$$ m _ {\alpha i} ^ {\mathrm{eff}} = \left(\Gamma_ {\alpha i}\right) ^ {2} m _ {\alpha} \quad \mathrm{(nosumover} \alpha \mathrm{)}. $$ If the effective masses of all modes are added in any particular direction, the sum should give the total mass of the model, except for mass at kinematically restrained degrees of freedom. Thus, if the effective masses of the modes used in the analysis add up to a value that is significantly less than the model's total mass, this suggests that modes that have significant participation in excitation in that direction have not been extracted. # Composite modal damping ABAQUS/Standard provides an option to define a composite damping factor for each material. These are assembled into fractions of critical damping values for each mode, $\left( \xi _ { \alpha } \right) _ { c }$ , according to $$ \left(\xi_ {\alpha}\right) _ {c} = \frac {1}{m _ {\alpha}} \phi_ {\alpha} ^ {N} \left(\sum_ {a} \xi_ {a} M _ {a} ^ {N M}\right) \phi_ {\alpha} ^ {M} \quad \mathrm{(nosumover} \alpha \mathrm{)}, $$ where $\xi _ { a }$ is the critical damping fraction given for material a and ${ \cal M } _ { a } ^ { N M }$ is the part of the structure's mass matrix made up of material a. # 2.5.3 Linear dynamic analysis using modal superposition Linear dynamic analysis using modal superposition is computationally inexpensive and can provide useful insight into the dynamic behavior of a system. With modern eigenvalue/eigenvector extraction techniques--such as the subspace method available in ABAQUS/Standard--the cost of obtaining a sufficient basis of eigensolutions is not excessive; and the subsequent computational effort involved in obtaining the dynamic response by modal superposition methods is relatively small, especially when compared to the cost of the direct integration methods used for general nonlinear analysis ( \`\`Implicit dynamic analysis,'' Section 2.4.1). The basic concept of modal superposition is that the response of the structure is expressed in terms of a relatively small number of eigenmodes of the system. The orthogonality of the eigenmodes uncouples this system. Furthermore, only eigenmodes that are close to the frequencies of interest are usually needed; for example, only the lowest few frequencies are usually required to obtain an accurate estimate of a structure's linear dynamic response to relatively long-term loading (for example, its steady-state response to low frequency excitation). The technique can be extended in a limited way into the nonlinear régime, but the superposition and orthogonality principles apply only to purely linear systems: for this reason the methods described in this section are implemented only for linear analysis. ABAQUS/Standard has two "subspace" procedures--one for nonlinear dynamic and the other one for steady-state dynamic analysis--that use some of the eigenmodes of the system on which the equilibrium equations are projected. In both cases the system's eigenmodes are used as a set of global basis vectors for computing the dynamic response, even though the system exhibits nonlinear or frequency-dependent effects during the dynamic response. These methods are cost-effective compared to fully nonlinear dynamic response analysis developed in terms of all the system's degrees of freedom. # Procedures The subspace projection method for steady-state response is described in \`\`Subspace-based steady-state dynamic analysis,'' Section 2.6.2. The time-domain subspace projection method is described in \`\`Subspace dynamics,'' Section 2.4.3. The procedures provided for modal dynamic analysis of linear systems are summarized below: a. Modal dynamic time history analysis (see \`\`Modal dynamic analysis,'' Section 2.5.5). This procedure can be used to obtain the time history response of a system to loading conditions that are given as functions of time. The response is integrated through