Reanalysis of linear dynamic systems using modified combined approximations with frequency shifts.
Weight reduction is very important in automotive design because of stringent demand on fuel economy. Structural optimization of dynamic systems using finite element (FE) analysis plays an important role in reducing weight while simultaneously delivering a product that meets all functional requirements for durability, crash and NVH. With advancing computer technology, the demand for solving large FE models has grown. Optimization is however costly due to repeated full-order analyses. Reanalysis methods can be used in structural vibrations to reduce the analysis cost from repeated eigenvalue analyses for both deterministic and probabilistic problems. Several reanalysis techniques have been introduced over the years including Parametric Reduced Order Modeling (PROM), Combined Approximations (CA) and the Epsilon algorithm, among others. It has been shown that the Modified Combined Approximations (MCA) method - an improvement over CA - is the most efficient reanalysis technique for problems with a large number of degrees of freedom. This paper proposes an improvement of the MCA method using frequency shifts. Numerical examples are presented and results are compared with existing methods.
CITATION: Haider, S. and Mourelatos, Z., "Reanalysis of Linear Dynamic Systems using Modified Combined Approximations with Frequency Shifts," SAE Int. J. Passeng. Cars - Mech. Syst. 9(1):2016.
Designs with minimum weight are sought while meeting all functional requirements in any industry. This is achieved using optimization. Finite element analysis is also a well-established numerical tool in the industry. It is frequently used for low frequency Noise, Vibration and Harshness (NVH) analysis. Reanalysis methods are used to estimate the natural frequencies and mode shapes of a modified structure without using a full eigenvalue analysis. This reduces the computational effort of an optimization method for NVH by eliminating repeated eigenvalue analyses. As real-life designs are becoming large and complex, the demand for efficient and accurate reanalysis methods has increased.
Reanalysis methods are based on local and global approximations. Local approximations are efficient but are effective only for small structural perturbations. On the other hand, global approximations are computationally expensive and preferable for larger structural perturbations. Rayleigh-Ritz is a popular local approximation method suited for small changes. Several reviews have been published on reanalysis methods [1, 2, 3, 4, 5, 6]. The first application of reanalysis methods was in linear static problems in early 1970s [1, 2]. Balmes [7, 8] proposed the Parametric Reduced Order Model (PROM) method by expanding the Rayleigh-Ritz method. He used the mode shapes from a few selected design configurations to predict the response at any design point throughout the design domain. Although, PROM is a local approximation method, it can handle larger structural changes because it uses information from multiple design points. The Combined Approximations (CA) method developed by Kirsch [9, 10, 11, 12, 13] combines the strength of both local and global approximations and is shown to be accurate even for large structural modifications. It uses a combination of binomial series (local) approximation (Neumann expansion) and a global reduced basis approximation. The method was originally applied to linear static problems and later extended to linear dynamics problems . An extended CA method was developed in  for large perturbations using the Rayleigh quotient. The use of the Epsilon algorithm for eigen-problem reanalysis using a Neumann series expansion was also demonstrated in .
A Modified Combined Approximation (MCA) method was introduced in [15, 16] for reanalysis of dynamic problems with many dominant modes. Recently, a Frequency Shift Combined Approximation (FSCA) method was proposed in  providing an improvement of the CA method based on frequency shifting. It was demonstrated that the use of the Epsilon algorithm  accelerates convergence. The authors in  claim that frequency shifting improves the accuracy and efficiency of the MCA method.
Reanalysis methods use a basis of eigenvectors from the baseline design, augmented sometimes by eigenvectors from other design points as in PROM , to estimate the eigenvalues and eigenvectors of a modified design without performing a new eigenvalue analysis. However, all basis eigenvectors are not usually linearly independent. To avoid numerical errors from this dependency, Gram-Schmidt orthogonalization must be performed. In general, this is very costly especially when the basis is large (many basis vectors) and/or the model size is large.
In this paper, we propose an improvement of the MCA method by introducing frequency shifting in order to eliminate the need for Gram Schmidt orthogonalization improving therefore, the efficiency of the MCA method. We present examples and demonstrate the superior efficiency and accuracy of the proposed method.
