Pulse-type bidirectional ground motion scaling method based on displacement response spectrum

Abstract

This invention discloses a scaling method for pulse-type bidirectional ground motion based on displacement response spectrum. The method first performs nonlinear dynamic analysis on the bidirectional ground motion, generating a displacement scalar time-history response spectrum representing the bidirectional ground motion input using a complex plane method. Then, it calculates the scaling factor that minimizes the sum of weighted squared errors. Finally, it obtains a displacement response spectrum that accurately reflects the pulse energy magnitude of the pulse-type ground motion. This method can more realistically and accurately represent the pulse energy magnitude of pulse-type ground motions.

Description

Technical Field

[0001] This invention belongs to the technical field of seismic design, specifically relating to a pulse-type bidirectional seismic motion scaling method based on displacement response spectrum. Background Technology

[0002] Seismic waves, as loads used in nonlinear dynamic time-history analysis of structures, directly affect the structural requirements under earthquakes. Pulsed ground motions possess unique pulse characteristics, containing significant energy within their velocity pulses, which substantially increases the seismic resistance requirements of structures. However, existing ground motion scaling methods often fail to accurately reflect the pulse energy magnitude of pulsed ground motions, leading to errors and uncertainties in structural seismic design. Therefore, a scaling method that accurately reflects the pulse energy magnitude of pulsed ground motions is needed to improve the accuracy and reliability of structural seismic design.

[0003] Currently, the conventional pulse-type bidirectional ground motion matching method selects several seismic waves and scales them within a certain period range (e.g., 0.2 to 1.5 times the fundamental period of the structure) using the acceleration response spectrum of the site with 5% damping as the target. Unidirectional seismic wave scaling methods include s... a (T1) Scaling methods and spectral matching methods within the range of the fundamental period, etc., while the scaling of bidirectional seismic waves requires combining the acceleration seismic wave components in two horizontal directions and then matching them with the response spectrum. The main methods include geometric mean (square root of the sum of squares, SRSS) method, D-scaling method, GMRotI50 method and Ozdemir method.

[0004] Due to the characteristics of pulse-type ground motion, the velocity pulse period is of great significance for assessing the seismic performance of structures. Commonly used methods for determining it include wavelet analysis, half-cycle velocity pulse method, and HHT method.

[0005] Currently, when designing seismic-resistant isolated bridges, the acceleration response spectrum is generally used as the target. Seismic wave matching is performed using the acceleration response spectrum value within the range of the bridge's fundamental period. This method involves performing nonlinear dynamic analysis to calculate the acceleration time history separately. , The maximum value of the acceleration time history is taken as the maximum acceleration of the equivalent single-degree-of-freedom system. Then, the acceleration time histories in both directions are converted using software-seismosignal to generate an acceleration response spectrum. , Subsequently, the two horizontal acceleration seismic wave components are combined using the geometric mean method (square root of the sum of squares, SRSS). The resulting acceleration scalar time history response spectrum is then processed using the Ozdemir method. The corresponding acceleration response spectrum can then be obtained. The specific process is as follows:Figure 3 As shown. In reality, isolated bridges differ from non-isolated bridges; their equivalent periods are generally larger, falling within the constant velocity or constant displacement range in the triple response spectrum. Their displacement demand is greatly affected by the velocity or displacement of the seismic input, but less affected by acceleration. Therefore, whether existing methods can truly reflect the pulse energy characteristics of pulse-type ground motions is debatable. Summary of the Invention

[0006] To address the shortcomings of existing technologies, this invention proposes a pulse-type bidirectional seismic motion scaling method based on displacement response spectrum. This method derives the bidirectional displacement time history of a specific support by performing nonlinear dynamic time history analysis on a seismically isolated bridge. Due to the difference between seismically isolated and non-isolated bridges, the equivalent period of the former is generally larger, placing it in the constant velocity or constant displacement range in the triple response spectrum. Its displacement demand is significantly affected by the velocity or displacement of the seismic input, but less so by acceleration. Therefore, the displacement response spectrum can more realistically and accurately reflect the pulse energy magnitude of pulse-type seismic motions.

