body modes and for .
Elastic modes are given by Eq. (54) with . Insertion of (54) in provides:
which reduces (54) to:
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Insertion of (74) into the boundary conditions provides:
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provides
Next, follows from Eq. (55). The eigenmode functions become:
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� Orthogonal Property of Eigenmodes
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Theorem: The eigenmode functions and to the eigenvalue problem (50) belonging to different circular eigenfrequencies and fulfill the orthogonality conditions:
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where is the modal mass in the th eigenvibration defined by:
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(78) and (79) are proved in much the same way as for a discrete MDOF system, cf. Lecture 5, Eqs. (55-58). The eigenvalue problem (50) is formulated for and . The differential equations forand are multiplied by and , respectively, followed by integrations over the interval . Next, integration by parts is performed on the stiffness term to obtain integrals symmetric inand , and the mechanical boundary conditions in (50) are applied in the boundary terms. The orthogonally conditions then follows upon withdrawing of the equations.
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Lecture 9
� Forced Vibrations of a Bernoulli-Euler Beam Element
Guided by the superposition of separated solutions (19) the solution of the boundary and initial value problem (39) is searched on the form:
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As precious the coefficients are referred to as the undamped modal coordinates. These are obtained as solutions to the following uncoupled ordinary differential equations:
where:
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Lecture 9
� : th modal load.
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� : th modal load.
� : th modal damping ratio.
(82) is proved by insertion of (81) into the partial differential equation of (39). Next, the equation is multiplied by followed by an integration over the interval , and integration by part is performed on the stiffness terms. Use of the orthogonality properties as given by (78) and (79) then provides the result, assuming that similar orthogonalityproperties apply to the damping terms (modal decoupling).
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Lecture 9
The modal equations (82) have exactly the same form as the modal equations of motion for a discrete MDOF system, cf. Lecture 5, Eq. (70). The only difference is that (82) refers to a continuous system, and consequently contains infinite many modal coordinates.
Although derived for a beam element, modal equations of exactly the same form can be derived for any continuous system of one, two or three
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same form can be derived for any continuous system of one, two or three dimension. One may say that the modal coordinate differential equations are structure independent. The specific dynamic system is only displayed indirectly via the modal parameters , , and .
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Lecture 9
� Example 7: Simply supported homogeneous beam with a moving load
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The bending stiffness , the mass per unit length and the distributed damping constant are constant along the beam.
At the time a vehicle with the constant velocity and the weightis entering the bridge which is assumed to be at rest. The inertial force
from the vertical motion of the vehicle is ignored, so is equal to the constant reaction from the vehicle on the bridge.
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constant reaction from the vehicle on the bridge.
The eigenmode function and the undamped angular eigenfrequencies are given as, cf. (61), (62):
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Lecture 9
The modal masses become, cf. (80):
The modal damping ratios become, cf. (84):
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The dynamic load can formally be written as:
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Upon insertion of (89) into (83) provides the following result for the modal loads:
Because the bridge starts at rest the initial values related to (82) becomes:
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Because the bridge starts at rest the initial values related to (82) becomes:
Then, the solution of (82) reads, cf. Lecture 3, Eq. (13):
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Lecture 9
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The integral in (92) can be evaluated analytically.
At least 40 terms need to be retained in the series solutions (81) to give a sufficiently accurate solution for the displacement mode. Even more modes need to be included, if the bending moment or the shear force is to be calculated.
Structural Dynamics
Lecture 9
Summary of Lecture 9
� Continuous System. This involves a continuous mass distribution. Elastic parameters are also continuously distributed.
� Vibrating string. Separation method. Non-dispersive wave propagation.
� Torsional vibration of rods. Same wave equation as for a vibrating string.
� Bernoulli-Euler beams. Non-dispersive wave propagation:
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� Bernoulli-Euler beams. Non-dispersive wave propagation:
. Eigenfrequencies of simply supported beams, cantilever
beams and free-free beams.