time: the integration method used is exact for loadings that vary piecewise linearly with time. Thus, the only approximations in this analysis procedure are the linearization of the problem, the spatial modeling (that is, the choice of the finite element model), the loading definitions, and the choice of the number of eigenmodes used to represent the system. b. Response spectrum analysis (see \`\`Response spectrum analysis,'' Section 2.5.6). Response spectrum analysis is often used to obtain an approximate upper bound to the peak significant response of a system to an input spectrum as a function of frequency: it gives the maximum response of a one degree of freedom system as a function of its fundamental frequency of vibration and of its damping ratio. The method has very low computational cost and gives useful information about the spectral behavior of a system with respect to frequency. c. Steady-state harmonic response analysis (see \`\`Steady-state linear dynamic analysis,'' Section 2.5.7, and \`\`Subspace-based steady-state dynamic analysis,'' Section 2.6.2). This procedure is used when the steady-state response of a system to harmonic excitation is required. The solution is given as the peak amplitudes and phase relationships of the solution variables (stress, displacement, etc.) as functions of frequency: postprocessing options are provided to display such results conveniently. A similar option is provided for direct harmonic response analysis without using the eigenmodes as a basis. The direct method is significantly more expensive computationally than the modal method: it is needed if the system is nonsymmetric (because ABAQUS presently does not have a nonsymmetric eigenvalue extraction capability) or if the system's behavior includes frequency-dependent parameters. The "subspace" method is typically less expensive than the direct method. It is generally used for nonsymmetric systems or when the system's behavior includes frequency-dependent parameters or discrete damping. d. Random response analysis (see \`\`Random response analysis,'' Section 2.5.8). This procedure is used when the structure is excited continuously and the loading can be expressed statistically in terms of a "Power Spectral Density Function." The response is calculated in terms of statistical quantities, such as the mean value and the standard deviation of nodal and element variables. # 2.5.4 Damping options for modal dynamics For linear dynamic analysis based on modal superposition, several options are provided in ABAQUS/Standard to introduce damping, as follows: # Critical damping factors The damping in each eigenmode can be given as a fraction of the critical damping for that mode. The equation of motion for a one degree of freedom system (one of the eigenmodes of the system) is $$ m \ddot {q} + c \dot {q} + k q = 0, $$ where m is the mass, c the damping, k the stiffness, and q the modal amplitude. The solution is of the form $$ q = A \exp \lambda t, $$ where A is a constant, and $$ \lambda = \frac {- c}{2 m} \pm \sqrt {\frac {c ^ {2}}{4 m ^ {2}} - \frac {k}{m}}. $$ The solution will be oscillatory if the expression under the root sign is negative. Critical damping is defined as the damping that makes this expression zero: $$ c _ {c r} = 2 \sqrt {m k}. $$ If the system is critically damped, after any disturbance the system will return to a static equilibrium state as rapidly as possibly without any oscillation. Typically, when damping is given as a fraction of critical damping associated with each mode, the values used are in the range of 1% to 10% of critical damping. This method of introducing damping has no physical basis in the finite element model: it is a purely mathematical concept introduced in association with the eigenmodes of the system. Thus, the concept cannot be extended to nonlinear applications where the equations of motion of the