Frequency response and system identification

Functions
void	c_frequency_response (int n, int nfreq, const double *mass, int ldm, const double *stiff, int ldk, double alpha, double beta, const double *freq, const c_modal_excite frc, double *modes, double *modeshapes, int ldms, double complex *rsp, int ldr, void *user_data)
	Compute a modal frequency response for a second-order system.
void	c_dynamic_stiffness_dense (int n, double omega, const double *mass, int ldm, const double *damp, int ldc, const double *stiff, int ldk, double complex *dyn_stiff, int ldd)
	Compute dense dynamic stiffness, K - omega^2 M + i omega C.
void	c_frf_general_damp_1 (int n, int nfreq, const double *mass, int ldm, const double *damp, int ldc, const double *stiff, int ldk, const double *freq, const c_modal_excite frc, double complex *rsp, int ldr, int *ranks, void *user_data)
	Compute a general-damping frequency response at explicit frequencies.
void	c_frf_general_damp_2 (int n, int nfreq, double freq1, double freq2, const double *mass, int ldm, const double *damp, int ldc, const double *stiff, int ldk, const c_modal_excite frc, double complex *rsp, int ldr, int *ranks, void *user_data)
	Compute a general-damping frequency response over an evenly spaced interval.
double	c_compute_modal_damping (double lambda, double alpha, double beta)
	Compute modal damping from Rayleigh coefficients.
double	c_chirp (double t, double amp, double span, double f1Hz, double f2Hz)
	Evaluate a swept-frequency chirp signal.
void	c_modal_response (int n, const double *mass, int ldm, const double *stiff, int ldk, double *freqs, double *modeshapes, int ldms)
	Compute modal frequencies and mass-normalized mode shapes.
void	c_normalize_mode_shapes (int n, double *x, int ldx)
	Normalize mode-shape columns.
void	c_frf_sweep (int n, int nfreq, c_harmonic_ode fcn, const double *freq, const double *iv, int solver, double complex *rsp, int ldr, const c_frequency_sweep_controls *opts, void *user_data)
	Perform a nonlinear harmonic frequency sweep.
void	c_set_frequency_sweep_defaults (c_frequency_sweep_controls *x)
	Fill frequency-sweep controls with defaults.
void	c_evaluate_accelerance_frf_model (int n, int norder, const double *mdl, const double *omega, double complex *h)
	Evaluate an accelerance rational FRF model.
void	c_evaluate_receptance_frf_model (int n, int norder, const double *mdl, const double *omega, double complex *h)
	Evaluate a receptance rational FRF model.
void	c_set_iteration_controls_defaults (c_iteration_controls *x)
	Fill nonlinear iteration controls with defaults.
void	c_fit_frf (int n, int norder, int method, const double *freq, const double complex *rsp, const double *maxp, const double *minp, const c_iteration_controls *controls, double *mdl, c_regression_statistics *stats)
	Fit an FRF model to measured data.
void	c_siso_frequency_response (int n, int nf, const double *x, const double *y, double fs, int winsize, c_window_function winfun, int method, double *freq, double complex *rsp, void *user_data)
	Estimate a SISO frequency response from input and output records.
void	c_cross_product (const double x[3], const double y[3], double z[3])
	Compute the cross product of two three-vectors.
void	c_to_skew_symmetric (const double x[3], double *y, int ldy)
	Form a skew-symmetric matrix from a three-vector.
double	c_vector_angle (const double x[3], const double y[3])
	Compute the angle between two vectors.
double	c_scalar_projection (const double x[3], const double y[3])
	Compute the scalar projection of one vector onto another.
void	c_vector_projection (const double x[3], const double y[3], double z[3])
	Project one vector onto another.
double	c_vector_magnitude (int n, const double *x)
	Compute a vector magnitude.
void	c_vector_normalize (int n, double *x)
	Normalize a vector in place.
double	c_dot_product (int n, const double *x, const double *y)
	Compute a dot product.
bool	c_is_symmetric (int m, int n, const double *a, int lda)
	Test whether a matrix is symmetric to within a small relative tolerance.
void	c_siso_model_fit_least_squares (int nsets, int nparams, int neqns, const c_ode_fit fcn, const c_dynamic_system_measurement *x, const double *ic, double *p, int integrator, int ind, const double *maxp, const double *minp, const c_iteration_controls *controls, const c_lm_solver_options *opts, int nconstraints, const double *xc, const double *yc, const c_constraint_equations constraints, int nweights, const double *weights, c_regression_statistics *stats, c_iteration_behavior *info, void *user_data)
	Fit an ODE model to dynamic-system measurements by constrained least squares.
void	c_set_lm_solver_options_defaults (c_lm_solver_options *x)
	Fill Levenberg-Marquardt options with defaults.
int	c_alloc_dynamic_system_measurement (int n, c_dynamic_system_measurement *x)
	Allocate one dynamic-system measurement record.
void	c_free_dynamic_system_measurement (c_dynamic_system_measurement *x)
	Release one dynamic-system measurement record.
c_dynamic_system_measurement *	c_alloc_dynamic_system_measurement_array (int n, const int *ptsper)
	Allocate an array of measurement records.
void	c_free_dynamic_system_measurement_array (int n, c_dynamic_system_measurement *x)
	Release an array of measurement records.

