frequency_response

private function frf_modal_prop_damp(mass, stiff, alpha, beta, freq, frc, modes, modeshapes, args) result(rst)

Computes the frequency response functions for a multi-degree-of-freedom system that uses proportional damping such that the damping matrix $C$ is related to the stiffness an mass matrices by proportional damping coefficients $\alpha$ and $\beta$ by $C = \alpha M + \beta K$ .

Arguments

Type	Intent	Optional	Attributes		Name
real(kind=real64),	intent(in),		dimension(:,:)	::	mass	The N-by-N mass matrix for the system. This matrix must be symmetric.
real(kind=real64),	intent(in),		dimension(:,:)	::	stiff	The N-by-N stiffness matrix for the system. This matrix must be symmetric.
real(kind=real64),	intent(in)			::	alpha	The mass damping factor, $\alpha$ .
real(kind=real64),	intent(in)			::	beta	The stiffness damping factor, $\beta$ .
real(kind=real64),	intent(in),		dimension(:)	::	freq	An M-element array of frequency values at which to evaluate the frequency response functions, in units of rad/s.
procedure(modal_excite),	intent(in),		pointer	::	frc	A pointer to a routine used to compute the modal forcing function.
real(kind=real64),	intent(out),	optional,	allocatable, dimension(:)	::	modes	An optional N-element allocatable array that, if supplied, will be used to retrieve the modal frequencies, in units of rad/s.
real(kind=real64),	intent(out),	optional,	allocatable, dimension(:,:)	::	modeshapes	An optional N-by-N allocatable matrix that, if supplied, will be used to retrieve the N mode shapes with each vector occupying its own column.
class(*),	intent(inout),	optional		::	args	An optional argument that can be used to communicate with the outside world.

Return Value type(frf)

The resulting frequency responses.

private function frf_modal_prop_damp_sparse(mass, stiff, alpha, beta, nmodes, freq, frc, modes, modeshapes, args) result(rst)

Computes a modal-truncated frequency response for a system with proportional damping using CSR sparse mass and stiffness matrices. The damping matrix is defined by $C=\alpha M+\beta K$ .

Arguments

Type	Intent	Optional	Attributes		Name
type(csr_matrix),	intent(in)			::	mass	The N-by-N symmetric positive-definite mass matrix.
type(csr_matrix),	intent(in)			::	stiff	The N-by-N symmetric stiffness matrix.
real(kind=real64),	intent(in)			::	alpha	The mass damping factor, $\alpha$ .
real(kind=real64),	intent(in)			::	beta	The stiffness damping factor, $\beta$ .
integer(kind=int32),	intent(in)			::	nmodes	The number of lowest-frequency modes to retain. This value must be greater than zero and less than N.
real(kind=real64),	intent(in),		dimension(:)	::	freq	An M-element array of frequency values in units of rad/s.
procedure(modal_excite),	intent(in),		pointer	::	frc	A pointer to the physical forcing function.
real(kind=real64),	intent(out),	optional,	allocatable, dimension(:)	::	modes	An optional NMODES-element array containing the retained modal frequencies in units of rad/s.
real(kind=real64),	intent(out),	optional,	allocatable, dimension(:,:)	::	modeshapes	An optional N-by-NMODES matrix containing the mass-normalized retained mode shapes.
class(*),	intent(inout),	optional		::	args	An optional argument passed to the forcing function.

Return Value type(frf)

The modal-truncated frequency responses.

private function frf_modal_prop_damp_2(mass, stiff, alpha, beta, nfreq, freq1, freq2, frc, modes, modeshapes, args) result(rst)

Computes the frequency response functions for a multi-degree-of-freedom system that uses proportional damping such that the damping matrix $C$ is related to the stiffness an mass matrices by proportional damping coefficients $\alpha$ and In modal coordinates, each mode has denominator $s^2+2\zeta_i\omega_i s+\omega_i^2,$ and the physical response is reconstructed from the mode shapes. $\beta$ by $C = \alpha M + \beta K$ .

