Contenido

Una historia de tiempo completa dará la respuesta de una estructura a lo largo del tiempo durante y después de la aplicación de una carga. Para encontrar la historia de tiempo completa de la respuesta de una estructura debe resolver la ecuación de movimiento de la estructura.

Example

A simple single degree of freedom system "Degree of Freedom (Engineering)") (a mass, M, in a spring of stiffness k, for example) has the following equation of motion:.

where is the acceleration (the second derivative of the displacement) and x is the displacement.

If the load F(t) is a Heaviside step function (the sudden application of a constant load), the solution to the equation of motion is:.

where and the fundamental natural frequency, .

The static deflection of a single degree of freedom system is:.

then we can write, combining the previous formulas:.

This gives the (theoretical) time history of the structure due to a load F(t), where the assumption is made that there is no damping.

Although this is too simplistic to apply to a real structure, the Heaviside step function is a reasonable model for the application of many real loads, such as the sudden addition of a piece of furniture or removal of a strut to a freshly cast concrete floor. However, in reality, charges are never applied instantly: they accumulate over a period of time (this can be very short). This time is called rise time.

As the number of degrees of freedom of a structure increases, it becomes difficult to calculate the time history manually: actual structures are analyzed using nonlinear finite element analysis software.

Any real structure will dissipate energy (mainly through friction). This can be modeled by modifying the DAF.

where y is typically 2 to 10% depending on the type of construction:.

• - Bolted steel ~6%.

• - Reinforced concrete ~5%.

• - Welded steel ~2%.

• - Brick masonry ~10%.

Methods to increase damping..

One of the most widely used methods to increase damping is to attach a layer of material with a high damping coefficient, for example, rubber, to a vibrating structure.

Un análisis modal calcula los modos naturales de un sistema dado, pero no necesariamente su respuesta de tiempo completa a una entrada dada. La frecuencia natural de un sistema depende únicamente de la rigidez de la estructura y la masa que participa con la estructura (incluido el peso propio). No depende de la función de carga.

Es útil conocer las frecuencias modales de una estructura, ya que le permite asegurarse de que la frecuencia de cualquier carga periódica aplicada no coincidirá con una frecuencia modal y, por lo tanto, causará resonancia, lo que provocará grandes oscilaciones .

El método es:.

1. • Encuentra los modos naturales (la forma que adopta una estructura) y las frecuencias naturales.

2. • Calcular la respuesta de cada modo.

3. • Opcionalmente, superponer la respuesta de cada modo para encontrar la respuesta modal completa a una carga dada.

Energy method

It is possible to calculate the system modes manually by the energy method. For a given mode shape of a multi-degree-of-freedom system, an “equivalent” mass, stiffness, and applied force can be found for a single-degree-of-freedom system. For simple structures, the basic mode shapes can be found by inspection, but it is not a conservative method. The Rayleigh principle states:

"The frequency ω of an arbitrary mode of vibration, calculated by the energy method, is always greater than or equal to the fundamental frequency ω."

For an assumed mode shape, of a structural system of mass M; flexural stiffness, EI (Young's modulus, E, multiplied by the second moment of area, I); and applied force, F(x):.

then, as above:.

Modal response

The complete modal response to a given load F(x, t) is . The addition can be carried out by one of the common methods:.

• - Overlay complete time histories of each mode (takes time but is accurate).

• - Superimpose the maximum amplitudes of each mode (fast but conservative).

• - Superimpose the square root of the sum of the squares (good estimate for widely spaced frequencies, but unsafe for closely spaced frequencies).

To manually superimpose the individual modal responses, having calculated them by the energy method:.

Assuming that the rise time t(T = 2π/ω) is known, it is possible to read the DAF from a standard graph. The static displacement can be calculated with . The dynamic displacement for the chosen mode and the applied force can be found from:.

For real systems, there is often a mass that participates in the force function (such as the mass of the ground in an earthquake) and a mass that participates in the inertial effects (the mass of the structure itself, M). The modal share factor Γ is a comparison of these two masses. For a system with a single degree of freedom Γ = 1.

• - DYSSOLVE: Dynamic System Solver - A free, lightweight, encrypted software that can be used to solve basic structural dynamics problems.

• - Structural Dynamics and Vibrations Laboratory at McGill University.

• - Frame3DD open source 3D structural dynamics analysis program.

• - Frequency response function from modal parameters.

• - Structural dynamics tutorials and Matlab scripts.

• - AIAA Exploring Structural Dynamics ( http://www.exploringstructuraldynamics.org/ ) – Structural Dynamics in Aerospace Engineering: interactive demonstrations, videos and interviews with practicing engineers.