[0007] The above objectives are achieved through the following technical solutions:

[0008] The present invention provides a pulse-type bidirectional seismic motion scaling method based on displacement response spectrum, which includes the following steps:

[0009] S1. (1) Perform nonlinear dynamic analysis on bidirectional ground motion, and obtain the acceleration time history data in the X and Y directions. and By combining the two real-valued sequences in the complex plane, a single time-varying complex ground acceleration vector is formed. Then the combined complex ground acceleration vector The displacement scalar time history response spectrum under bidirectional seismic input was generated using the Newmark-β method. Specifically, as shown in equation (4): (4) In the formula, i is the imaginary unit. The acceleration time histories are for the X and Y directions, respectively. This complex time histories fully preserve the amplitude and phase coupling information of the bidirectional ground motion.

[0010] For a given single-degree-of-freedom system, assuming it has the same dynamic characteristics in the X and Y directions, the dynamic characteristics include the natural period T and the damping ratio. For a single-degree-of-freedom system, the excitation results obtained at different natural periods T are different; Under the excitation, the complex relative displacement response of this single-degree-of-freedom system Satisfying the differential equation shown in equation (5): +2 + = - (5) in, These are the time histories of the relative displacement, relative velocity, and relative acceleration of the structure, respectively. The undamped natural circular frequency of the structure;

[0011] Using Newmark- The above complex differential equations are solved numerically using the successive integration method (taking the control numerical damping). Controlled acceleration distribution assumption ), to obtain the complex displacement response sequence of the structure throughout the entire earthquake duration. .

[0012] A single-degree-of-freedom system is a linear system, which in The displacement response under excitation can be considered as a series of infinitely short pulses, and the corresponding unit impulse response function can be expressed as follows. The result is obtained by performing a convolution integral, as shown in equation (6): (6) in, Damped natural frequency

[0013] According to the superposition principle of linear systems, the analytical solution of the differential equation in equation (5) can be expressed as the complex convolution integral shown in equation (7): (7) in These are the time histories of the structure's relative displacement, relative velocity, and relative acceleration response in the X direction, respectively. The time histories of the structure's relative displacement, relative velocity, and relative acceleration in the Y direction are given. According to the superposition principle, the complex displacement response... The general integral form should be as shown in equation (8): (8)

[0014] Substituting the complex excitation defined in equation (4) into the integral of equation (8): (9) in As the integral time variable, due to the unit impulse response function It is a number that contains no imaginary numbers. For a real function, based on the linearity of integrals, we can directly expand the parentheses in formula (9) and split the integral into two parts of formula (10): (10) Therefore, complex displacement response The real part is the displacement in the X direction. The imaginary part is the displacement in the Y direction. As shown in formula (11): (11)

[0015] (3) Definition of trajectory envelope response spectrum: displacement response spectrum value under bidirectional seismic action. Defined as the maximum radial distance by which the structure deviates from its equilibrium position in the complex plane motion trajectory, i.e., the maximum value of the complex displacement modulus, as shown in formula (12): (12)

[0016] The formula (12) yields This only represents the displacement response spectrum of the structure at a single specific natural period T. To obtain the complete requirements of the structure over a wide frequency range, a target period range is set (e.g., from 0.01s to 4.0s), and divided into n discrete period points (denoted as ) with a specified step size. , ).

[0017] By sequentially changing the undamped natural circle frequency of the structure By repeatedly calculating the complex displacement response and extracting the envelope extrema based on the natural period T in the data, a series of corresponding displacement response spectrum values ​​under bidirectional seismic motion can be obtained. ;

[0018] These discrete displacement response spectrum values By combining them in a periodic order, a displacement scalar time history response spectrum can be constructed. ;

[0019] The generated displacement scalar time history response spectrum In, the i-th element it contains That is, equivalent to the system in the i-th target period. The displacement response spectrum value calculated under bidirectional seismic motion. The displacement scalar time history response spectrum This will be used as baseline data and directly input into subsequent steps to calculate the scaling factor that minimizes the error. .