Arguments

Type	Intent	Optional	Attributes		Name
real(kind=real64),	intent(in),		dimension(:,:)	::	mass	The N-by-N mass matrix for the system. This matrix must be symmetric.
real(kind=real64),	intent(in),		dimension(:,:)	::	stiff	The N-by-N stiffness matrix for the system. This matrix must be symmetric.
real(kind=real64),	intent(in)			::	alpha	The mass damping factor, $\alpha$ .
real(kind=real64),	intent(in)			::	beta	The stiffness damping factor, $\beta$ .
integer(kind=int32),	intent(in)			::	nfreq	The number of frequency values to analyze. This value must be at least 2.
real(kind=real64),	intent(in)			::	freq1	The starting frequency, in units of rad/s.
real(kind=real64),	intent(in)			::	freq2	The ending frequency, in units of rad/s.
procedure(modal_excite),	intent(in),		pointer	::	frc	A pointer to a routine used to compute the modal forcing function.
real(kind=real64),	intent(out),	optional,	allocatable, dimension(:)	::	modes	An optional N-element allocatable array that, if supplied, will be used to retrieve the modal frequencies, in units of rad/s.
real(kind=real64),	intent(out),	optional,	allocatable, dimension(:,:)	::	modeshapes	An optional N-by-N allocatable matrix that, if supplied, will be used to retrieve the N mode shapes with each vector occupying its own column.
class(*),	intent(inout),	optional		::	args	An optional argument that can be used to communicate with the outside world.

Return Value type(frf)

The resulting frequency responses.

private function frf_modal_prop_damp_sparse_2(mass, stiff, alpha, beta, nmodes, nfreq, freq1, freq2, frc, modes, modeshapes, args) result(rst)

Computes a modal-truncated frequency response for a system with proportional damping using CSR sparse mass and stiffness matrices. The damping matrix is defined by $C=\alpha M+\beta K$ .

Arguments

Type	Intent	Optional	Attributes		Name
type(csr_matrix),	intent(in)			::	mass	The N-by-N symmetric positive-definite mass matrix.
type(csr_matrix),	intent(in)			::	stiff	The N-by-N symmetric stiffness matrix.
real(kind=real64),	intent(in)			::	alpha	The mass damping factor, $\alpha$ .
real(kind=real64),	intent(in)			::	beta	The stiffness damping factor, $\beta$ .
integer(kind=int32),	intent(in)			::	nmodes	The number of lowest-frequency modes to retain. This value must be greater than zero and less than N.
integer(kind=int32),	intent(in)			::	nfreq	The number of frequency values to analyze. This value must be at least 2.
real(kind=real64),	intent(in)			::	freq1	The starting frequency, in units of rad/s.
real(kind=real64),	intent(in)			::	freq2	The ending frequency, in units of rad/s.
procedure(modal_excite),	intent(in),		pointer	::	frc	A pointer to the physical forcing function.
real(kind=real64),	intent(out),	optional,	allocatable, dimension(:)	::	modes	An optional NMODES-element array containing the retained modal frequencies in units of rad/s.
real(kind=real64),	intent(out),	optional,	allocatable, dimension(:,:)	::	modeshapes	An optional N-by-NMODES matrix containing the mass-normalized retained mode shapes.
class(*),	intent(inout),	optional		::	args	An optional argument passed to the forcing function.

Return Value type(frf)

The modal-truncated frequency responses.

private function frf_general_damp_1(mass, damp, stiff, freq, frc, ranks, args) result(rst)

Computes the frequency response functions for a multi-degree-of-freedom system that has a general damping matrix, and is not necessarily symmetric. The problem is treated as the solution to the linear system $\left( K - \omega^{2} M + j \omega C \right) H(\omega) = F(\omega)$ .

Arguments

Type	Intent	Optional	Attributes		Name
real(kind=real64),	intent(in),		dimension(:,:)	::	mass	The N-by-N mass matrix.
real(kind=real64),	intent(in),		dimension(:,:)	::	damp	The N-by-N damping matrix.
real(kind=real64),	intent(in),		dimension(:,:)	::	stiff	The N-by-N stiffness matrix.
real(kind=real64),	intent(in),		dimension(:)	::	freq	An M-element array of frequency values at which to evaluate the frequency response functions, in units of rad/s.
procedure(modal_excite),	intent(in),		pointer	::	frc	A pointer to a routine used to compute the modal forcing function.
integer(kind=int32),	intent(out),	optional,	dimension(:)	::	ranks	Provides information on the rank of the dynamic stiffness matrix for each frequency. If provided, this array must be the same length as freq.
class(*),	intent(inout),	optional		::	args	An optional argument that can be used to communicate with the outside world.

Return Value type(frf)

The resulting frequency responses.

private function frf_general_damp_2(mass, damp, stiff, nfreq, freq1, freq2, frc, ranks, args) result(rst)

Computes the frequency response functions for a multi-degree-of-freedom system that has a general damping matrix, and is not necessarily symmetric. The problem is treated as the solution to the linear system $\left( K - \omega^{2} M + j \omega C \right) H(\omega) = F(\omega)$ .