system are integrated directly and where the natural frequencies of the system are constantly changing because of nonlinearities. # Rayleigh damping Rayleigh damping is defined by a damping matrix formed as a linear combination of the mass and the stiffness matrices: $$ C ^ {M N} = \alpha M ^ {M N} + \beta K ^ {M N}. $$ With Rayleigh damping the eigenvectors of the damped system are the same as the eigenvectors of the undamped system. Rayleigh damping can, therefore, be converted into critical damping fractions for each mode: this is the way Rayleigh damping is handled in ABAQUS/Standard. A form of Rayleigh damping is also provided in ABAQUS for nonlinear analysis. When the problem is nonlinear the mass damping factor can be used directly: the stiffness damping factor is interpreted as creating viscoelastic behavior in which the viscosity is proportional to the elasticity, which gives exactly the stiffness proportional damping effect defined above for the linear case. # Composite modal damping When composite modal damping is used, a damping value is defined for each material as a fraction of critical damping to be associated with that material. These values are converted into a weighted average for each eigenmode, weighted by the mass matrix according to the equation $$ \xi_ {\alpha} = \frac {1}{m _ {\alpha}} \phi_ {\alpha} ^ {M} \xi_ {m} M _ {m} ^ {M N} \phi_ {\alpha} ^ {N} \quad \mathrm{(nosumover} \alpha \mathrm{)}, $$ where $\xi _ { \alpha }$ is the critical damping ratio used in mode $\alpha ; \xi _ { m }$ is the critical damping fraction defined for material m; $M _ { m } ^ { M N }$ is the mass matrix associated with material m; $\phi _ { \alpha } ^ { M }$ is the eigenvector of the ®th mode; and $m _ { \alpha }$ is the generalized mass associated with the ®th mode $( m _ { \alpha } = \phi _ { \alpha } ^ { M } M ^ { M N } \phi _ { \alpha } ^ { N }$ , no sum over ®). # Structural damping Structural damping assumes that the damping forces are proportional to the forces caused by stressing of the structure and are opposed to the velocity. This form of damping can be used only if the displacement and velocity are exactly $9 0 ^ { \circ }$ out of phase, which is the case when the excitation is sinusoidal, so structural damping can be used only in steady-state and random response analysis. The damping forces are then $$ F _ {D} ^ {N} = i s I ^ {N}, $$ where $I ^ { N }$ are the forces caused by stressing of the structure, $F _ { D } ^ { N }$ are the damping forces, s is the structural damping factor, and $i = \sqrt { - 1 }$ . Any combination of damping options can be used in an analysis: the effects will be added if several damping definitions are chosen. # 2.5.5 Modal dynamic analysis The modal dynamic procedure provides time history analysis of linear systems. The excitation is given as a function of time: it is assumed that the amplitude curve is specified so that the magnitude of the excitation varies linearly within each increment. When the model is projected onto the eigenmodes used for its dynamic representation, we obtain a set of uncoupled one degree of freedom systems, for any of which the equilibrium equation at time t is Equation 2.5.5-1 $$ \ddot {q} + 2 \xi \omega \dot {q} + \omega^ {2} q = f _ {t} = f _ {t - \Delta t} + \frac {\Delta f}{\Delta t} \Delta t, $$ where $\xi$ is the critical damping ratio (the ratio of the damping term in this equation to that damping that would cause critical damping of the equation); $q$ is the "generalized coordinate" of the mode (the amplitude of the response in this mode); $\omega = \sqrt { k / m }$ is the natural frequency of the undamped mode (obtained as the square root of the eigenvalue in the \*FREQUENCY step that precedes the modal dynamic time history analysis); $f$ is the magnitude of the loading projected onto this mode (the "generalized load" for the mode); and $\Delta f$ is the change in $f$ over the time increment, which is $\Delta t .