REVIEW OF EXISTING REANALYSIS METHODS
In this section, we briefly review the most popular reanalysis methods PROM, CA, MCA and FSCA.
Parametric Reduced Order Modeling (PROM)
The PROM method approximates the mode shapes of a new design in the subspace spanned by the dominant mode shapes of some representative designs, which are selected so as the formed basis captures the dynamic characteristics in each dimension of the parameter space. Balmes [7, 8] suggested that each of these representative designs should be at the mid-range of each design parameter in the parameter space.
For a structure with m design variables, Zhang  suggested that the representative designs include a baseline design for which all parameters are at their lower limits plus m designs obtained by perturbing the design variables from their lower limits to their upper limits, one at a time. The points representing these designs in the space of the design parameters are called corner points. This resulted in a more accurate PROM algorithm compared with the original Balmes [7, 8] algorithm.
In PROM, the mode shapes of a new design are approximated in the space of the mode shapes of the corner points as
[PHI] [approximately equal to] [[??].sub.p] = P[THETA] (1)
where the modal matrix P contains the basis vectors as in Equation (2)
P=[[[PHI].sub.0] [[PHI].sub.1] ... [[PHI].sub.m]] (2)
and [THETA] contains the coefficients of these vectors. The columns of P are the dominant mode shapes of the above (m + 1) designs. [[PHI].sub.0] is the modal matrix composed of the dominant mode shapes of the baseline design, and [[PHI].sub.i] is the modal matrix of the [it.sup.h] corner point. The mode shapes of the new design satisfy the following eigenvalue problem,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)
where [LAMBDA] is a diagonal matrix of the first s eigenvalues.
A reduced eigenvalue problem is obtained by pre-multiplying both sides of Equation (3) by [P.sup.T] to obtain
[K.sub.R][THETA] = [LAMBDA][M.sub.R][THETA] (4)
where the reduced stiffness and mass matrices are
[K.sub.R] = [P.sup.T]KP and [M.sub.R] = [P.sup.T]MP. (5)
The matrix [THETA] in Equation (4) contains the eigenvectors of the reduced stiffness and mass matrices [K.sub.R] and [M.sub.R].
For m design variables, (m + 1) eigenvalue problems must be solved in order to form the basis P in Equation (2). Therefore, both the cost of obtaining the modal matrices [[PHI].sub.i] and the size of matrix P increase linearly with m. The PROM approach uses the following algorithm to compute the mode shapes of a new design:
1. Find the mode shapes of the baseline design and the designs corresponding to the m corner points in the design space, and form subspace basis P.
2. Find the reduced stiffness and mass matrices [K.sub.R] and [M.sub.R] from Equation (5).
3. Solve eigen-problem (4) for matrix [THETA].
4. Reconstruct the approximated eigenvectors in [[??].sub.p] using Equation (1).
In the above procedure, step 1 is performed only once. A reanalysis requires only steps 2 to 4. For a small number of mode shapes and a small number of design parameters, the cost of steps 2 to 4 is much smaller compared to the cost of a full analysis.
The computational cost of PROM consists of a) the cost of performing (m + 1) full eigen-analyses to form subspace basis P in Equation (7), and b) the cost of reanalysis of each new design in steps 2 to 4. The former is considered the fixed cost of PROM because it does not depend on the number of reanalyses and the latter represents the variable cost of PROM because it is proportional to the number of reanalyses. The fixed cost is not attributed to the calculation of the response of a particular design. Simply, it is required to obtain the information needed to apply PROM for a given problem. The variable cost (cost of reanalysis of a new design in part b) is small compared to the fixed cost.
The fixed cost of PROM is proportional to the number of design variables m since the basis P consists of the dominant eigenvectors [[PHI].sub.0] of the baseline design, and the dominant eigenvectors [[PHI].sub.i], i = 1, ..., m of the m corner design points as shown in Equation (2). As the size of P increases, so does the fixed cost because more eigenvalue problems and mode shapes must be calculated. The PROM method results in significant cost savings when applied to problems that involve few design variables (less than 20) and require many analyses (e.g. Monte Carlo simulation or gradient-free optimization using genetic algorithms).