[0020] S2. Calculate the sum of weighted squared errors. Minimize scaling factor ; Where n represents the time history of a seismic wave, i.e., duration / time step. Indicates the first Displacement response spectrum value at time 1 The corresponding value, Indicates the first At any given time, the seismic response spectrum specified in the design or code. The value; Indicates the first Time-displacement scalar time-history response spectrum The value, Indicates the first The seismic response spectrum specified in the design or code. The value;

[0021] S3. The displacement response spectrum that ultimately accurately reflects the magnitude of the pulse energy of pulse-type seismic ground motion. The details are as follows: (3)

[0022] Furthermore, the sum of weighted squared errors described in step S2 At the selected target period, the seismic response spectrum specified in the design or code. The value and the displacement scalar time history response spectrum obtained in step S1 The differences between them were calculated, and the specific results are as follows: (2)

[0023] Beneficial effects:

[0024] 1. The method of this invention focuses on the displacement response spectrum, emphasizing the maximum displacement response of the structure during an earthquake. By extracting and processing the displacement response spectrum, the magnitude of the pulse energy of the seismically isolated bridge under pulse-type ground motion can be determined.

[0025] 2. In processing displacement time histories, this invention differs from existing solutions that first convert and then combine the data. Instead, it adapts to different situations by differentiating the processing methods based on whether or not pulses are present. Attached Figure Description

[0026] Figure 1 This is the displacement seismic wave component conversion process according to an embodiment of the present invention;

[0027] Figure 2 This is a comparison diagram of the displacement response spectrum before and after scaling in an embodiment of the present invention;

[0028] Figure 3 This is a flowchart of the ground motion scaling method for acceleration response spectrum mentioned in the background section;

[0029] Figure 4This is a comparison of the displacement response spectra of seismic waves in soft soil fields and seismic waves in hard soil fields according to the present invention; Figure 4 (a) is the displacement response spectrum of seismic waves in the soft soil field. Comparison results with standard spectrum Figure 4 (b) is the seismic wave displacement response spectrum of a hard soil field. Results compared with standard spectra;

[0030] Figure 5 This invention relates to the displacement response spectrum calculated from seismic waves in a soft soil field. Comparison of the Ozdemir method with the standard spectrum. Figure 5 (a) is the displacement response spectrum calculated from seismic waves in the soft soil field. The comparison results between the Ozdemir method and the standard spectrum Figure 5 (b) is the displacement response spectrum calculated from seismic waves in a hard soil field. Comparison of the Ozdemir method with the standard spectrum;

[0031] Figure 6 This is a flowchart of the complex plane method according to an embodiment of the present invention. Detailed Implementation

[0032] The present invention will be further described below with reference to the accompanying drawings and specific embodiments, but the scope of protection of the present invention is not limited thereto.

[0033] like Figure 1 As shown, the pulse-type bidirectional seismic motion scaling method based on displacement response spectrum in this embodiment includes the following steps:

[0034] S1. Perform nonlinear dynamic analysis on bidirectional seismic waves. Incorporate the seismic waves (using 22 seismic waves including CC101 as examples) into the complex plane calculation method to perform nonlinear dynamic analysis on bidirectional ground motion, generating a displacement scalar time-history response spectrum representing the bidirectional ground motion input. The details are as follows:

[0035] Nonlinear dynamic analysis of bidirectional ground motion was performed, and the acceleration time history data in the X and Y directions were analyzed. and By combining the two real-valued sequences in the complex plane, a single time-varying complex ground acceleration vector is formed. Then the combined complex ground acceleration vector The displacement scalar time history response spectrum under bidirectional seismic input was generated using the Newmark-β method. Specifically, as shown in equation (4): (4) In the formula, i is the imaginary unit. The acceleration time histories are for the X and Y directions, respectively. This complex time histories fully preserve the amplitude and phase coupling information of the bidirectional ground motion.

[0036] For a given single-degree-of-freedom system, assuming it has the same dynamic characteristics in the X and Y directions, the dynamic characteristics include the natural period T and the damping ratio. For a single-degree-of-freedom system, the excitation results obtained at different natural periods T are different; Under the excitation, the complex relative displacement response of this single-degree-of-freedom system Satisfying the differential equation shown in equation (5): +2 + = - (5) in, These are the time histories of the relative displacement, relative velocity, and relative acceleration of the structure, respectively. The undamped natural circular frequency of the structure;

[0037] Using Newmark- The above complex differential equations are solved numerically using the successive integration method (taking the control numerical damping). Controlled acceleration distribution assumption ), to obtain the complex displacement response sequence of the structure throughout the entire earthquake duration. .