$ . The solution to this equation is readily obtained as a particular integral for the loading and a solution to the homogeneous equation (with no right-hand side). These solutions can be combined and written in the general form Equation 2.5.5-2 $$ \left\{ \begin{array}{c} q _ {t + \Delta t} \\ \dot {q} _ {t + \Delta t} \end{array} \right\} = \left[ \begin{array}{c c} a _ {1 1} & a _ {1 2} \\ a _ {2 1} & a _ {2 2} \end{array} \right] \left\{ \begin{array}{c} q _ {t} \\ \dot {q} _ {t} \end{array} \right\} + \left[ \begin{array}{c c} b _ {1 1} & b _ {1 2} \\ b _ {2 1} & b _ {2 2} \end{array} \right] \left\{ \begin{array}{c} f _ {t} \\ f _ {t + \Delta t} \end{array} \right\}, $$ where $a _ { i j }$ and $b _ { i j } , i , j = 1 , 2$ , are constants, since we have assumed that the loading only varies linearly over the time increment (that is, $\Delta f / \Delta t$ is constant). There are three cases of this solution for nonrigid-body motion $( \omega \neq 0 )$ , depending on whether the damping in the modal equilibrium equation is greater than, equal to, or less than critical damping (that is, depending on whether $( \xi ^ { 2 } - 1 )$ is positive, zero, or negative). For convenience, we define $$ \bar {\omega} = \omega \sqrt {1 - \xi^ {2}}. $$ # Damping less than critical This case is the most common and gives $$ a _ {1 1} = \exp (- \xi \omega \Delta t) \left(\xi \frac {\omega}{\bar {\omega}} \sin \bar {\omega} \Delta t + \cos \bar {\omega} \Delta t\right) $$ $$ a _ {1 2} = \exp (- \xi \omega \Delta t) \frac {1}{\bar {\omega}} \sin \bar {\omega} \Delta t $$ $$ a _ {2 1} = - \exp (- \xi \omega \Delta t) \frac {\omega}{\sqrt {1 - \xi^ {2}}} \sin \bar {\omega} \Delta t $$ $$ a _ {2 2} = \exp (- \xi \omega \Delta t) \left(\cos \bar {\omega} \Delta t - \frac {\xi \omega}{\bar {\omega}} \sin \bar {\omega} \Delta t\right) $$ $$ b _ {1 1} = - \exp \left(- \xi \omega \Delta t\right) \left\{\left(\frac {\xi}{\omega \bar {\omega}} + \frac {2 \xi^ {2} - 1}{\omega^ {2} \bar {\omega} \Delta t}\right) \sin \bar {\omega} \Delta t + \left(\frac {1}{\omega^ {2}} + \frac {2 \xi}{\omega^ {3} \Delta t}\right) \cos \bar {\omega} \Delta t \right\} + \frac {2 \xi}{\omega^ {3} \Delta t} _ {1 2} = $$ $$ \exp \left(- \xi \omega \Delta t\right) \left\{\frac {2 \xi^ {2} - 1}{\omega^ {2} \bar {\omega} \Delta t} \sin \bar {\omega} \Delta t + \frac {2 \xi}{\omega^ {3} \Delta t} \cos \bar {\omega} \Delta t \right\} + \frac {1}{\omega^ {2}} - \frac {2 \xi}{\omega^ {3} \Delta t} $$ # Procedures $$ \begin{array}{l} b _ {2 1} = - \exp (- \xi \omega \Delta t) \left\{\left(\bar {\omega} \cos \bar {\omega} \Delta t - \xi \omega \sin \bar {\omega} \Delta t\right) \left(\frac {2 \xi^ {2} - 1}{\omega^ {2} \bar {\omega} \Delta t} + \frac {\xi}{\bar {\omega} \omega}\right) \right. \\ - \left(\bar {\omega} \sin \bar {\omega} \Delta t + \xi \omega \cos \bar {\omega} \Delta t\right) \left(\frac {1}{\omega^ {2}} + \frac {2 \xi}{\omega^ {3} \Delta t}\right) \Biggr \} - \frac {1}{\omega^ {2} \Delta t} _ {2 2} = \\ - \exp (- \xi \omega \Delta t) \left\{- (\bar {\omega} \cos \bar {\omega} \Delta t - \xi \omega \sin \bar {\omega} \Delta t) \frac {2 \xi^ {2} - 1}{\bar {\omega} \omega^ {2} \Delta t} + (\bar {\omega} \sin \bar {\omega} \Delta t + \xi \omega \cos \bar {\omega} \Delta t) \frac {2 \xi}{\omega^ {3} \Delta t} \right\} \\ + \frac {1}{\omega^ {2} \Delta t} \\ \end{array} $$ # Damping equal to critical In this case $$ a _ {1 1} = \exp (- \xi \omega \Delta t) (1 + \xi \omega \Delta t) $$ $$ a _ {1 2} = \exp (- \xi \omega \Delta t) \Delta t $$ $$ a _ {2 1} = - \exp (- \xi \omega \Delta t) (\xi \omega) ^ {2} \Delta t $$ $$ a _ {2 2} = \exp (- \xi \omega \Delta t) (1 - \xi \omega \Delta t) $$ $$ b _ {1 1} = - \exp (- \xi \omega \Delta t) \left\{- (1 + \xi \omega \Delta t) \left(\frac {1}{\omega^ {2}} + \frac {2 \xi}{\omega^ {3} \Delta t}\right) + \frac {1}{\omega^ {2}} \right\} $$ $$ b _ {1 2} = \exp \left(- \xi \omega \Delta t\right) \left\{\left(1 + \xi \omega \Delta t\right) \left(\frac {2 \xi}{\omega^ {3} \Delta t} - \frac {1}{\omega^ {2}}\right) + \frac {1}{\omega^ {2}} \right\} - \frac {2 \xi}{\omega^ {3} \Delta t} _ {2 1} = $$ $$ - \exp (- \xi \omega \Delta t) \left\{(\xi \omega) ^ {2} \Delta t \left(\frac {1}{\omega^ {2}} + \frac {2 \xi}{\omega^ {3} \Delta t}\right) + \frac {1 - \xi \omega \Delta t}{\omega^ {2} \Delta t} \right\} - \frac {1}{\Delta t \omega^ {2}} $$ $$ b _ {2 2} = - \exp (- \xi \omega \Delta t) \left\{- (\xi \omega) ^ {2} \Delta t \frac {2 \xi}{\omega^ {3} \Delta t} - \frac {1 - \xi \omega \Delta t}{\Delta t \omega^ {2}} \right\} + \frac {1}{\omega^ {2} \Delta t} $$ # Damping higher than critical In this case $$ a _ {1 1} = \exp (- \xi \omega \Delta t) \left(\xi \frac {\omega}{\bar {\omega}} \sinh \bar {\omega} \Delta t + \cosh \bar {\omega} \Delta t\right) $$ $$ a _ {1 2} = \exp (- \xi \omega \Delta t) \frac {1}{\bar {\omega}} \sinh \bar {\omega} \Delta t $$ $$ a _ {2 1} = \exp (- \xi \omega \Delta t) \frac {\omega}{\sqrt {\xi^ {2} - 1}} \sinh \bar {\omega} \Delta t $$ $$ a _ {2 2} = \exp (- \xi \omega \Delta t) \left(\cosh \bar {\omega} \Delta t - \frac {\xi \omega}{\bar {\omega}} \sinh \bar {\omega} \Delta t\right) $$ $$ b _ {1 1} = - \exp \left(- \xi \omega \Delta t\right) \left\{\left(\frac {\xi}{\omega \bar {\omega}} + \frac {2 \xi^ {2} - 1}{\omega^ {2} \bar {\omega} \Delta t}\right) \sinh \bar {\omega} \Delta t + \left(\frac {1}{\omega^ {2}} + \frac {2 \xi^ {2}}{\omega^ {3} \Delta t}\right) \cosh \bar {\omega} \Delta t \right\} + \frac {2 \xi}{\omega^ {3} \Delta t} $$ $$ b _ {1 2} = \exp (- \xi \omega \Delta t) \left\{\frac {2 \xi^ {2} - 1}{\omega^ {2} \bar {\omega} \Delta t} \sinh \bar {\omega} \Delta t + \frac {2 \xi}{\omega^ {3} \Delta t} \cosh \bar {\omega} \Delta t \right\} + \frac {1}{\omega^ {2}} - \frac {2 \xi}{\omega^ {3} \Delta t} $$ # Procedures $$ \begin{array}{l} b _ {2 1} = - \exp (- \xi \omega \Delta t) \left\{(\bar {\omega} \cosh \bar {\omega} \Delta t - \xi \omega \sinh \bar {\omega} \Delta t) \left(\frac {2 \xi^ {2} - 1}{\omega^ {2} \bar {\omega} \Delta t} + \frac {\xi}{\bar {\omega} \omega}\right) \right. \\ - \left(- \bar {\omega} \sinh \bar {\omega} \Delta t + \xi \omega \cosh \bar {\omega} \Delta t\right) \left(\frac {1}{\omega^ {2}} + \frac {2 \xi}{\omega^ {3} \Delta t}\right) \Biggr \} - \frac {1}{\omega^ {2} \Delta t} \\ \end{array} $$ $$ b _ {2 2} = - \exp (- \xi \omega \Delta t) \left\{- (\bar {\omega} \cosh \bar {\omega} \Delta t - \xi \omega \sinh \bar {\omega} \Delta t) \frac {2 \xi^ {2} - 1}{\bar {\omega} \omega^ {2} \Delta t} \right. $$ $$ \left. + \left(- \bar {\omega} \sinh \bar {\omega} \Delta t + \xi \omega \cosh \bar {\omega} \Delta t\right) \frac {2 \xi}{\omega^ {3} \Delta t} \right\} + \frac {1}{\omega^ {2} \Delta t} $$ # Rigid body mode with damping If there are rigid body modes in the finite element model, there will be one or several eigenvalues that are zero. The equation of motion (Equation 2.5.5-1) is reduced to Equation 2.5.5-3 $$ \ddot {q} + \alpha \dot {q} = f _ {t - \Delta t} + \frac {\Delta f}{\Delta t} \Delta t. $$ Only Rayleigh damping can be specified for a rigid body mode, since the critical damping is zero. Furthermore, since it is a rigid body mode, only the mass damping factor, ®, appears (stiffness damping requires that there be straining of the body). For this case $$ a _ {1 1} = 1 $$ $$ a _ {1 2} = \frac {1}{\alpha} (1 - \exp (- \alpha \Delta t)) $$ $$ a _ {2 1} = 0 $$ $$ a _ {2 2} = \exp (- \alpha \Delta t) $$ $$ b _ {1 1} = - \frac {1}{\alpha^ {2}} \left(1 - \exp \left(- \alpha \Delta t\right)\right) \left(1 + \frac {1}{\alpha \Delta t}\right) + \Delta t \left(1 + \frac {1}{\alpha \Delta t}\right) - \frac {\Delta t}{2 \alpha} $$ $$ b _ {1 2} = \frac {1}{\alpha} \left(1 - \exp \left(- \alpha \Delta t\right)\right) \frac {1}{\alpha^ {2} \Delta t} - \frac {1}{\alpha^ {2}} + \frac {\Delta t}{2 \alpha} $$ $$ b _ {2 1} = \frac {1}{\alpha} \left(1 + \frac {1}{\alpha \Delta t}\right) \left(1 - \exp \left(- \alpha \Delta t\right)\right) - \frac {1}{\alpha} $$ $$ b _ {2 2} = - \frac {1}{\alpha^ {2} \Delta t} \left(1 - \exp \left(- \alpha \Delta t\right)\right) + \frac {1}{\alpha} $$ # Rigid body mode without damping For the particular case of a rigid body mode without damping, the equation of motion ( Equation 2.5.5-1) is reduced to $$ \ddot {q} = f _ {t - \Delta t} + \frac {\Delta f}{\Delta t} \Delta t. $$ For this case $$ a _ {1 1} = 1 $$ $$ a _ {1 2} = \Delta t $$ $$ a _ {2 1} = 0 $$ $$ a _ {2 2} = 1 $$ $$ b _ {1 1} = \frac {\Delta t ^ {2}}{3} $$ $$ b _ {1 2} = \frac {\Delta t ^ {2}}{6} $$ $$ b _ {2 1} = \frac {\Delta t}{2} $$ $$ b _ {2 2} = \frac {\Delta t}{2} $$ # Response of nodal and element variables The time integration is done in terms of the generalized coordinates, and the response of the physical variables is then immediately available by summation: $$ u = \sum_ {\alpha} \phi_ {\alpha} q _ {\alpha} $$ $$ \varepsilon = \sum_ {\alpha} \varepsilon_ {\alpha} q _ {\alpha} $$ $$ \sigma = \sum_ {\alpha} \sigma_ {\alpha} q _ {\alpha} $$ $$ R = \sum_ {\alpha} R _ {\alpha} q _ {\alpha}, $$ where $\phi _ { \alpha }$ are the modes, $\varepsilon _ { \alpha } .$ , are the modal strain amplitudes, $\sigma _ { \alpha }$ are the modal stress amplitudes, and $R _ { \alpha }$ are the modal reaction force amplitudes corresponding to each eigenvector ®. # Initial conditions At the beginning of the step initial displacements and initial velocities must be converted to equivalent # Procedures values of the generalized coordinates, which can only be done exactly if the number of eigenvectors equals the number of degrees of freedom. Since this is usually not the case, the initial values of the generalized coordinate displacement and velocity are calculated as $$ q _ {\alpha} = \frac {1}{m _ {\alpha}} \phi_ {\alpha} ^ {M} M ^ {M N} x _ {0} ^ {N}, $$ where $m _ { \alpha }$ is the generalized mass for eigenvector ®, $\phi _ { \alpha }$ is the eigenvector, $M ^ { M N }$ is the mass matrix, and $x _ { 0 } ^ { N }$ are the initial displacements. Similarly, for initial velocities $$ \dot {q} _ {\alpha} = \frac {1}{m _ {\alpha}} \phi_ {\alpha} ^ {M} M ^ {M N} \dot {x} _ {0} ^ {N}. $$ # Base motion definition Many linear dynamic problems involve finding the response of a structure to a "base motion": a time history of displacement, velocity, or acceleration given for the points where the displacements of the structure are prescribed. In all cases these base motions are converted into an acceleration history. If velocities are given, the acceleration is defined by the backward difference rule $$ \ddot {z} _ {t + \Delta t} = \frac {\dot {z} _ {t + \Delta t} - \dot {z} _ {t}}{\Delta t}. $$ If displacements are given, the acceleration is defined by the rule $$ \ddot {z} _ {t + \Delta t} = \frac {z _ {t + \Delta t} - 2 z _ {t} + z _ {t - \Delta t}}{\Delta t ^ {2}}. $$ The response is calculated relative to the base. If total values of nodal variables are required, the motion at the base is added to the relative values: $$ \ddot {u} _ {\mathrm{abs}} = \ddot {z} + \ddot {x}, $$ $$ \dot {u} _ {\mathrm{abs}} = \dot {z} + \dot {x}, $$ $$ u _ {\mathrm{abs}} = z + x, $$ where $$ \dot {z} _ {t + \Delta t} = \dot {z} _ {t} + \frac {\Delta t}{2} \left(\ddot {z} _ {t} + \ddot {z} _ {t + \Delta t}\right), $$ $$ z _ {t + \Delta t} = z _ {t} + \dot {z} _ {t} \Delta t + \frac {\Delta t ^ {2}}{3} \left(\ddot {z} _ {t} + \frac {1}{2} \ddot {z} _ {t + \Delta t}\right). $$ # 2.5.6 Response spectrum analysis Response spectrum analysis is intended to provide an inexpensive approach to estimating the peak response of a model (usually