Combined Approximations (CA)
In the CA method [9, 10, 11, 12, 13], the eigenvectors at a new design are approximated using a linear combination of basis vectors. A subspace basis is formed as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)
where s is the number of basis vectors normally in the range of 3-6 depending on accuracy and efficiency [10, 20].
For a modified design, the stiffness and mass matrices are given by
[K.sub.p]=[K.sub.0]+[LAMBDA]K, [M.sub.p],=[M.sub.0]+[LAMBDA]M (7)
where [K.sub.0] and [M.sub.0] are the stiffness and mass matrices of the baseline (original) design and [DELTA]K and [DELTA]M represent large perturbations. The basis vectors in Equation (6) are given by Equations (8) and (9)  as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)
where [[PHI].sub.0] are the eigenvectors of the baseline design. For the perturbed (new) design, a reduced model is obtained with the following stiffness and mass matrices
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)
and an eigenvalue problem is solved using [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] of Equation (10) to calculate the eigenvectors [THETA] of the reduced model. These eigenvectors are then projected back to the physical domain to obtain the following approximate eigenvectors [[PHI].sub.p] of the new design as
[[PHI].sub.p] =[R.sub.nxs] [THETA]. (11)
Because the basis vectors are linearly dependent, Gram Schmidt orthogonalization is performed to avoid numerical errors. The CA method is efficient for problems where the number of desired modes is low. In such a case, the computational effort of stiffness matrix decomposition (Equations 8 and 9) dominates.
The CA method has the following three main advantages:
1. It only requires a single matrix decomposition of stiffness matrix [K.sub.0] to calculate the subspace basis R,
2. It is accurate because the basis is updated for every new design, and
3. The eigenvectors of a new design are efficiently approximated using Equation (11) where the eigenvectors [THETA] correspond to a much smaller eigen-problem.
Kirsch  has suggested using frequency shifts to improve the estimation accuracy of higher modes. By introducing a frequency shift [mu] as
[??] = [lambda] - [mu] (12)
the modified stiffness becomes
[??] = K - [mu]M = [K.sub.0] + ([DELTA]K - [mu]M) (13)
The shift [mu] can be calculated as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)
Due to the increased computational cost however, the Rayleigh quotient formula is used instead
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)
The concept of shifts is utilized to improve the quality of basis vectors in order to improve the estimation of the eigenvectors of the modified design. The eigenvectors computed using the stiffness with shifting (Equation 13) are the same without shifting (Equation 4).
It should be noted that the CA method is more suitable than PROM, if the number of reanalyses is less than the number of design parameters. However, it is efficient only for problems where the number of retained modes is small.
Modified Combined Approximations (MCA)
A modified combined approximations (MCA) method was presented in [15, 16] where the columns of the subspace basis are obtained from by the following recursive equations
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)
where [K.sub.p] and [M.sub.p] are the stiffness and mass matrices of the modified design. The subspace basis is defined as
T = [[[PHI].sub.0] [T.sub.1] ... [T.sub.s],] (18)
where s is equal to 1 or 2. Gram Schmidt orthogonalization is again performed to avoid numerical errors from linear dependency. A reduced model is defined using the following reduced stiffness and mass matrices
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19)
and the eigenvalue problem [K.sup.R][THETA] = [[lambda].sub.0][M.sup.R][THETA]is then solved to obtain the eigenvectors [THETA] which are then used to approximate the eigenvectors of the modified design as
[[PHI].sub.p] = T[THETA] (20)
Although the MCA method has a slightly higher matrix decomposition cost as the matrix [K.sub.p] must be decomposed for each new design, the overall computational effort of MCA is lower than CA because the forward-backwards cost in solving Equations (16) and (17) is much lower than that of CA. For this reason, MCA is computationally more efficient than CA for large size models with many modes.