[0038] A single-degree-of-freedom system is a linear system, which in The displacement response under excitation can be considered as a series of infinitely short pulses, and the corresponding unit impulse response function can be expressed as follows. The result is obtained by performing a convolution integral, as shown in equation (6): (6) in, Damped natural frequency

[0039] According to the superposition principle of linear systems, the analytical solution of the differential equation in equation (5) can be expressed as the complex convolution integral shown in equation (7): (7) in These are the time histories of the structure's relative displacement, relative velocity, and relative acceleration response in the X direction, respectively. The time histories of the structure's relative displacement, relative velocity, and relative acceleration in the Y direction are given. According to the superposition principle, the complex displacement response... The general integral form should be as shown in equation (8): (8)

[0040] Substituting the complex excitation defined in equation (4) into the integral of equation (8): (9) in As the integral time variable, due to the unit impulse response function It is a number that contains no imaginary numbers. For a real function, based on the linearity of integrals, we can directly expand the parentheses in formula (9) and split the integral into two parts of formula (10): (10)

[0041] Therefore, complex displacement response The real part is the displacement in the X direction. The imaginary part is the displacement in the Y direction. As shown in formula (11): (11)

[0042] (4) Definition of trajectory envelope response spectrum: displacement response spectrum value under bidirectional seismic action. Defined as the maximum radial distance by which the structure deviates from its equilibrium position in the complex plane motion trajectory, i.e., the maximum value of the complex displacement modulus, as shown in formula (12):

[0043] (12) The formula (12) yields This only represents the displacement response spectrum of the structure at a single specific natural period T. To obtain the complete requirements of the structure over a wide frequency range, a target period range is set (e.g., from 0.01s to 4.0s), and divided into n discrete period points (denoted as ) with a specified step size. , ).

[0044] By sequentially changing the undamped natural circle frequency of the structure By repeatedly calculating the complex displacement response and extracting the envelope extrema based on the natural period T in the data, a series of corresponding displacement response spectrum values ​​under bidirectional seismic motion can be obtained. ;

[0045] These discrete displacement response spectrum values By combining them in a periodic order, a displacement scalar time history response spectrum can be constructed. ;

[0046] The generated displacement scalar time history response spectrum In, the i-th element it contains That is, equivalent to the system in the i-th target period. The displacement response spectrum value calculated under bidirectional seismic motion. The displacement scalar time history response spectrum This will be used as baseline data and directly input into subsequent steps to calculate the scaling factor that minimizes the error. .

[0047] S2. Calculate the sum of weighted squared errors. Minimize scaling factor Sum of weighted squared errors At the selected target period, the seismic response spectrum specified in the design or code. The value and the displacement scalar time history response spectrum obtained in step S1 The difference between them was calculated. The calculation employs the Ozdemir method, an amplitude scaling method (not involving spectral matching) designed to minimize the displacement scalar time-history response spectrum. Seismic response spectrum with different periods as specified in design or code The sum of weighted squared errors between the values The specific results are as follows: (2)

[0048] n represents the time history number (duration / time step) of a certain seismic wave. Indicates the first Time-displacement scalar time-history response spectrum The value, Indicates the first At any given time, the seismic response spectrum specified in the design or code. The value of .

[0049] The Ozdemir method is essentially the Least Squares Method, a mathematical optimization technique used to find the best-fit curve (or straight line) between data points and a mathematical model (usually a function). Its core idea is to minimize the sum of squares of the vertical distances (residuals) between all data points and the model's predicted values.

[0050] At the selected target period, the seismic response spectrum specified in the design or code. The value of the displacement scalar time history response spectrum The difference between them is calculated. Due to the long-period characteristics of seismic isolation bridges, 0 seconds (s) to 4 seconds (s) are selected as the target period for matching.

[0051] To find Minimize scaling factor The specific steps are as follows:

[0052] First, as can be seen from equation (2), After expansion, it becomes a quadratic polynomial in terms of a. Therefore, when we differentiate equation (2) with respect to the scaling factor a, the scaling factor a is 0 when the result is 0. At this point, the scaling factor a is the sum of the weighted squared errors. The minimized value of 'a' is calculated as follows: (1)

[0053] S3. Adjust the scaling factor Substituting the displacement scalar time history response spectrum In this process, the displacement response spectrum that accurately reflects the magnitude of the pulse energy of the pulse-type seismic motion can be obtained. The details are as follows:

[0054] (3)

[0055] The final displacement response spectrum can accurately reflect the magnitude of the pulse energy of pulse-type ground motion. like Figure 5 As shown in Figure 10.