a model of a structure) subjected to "base motion": the simultaneous motion of all nodes fixed with \*BOUNDARY conditions. The approach assumes that the system's response is linear, so it can be analyzed in the frequency domain using its lowest eigenmodes-- $\Phi _ { \alpha } \mathrm { - } \mathrm { - } \mathrm { a n d }$ eigenfrequencies $\omega _ { \alpha }$ --extracted in a previous \*FREQUENCY step. The method is typically used to estimate the response of a building or of a piping system in a building to an earthquake. The method is not appropriate if the excitation is so severe that nonlinear effects in the system are important. In such a case the time history of the base excitation must be known and used with the \*DYNAMIC procedure to obtain the system's response. Even for a linear system the response spectrum method provides only estimates of the peak response. If more precise values are required, the \*MODAL DYNAMIC procedure can be used to integrate the system through time and, thus, develop its response to the given base excitation. In \*RESPONSE SPECTRUM analysis the estimates of peak values are obtained by combining the peak responses of the participating modes corresponding to user-specified spectra definitions. Several approximations are introduced by response spectrum analysis. The conversion from a time history of excitation into an equivalent frequency domain spectrum is based on the behavior of a single degree of freedom system. Different spectra are often applied in different excitation directions. Once the spectra are known, the peak modal responses can be calculated. The manner in which these peak modal responses are combined to estimate the peak physical response, together with the manner in which multidirectional excitations are combined, introduces approximations. Since no one method gives good approximations for all cases, several methods are offered. These methods are discussed in the Regulatory Guide 1.92 (1976) of the U.S. Nuclear Regulatory Commission, in the papers by Anagnastopoulos (1981), Der Kiureghian (1981), and Smeby (1984) and in the book by A. K. Gupta (1990). The choice of the summation rules depends on the particular case and is a matter of the user's judgment. Since response spectrum analysis is commonly used as a basic design tool, spectra are defined in many design codes for such applications as seismic analysis of buildings. In such cases the user works from the given spectra. In other cases the time history of a known base excitation must first be converted into a response spectrum by considering the response of a single degree of freedom system excited by the known base motion. For this purpose the single degree of freedom system is characterized by its undamped natural frequency, $\omega ,$ and the fraction of critical damping present in the system, ». The equation of motion of the system is integrated through time to find peak values of relative displacement, relative velocity, and acceleration. The integration described in \`\`Modal dynamic analysis,'' Section 2.5.5, can be used for this purpose, since it is exact when the base motion record varies linearly with time. Thus, the maximum relative and absolute values of displacement, velocity, and acceleration are found for the linear, one degree of freedom system. This process is repeated for all frequency and damping values in the range of interest to construct displacement, velocity, and acceleration spectra, $S ^ { D } ( \omega , \xi ) , S ^ { V } ( \omega , \xi )$ ; and $S ^ { A } ( \omega , \xi )$ . A FORTRAN program to build spectra in this way from an acceleration record is given in \`\`Analysis of a cantilever subject to earthquake motion, '' Section 1.4.13 of the ABAQUS Benchmarks Manual (file cantilever\_spectradata.f.) If there is no damping, the relationship between $S ^ { D } , S ^ { V }$ , and $S ^ { A }$ is given by