Frequency Shift Combined Approximations (FSCA)
Xu et al.  has proposed a new reanalysis method, called FSCA, based on an inverse iteration with frequency shifts and convergence acceleration using the Epsilon algorithm. Two algorithms are proposed namely, FSCA I and FSCA II. The authors claim that FSCA is more accurate and efficient compared to the CA and MCA methods for large modifications and a large number of retained modes.
Recall that the eigenproblem formulation is given by
K[PHI] = [LAMBDA]M[PHI] (21)
where [PHI] is the modal basis (matrix of modes) and [LAMBDA] is a diagonal matrix of the corresponding eigenvalues. We will briefly present the FSCA I algorithm and then provide the main difference between the FSCA II and FSCA I algorithms for completeness.
Subtracting [[mu].sup.-1]K[PHI] from both sides of Equation (21) yields
K[PHI] - [[mu].sup.-1] K[PHI] = [LAMBDA]M[PHI] -[mu]-1 K[PHI] (22)
or after some algebra
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (23)
To accelerate the convergence of obtaining [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Xu et al.  proposed to express [[PHI].sub.nxm] as a linear combination of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], i = 0, 1, 2, ... ., s-1 as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (24)
where, the basis vector R is
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (25)
Similarly we get
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (26)
Using [[PHI].sub.nxm] from Equation (24) in place of [PHI] in Equation (21) and premultiplying with [R.sup.T], we get the following reduced model
[R.sup.T]KRX = [LAMBDA][R.sup.T]MRX. (27)
The above equation is solved for [X.sub.msxm]. The original eigenvectors are then obtained as
[[PHI].sub.nxm] = [R.sub.nxms] [X.sub.msxm]. (28)
The FSCA II algorithm is obtained by subtracting [mu]M[PHI] from both sides of Equation (21); i.e.,
K[PHI] - [mu]M[PHI] = [LAMBDA]M[PHI] - [mu]M[PHI] (29)
Equations similar to (23) through (28) are then used to define the FSCA II method.
To improve efficiency, the convergence of the series in Equation (24) can be accelerated. Pade approximation and Shanks transformation [21, 22] have been shown to improve the accuracy of reanalysis as they can accelerate convergence of a series. Recently, the Epsilon algorithm has been used to accelerate the convergence of a series [23, 24]. In several publications, the Epsilon algorithm has been integrated with the CA method for reanalysis of static and vibratory problems [14, 18, 21]. The Epsilon algorithm has been used to accelerate the convergence of the CA sequence of Equations (8) and (9) improving accuracy and efficiency.
From Equation (25), we obtain
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (30)
where, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the [j.sup.th] (j = 1, 2, ...,m) eigenvector for the [i.sup.th] iteration (i = 0,1,...,s-1). Based on Equation (30), the partial sums of the sequence are given as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (31)
According to the Epsilon algorithm, for a given sequence [s.sub.0], [s.sub.1], [s.sub.2], ..., an iterative process is formed as in Equation (32) using the Epsilon algorithm iterative table of Figure 1. This iterative process is used to generate the basis vectors.
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (32)
Entries with even subscripts such as in [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are only useful for extrapolation. Usually, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] converges to the exact solution . Using the following definition of the inverse of a real vector [17, 25]
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (33)
the term [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] in Equation (32) is expressed as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (34)
An advantage of the FSCA method is the reduction of the computational effort for orthonormalization because the size of the basis is reduced resulting in a faster runtime. This can be understood by the following example. Assume we have 50 modes (eigenvectors) for a particular iteration. If we need 5 iterations, the total number of eigenvectors in the basis will be 250. Using the Epsilon algorithm we reduce the basis size to 50, reducing therefore, the effort for orthonormalization.
PROPOSED METHOD - MODIFIED COMBINED APPROXIMATIONS WITH FREQUENCY SHIFTING (MCAS)
It was shown in  that MCA is accurate and more efficient than CA when the number of desired modes is high. However, as the size of the basis increases by increasing s in Equations (17) and (18), the MCA method becomes costly due to the orthonormalization of a larger basis.