Detailed Description

Function Documentation

c_frequency_response()

void c_frequency_response	(	int	n,
		int	nfreq,
		const double *	mass,
		int	ldm,
		const double *	stiff,
		int	ldk,
		double	alpha,
		double	beta,
		const double *	freq,
		const c_modal_excite	frc,
		double *	modes,
		double *	modeshapes,
		int	ldms,
		double complex *	rsp,
		int	ldr,
		void *	user_data
	)

Compute a modal frequency response for a second-order system.

Parameters

n	System order.
nfreq	Frequency count.
mass	Mass matrix.
ldm	Leading dimension of mass.
stiff	Stiffness matrix.
ldk	Leading dimension of stiff.
alpha	Mass-proportional damping coefficient.
beta	Stiffness-proportional damping coefficient.
freq	Frequencies.
frc	Modal force callback.
modes	Output modal frequencies.
modeshapes	Output mode shapes.
ldms	Leading dimension of modeshapes.
rsp	Output complex response.
ldr	Leading dimension of rsp.
user_data	Opaque caller data forwarded to frc.

c_dynamic_stiffness_dense()

void c_dynamic_stiffness_dense	(	int	n,
		double	omega,
		const double *	mass,
		int	ldm,
		const double *	damp,
		int	ldc,
		const double *	stiff,
		int	ldk,
		double complex *	dyn_stiff,
		int	ldd
	)

Compute dense dynamic stiffness, K - omega^2 M + i omega C.

Matrices use column-major storage.

Parameters

n	Matrix order.
omega	Excitation frequency in radians per second.
mass	Mass matrix.
ldm	Leading dimension of mass.
damp	General damping matrix.
ldc	Leading dimension of damp.
stiff	Stiffness matrix.
ldk	Leading dimension of stiff.
dyn_stiff	Output complex dynamic stiffness matrix.
ldd	Leading dimension of dyn_stiff.

c_frf_general_damp_1()

void c_frf_general_damp_1	(	int	n,
		int	nfreq,
		const double *	mass,
		int	ldm,
		const double *	damp,
		int	ldc,
		const double *	stiff,
		int	ldk,
		const double *	freq,
		const c_modal_excite	frc,
		double complex *	rsp,
		int	ldr,
		int *	ranks,
		void *	user_data
	)

Compute a general-damping frequency response at explicit frequencies.

Matrices and response use column-major storage.

Parameters

n	System order.
nfreq	Frequency count.
mass	Mass matrix.
ldm	Leading dimension of mass.
damp	General damping matrix.
ldc	Leading dimension of damp.
stiff	Stiffness matrix.
ldk	Leading dimension of stiff.
freq	Frequencies in radians per second.
frc	Force callback.
rsp	Output complex response matrix.
ldr	Leading dimension of rsp.
ranks	Output rank of each dynamic stiffness matrix.
user_data	Opaque caller data forwarded to frc.

c_frf_general_damp_2()

void c_frf_general_damp_2	(	int	n,
		int	nfreq,
		double	freq1,
		double	freq2,
		const double *	mass,
		int	ldm,
		const double *	damp,
		int	ldc,
		const double *	stiff,
		int	ldk,
		const c_modal_excite	frc,
		double complex *	rsp,
		int	ldr,
		int *	ranks,
		void *	user_data
	)

Compute a general-damping frequency response over an evenly spaced interval.

Matrices and response use column-major storage.

Parameters

n	System order.
nfreq	Frequency count, at least 2.
freq1	Starting frequency in radians per second.
freq2	Ending frequency in radians per second.
mass	Mass matrix.
ldm	Leading dimension of mass.
damp	General damping matrix.
ldc	Leading dimension of damp.
stiff	Stiffness matrix.
ldk	Leading dimension of stiff.
frc	Force callback.
rsp	Output complex response matrix.
ldr	Leading dimension of rsp.
ranks	Output rank of each dynamic stiffness matrix.
user_data	Opaque caller data forwarded to frc.

c_compute_modal_damping()

double c_compute_modal_damping	(	double	lambda,
		double	alpha,
		double	beta
	)

Compute modal damping from Rayleigh coefficients.