Contenido

Una historia de tiempo completa dará la respuesta de una estructura a lo largo del tiempo durante y después de la aplicación de una carga. Para encontrar la historia de tiempo completa de la respuesta de una estructura debe resolver la ecuación de movimiento de la estructura.

Example

A simple single degree of freedom system "Degree of Freedom (Engineering)") (a mass, M, in a spring of stiffness k, for example) has the following equation of motion:.

where is the acceleration (the second derivative of the displacement) and x is the displacement.

If the load F(t) is a Heaviside step function (the sudden application of a constant load), the solution to the equation of motion is:.

where and the fundamental natural frequency, .

The static deflection of a single degree of freedom system is:.

then we can write, combining the previous formulas:.

This gives the (theoretical) time history of the structure due to a load F(t), where the assumption is made that there is no damping.

Although this is too simplistic to apply to a real structure, the Heaviside step function is a reasonable model for the application of many real loads, such as the sudden addition of a piece of furniture or removal of a strut to a freshly cast concrete floor. However, in reality, charges are never applied instantly: they accumulate over a period of time (this can be very short). This time is called rise time.

As the number of degrees of freedom of a structure increases, it becomes difficult to calculate the time history manually: actual structures are analyzed using nonlinear finite element analysis software.

Any real structure will dissipate energy (mainly through friction). This can be modeled by modifying the DAF.

where y is typically 2 to 10% depending on the type of construction:.

• - Bolted steel ~6%.

• - Reinforced concrete ~5%.

• - Welded steel ~2%.

• - Brick masonry ~10%.

Methods to increase damping..

One of the most widely used methods to increase damping is to attach a layer of material with a high damping coefficient, for example, rubber, to a vibrating structure.

Un análisis modal calcula los modos naturales de un sistema dado, pero no necesariamente su respuesta de tiempo completa a una entrada dada. La frecuencia natural de un sistema depende únicamente de la rigidez de la estructura y la masa que participa con la estructura (incluido el peso propio). No depende de la función de carga.

Es útil conocer las frecuencias modales de una estructura, ya que le permite asegurarse de que la frecuencia de cualquier carga periódica aplicada no coincidirá con una frecuencia modal y, por lo tanto, causará resonancia, lo que provocará grandes oscilaciones .

El método es:.

1. • Encuentra los modos naturales (la forma que adopta una estructura) y las frecuencias naturales.

2. • Calcular la respuesta de cada modo.

3. • Opcionalmente, superponer la respuesta de cada modo para encontrar la respuesta modal completa a una carga dada.

Energy method

It is possible to calculate the system modes manually by the energy method. For a given mode shape of a multi-degree-of-freedom system, an “equivalent” mass, stiffness, and applied force can be found for a single-degree-of-freedom system. For simple structures, the basic mode shapes can be found by inspection, but it is not a conservative method. The Rayleigh principle states:

"The frequency ω of an arbitrary mode of vibration, calculated by the energy method, is always greater than or equal to the fundamental frequency ω."

For an assumed mode shape, of a structural system of mass M; flexural stiffness, EI (Young's modulus, E, multiplied by the second moment of area, I); and applied force, F(x):.

then, as above:.

Modal response

The complete modal response to a given load F(x, t) is . The addition can be carried out by one of the common methods:.

• - Overlay complete time histories of each mode (takes time but is accurate).

• - Superimpose the maximum amplitudes of each mode (fast but conservative).

• - Superimpose the square root of the sum of the squares (good estimate for widely spaced frequencies, but unsafe for closely spaced frequencies).

To manually superimpose the individual modal responses, having calculated them by the energy method:.

Assuming that the rise time t(T = 2π/ω) is known, it is possible to read the DAF from a standard graph. The static displacement can be calculated with . The dynamic displacement for the chosen mode and the applied force can be found from:.

For real systems, there is often a mass that participates in the force function (such as the mass of the ground in an earthquake) and a mass that participates in the inertial effects (the mass of the structure itself, M). The modal share factor Γ is a comparison of these two masses. For a system with a single degree of freedom Γ = 1.

• - DYSSOLVE: Dynamic System Solver - A free, lightweight, encrypted software that can be used to solve basic structural dynamics problems.

• - Structural Dynamics and Vibrations Laboratory at McGill University.

• - Frame3DD open source 3D structural dynamics analysis program.

• - Frequency response function from modal parameters.

• - Structural dynamics tutorials and Matlab scripts.

• - AIAA Exploring Structural Dynamics ( http://www.exploringstructuraldynamics.org/ ) – Structural Dynamics in Aerospace Engineering: interactive demonstrations, videos and interviews with practicing engineers.