Because the stiffness matrix of a free-free model is singular, we propose using frequency shifting to eliminate the singularity. The main advantage is the elimination of the costly orthonormalization process. We should note that as the basis is formed iteratively, the estimated eigenvectors from all iterations are linearly dependent. To remove this dependency, Gram Schmidt orthonormalization must be employed which becomes costlier as the basis size increases. The proposed modified combined approximation with frequency shift reduces the computational cost by eliminating the costly orthonormalization increasing therefore, efficiency especially for problems with many modes. We have also observed that only one iteration (s = 1) suffices for either a constrained or a free-free model using the MCA method to get accurate results. This is not true for the CA and FSCA methods. Orthonormalization is also needed for the latter.
Assuming a shift factor [mu], a modified stiffness is defined as
K = [K.sub.p] - [mu][M.sub.p]. (35)
The shift factor can be either a constant (e.g., 100) or it can be determined from the Rayleigh quotient of Equation (15). The basis is then generated similarly to the original MCA method as
T = [T.sub.1] = [K.sup.-1] [M.sub.p] [[PHI].sub.0] (36)
As we mentioned, a single iteration suffices to obtain accurate results with the added benefit of eliminating the orthonormalization. The accuracy of this approach is demonstrated in the next section.
In this section, we demonstrate the superior efficiency and accuracy performance of the proposed MCAS method using two examples; a small size model (less than 50,000 DOF) and a large size model (more than 1,000,000 DOF). MSC/NASTRAN with DMAP (Direct Matrix Abstraction Programming) has been used to obtain the results for both examples. We show that the proposed MCAS method is the most efficient and accurate reanalysis method for both small and large size models.
Small Size Truck Model
The model we use in this section is a Chevy C2500 pickup truck with 43,116 DOF (Figure 2). The model is free-free. We develop a modal model with eigenvectors up to 200 Hz to calculate the frequency response in the range of 5 to 85 Hz. Figure 3 shows a schematic of the parts we modify. The arrows indicate the locations where a load is applied and the response is measured.
Table 1 lists the design variables and their baseline and modified (perturbed) values. The modified design is a large perturbation of the baseline design. The design variables include plate thickness, mass, density and Young's modulus. The baseline and perturbed values of density and Young's modulus correspond to steel and aluminum, respectively.
A basis is constructed using the baseline values, and the frequency response is calculated for a modified design with perturbed values using the CA, FSCA, MCA, and the proposed MCAS reanalysis techniques.
A unit lateral load is applied at the center of the doors and the response is measured at the same point. Similarly, unit vertical loads are applied at the cab and rails and the responses are measure at the same points (arrows in Figure 3).
Figures 4 to 6 compare the frequency response at the left door, truck bed and left rail respectively, for all reanalysis methods.
Figure 4 shows that the left door outer response calculated by the proposed MCAS method is almost exact. The CA method is also very accurate. However, the accuracy of both FSCA I and FSCA II methods is not good. The responses at the truck bed (Figure 5) and left rail (Figure 6) locations show similar trends.
Table 2 compares the computational effort among all reanalysis methods. The CA method with shifting is more efficient than MCA. Also, the FSCA I and FSCA II methods using the Epsilon algorithm, are more efficient that CA and MCA. However, the proposed MCAS method is by far the most efficient reanalysis method. It should be noted that two iterations (s = 2) are used to build the basis vectors for the CA and MCA methods. The FSCA methods required a minimum of 5 iterations to generate the basis (Refer to Figure 1). The proposed MCAS method required only one iteration to generate the basis. Therefore, for a small model size and a few dominant modes (58 modes up to 90 Hz in this example), the proposed MCAS method is the most efficient and most accurate (Figures 4 to 6) among all reanalysis methods.
In this section, we use the large size truck model of Figure 7. The Checy C2500 pickup truck model of the previous section was modified by adding trim components to make it more realistic. The model has 3,953,837 DOF and is free-free. Modes up to 90 Hz (300 modes) were utilized to build a modal model. Figure 8 shows the selected parts to be modified.
Table 3 shows the design variables and their baseline and modified (perturbed) values. The perturbed values of density and Young's modulus correspond to aluminum.