Parameters

lambda	Modal eigenvalue.
alpha	Mass-proportional coefficient.
beta	Stiffness-proportional coefficient.

Returns

Modal damping ratio.

c_chirp()

double c_chirp	(	double	t,
		double	amp,
		double	span,
		double	f1Hz,
		double	f2Hz
	)

Evaluate a swept-frequency chirp signal.

Parameters

t	Time.
amp	Amplitude.
span	Sweep span.
f1Hz	Initial frequency in hertz.
f2Hz	Final frequency in hertz.

Returns

Signal value.

c_modal_response()

void c_modal_response	(	int	n,
		const double *	mass,
		int	ldm,
		const double *	stiff,
		int	ldk,
		double *	freqs,
		double *	modeshapes,
		int	ldms
	)

Compute modal frequencies and mass-normalized mode shapes.

Parameters

n	System order.
mass	Mass matrix.
ldm	Leading dimension of mass.
stiff	Stiffness matrix.
ldk	Leading dimension of stiff.
freqs	Output modal frequencies.
modeshapes	Output mode shapes.
ldms	Leading dimension of modeshapes.

c_normalize_mode_shapes()

void c_normalize_mode_shapes	(	int	n,
		double *	x,
		int	ldx
	)

Normalize mode-shape columns.

Parameters

n	Number of rows or modes.
x	Mode-shape matrix, updated in place.
ldx	Leading dimension of x.

c_frf_sweep()

void c_frf_sweep	(	int	n,
		int	nfreq,
		c_harmonic_ode	fcn,
		const double *	freq,
		const double *	iv,
		int	solver,
		double complex *	rsp,
		int	ldr,
		const c_frequency_sweep_controls *	opts,
		void *	user_data
	)

Perform a nonlinear harmonic frequency sweep.

Parameters

n	State dimension.
nfreq	Frequency count.
fcn	Harmonic ODE callback.
freq	Frequencies.
iv	Initial values.
solver	Integration method.
rsp	Output complex response.
ldr	Leading dimension of rsp.
opts	Sweep controls.
user_data	Opaque caller data forwarded to fcn.

c_set_frequency_sweep_defaults()

void c_set_frequency_sweep_defaults

(

c_frequency_sweep_controls *

x

)

Fill frequency-sweep controls with defaults.

Parameters

x	Controls updated in place.

c_evaluate_accelerance_frf_model()

void c_evaluate_accelerance_frf_model	(	int	n,
		int	norder,
		const double *	mdl,
		const double *	omega,
		double complex *	h
	)

Evaluate an accelerance rational FRF model.

Parameters

n	Frequency count.
norder	Model order.
mdl	Model coefficients.
omega	Angular frequencies.
h	Output complex FRF values.

c_evaluate_receptance_frf_model()

void c_evaluate_receptance_frf_model	(	int	n,
		int	norder,
		const double *	mdl,
		const double *	omega,
		double complex *	h
	)

Evaluate a receptance rational FRF model.

Parameters

n	Frequency count.
norder	Model order.
mdl	Model coefficients.
omega	Angular frequencies.
h	Output complex FRF values.

c_set_iteration_controls_defaults()

void c_set_iteration_controls_defaults

(

c_iteration_controls *

x

)

Fill nonlinear iteration controls with defaults.

Parameters

x	Controls updated in place.

c_fit_frf()

void c_fit_frf	(	int	n,
		int	norder,
		int	method,
		const double *	freq,
		const double complex *	rsp,
		const double *	maxp,
		const double *	minp,
		const c_iteration_controls *	controls,
		double *	mdl,
		c_regression_statistics *	stats
	)

Fit an FRF model to measured data.

Parameters

n	Data count.
norder	Model order.
method	Model or update method.
freq	Frequencies.
rsp	Measured complex response.
maxp	Parameter upper bounds.
minp	Parameter lower bounds.
controls	Iteration controls.
mdl	Output fitted parameters.
stats	Output regression statistics.

c_siso_frequency_response()

void c_siso_frequency_response	(	int	n,
		int	nf,
		const double *	x,
		const double *	y,
		double	fs,
		int	winsize,
		c_window_function	winfun,
		int	method,
		double *	freq,
		double complex *	rsp,
		void *	user_data
	)

Estimate a SISO frequency response from input and output records.

Parameters

n	Sample count.
nf	Frequency-bin count.
x	Input samples.
y	Output samples.
fs	Sampling frequency.
winsize	Window size.
winfun	Window callback.
method	H1 or H2 estimator.
freq	Output frequencies.
rsp	Output complex response.
user_data	Opaque caller data forwarded to winfun.

c_cross_product()

void c_cross_product	(	const double	x[3],
		const double	y[3],
		double	z[3]
	)

Compute the cross product of two three-vectors.