[0056] For the critical response period range of the seismic isolation system, i.e. The following conditions must be met: ,

[0057] In the formula, This refers to the seismic response spectrum specified in the design or code. 1.3 is the amplification factor specified in the American Society of Civil Engineers (ASCE) code, used to cover uncertainties; 0.9 is the minimum allowable matching ratio. Indicates the effective period of the design earthquake , This represents all periods between the effective periods of the maximum considered earthquake.

[0058] We categorized the 22 selected seismic waves into two types: hard soil waves and soft soil waves. The seismic wave scaling factors in the final calculation example are as follows:

[0059] The seismic wave scaling factors are shown in Table 1:

[0060] The final displacement response spectrum can accurately reflect the magnitude of the pulse energy of pulse-type ground motion. like Figure 2 , Figure 4 , Figure 5 As shown, from Figure 4 As can be seen, the response spectrum of Example 1 during the target period (0s-4s) is similar to the seismic response spectrum specified in the design or code. It has a good matching effect and can accurately reflect the displacement effect caused by ground vibrations during the target period. And from... Figure 5 As can be seen, compared with the Ozdemir method, the pulse-type bidirectional ground motion scaling method based on displacement response spectrum of the present invention can more accurately reflect the actual situation of ground motion, and provide a more accurate method for improving the accuracy and reliability of seismic design of structures.

[0061] With the increasing popularity of seismic isolation bridges, the conventional seismic design of seismic isolation bridges in existing technologies generally uses the acceleration response spectrum as the target, and matches the seismic waves with the acceleration response spectrum value in the vicinity of the bridge's fundamental period.

[0062] Seismic waves, used as loads in nonlinear dynamic time-history analysis of structures, directly affect the structure's seismic requirements. Pulse-type ground motions possess unique pulse characteristics, containing significant energy within velocity pulses and releasing it concentrated within seconds, unlike the continuous vibrations of ordinary ground motions. This characteristic significantly increases the seismic resistance requirements of structures.

[0063] However, due to the increasing prevalence of seismically isolated bridges, their equivalent periods, which differ from those of non-seismically isolated bridges, often fail to accurately reflect the magnitude of pulse energy when dealing with pulse-type ground motions (e.g., Figure 5 As shown in (a) in the figure, this leads to certain errors and uncertainties in the seismic design of structures. Therefore, a scaling method that can accurately reflect the magnitude of pulse energy of pulse-type ground motion is needed to improve the accuracy and reliability of seismic design of structures.

[0064] In reality, seismically isolated bridges differ from non-seismically isolated bridges. Seismically isolated bridges are long-period structures with generally large equivalent periods, falling within the constant velocity or constant displacement range in the triple response spectrum. In contrast, the energy of pulse-type seismic ground motions is concentrated at lower frequencies. This means the structural response is more significantly influenced by the velocity and displacement spectra. Therefore, its displacement demand is greatly affected by the velocity or displacement of the seismic input, but less so by acceleration. Whether existing methods using acceleration response spectra can accurately reflect the pulse energy characteristics of pulse-type seismic ground motions is debatable. The method proposed in this invention, which uses displacement response spectra to represent the pulse energy in pulse-type seismic ground motions, provides a more realistic and accurate reflection of the energy magnitude of pulse-type seismic ground motions compared to existing technologies.

[0065] The above embodiments are preferred implementations of the nodes of the present invention, but the implementation of the present invention is not limited to the above embodiments. Any changes, modifications, substitutions, combinations, or simplifications made without departing from the spirit and principle of the present invention shall be considered equivalent substitutions and shall be included within the protection scope of the present invention.