Arguments

Type	Intent	Optional	Attributes		Name
real(kind=real64),	intent(in),		dimension(:,:)	::	mass	The N-by-N mass matrix.
real(kind=real64),	intent(in),		dimension(:,:)	::	damp	The N-by-N damping matrix.
real(kind=real64),	intent(in),		dimension(:,:)	::	stiff	The N-by-N stiffness matrix.
integer(kind=int32),	intent(in)			::	nfreq	The number of frequency values to analyze. This value must be at least 2.
real(kind=real64),	intent(in)			::	freq1	The starting frequency, in units of rad/s.
real(kind=real64),	intent(in)			::	freq2	The ending frequency, in units of rad/s.
procedure(modal_excite),	intent(in),		pointer	::	frc	A pointer to a routine used to compute the modal forcing function.
integer(kind=int32),	intent(out),	optional,	dimension(:)	::	ranks	Provides information on the rank of the dynamic stiffness matrix for each frequency. If provided, this array must be the same length as freq.
class(*),	intent(inout),	optional		::	args	An optional argument that can be used to communicate with the outside world.

Return Value type(frf)

The resulting frequency responses.

private function siso_freqres(x, y, fs, win, method) result(rst)

Estimates the frequency response of a single-input, single-output (SISO) system.

Arguments

Type	Intent	Optional	Attributes		Name
real(kind=real64),	intent(in),		dimension(:)	::	x	An N-element array containing the excitation signal.
real(kind=real64),	intent(in),		dimension(:)	::	y	An N-element array containing the response signal.
real(kind=real64),	intent(in)			::	fs	The sampling frequency, in Hz.
class(window),	intent(in),	optional,	target	::	win	The window to apply to the data. If nothing is supplied, no window is applied.
integer(kind=int32),	intent(in),	optional		::	method	Enter 1 to utilize an H1 estimator; else, enter 2 to utilize an H2 estimator. The default is an H1 estimator. An H1 estimator is defined as the cross-spectrum of the input and response signals divided by the energy spectral density of the input. An H2 estimator is defined as the energy spectral density of the response divided by the cross-spectrum of the input and response signals. $H_{1} = \frac{P_{xy}}{P_{xx}}$ $H_{2} = \frac{P_{yy}}{P_{xy}}$

Return Value type(frf)

The resulting frequency response function.

private function mimo_freqres(x, y, fs, win, method) result(rst)

Estimates the frequency responses of a multiple-input, multiple-output (MIMO) system.

Arguments

Type	Intent	Optional	Attributes		Name
real(kind=real64),	intent(in),		dimension(:,:)	::	x	An N-by-P array containing the P inputs to the system.
real(kind=real64),	intent(in),		dimension(:,:)	::	y	An N-by-M array containing the M outputs from the system.
real(kind=real64),	intent(in)			::	fs	The sampling frequency, in Hz.
class(window),	intent(in),	optional,	target	::	win	The window to apply to the data. If nothing is supplied, no window is applied.
integer(kind=int32),	intent(in),	optional		::	method	Enter 1 to utilize an H1 estimator; else, enter 2 to utilize an H2 estimator. The default is an H1 estimator. An H1 estimator is defined as the cross-spectrum of the input and response signals divided by the energy spectral density of the input. An H2 estimator is defined as the energy spectral density of the response divided by the cross-spectrum of the input and response signals. $H_{1} = \frac{P_{xy}}{P_{xx}}$ $H_{2} = \frac{P_{yy}}{P_{xy}}$

Return Value type(mimo_frf)

The resulting frequency response functions.

frequency_response Interface

Contents

Module Procedures

public interface frequency_response

Module Procedures

private function frf_modal_prop_damp(mass, stiff, alpha, beta, freq, frc, modes, modeshapes, args) result(rst)

Arguments

Return Value type(frf)

private function frf_modal_prop_damp_sparse(mass, stiff, alpha, beta, nmodes, freq, frc, modes, modeshapes, args) result(rst)

Arguments

Return Value type(frf)

private function frf_modal_prop_damp_2(mass, stiff, alpha, beta, nfreq, freq1, freq2, frc, modes, modeshapes, args) result(rst)

Arguments

Return Value type(frf)

private function frf_modal_prop_damp_sparse_2(mass, stiff, alpha, beta, nmodes, nfreq, freq1, freq2, frc, modes, modeshapes, args) result(rst)

Arguments

Return Value type(frf)

private function frf_general_damp_1(mass, damp, stiff, freq, frc, ranks, args) result(rst)

Arguments

Return Value type(frf)

private function frf_general_damp_2(mass, damp, stiff, nfreq, freq1, freq2, frc, ranks, args) result(rst)

Arguments

Return Value type(frf)

private function siso_freqres(x, y, fs, win, method) result(rst)

Arguments

Return Value type(frf)

private function mimo_freqres(x, y, fs, win, method) result(rst)

Arguments

Return Value type(mimo_frf)