Parameters

x	First vector.
y	Second vector.
z	Output vector.

c_to_skew_symmetric()

void c_to_skew_symmetric	(	const double	x[3],
		double *	y,
		int	ldy
	)

Form a skew-symmetric matrix from a three-vector.

Parameters

x	Input vector.
y	Output matrix.
ldy	Leading dimension of y.

c_vector_angle()

double c_vector_angle	(	const double	x[3],
		const double	y[3]
	)

Compute the angle between two vectors.

Parameters

x	First vector.
y	Second vector.

Returns

Angle in radians.

c_scalar_projection()

double c_scalar_projection	(	const double	x[3],
		const double	y[3]
	)

Compute the scalar projection of one vector onto another.

Parameters

x	Vector being projected.
y	Reference vector.

Returns

Scalar projection.

c_vector_projection()

void c_vector_projection	(	const double	x[3],
		const double	y[3],
		double	z[3]
	)

Project one vector onto another.

Parameters

x	Vector being projected.
y	Reference vector.
z	Output projection.

c_vector_magnitude()

double c_vector_magnitude	(	int	n,
		const double *	x
	)

Compute a vector magnitude.

Parameters

n	Vector length.
x	Vector.

Returns

Euclidean magnitude.

c_vector_normalize()

void c_vector_normalize	(	int	n,
		double *	x
	)

Normalize a vector in place.

Parameters

n	Vector length.
x	Vector updated in place.

c_dot_product()

double c_dot_product	(	int	n,
		const double *	x,
		const double *	y
	)

Compute a dot product.

Parameters

n	Vector length.
x	First vector.
y	Second vector.

Returns

Dot product.

c_is_symmetric()

bool c_is_symmetric	(	int	m,
		int	n,
		const double *	a,
		int	lda
	)

Test whether a matrix is symmetric to within a small relative tolerance.

Parameters

m	Row count.
n	Column count.
a	m-by-n matrix.
lda	Leading dimension of a.

Returns

true when a is square and symmetric; else, false.

c_siso_model_fit_least_squares()

void c_siso_model_fit_least_squares	(	int	nsets,
		int	nparams,
		int	neqns,
		const c_ode_fit	fcn,
		const c_dynamic_system_measurement *	x,
		const double *	ic,
		double *	p,
		int	integrator,
		int	ind,
		const double *	maxp,
		const double *	minp,
		const c_iteration_controls *	controls,
		const c_lm_solver_options *	opts,
		int	nconstraints,
		const double *	xc,
		const double *	yc,
		const c_constraint_equations	constraints,
		int	nweights,
		const double *	weights,
		c_regression_statistics *	stats,
		c_iteration_behavior *	info,
		void *	user_data
	)

Fit an ODE model to dynamic-system measurements by constrained least squares.

Parameters

nsets	Number of measurement sets.
nparams	Number of parameters.
neqns	Number of model equations.
fcn	ODE callback.
x	Measurement records.
ic	Initial conditions.
p	Parameters, updated in place.
integrator	ODE integration method.
ind	Integration direction or index.
maxp	Upper parameter bounds.
minp	Lower parameter bounds.
controls	Iteration controls.
opts	Levenberg-Marquardt options.
nconstraints	Constraint count.
xc	Constraint inputs.
yc	Constraint outputs.
constraints	Constraint callback.
nweights	Weight count.
weights	Residual weights.
stats	Output regression statistics.
info	Output iteration statistics.
user_data	Opaque caller data forwarded to fcn and constraints.

c_set_lm_solver_options_defaults()

void c_set_lm_solver_options_defaults

(

c_lm_solver_options *

x

)

Fill Levenberg-Marquardt options with defaults.

Parameters

x	Options updated in place.

c_alloc_dynamic_system_measurement()

int c_alloc_dynamic_system_measurement	(	int	n,
		c_dynamic_system_measurement *	x
	)

Allocate one dynamic-system measurement record.

Parameters

n	Sample count.
x	Record initialized by the routine.

Returns

Zero on success; nonzero on allocation failure.

c_free_dynamic_system_measurement()

void c_free_dynamic_system_measurement

(

c_dynamic_system_measurement *

x

)

Release one dynamic-system measurement record.

Parameters

x	Record to release.

c_alloc_dynamic_system_measurement_array()

c_dynamic_system_measurement * c_alloc_dynamic_system_measurement_array	(	int	n,
		const int *	ptsper
	)

Allocate an array of measurement records.

Parameters

n	Number of records.
ptsper	Samples per record.

Returns

Allocated array, or NULL on failure.

c_free_dynamic_system_measurement_array()

void c_free_dynamic_system_measurement_array	(	int	n,
		c_dynamic_system_measurement *	x
	)

Release an array of measurement records.

Parameters

n	Number of records.
x	Array to release.