Claims

1. A pulse-type bidirectional ground motion scaling method based on displacement response spectrum, characterized by, The method includes the following steps: S1. Perform nonlinear dynamic analysis on bidirectional ground motion, and use the complex plane method to generate a displacement scalar time history response spectrum representing the bidirectional ground motion input from the displacement data in the X and Y directions. ; S2. Calculate the sum of weighted squared errors. Minimize scaling factor As shown in equation (1); （1） Where n represents the time history of a certain seismic wave, i.e., duration / time step; Indicates the first Time-displacement scalar time-history response spectrum The value, Indicates the first The seismic response spectrum specified in the design or code. The value; The sum of the weighted squared errors At the selected target period, the seismic response spectrum specified in the design or code. The value and the displacement scalar time history response spectrum obtained in step S1 The difference between them was calculated, and the specific result is shown in equation (2): （2） S3. The displacement response spectrum that ultimately accurately reflects the magnitude of the pulse energy of pulse-type seismic ground motion. Specifically, as shown in equation (3): （3）。 2. The pulse-type bidirectional seismic motion scaling method based on displacement response spectrum according to claim 1, characterized in that, Step S1 involves using the displacement data in the X and Y directions to generate a displacement scalar time history response spectrum representing bidirectional seismic input through a complex plane method. The specific method is as follows: Nonlinear dynamic analysis of bidirectional ground motion was performed, and the acceleration time history data in the X and Y directions were analyzed. and By combining the two real-valued sequences in the complex plane, a single time-varying complex ground acceleration vector is formed. Then the combined complex ground acceleration vector The displacement scalar time history response spectrum under bidirectional seismic input was generated using the Newmark-β method. Specifically, as shown in equation (4): （4） In the formula, i is the imaginary unit. The acceleration time histories are for the X and Y directions, respectively. This complex time histories fully preserve the amplitude and phase coupling information of the bidirectional ground motion. For a given single-degree-of-freedom system, assuming it has the same dynamic characteristics in the X and Y directions, the dynamic characteristics include the natural period T and the damping ratio. For a single-degree-of-freedom system, the excitation results obtained at different natural periods T are different; Under the excitation, the complex relative displacement response of this single-degree-of-freedom system Satisfying the differential equation shown in equation (5): +2 + = - （5） in, These are the time histories of the relative displacement, relative velocity, and relative acceleration of the structure, respectively. The undamped natural circular frequency of the structure; Using Newmark- The complex differential equation above is solved numerically using the step-by-step integration method, taking control numerical damping as an example. Controlled acceleration distribution assumption The complex displacement response sequence of the structure during the entire earthquake duration was obtained. ; A single-degree-of-freedom system is a linear system, which in The displacement response under excitation can be considered as a series of infinitely short pulses, and the corresponding unit impulse response function can be expressed as follows. The result is obtained by performing a convolution integral, as shown in equation (6): （6） in, The damped natural frequency; According to the superposition principle of linear systems, the analytical solution of the differential equation in equation (5) can be expressed as the complex convolution integral shown in equation (7): （7） in These are the time histories of the structure's relative displacement, relative velocity, and relative acceleration response in the X direction, respectively. The time histories of the structure's relative displacement, relative velocity, and relative acceleration in the Y direction are given. According to the superposition principle, the complex displacement response... The general integral form should be as shown in equation (8): （8） Substituting the complex excitation defined in equation (4) into the integral of equation (8): （9） in As the integral time variable, due to the unit impulse response function It is a number that contains no imaginary numbers. For the real function, based on the linearity property of the integral, we can directly expand the parentheses in formula (9) and split the integral into two parts of formula (10): （10） Therefore, complex displacement response The real part is the displacement in the X direction. The imaginary part is the displacement in the Y direction. As shown in formula (11): （11） (3) Definition of trajectory envelope response spectrum: displacement response spectrum value under bidirectional seismic action. Defined as the maximum radial distance by which the structure deviates from its equilibrium position in the complex plane motion trajectory, i.e., the maximum value of the complex displacement modulus, as shown in formula (12): （12） The formula (12) yields This only represents the displacement response spectrum of the structure under a single, specific natural vibration period T. To obtain the complete requirements of the structure over a wide frequency range, a target period range is set and divided into n discrete period points according to a specified step size, denoted as . , ; By sequentially changing the undamped natural circle frequency of the structure By repeatedly calculating the complex displacement response and extracting the envelope extrema based on the natural period T in the data, a series of corresponding displacement response spectrum values ​​under bidirectional seismic motion can be obtained. ; These discrete displacement response spectrum values By combining them in a periodic order, a displacement scalar time history response spectrum can be constructed. ; The generated displacement scalar time history response spectrum In, the i-th element it contains That is, equivalent to the system in the i-th target period. The displacement response spectrum value calculated under bidirectional seismic motion. The displacement scalar time history response spectrum This will be used as baseline data and directly input into subsequent steps to calculate the scaling factor that minimizes the error. .

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