Similarly to the previous section, the design variables include plate thicknesses, and density and Young's modulus change from steel to aluminum. The model is excited by a unit lateral load on the left door, a unit vertical load on the truck bed and a unit vertical load on the left rail. The responses are calculated at the same points. Figures 9, 10 and 11 compare the frequency response among CA, MCA, FSCA and MCAS methods. All three figures indicate that the MCA and the proposed MCAS method are much more accurate than the CA with shifting or the most recent FSCA I and FSCA II methods.
Table 4 compares the computational effort among all reanalysis techniques. The MCA and the proposed MCAS methods are much more efficient compared to the other reanalysis methods. They are also 25% and 34% respectively, more efficient than exact analysis. The efficiency of MCAS becomes more evident if more modes are retained.
SUMMARY AND CONCLUSIONS
An efficient and accurate reanalysis approach was presented for repeated analyses (e.g., deterministic or probabilistic optimization) of dynamic systems. The new method, called MCAS (Modified Combined Approximations with frequency Shifts) enhances the existing MCA reanalysis method by adding frequency shifting. This eliminates the numerical issues associated with singularities in the stiffness and mass matrices removing therefore, the need of using the computationally inefficient Gram Schmidt orthonormalization to obtain linearly independent eigenvectors. This in turn speeds up the reanalysis process showing significant reduction in the computational effort without any adverse effect on accuracy. Two examples involving small and large size models demonstrated the superior efficiency and accuracy of the proposed MCAS reanalysis approach compared with all previously available reanalysis methods.
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Syed F. Haider
Zissimos P. Mourelatos
Professor, Mechanical Engineering Department
Oakland University, Rochester, MI
Syed F. Haider
PhD Candidate, Mechanical Engineering Department
Oakland University, Rochester, MI
Table 1. Design variables for small size truck model Thickness (mm) Baseline value Perturbed value Hood 0.866 1.125 Cabin 0.866 1.25 Door outer Panel 0.866 1.125 Truck Bed 1.5 2.5 Rails 3.0 4.0 Mass (Kg) Baseline value Perturbed value Side view mirror 0.2 0.4 Spare wheel 15.0 30.0 Density (Mg/mm3) Baseline value Perturbed value Hood 7.85e-9 2.71e-9 Cabin 7.85e-9 2.71e-9 Door outer Panel 7.85e-9 2.71e-9 Young's Modulus Baseline value Perturbed value Hood 210000 71000 Cabin 210000 71000 Door outer Panel 210000 71000 Table 2. Computational effort comparison for small size truck model Method Time (sec) CA with shifting (s = 2) 646 MCA (s = 2) 891 MCAS(s=1) 93 FSCA 1 (s = 5) 380 FSCA 2 (s = 5) 387 Table 3. Design variables for large size truck model Thickness (mm) Baseline value Perturbed value Door inner Panel 0.866 1.5 Roof 0.866 1.732 Door outer Panel 0.866 1.732 Truck Bed 0.866 1.732 IP panel 0.866 1.732 Front inner rail 3.0 4.0 Front outer rail 3.0 4.0 Rear inner rail 3.0 5.0 Mass (Kg) Baseline value Perturbed value Spare wheel 15.0 20.0 Density Baseline value Perturbed value (Mg/[mm.sup.3]) Door inner Panel 7.85e-9 2.71e-9 Door outer Panel 7.85e-9 2.71e-9 Young's Baseline value Perturbed value Modulus (MPa) Door inner Panel 206800 71800 Door outer Panel 206800 71800 Table 4. Computational effort comparison for large size truck model Method Time (sec) Exact analysis 6298 CA with Shifting (s = 2) 13175 MCA with (s = 2) 4696 MCAS(s = 1) 4167 FSCA 1 (s = 5) 12810 FSCA 2 (s = 5) 13440
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|Author:||Haider, Syed F.; Mourelatos, Zissimos|
|Publication:||SAE International Journal of Passenger Cars - Mechanical Systems|
|Date:||Apr 1, 2016|
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